Navier-Stokes, LLMs and the Ownership of Ideas

Mathematics · Artificial intelligence · Open science

Navier–Stokes, LLMs, And The Ownership of Ideas

What OpenAI claims to have proved, why external forcing does count, and what the dispute reveals about the social future of mathematics.

José Mauricio Gómez Julián · September 9, 2026 · 16 min read

One of the most important and unusual mathematical stories in recent years began with a rumor. Two researchers, Tristan Buckmaster and Levent Alpöge, had spent months developing a line of work on singularities in fluid equations. OpenAI learned that something important was happening, deployed thousands of artificial-intelligence agents, and, a few days later, announced a proof of finite-time blowup for the three-dimensional Navier–Stokes equations.

The story contains every ingredient needed to produce confusion: one of the seven Millennium Prize Problems, a 166-page proof, a Lean formalization, about 10,000 agents, 130 billion tokens, allegations concerning scientific credit and private data, and a conversation that Buckmaster interpreted as pressure against his career. Understanding it requires resisting two simplifications. The first is that a large amount of computation automatically turns a result into scientific understanding. The second is that the presence of an external force makes the proof irrelevant. Neither is correct.

1. The fundamental question

The Navier–Stokes equations describe the motion of viscous fluids such as water and air. They do not track molecules one by one. They treat the fluid as a continuous medium and relate its velocity, pressure, viscosity, and external forces:

∂u/∂t + (u · ∇)u = −∇p + νΔu + f,    ∇ · u = 0 u is velocity; p, pressure; ν, viscosity; and f, an external force. The second equality says that the fluid is incompressible.

In three dimensions, we know how to construct global weak solutions, meaning solutions that satisfy the equations in a generalized sense. What we did not know was whether smooth initial data always produce smooth solutions for all time or whether, under some conditions, velocity can grow without bound in finite time. Such a loss of regularity is called a singularity or blowup.

Viscosity tends to smooth differences in velocity. The nonlinearity represented by (u · ∇)u transports and reorganizes motion. The problem arises from the competition between these effects: is viscous dissipation always enough to prevent infinite concentration, or can three-dimensional dynamics overcome it?

A physical clarification

Mathematically infinite velocity does not mean that a real fluid literally reaches that speed. It means that the continuum model loses regularity and ceases to describe the process in the classical way. The question is mathematical, although it is physically motivated.

2. What Clay actually requires

The first indispensable correction belongs here. The official statement written by Charles Fefferman offers four routes. Alternatives A and B ask for global existence and smoothness when the external force is zero, respectively in the whole space and on the three-dimensional torus. Alternatives C and D permit a disproof of global regularity using smooth initial data and an external force that is also smooth, on those same two domains.

Alternative	Domain	External force	What must be shown
A	ℝ³	f = 0	Global existence and smoothness for every admissible initial datum.
B	𝕋³	f = 0	The same global regularity in the periodic case.
C	ℝ³	smooth f	Existence of data and a force for which there is no global classical solution of bounded energy.
D	𝕋³	smooth f	The corresponding breakdown in the periodic case.

Calling smooth forcing a “loophole” foreign to the prize is therefore technically incorrect. It is written into alternatives C and D. If OpenAI’s proof is correct and satisfies the stated hypotheses exactly, it would resolve the official problem through those routes.

That does not make all results equivalent. Producing a singularity with a specially constructed smooth force is a more limited claim, physically and conceptually, than proving a singularity without external forcing. It resolves the institutional disjunction formulated by Clay, but it does not answer the stronger question of whether unforced dynamics can break down by themselves. Both statements can be true at once, without verbal tricks.

External forcing does not formally invalidate the result; it precisely limits what the result teaches us about fluids.

3. What OpenAI claims to have proved

OpenAI’s paper constructs, for every positive viscosity, velocity and pressure fields that remain smooth up to a critical time, together with a smooth, compactly supported force f(x,t). The fluid starts from rest. As time approaches t = 1, the maximum velocity grows without bound, while total kinetic energy remains uniformly bounded.

The intuitive picture is a vortex that spins, stretches, and concentrates around an increasingly small region. Velocity can rise indefinitely without total energy blowing up because the volume occupied by the extreme motion contracts quickly enough. An imperfect but useful analogy is to concentrate ever greater intensity inside ever less space.

It is not enough to draw a singular flow and then define an infinite force to sustain it. That would be trivial and inadmissible. The difficulty is to make acceleration, pressure, nonlinear transport, and viscosity develop large terms that cancel with extraordinary precision, leaving a perfectly smooth external force as the residual. The paper introduces oscillatory pulses whose internal momentum transport corrects the singular parts of that residual.

Exact scope of the announced theorem

• Yes: finite-time velocity blowup for three-dimensional Navier–Stokes with positive viscosity and a smooth external force.
• Yes: uniformly bounded total kinetic energy throughout the evolution.
• Yes, conditional on the proof being correct: alternatives C and D of the official formulation.
• No: a proof of blowup for unforced Navier–Stokes.
• Not yet: stable, independent acceptance by the mathematical community.

4. How the agents worked

According to OpenAI, on September 1 it assigned groups of agents to the different alternatives of several Millennium Prize Problems. The agents could consult a cached copy of the internet, run code, divide into groups, and communicate results. Nearly one hundred agents reportedly first produced a solution for unforced Euler. From that result, the company concentrated its resources on Navier–Stokes.

The decisive group eventually involved on the order of 10,000 concurrent agents. OpenAI reports that the result appeared after about 88 hours; Lean formalization and verification took an additional 17 hours using GPT-6 Astra. The Navier–Stokes effort reportedly involved 2.7 million messages and approximately 130 billion generated tokens.

These numbers neither prove understanding nor disprove it. They describe a new material organization of intellectual labor: massively parallel search, route selection, circulation of partial results, automated consolidation, and formal verification. Reducing this to “mere brute force” hides the coordination architecture; presenting it as an autonomous mathematical consciousness hides the human, bibliographical, and computational scaffolding that made the search possible.

5. The mathematical genealogy

The route did not emerge in a vacuum. Buckmaster attributes the basic idea of the program to Diego Córdoba and Luis Martínez-Zoroa, who had spent years studying forced blowups and mechanisms of amplification across scales. Their work had shown how increasingly concentrated layers of vorticity could amplify while the regularity of the force remained controlled.

Buckmaster and Alpöge took that program as their starting point. With extensive help from Claude and Codex, they say they pushed constructions with rougher forcing to smooth forcing for incompressible porous media, Boussinesq, and three-dimensional incompressible Euler. According to their statement, they obtained the Boussinesq and Euler results on August 15 and verified the first model-generated proof in Lean on August 22.

OpenAI’s announced contribution would be different and later: a construction for viscous Navier–Stokes. Even if it was obtained independently, its historical meaning cannot be narrated as if 10,000 agents had discovered both the problem and the strategy from a blank page. In mathematics, the final proof is only one part of discovery. Recognizing which simplification to attack, which literature matters, and which mechanism may survive new constraints is genuine intellectual work.

6. Timeline of the dispute

Date in 2026	Reported event	Source of the claim
August 15	Buckmaster and Alpöge obtain smooth-forcing blowup for Boussinesq and Euler.	Buckmaster’s statement.
August 22	Lean verification of their initial proof is completed.	Buckmaster’s statement.
September 1	OpenAI hears rumors and launches agents on the Millennium Prize Problems.	OpenAI’s account.
September 3	Buckmaster privately writes to a mathematician at OpenAI to clarify the rumor.	Email reproduced by Buckmaster.
September 5	OpenAI’s agents reach the Navier–Stokes construction, about 88 hours after starting.	OpenAI’s account.
September 6	OpenAI’s formal verification is completed and two calls with Buckmaster take place.	OpenAI and Buckmaster’s statement.
September 8	OpenAI publishes the article, proof, and formalization; Buckmaster makes his statement public.	The respective publications.

The temporal proximity fuels the controversy. Buckmaster says that almost nobody was working on this route and that the sudden focus on the forced problem was a warning sign to him. OpenAI responds that it distributed agents among all four official alternatives, first obtained an unforced Euler result, and only then identified Navier–Stokes as the most promising direction.

7. What is established and what is disputed

The public discussion mixes verifiable documents, incompatible accounts, and inferences. They should be separated.

Evidence map

• Publicly documented: an analytical manuscript, a Lean formalization, Buckmaster’s statement, and OpenAI’s recognition of Buckmaster and Alpöge’s priority on forced Euler all exist.
• Claimed by OpenAI: no researcher or agent accessed their specific work before publication; the agents reached different results, and the company does not intend to claim the monetary prize.
• Acknowledged by OpenAI as a remote possibility: it cannot rule out that de-identified data derived from their product usage helped improve its models.
• Claimed by Buckmaster: he received no clear answer about training; he was offered an arrangement in which he would present OpenAI’s result alone and Alpöge would be excluded because he works at Anthropic; he also received statements he interpreted as professional threats.
• Disputed: Sébastien Bubeck rejects Buckmaster’s characterization of the interaction, according to the cited public summary.
• Not publicly established: that private drafts were used to produce OpenAI’s proof or that deliberate appropriation occurred.

Buckmaster himself draws the boundary: he has not seen how the model worked, does not know whether his data were used, and does not present that accusation as an established fact. His stated purpose is to record what he was told, when he was told it, and what arrangements were proposed. This caution does not erase the questions; it prevents them from being answered through condemnation without evidence.

OpenAI’s sentence about de-identified data also does not demonstrate a causal transmission from those drafts to the proof. It does reveal a governance problem: for a researcher placing unpublished work in a tool, an inability to exclude aggregate influence may be scientifically unacceptable even if the handling complied with the service’s contractual terms.

8. What Lean verifies and what it does not

Lean is a proof assistant. A formalized proof is decomposed into steps that a small kernel checks against precise logical rules. This sharply reduces the chance that an algebraic leap, forgotten hypothesis, or invalid inference survives. In a manuscript over one hundred pages long, that guarantee is extraordinarily valuable.

Lean does not, by itself, answer four separate questions:

1. Correspondence: whether the formal theorem exactly represents the informal mathematical claim being announced.
2. Interpretation: what physical meaning the construction has and how much it depends on a designed force.
3. Novelty: which ideas are new and which come from prior work.
4. Credit: who formulated the route, supplied the decisive mechanisms, directed the search, and should be named as an author.

A logical certificate does not replace a human exposition. The community still needs to read the manuscript, check that the formalization matches the intended theorem, reconstruct the conceptual mechanism, and place it within the literature. Formal verification strengthens the evidence; it does not exhaust scientific practice.

9. The Deep Blue–Kasparov moment

Buckmaster calls the episode a “Deep Blue–Kasparov moment.” The comparison does not mean that mathematics is simply another game with closed rules. It points to a change in the division of labor: a machine can explore, combine, and verify on a timescale inaccessible to a person, but its productivity depends on a search space structured by historically accumulated problems, theories, formal libraries, and strategies.

In a major problem, the scarce resource is not always writing the last line of a proof. It may be identifying the promising route after years of failed attempts. If revealing that route allows thousands of agents to be mobilized immediately, then selecting the correct intermediate question acquires enormous economic and scientific value.

When proving becomes massively parallelizable, knowing what should be proved may become the scarcest part of discovery.

This could make open mathematics more guarded. A seminar, an informal draft, or a query submitted to a platform would cease to be only an act of cooperation and could become an exploitable signal for institutions with incomparable computing capacity. The danger is not that artificial intelligence participates in science, but that openness becomes unilateral: open knowledge from below and closed appropriative capacity from above.

10. A materialist reading

The controversy cannot be understood solely as a moral conflict between individuals. It expresses a transformation in the productive forces of science. Models, data centers, multi-agent systems, and proof assistants turn enormous quantities of accumulated mathematical labor into an infrastructure capable of producing new combinations at great speed.

At the same time, this productive force is concentrated under private ownership. Researchers contribute questions, drafts, corrections, usage, and tacit knowledge; the platform aggregates those interactions, improves closed models, and can return results at a scale no academic group can match. Even if no concrete appropriation occurred in this case, the structural asymmetry is real.

The decisive question is therefore not whether the machine “really thought,” a metaphysical dispute of little use. We should ask who controlled the means of mathematical production, who chose the direction of the search, which prior labor it depended on, who could audit the process, and how authorship, prestige, and benefits were distributed.

Computation is not an epistemological sin. It can be a genuine force of discovery. But a new productive force organized through opaque relations of ownership and credit can simultaneously expand human capacity and deepen the subordination of scientific labor.

11. The rules we need

The episode suggests a minimum program for AI-assisted research:

1. Verifiable confidential mode: unpublished drafts should be excluded from training and any later retrieval, with auditable guarantees rather than contractual language alone.
2. Provenance: every major result should retain a record of models, versions, consulted sources, decisive prompts, human intervention, and circulation among agents.
3. Compute disclosure: the number of agents, tokens, time, tools, and selection criteria should be treated as part of the method.
4. Conceptual genealogy: attribution should recognize not only the final proof but also the prior program that made the route visible.
5. Authorship independent of corporate rivalry: a researcher’s affiliation cannot justify excluding a scientifically relevant contribution.
6. Open review: the manuscript, formalization, and correspondence between them should remain available for independent examination.
7. Public capacity: universities and scientific consortia need shared infrastructure so frontier mathematics does not depend exclusively on private laboratories.

These rules would not settle every priority dispute, but they would make verifiable questions that currently depend on conflicting institutional statements.

12. Conclusion

Did OpenAI solve Navier–Stokes? The rigorous answer is conditional. The announced theorem targets precisely alternatives C and D of Clay’s official statement. The use of a smooth external force is not a trick, although it leaves the stronger unforced question untouched. If the analytical proof and its formalization withstand expert review, the mathematical achievement would be historic.

But the result did not arise outside history. It depends on decades of analysis, on the route opened by Córdoba and Martínez-Zoroa, on the work of Buckmaster and Alpöge, and on extraordinary computational infrastructure. Nor have the questions about data, priority, and institutional pressure been resolved. The most serious allegations should be investigated, neither repeated as facts nor dissolved through public relations.

The deepest lesson is that artificial intelligence does not abolish the social relations of science. It makes them more visible. The more powerful automated proving becomes, the more important it will be to protect openness, record the provenance of ideas, and distribute credit fairly. The future of mathematics will depend not only on what machines can prove, but on the institutions through which we decide what it means to discover.

· · ·

References

Specified sources

1. Julien Cadot, public summary of the controversy.https://x.com/juliencdt/status/2097582940863471900
2. Mehdi, technical, epistemological, and political criticism of the announcement.https://x.com/bettercallmedhi/status/2097472336257863722
3. Susan Zhang, reproduction of the manuscript and OpenAI’s statement on concurrent work.https://x.com/suchenzang/status/2097376584814432392
4. Tristan Buckmaster, statement on the results and conversations with OpenAI.https://cims.nyu.edu/~tristanb/statement.pdf
5. Jasper, mathematical analysis and summary of the dispute.https://x.com/zjasper/status/2097547276755619988

Primary technical documents

6. OpenAI, “On the Navier–Stokes Millennium Prize Problem.”https://openai.com/index/navier-stokes-solution/
7. OpenAI, “Finite Time Blowup for Navier–Stokes.”Analytical manuscript
8. Charles L. Fefferman, official formulation of the Millennium Prize Problem.Existence and Smoothness of the Navier–Stokes Equation
9. OpenAI, Lean formalization.https://github.com/openai/NavierStokesAndEuler

Read Buckmaster’s statement

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9 September, 2026

GENERALITIES ON THE STATISTICAL THEORY OF SAMPLING SURVEYS

SURVEY THEORY · SAMPLING · INFERENCE

GENERAL REMARKS ON THE STATISTICAL THEORY OF SAMPLE SURVEYS

José Mauricio Gómez Julián · 2021

Conceptual and technical notes on sampling design, frames, inclusion probabilities, design-based inference, nonresponse, total survey error, and estimation.

The statistical theory of sample surveys is often learned as a collection of procedures: how to select a sample, how to estimate a total or a mean, how to calculate a standard error, and how to determine a sample size. Yet behind every procedure lies a prior conceptual question. What, exactly, is the population to which one intends to infer? What entity is selected at each stage? What makes a selection probabilistic? Where does the randomness that allows us to speak of precision come from? What part of survey error can be quantified by the sampling design, and what part arises from coverage, nonresponse, measurement, or processing?

The notes that follow are organized around problems that appear in the classical sampling literature—in particular Cochran (1977), Kish (1965), and Lohr (2021)—and develop them using tools from contemporary survey theory. The purpose is not to replace the reading of those authors, but to make explicit the conditions under which their statements acquire meaning and the operational consequences that follow from them. When more recent terminology is introduced, the discussion draws especially on Särndal, Swensson, and Wretman (1992), Groves et al. (2009), Biemer and Lyberg (2003), AAPOR’s definitions, and the guidelines of the United Nations, Statistics Canada, and the U.S. Census Bureau.

· · ·

1. In a survey, are costs high and administration more complex?

The answer necessarily depends on the basis of comparison. Relative to a census, a sample survey is usually cheaper and administratively less demanding because it observes only part of the population. Indeed, one of the historical reasons for the development of probability sampling was to obtain sufficiently precise information at a cost far below that of a complete enumeration.

The comparison changes, however, if the reference point is a smaller-scale qualitative procedure, such as a focus group, an in-depth interview, or intensive observation. A national probability survey may require a frame, sample selection, instrument programming, training, supervision, callbacks, weighting, quality control, variance estimation, and documentation. A focus group may be far less expensive, but it does not perform the same inferential function. The choice should not be made by asking which technique is “cheaper” in the abstract, but which technique can answer the research question with the kind of evidence required.

A focus group, for example, can be extraordinarily useful for understanding vocabularies, categories of meaning, mechanisms, and problems in questionnaire comprehension. It is not, however, a direct substitute for a probability survey when the objective is to estimate parameters of a defined population. Cost must therefore be judged in relation to the objective, the required precision, and the type of inference one intends to make.

2. What is a double-barreled question?

It is a question that asks for a single response concerning two or more propositions that could receive different answers. The problem is not that the respondent has “two response options”; almost every closed question does. The problem is that the wording fuses two objects of judgment and forces the respondent to compress them into a single answer.

For example:

Are you satisfied with your salary and with your working conditions?

A person may be satisfied with the salary and dissatisfied with the conditions, or the reverse. If the scale offers only one response, the resulting datum has no univocal interpretation. The solution is to separate the components whenever each constitutes a substantively distinguishable dimension.

This problem points to something more general: a questionnaire is not a neutral container into which already constituted variables are simply “placed.” The concrete wording helps determine the cognitive operation performed by the respondent and, consequently, the datum that is ultimately recorded. Instrument design is therefore part of measurement and not a purely editorial task.

3. When is a questionnaire too long?

There is no universal number of questions or minutes beyond which a questionnaire becomes too long. Burden depends on mode of administration, the cognitive complexity of the questions, the sensitivity of the topics, the population being interviewed, the need to consult records, the number of skip patterns, and the repetition of scales, among other factors.

A face-to-face interview may tolerate a different duration from a telephone interview or a self-administered online questionnaire. Likewise, twenty questions that require recalling amounts, dates, or episodes may impose a greater burden than fifty very brief and homogeneous items.

Pilot testing is important, but it should be distinguished from cognitive pretesting. A pilot reproduces field procedures on a smaller scale and makes it possible to study duration, contact rates, skip patterns, logistics, and implementation problems. A cognitive interview, by contrast, seeks to understand how the respondent interprets the question, retrieves the information, and selects an answer. The U.S. Census Bureau also includes usability testing, behavior coding, debriefing, and split-panel designs among questionnaire evaluation tools.

The methodological rule, therefore, is not “make it as short as possible,” but retain everything needed to measure the defined concepts and remove what adds burden without adding substantive information. Brevity and quality are not synonyms.

4. What does it mean for a frame to contain duplicates?

A sampling frame is the operational device through which sampling units can be identified or reached in order to carry out selection. Sometimes it takes the form of a list; at other times it consists of geographic areas, addresses, administrative registers, potential telephone numbers, or procedures that make it possible to generate selectable units.

Duplication occurs when the same population unit is represented more than once in the selection mechanism. If records are selected with equal probability and one person appears twice, that person may acquire a higher inclusion probability than another who appears only once. But the problem should not be described by saying that any inequality in probabilities makes the sample “nonrandom.” There are deliberately unequal-probability probability designs.

What matters is to distinguish between inequality that is designed and known and multiplicity that is accidental or unknown. A frame may suffer, among other problems, from:

• undercoverage: units of the target population that cannot be reached through the frame;
• overcoverage: records that do not belong to the target population;
• duplicates or multiplicity;
• outdated records;
• units whose status or location has changed;
• many-to-one or one-to-many relationships between records and target units.

The statistical consequence depends on the specific selection mechanism and on whether multiplicity can be measured and corrected in the inclusion probabilities or in weighting.

5. What does data editing mean, and what should be done with nonresponse?

In survey work, “editing” should not be understood as deleting observations that look strange. Editing consists of applying explicit validity, consistency, and plausibility rules to identify records that require review. An improbable answer is not automatically false, and an inconsistent answer should not disappear without a trace.

A reasonable workflow always preserves the raw information and separates the stages:

1. captured datum;
2. validation of range, type, and structure;
3. verification of skips and logical consistency;
4. possible recontact, when feasible and planned;
5. documented editing rules;
6. marking of missing values;
7. possible imputation;
8. production of the analytical dataset without destroying the original dataset.

Suppose that a section of the questionnaire applies only to households that reported a certain economic activity. If answers appear in that section for households that, according to the filter question, do not perform that activity, several possibilities exist: the filter was captured incorrectly, the later response was captured incorrectly, the interviewer failed to follow the skip, the definition was misunderstood, or there is a real situation that the design did not anticipate. The datum must be investigated; it is not enough to declare it “obviously wrong.”

It is also necessary to distinguish states that are often collapsed under a single missing-value label: Not applicable, Don’t know, No answer, explicit refusal, failure to contact, structural skip, capture error, and value lost during processing. “Not applicable” is not simply another form of DK/NR: it means that the question does not belong to the logical universe of that unit.

Unit nonresponse and item nonresponse

Unit nonresponse occurs when no substantive information is obtained from a selected unit; item nonresponse occurs when an interview exists but particular answers are missing. The former is often addressed through weighting adjustments and the latter may require imputation or other methods, although the specific techniques depend on the missingness mechanism and the analytical objective (Kalton and Kasprzyk, 1986).

A high nonresponse rate does not mechanically imply that a variable is unusable, just as a high response rate does not demonstrate the absence of bias. Nonresponse bias depends on the relationship between response propensity and the variables to be estimated.

AAPOR distinguishes, among other measures, response rate, cooperation rate, contact rate, and refusal rate. They are not synonyms. The response rate relates completed interviews to the set of eligible cases—with variants concerning the treatment of unknown eligibility and partial interviews; cooperation conditions on contacted units; the contact rate studies the success of contact; and the refusal rate quantifies a specific form of nonresponse.

Finally, interviewer probing must be standardized. The objective is not to press until “no survey is ever left incomplete,” but to facilitate a valid answer without inducing it and while respecting the participant’s consent.

6. Efficiency, repeated sampling, and confidence intervals

Cochran frames sampling theory as a joint problem of precision and cost: to design selection and estimation procedures that produce the required accuracy at the lowest possible cost. The underlying idea remains fundamental. There is no “best” design outside an objective; there is a design that is more or less efficient for particular estimands, constraints, and costs.

The precision of a procedure is studied through its distribution under hypothetical repetitions of the sampling mechanism. From a design-based perspective, the values of the finite population may be regarded as fixed, while randomness arises from the set of samples that the design could have selected.

Here it is important to state the interpretation of a confidence interval precisely. If a procedure constructs intervals with a 95% confidence level, that does not mean that 95 out of 100 samples will produce “values similar” to the value observed in our sample. It means that, under the assumptions of the procedure and under hypothetical repetitions of the design, approximately 95% of the intervals constructed in that way would contain the fixed population parameter.

Once a particular frequentist interval has been observed, the parameter does not thereby become random. The randomness belongs to the procedure that would have produced different intervals under other possible samples.

Sample size for a proportion

In an elementary formulation under simple random sampling, if an approximate margin of error \(e\) is desired for a proportion at a level associated with \(z_{1-\alpha/2}\), one may begin with

\[ n_0=\frac{z_{1-\alpha/2}^2\,p(1-p)}{e^2}. \]

When there is no useful prior information about \(p\) and a conservative case is sought, \(p=0.5\) is used because the function

\[ p(1-p) \]

attains its maximum at \(p=0.5\). That is the mathematical reason. It is not a consequence of the central limit theorem.

If the population is finite and the sampling fraction is no longer negligible, sample size can be adjusted, within the same simple framework, by

\[ n=\frac{Nn_0}{N+n_0-1}. \]

In complex designs, the calculation must incorporate stratification, clustering, unequal probabilities, domain objectives, anticipated nonresponse, and, where appropriate, an expected design effect. Sample size should not be calculated in isolation from the design.

7. Are there “common sample sizes”?

Recurring sizes—400, 800, 1,000 interviews, for example—do appear in professional practice because certain combinations of precision, budget, and mode recur. But those numbers are not statistical constants.

For a proportion near 0.5, a simple random sample of about 1,067 units yields, under a normal approximation and without finite population correction, a sampling margin close to three percentage points at 95%. This accounts for part of the recurrence of sample sizes near one thousand in polls. The calculation, however, does not by itself demonstrate that a survey is nationally representative or that its total error is ±3 percentage points.

If there is clustering, highly variable weights, small domains, unequal response rates, or incomplete coverage, the same \(n\) may yield very different precision and quality. The proper question is not “what is the traditional size?” but “what precision do I need for which estimand, under what design, and at what cost?”

8. Two differences between classical theory and survey theory

8.1. Distributions, models, and design

A classical observation by Cochran contrasts part of model-based statistical inference with survey theory for finite populations. In many classical problems, observations are assumed to arise from a distribution with a specified mathematical form and its parameters are estimated. In survey sampling, by contrast, a central tradition develops inference from the selection mechanism without requiring a parametric distribution for the population values.

The expression “few measurements on each unit” does not refer to a sample with fewer than 30 or 50 cases. It refers to the number and type of variables measured on each unit. A large survey may collect many variables with very different distributions; there is no reason to impose the same parametric family on all of them merely because the dataset is large.

Computing power does not eliminate this conceptual difference. Today at least three perspectives coexist:

• design-based inference: randomness is attributed to the selection mechanism;
• model-assisted inference: models are used to improve estimators, calibrate, or exploit auxiliary information while retaining properties justified by the design;
• model-based inference: inference rests more directly on a probability model for the observed values or mechanisms.

Särndal, Swensson, and Wretman (1992) systematically develop the model-assisted perspective. The distinction is not a relic from before modern computers; it answers different questions about the source of randomness and the assumptions that support inference.

8.2. Finite populations and the finite population correction

A population is finite if it contains \(N<\infty\) units. This does not depend on whether the researcher knows \(N\) exactly. A finite population can exist even when its size is unknown or uncertain.

Under simple random sampling without replacement, for the population mean \(\bar Y\), if

\[ S^2=\frac{1}{N-1}\sum_{i=1}^{N}(Y_i-\bar Y)^2, \]

then

\[ \operatorname{Var}(\bar y) =\left(1-\frac{n}{N}\right)\frac{S^2}{n}. \]

The factor

\[ 1-f,\qquad f=\frac{n}{N}, \]

is the finite population correction in the variance. When \(f\) is very small, \(1-f\approx1\) and its effect may be negligible. When the sample covers a large proportion of the population, ignoring it can substantially overestimate sampling variance.

The reason is materially simple: if an increasing proportion of a finite population is observed without replacement, progressively less uncertainty remains about the part that was not observed. When \(n=N\), a census of that population has been conducted and the variance due to the sampling mechanism is zero, although measurement, coverage, processing, and other errors may still exist.

9. What does it mean to be sensitive to respondents’ expectations?

The interviewer must maintain a combination that is not always easy: rapport without leading. The interviewer should establish a relationship that facilitates communication, explain the purpose of the interview, listen, and use permitted probes; but should not guide the respondent toward an answer that appears socially desirable or compatible with the researcher’s expectations.

Empathy does not mean abandoning standardization. An interview can be humane, respectful, and sensitive to context without arbitrarily altering the meaning of the questions. In standardized instruments, an important part of training consists precisely in learning which clarifications are permitted and how to use neutral probes.

Sensitivity also has an ethical dimension. The interviewer must recognize discomfort, fatigue, privacy, risks, and the right not to answer. Data quality cannot be pursued through coercion.

10. Courtesy, tone, social desirability, consent, and position

The interaction between interviewer and respondent can affect measurement. Tone of voice, the way response options are presented, verbal or gestural reactions, the respondent’s perception of the interviewer’s status, and topic sensitivity can alter the observed response.

Several mechanisms should be distinguished:

• social desirability: a tendency to provide an answer perceived as socially acceptable;
• acquiescence: a tendency to agree with formulations regardless of content;
• interviewer effect: systematic variation associated with interviewer characteristics or behavior;
• mode effect: differences associated with face-to-face, telephone, web, paper, or other modes;
• context and order effect: responses conditioned by preceding questions or by the structure of the instrument.

Informed consent, however, is not a bias. It is an ethical condition of research involving human participants. A different question is whether the decision to participate may generate self-selection or differential nonresponse. That possible source of bias should be studied as part of the participation process, not presented as a reason to weaken consent.

11. Survey design and absolute sample size

Lohr insists on an idea that should be formulated carefully: an enormous sample does not rescue a defective design. The relevant quantity in her discussion is the absolute size of the sample, not the absolute size of the population.

For large populations, the precision of an estimator under simple random sampling depends primarily on \(n\); \(N\) appears in the finite population correction and has little effect when \(n/N\) is small. Thus, a sample of one thousand people may have a similar sampling error whether the population contains one million or one hundred million people, provided that the selection mechanism and other assumptions are comparable.

But this statement does not mean that any sample of one thousand units is adequate. A convenience sample of one hundred thousand people may have more selection bias than a probability sample of one thousand. Increasing \(n\) reduces random variability under the assumed mechanism; it does not automatically erase systematic errors of coverage, selection, measurement, or nonresponse.

Hence a rule that should remain present whenever polls are read:

\[ \text{large sample}\;\nRightarrow\;\text{representativeness}. \]

And likewise:

\[ \text{small sampling margin}\;\nRightarrow\;\text{small total error}. \]

12. Content, units, extent, and time when defining a population

A study population must be defined precisely enough to decide who or what belongs to it. It is useful to specify its substantive content, the units that compose it, its spatial extent, and the time or reference period.

Content is not simply spatiotemporal location; that belongs mainly to extent and time. Content determines what kind of entity and what substantive condition make a unit part of the population. For example: “persons aged 18 or older who are usual residents of private households in Costa Rica during the reference period.” There we find an entity, an age condition, a residence condition, a household delimitation, and a territorial-temporal extent.

The different units

The word “unit” has a general meaning—something that may be considered an entity for a given analysis—but in surveys it acquires different technical functions. They should not be confused.

Element or target unit. The population entity about which information is to be produced: a person, a household, a firm, a plot of land, a school.

Unit of analysis or study unit. The entity to which the analyzed variables are attributed. It may coincide with the target element, though not always.

Observation unit. The entity on which observation or measurement is directly performed.

Reporting unit. The person or entity that provides the information. In a household survey, one adult may report on other members; the reporting unit and the analytical unit are then not identical.

Sampling unit. A selectable unit at a particular stage of the design. At a first stage it may be a geographic segment; at a second, a dwelling; at a third, a person.

Primary sampling unit (PSU). The unit selected at the first stage of a multistage design. It is not defined by being “a place” or by necessarily containing only one final unit.

Listing unit. A unit identified during a listing operation in order to construct or update the frame for a later stage. Its function is operational.

The same entity may perform several functions. If a complete list of students is available and students are directly selected and asked about themselves, a student may simultaneously be the element, unit of analysis, observation unit, reporting unit, and sampling unit. But this empirical coincidence does not make the concepts synonymous.

13. Domain, stratum, class, and subclass

An estimation domain is a subpopulation for which separate estimates are desired. It may be defined by region, sex, age, economic sector, or another relevant condition.

A stratum, by contrast, is a partition used by the sampling design before the sample is selected. Strata are generally intended to control composition, improve efficiency, or ensure sufficient representation of relevant groups.

A domain may coincide with a stratum, but it need not. It may cut across several strata or be defined using information obtained after selection. For example, the population may be stratified by province and, later, a domain may be estimated consisting of persons employed in a particular economic industry across all provinces.

The notion of a class as an interval in a frequency distribution belongs to a different problem. It is not necessary for defining an estimation domain. Likewise, “subclass” should not be used as a general synonym for domain unless a source explicitly defines that terminology for a particular scheme.

14. What does \(Y_i\) represent?

If the finite population is

\[ U=\{1,2,\ldots,N\}, \]

and \(Y\) is a study variable, then \(Y_i\) represents the value taken by that variable for unit \(i\). It does not represent “the characteristics” of the unit in the plural, but one particular realization of a defined variable.

The population total is

\[ T_Y=\sum_{i=1}^{N}Y_i, \]

and the population mean is

\[ \bar Y=\frac{1}{N}\sum_{i=1}^{N}Y_i. \]

In design-based inference, a particularly important idea is that the vector

\[ (Y_1,\ldots,Y_N) \]

may be treated as fixed. The distribution used to study an estimator arises from the sampling design: which samples \(s\) could have been selected and with what probabilities.

This prevents a common confusion. The use of probability does not require us to think that every material characteristic of the population was “generated” by a known parametric distribution. Probability may reside in the act of selection.

15. What does MESIP selection mean?

In terminology used in various Latin American texts, MESIP commonly denotes selection with equal probability of the elements. It should be understood as a particular class of probability design, not as the definition of all probability sampling.

If \(\pi_i\) is the inclusion probability of unit \(i\),

\[ \pi_i=P(i\in s), \]

an equal-probability design satisfies, for comparable units within the design,

\[ \pi_i=\pi_j. \]

But a general probability design may have

\[ \pi_i\neq\pi_j. \]

The inequality may be deliberate, for example when selection is made with probability proportional to a measure of size.

16. What does “unrestricted random sampling” mean?

The expression is used in part of the Spanish-language literature to refer to simple random sampling without replacement: from a population of size \(N\), a fixed-size sample \(n\) is selected so that every possible subset of \(n\) units has the same probability of selection.

If \(S\) is the set of all possible samples of size \(n\), then for every \(s\in S\),

\[ P(S=s)=\binom{N}{n}^{-1}. \]

It follows that every unit has inclusion probability

\[ \pi_i=\frac{n}{N}. \]

It is a fundamental design precisely because it permits transparent derivations and serves as a benchmark. But it is not “random sampling as such” in an exhaustive sense: probability sampling includes stratified, systematic, cluster, multistage, and unequal-probability designs.

17. Mean squared error, variance, and bias

Let \(\hat\theta\) be an estimator of the parameter \(\theta\). Its mean squared error is

\[ \operatorname{MSE}(\hat\theta) =E\left[(\hat\theta-\theta)^2\right]. \]

It can be decomposed as

\[ \operatorname{MSE}(\hat\theta) =\operatorname{Var}(\hat\theta) +\operatorname{Bias}(\hat\theta)^2, \]

where

\[ \operatorname{Bias}(\hat\theta) =E(\hat\theta)-\theta. \]

The identity can be seen by writing

\[ \hat\theta-\theta =\bigl(\hat\theta-E(\hat\theta)\bigr) +\bigl(E(\hat\theta)-\theta\bigr), \]

squaring, and taking expectations. The cross term vanishes because

\[ E\left[\hat\theta-E(\hat\theta)\right]=0. \]

Bias is not the difference between one observed realization of \(\hat\theta\) and its expectation. That difference is a random deviation of the realization from the center of its distribution. Bias is a property of the estimation procedure: the difference between the estimator’s expectation and the parameter it is intended to estimate.

The decomposition is conceptually important because it displays a possible trade-off between variability and bias. An estimator with slightly larger variance may have smaller MSE if it reduces bias enough, and vice versa. “Precision” and “accuracy” should not be collapsed into a single colloquial intuition when the analysis requires their components to be distinguished.

18. What are clusters, and how are they related to PSUs?

A cluster in sampling is a grouping of elementary units that may be used as a selection unit. It should not be confused with a cluster produced by a machine-learning algorithm. In clustering, the grouping is usually the result of an algorithmic criterion of similarity or distance; in sampling, the cluster belongs to the design and may be defined by geography, administrative organization, dwellings, schools, establishments, or other structures.

A PSU is simply the unit selected at the first stage. In many household designs the PSU is a geographic cluster—for example, a census segment—so the two notions coincide empirically. They are not, however, synonymous by definition.

Design	First unit selected	Stages	Are equal probabilities mandatory?
Simple random	Element	1	Yes, in the classical simple design
Stratified	Element within stratum	1	Not necessarily across strata
One-stage cluster	Cluster	1	No
Multistage	PSU; then secondary units, etc.	2 or more	No
PPS	Unit with probability proportional to size	1 or more	No

Clustering matters because units within the same cluster often resemble one another. If there is positive intraclass correlation, interviewing twenty people within the same cluster may provide less information than interviewing twenty people spread across many clusters.

This relative loss or gain in precision is often summarized through a design effect,

\[ \operatorname{deff} =\frac{\operatorname{Var}_{\text{design}}(\hat\theta)}{\operatorname{Var}_{\text{SRS}}(\hat\theta)}, \]

which compares the variance under the actual design with a simple-random-sampling reference of comparable size. A \(\operatorname{deff}>1\) indicates greater variance than the reference; it is not an “error” of an algorithm, but a property that may arise from the design, intraclass correlation, and weight variability.

19. What does “margin of error” mean?

In the language of polls, the margin of error is usually the half-width of a confidence interval associated with sampling uncertainty under a particular design and estimation method. It is not “the total amount of error in the survey.”

For an estimate \(\hat\theta\), a generic form is

\[ \hat\theta\pm z_{1-\alpha/2}\,\widehat{SE}(\hat\theta), \]

so that the margin is

\[ MOE=z_{1-\alpha/2}\,\widehat{SE}(\hat\theta). \]

In complex designs, \(\widehat{SE}\) must correspond to the design: strata, clusters, unequal probabilities, calibration, or replicate weights, as applicable. Calculating a margin as though the sample had been simple random when it was not can understate or overstate uncertainty.

Sampling margin and total survey error

The Total Survey Error paradigm requires separating sampling variability from other sources of error. These include:

• coverage error;
• nonresponse error;
• measurement error;
• construct or specification error;
• processing and coding error;
• editing and imputation error;
• frame and linkage error;
• errors arising from models or adjustments.

Thus a phrase such as “the survey has a margin of error of ±3%” should be understood, unless otherwise specified, as a statement about one component of uncertainty. It does not demonstrate that the true value lies within three points or that every other source of error is small.

20. Why is probability sampling preferred? Can only probability samples be useful?

A probability design provides a known selection distribution and therefore permits design-based inferences and estimation of sampling variability under explicit conditions. This is an enormous advantage when the objective is to infer to a defined population.

But “nonprobability” does not automatically mean “useless.” Convenience, judgment, quota, network, or self-selected samples may be appropriate for exploration, qualitative research, instrument development, mechanism studies, hard-to-reach populations, or questions for which design-based population inference is not the objective.

The problem arises when such samples are attributed properties that their selection mechanism does not justify. If a researcher studies a localized group because it possesses a rare characteristic, valid knowledge may be produced about that group or about particular mechanisms. One cannot infer without additional assumptions that the prevalence of that characteristic in the country is equal to the prevalence observed in the group.

The word representativeness should be used cautiously. A probability sample may suffer from undercoverage, differential nonresponse, or measurement error. It is therefore preferable to state exactly what property has been secured: a defined coverage frame, known probabilities, calibration to particular totals, balance on auxiliary variables, or inferential capacity under a specified design.

Methodological ethics require documenting the selection method and the limits of inference. A nonprobability sample should not be presented as probabilistic; a probability sample should likewise not be presented as free from the remaining sources of survey error.

21. “Randomization is a physical operation”: what does that mean?

Kish insists that a probability sample requires more than writing a distribution on paper. There must be an effective selection procedure whose operation is congruent with the specified probability model.

The word “physical” should not be understood as opposed to computation. An algorithm implemented on a computer can perfectly well form part of the material operation of selection. What Kish contrasts is an actually executed chance mechanism with a mere belief, assumption, or declaration that the units “are as if they had been random.”

We can separate two levels:

\[ \text{probability model}\neq\text{randomization procedure}, \]

but a probability design requires correspondence between them. If the protocol states that each of \(N\) units has equal probability of selection, the actual procedure—a random-number table, an audited pseudorandom algorithm, a draw, or another mechanism—must implement that property.

This distinction is philosophically interesting. Probability in a probability survey is not merely a formal property of the language used to describe selection; it is linked to an objective, reproducible practice. The theory acquires inferential capacity because a controlled relation exists between the material mechanism and the mathematically defined sample space.

22. What does it mean that the frame includes “physical lists” and procedures that do not literally list all units?

Kish uses frame in a broader sense than “a file containing every name.” A physical list may be a register of persons, establishments, addresses, plots, or administrative units. But there may also be procedures that allow units to be selected without constructing an exhaustive list of all final elements.

An area frame, for example, may divide a territory into selectable segments; dwellings are then listed within the selected segments. In random-digit dialing, the frame may be conceptual or algorithmic: the space of eligible numbers is defined and selections are generated without previously possessing a file of all subscribers.

“Actually listing” means explicitly enumerating all relevant units in an operational list. Avoiding that effort does not mean replacing the physical by “the computational” in the abstract; it means constructing a coverage and selection mechanism that does not require an exhaustive prior enumeration of all final units.

23. Why is it so difficult to construct a frame for human populations?

Because target population, frame, and field location are not the same thing. An electoral register, for example, may be excellent for a population of registered voters and at the same time inadequate for a population defined as “all resident persons” if it excludes minors, unregistered persons, or other groups.

Human populations also change. People are born, die, migrate, move homes, form or dissolve households, use multiple telephones, lose numbers, share devices, and may reside temporarily in different places. A perfect frame would be costly not because we need to know where every person is at every hour, but because it must maintain a sufficiently good correspondence between target units and selectable units.

Large statistical systems address this problem through different strategies: address frames, integrated administrative registers, area frames, dwelling listings within PSUs, dual or multiple frames, and periodic updates. Each solution distributes costs and risks of undercoverage, duplication, and obsolescence differently.

The correct methodological question is not “does a complete list of every inhabitant exist?” but “what frame or combination of frames provides a documented path by which units of the target population have inclusion probabilities that are known or can be modeled in accordance with the design?”

24. What is multistage sampling?

It is a design in which selection occurs in two or more stages. In a household survey, for example, one might select:

1. PSUs or geographic segments;
2. dwellings within the PSUs;
3. one person within each dwelling.

If the selections are conditional, the overall inclusion probability of person \(i\) may be written schematically as the product of the probabilities at each stage:

\[ \pi_i =\pi_{h}\,\pi_{j\mid h}\,\pi_{i\mid j,h}, \]

depending on the notation adopted for PSU, dwelling, and person.

The basic design weight is then

\[ w_i=\frac{1}{\pi_i}. \]

In practice, that weight may later be modified for nonresponse, calibration, poststratification, weight trimming, or other procedures. The basic design weight must always be distinguished from final analytical weights.

Multistage sampling makes it possible to work with enormous populations without first constructing a national list of every final element. Its cost is usually much lower, although geographic concentration can increase variance relative to a simple random sample of the same size.

25. Multiplicity, unequal probabilities, and Horvitz-Thompson estimation

Suppose a telephone frame allows a household with two lines to appear twice while another household with one line appears once. If lines are selected with equal probability and households are the final target, household inclusion probabilities may be unequal. If that multiplicity is unknown and ignored, it may produce bias.

But it does not follow that all random sampling requires equal probability. Survey theory explicitly includes unequal-probability designs. What is fundamental is that the probabilistic mechanism be specified and that the probabilities required for estimation be known or calculable.

For a population \(U\) and a total

\[ T_Y=\sum_{i\in U}Y_i, \]

if each unit has positive inclusion probability \(\pi_i>0\), the Horvitz-Thompson estimator is

\[ \widehat T_{HT} =\sum_{i\in s}\frac{Y_i}{\pi_i}. \]

Its logic is immediate: a unit that is difficult to select represents more units of the population than a unit with a higher inclusion probability. Hence the inverse of \(\pi_i\) appears.

Under the appropriate design,

\[ E_p\left(\widehat T_{HT}\right)=T_Y, \]

where \(E_p\) denotes expectation with respect to the sampling design. The historical importance of the Horvitz and Thompson (1952) result lies precisely in showing how samples without replacement and with unequal probabilities can be handled without thereby turning them into “nonrandom” samples.

This allows us to return to the problem of duplicates with greater precision. If multiplicity changes \(\pi_i\) in a known way and the estimator incorporates it correctly, the inequality can be handled. If multiplicity is unknown, it becomes a frame problem that prevents the probabilities from being properly known and can distort inference.

· · ·

Some General Consequences

Four principles that cut across survey theory can be drawn from the preceding problems.

First, design and estimation form a unity. There is no estimator whose inferential meaning can be completely separated from the way the sample was produced. Weighting, variance estimation, and the interpretation of intervals depend on the design.

Second, randomness does not necessarily mean equality. A design may be probabilistic and assign unequal probabilities; what matters is that the mechanism be probabilistically defined and that its relevant probabilities can be incorporated into inference.

Third, sampling error is not total survey error. A narrow interval may coexist with undercoverage, differential nonresponse, poor measurement, or processing errors. The mathematical precision of one component should not be confused with overall validity.

Fourth, the statistical operation is also a material operation. Frames, listings, interviews, selection algorithms, questionnaire skips, and editing rules are concrete practices. Theory does not float above them: its assumptions are realized—or violated—in the actual process by which data are produced.

References

AAPOR. (2023). Standard Definitions: Final Dispositions of Case Codes and Outcome Rates for Surveys (10th ed.). American Association for Public Opinion Research. https://aapor.org/standards-and-ethics/standard-definitions/

Biemer, P. P., & Lyberg, L. E. (2003). Introduction to Survey Quality. Wiley. https://doi.org/10.1002/0471458740

Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley.

Groves, R. M. (1989). Survey Errors and Survey Costs. Wiley. https://doi.org/10.1002/0471725277

Groves, R. M., Fowler, F. J. Jr., Couper, M. P., Lepkowski, J. M., Singer, E., & Tourangeau, R. (2009). Survey Methodology (2nd ed.). Wiley.

Horvitz, D. G., & Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47(260), 663-685. https://doi.org/10.1080/01621459.1952.10483446

Kalton, G., & Kasprzyk, D. (1986). The treatment of missing survey data. Survey Methodology, 12(1), 1-16.

Kish, L. (1965). Survey Sampling. Wiley.

Lohr, S. L. (2021). Sampling: Design and Analysis (3rd ed.). CRC Press.

Särndal, C.-E., Swensson, B., & Wretman, J. (1992). Model Assisted Survey Sampling. Springer.

Statistics Canada. (2003). Survey Methods and Practices. Catalogue no. 12-587-X. https://www150.statcan.gc.ca/n1/en/pub/12-587-x/12-587-x2003001-eng.pdf

United Nations Statistics Division. (2008). Designing Household Survey Samples: Practical Guidelines. Studies in Methods, Series F, No. 98. https://unstats.un.org/unsd/publication/seriesf/seriesf_98e.pdf

U.S. Census Bureau. (2021). Questionnaire Testing and Evaluation Methods for Censuses and Surveys. https://www.census.gov/about/policies/quality/standards/appendixa2.html

Additional Resources of Interest

The following resources may be useful for quick orientation, discussion, examples, or bibliographic navigation. They do not replace the methodological sources cited above.

• Cross Validated — survey-sampling tag: technical user discussions about weighting, complex designs, variances, and practical problems. https://stats.stackexchange.com/questions/tagged/survey-sampling
• Wikipedia — “Survey sampling”: an introductory map of concepts and bibliographic links for locating topics. https://en.wikipedia.org/wiki/Survey_sampling
• CRAN — survey package: documentation for the R ecosystem for analysis of complex samples; particularly useful for connecting theory to reproducible implementations. https://cran.r-project.org/package=survey
• Statistics Canada — Survey Methodology: a specialized journal publishing theoretical and applied research on sampling, weighting, nonresponse, integration, and quality. https://www150.statcan.gc.ca/n1/en/catalogue/12-001-x
• United Nations — household survey handbook collection: guides on design, frames, nonsampling errors, and processing. https://unstats.un.org/iswghs/SurveyHandbooksRevision

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7 September, 2026

NATIONAL SURVEY ON THE ASPECTS OF VIRTUALITY RELATED TO THE COVID-19 PANDEMIC (ENAVIRPA 2021)

Sample Surveys · Economics · Generalized Linear Models · Statistical Learning

National Survey on Aspects of Virtuality Linked to the COVID-19 Pandemic (ENAVIRPA 2021)

Economic Impact Module, descriptive results, inferential design, and reproducible evaluation

José Mauricio Gómez Julián · 2021

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Authorship and scope. ENAVIRPA 2021 was a collective project carried out by the students of Introduction to Sample Surveys. The authorship indicated here corresponds to the Economic Impact module, the analytical strategy developed around it, and the writing of this report. The remaining modules are retained as components of the collective project and are attributed to their respective authors.

PROLOGUE

The National Survey on Aspects of Virtuality Linked to the COVID-19 Pandemic (ENAVIRPA 2021) was a collective effort carried out by the students enrolled in Introduction to Sample Surveys at the University of Costa Rica. Fieldwork, questionnaire integration, and the different thematic modules belong to that collective project. This document focuses primarily on the Economic Impact (AE) module, whose design and analysis correspond to José Mauricio Gómez Julián, and makes complementary use of questions from other modules developed by other members of the course. The presence of those items in the analysis does not imply that their construction is attributed to the author of the economic module.

The exercise had a twofold purpose. First, to carry out a real telephone survey during the health crisis and describe material aspects associated with the pandemic. Second, to use the data as a pedagogical setting for studying what can —and cannot— be done with a sample survey when moving from description to inference, generalized linear models, and predictive evaluation.

The scope must be established from the outset. The 235 interviews constitute a real data set, but the number of observations alone is not enough to turn every sample statistic into an unbiased national estimate. Representativeness also depends on the selection mechanism, frame coverage, inclusion probabilities, nonresponse, weighting, measurement error, and questionnaire constraints. The design was carried out under the time and coordination limitations inherent in an academic exercise; moreover, the modules did not originate from a single common theoretical framework. The results are therefore interpreted primarily as results from this sample and as methodological evidence concerning the procedures employed.

It is also useful to distinguish a survey from a psychometric instrument. A survey is a procedure for collecting information and may contain demographic, labor, economic, attitudinal, or psychometric-scale variables. Not every questionnaire is, merely by virtue of being a questionnaire, a psychometric instrument. When a module seeks to measure a latent construct, additional problems of reliability, validity, and measurement error arise and require specific procedures. ENAVIRPA did not undertake a global psychometric validation of the questionnaire; this constitutes a limitation only for interpretations that seek to attribute the measurement of latent constructs to sets of items.

The criterion for incorporating complementary modules was to study the multidimensional distribution of social losses associated with the pandemic: work, income, access to virtual environments, food, hydration, physical activity, and other living conditions. The economy does not exist separately from these concrete determinations, and for that reason the AE module is also analyzed in relation to sociodemographic information and variables from other modules.

One final methodological clarification applies throughout the document: whenever the questionnaire, analytical database, and a published table appear to suggest different numerical codings, the unit of interpretation will always be the substantive category, not the number arbitrarily assigned to it. The ENAVIRPA2021csv.csv database stores AE3-AE9 as textual labels —“Aumentó”, “Disminuyó”, “Sin cambios”, “No es una fuente de ingresos/apoyo”, and “No responde”—. The reproducible analysis in this report uses those labels directly and does not depend on an undocumented numerical recoding.

I. OBJECTIVES

I.1. General objective

To plan, conduct, and analyze a sample survey associated with the health crisis caused by COVID-19, with particular emphasis on its economic impact.

I.2. Specific objectives

1. To describe the main results of the Economic Impact module.
2. To integrate, when conceptually appropriate, information from the sociodemographic, virtual education, technology, telework, and health-habits modules.
3. To examine what types of inference can be formulated from the collected data and under which restrictions.
4. To study generalized linear models, particularly probit and logit, as tools for analyzing binary variables.
5. To show the complementarity between statistical inference and supervised learning, distinguishing fitting, inference, and out-of-sample generalization.
6. To make explicit the restrictions imposed by the sampling design, nonresponse, absence of weighting, coding, skip patterns, and sample size.

II. DESCRIPTIVE METHODOLOGY

II.1. General summary

Element	Description
Study population	Persons aged 18 or older, cell-phone users, and residents of Costa Rica.
Operational frame	Active cellular-number banks constructed from operator prefixes, according to the information used by the course.
Data-collection mode	Computer-assisted telephone interviewing (CATI), using CSPro 7.4.
Selection procedure	Sampling from active cellular-number banks using the Waksberg procedure as a reference.
Originally planned sample size	300 interviews, 30 for each of the ten participating students.
Final sample size	235 completed interviews.
Fieldwork	June–July 2021, according to the course chronology.
Reference sampling error	±6.39 percentage points for a proportion under the idealized SRS calculation with p=0.5; this is not a measure of total survey error.
Weighting	A complete system of design weights, nonresponse adjustments, and calibration was not available.
Scope	Academic exercise with real data; descriptive results are presented primarily as sample results.

II.2. General description of the research process

Fieldwork was carried out by ten students from a graduate program at the University of Costa Rica. The group received training in CSPro 7.4; the modules prepared by the different participants were subsequently assembled, a pilot test of approximately three interviews per student was conducted, and fieldwork then proceeded. The collective target was 300 interviews. A total of 235 were completed.

For the fieldwork carried out by José Mauricio Gómez Julián, 332 calls and approximately 18 hours of telephone contact were recorded. The summary log reports 34 busy numbers, 94 unanswered calls, and 183 calls after inactive numbers were excluded. From those three figures one can compute 1-(128/183)=0.300546; however, that quantity should not automatically be called a “response rate.” The 55 residual cases are not necessarily completed interviews, and the denominator does not fully classify eligibility, refusal, contact, and unknown eligibility. A standardized response rate would require final dispositions for all cases under an explicit definition, such as those used by AAPOR.

Accordingly, this report retains the raw call counts but does not assign the 30.0546% figure the meaning of a standardized response rate. The verifiable fact for the author of the economic module is that the 30 interviews assigned to him were completed; the verifiable collective fact is that 235 of the 300 planned interviews were completed, or 78.3% of the production target. Neither ratio is, by itself, a probabilistic response rate.

II.2.1. Activity schedule

II.2.2. Telephone-number banks

II.2.3. General chronology of fieldwork

II.2.4. Call log

II.3. Presentation of the Economic Impact (AE) module

ECONOMIC IMPACT

AE1	Have you received, or did you receive, any financial benefit or support from the national or local government since the health emergency/quarantine caused by COVID-19 began? YES 2. NO (GO TO AE3) 3. DOES NOT KNOW/NO ANSWER (GO TO AE3)
AE2	Please indicate which of the following benefits/support you receive or received (Multiple response) Yes, food/groceries Yes, financial resources Yes, medical prevention supplies (gloves, masks, disinfectant, etc.) Yes, personal-hygiene supplies (sanitary pads, baby diapers, etc.) NO DOES NOT KNOW/NO ANSWER

II.3.1. Note on the coding of AE3-AE9

In the data-collection instrument, AE3-AE9 appear with the codes 4 = Increased, 3 = No change, 2 = Decreased, and 1 = Not a source of income/support. A later table may order the same categories using a different numbering solely for presentation. A reader could quite reasonably believe, when comparing the two places, that the data were interpreted using incompatible codes. That need not be the case, because the number is only a label and can be recoded; what would be problematic is being unable to determine which substantive category each value represents during analysis.

The available analytical database removes that ambiguity because AE3-AE9 are stored directly as text. Consequently, every table and constructed variable here uses the semantic categories rather than a numerical sequence. Reproducibility therefore does not depend on reconstructing an intermediate recoding that was not documented.

II.4. Items used from other modules

II.4.1. SOCIODEMOGRAPHIC MODULE (SD)

II.4.2. VIRTUAL EDUCATION MODULE (E)

II.4.3. TECHNOLOGY MODULE (TC)

II.4.4. TELEWORK MODULE (TE)

II.4.5. PHYSICAL AND MENTAL HEALTH MODULE (FN)

III. PRESENTATION OF DESCRIPTIVE RESULTS

III.1. AE module: economic impact

III.1.1. AE1. Government financial support

Fifty-six people, 23.8% of the sample, reported having received some benefit or financial support from the national or local government since the beginning of the emergency; 179, or 76.2%, answered that they had not.

III.1.2. AE2. Type of support received

AE2 was conditional on AE1: only the 56 people who answered “yes” to AE1 were supposed to answer it. In the database, the other 179 records correctly appear as missing values generated by the questionnaire skip. They are not “does not know/no answer”; they are not applicable. Since AE2 allowed multiple responses, each percentage must be calculated over the 56 eligible people, and the percentages need not add to 100%.

Thus, among those who received some form of support, 41.1% reported food and 85.7% financial resources. It is not correct to divide each selection by the total number of selections made and describe the resulting figure as a percentage of people.

III.1.3. AE3-AE9. Changes in sources of income and support

Paid work was the source with the greatest presence and also the largest number of contractions: 87 people, 37.0% of the entire sample, reported a decrease; nine, 3.8%, an increase; and 90, 38.3%, no change. In AE6, 192 people, 81.7%, indicated that income from rental property, investments, or savings was not a source of income/support during the observed period.

Means of production as capital: scope of AE6

In the specific sense used in this analysis, the possession of means of production does not refer merely to ownership of instruments personally used by an independent producer, but to their possession as capital, that is, as property generating income separable from the owner’s direct labor.

The distinction is substantive. A self-employed worker may own tools, a vehicle, or a machine and reproduce materially through his or her own labor without the observed income thereby arising from capital in the sense defined here. AE6, by contrast, asks about income from rental property, investments, or savings and therefore serves as an empirical indicator of the presence of property or patrimonial income. Under this operationalization, the 192 cases stating that AE6 is not a source are classified as an observed absence of income associated with property-capital; the 39 cases reporting increase, decrease, or stability are classified as presence of that source; four cases remain indeterminate because of nonresponse.

This is not equivalent to a legal inspection of assets. A person may own an asset that temporarily produces no rent or whose profits are reinvested. That is an observational limitation, not an objection to the concept. The inference should therefore be formulated as the observed absence or presence of income indicative of property-capital within the categories captured by AE6.

TE1 separately retains information on labor status and business ownership. It is useful for distinguishing wage workers, employers, and independent producers, but it should not be mechanically confused with AE6: “own business without employees” and “receives income from property-capital” are related but nonidentical variables.

III.1.4. AE10. Income change among other household members

A total of 44.3% indicated that another member of the household had experienced a decrease in income; 47.7% reported no change and 3.4% reported an increase.

III.1.5. AE11. Sufficiency of family income

A total of 34.5% reported that income was not enough to live on —10.2% with great difficulties and 24.3% with difficulties—; 47.2% said it was just enough, and 17.0% said it was enough and allowed them to save. AE11 directly measures subjective income sufficiency and is kept separate from a formal poverty line.

III.2. TE module: work and telework

III.2.1. TE1. Main activity

The sample contains 105 wage workers, 33 producers with their own business and no employees, and 11 business owners who employ other people. Only the last category directly identifies an employer relation involving the labor of others; the own-business-without-employees category describes independent production. This distinction prevents every ownership of instruments or of a small business from being transformed into capitalist ownership in the substantive sense used for AE6.

III.2.2. TE6. Change in working hours or remuneration

TE6 produced 38 substantive responses and 197 cases that were not applicable because of the module flow. Among the 38 people exposed to the question, 16 —42.1%— reported working more hours. This percentage should not be projected onto all 235 interviews because the question was not asked of every participant.

III.3. FN module: physical-health and nutrition habits

III.3.1. FN2. Hydration

III.3.2. FN6-FN8. Frequency of meals

III.3.3. FN9-FN15. Changes in food groups

A change in frequency does not by itself have a nutritional sign. Reducing sugary drinks or fast food does not have the same meaning as reducing fruits, vegetables, or protein sources. Any index of “diet deterioration” must therefore rest on an explicit nutritional rule rather than on the simple operation “decreased = worsened.”

III.3.4. FN17. Hours of sleep during the pandemic

The data do not show that 57% slept more: 35 people —14.9%— reported an increase in hours of sleep, while 63 —26.8%— reported a decrease and 134 —57.0%— stability. The 57% figure corresponds to “remained the same.”

III.3.5. FN18. Physical activity

Physical activity increased for 61 people —26.0%— and decreased for 57 —24.3%—.

III.4. E module: virtual education

E6 has 161 applicable responses and 74 nonapplicable cases.

Lack of money was reported by 11 people, 6.8% of applicable cases and 4.7% of the entire sample. Lack of time was reported by 27, 16.8% of applicable cases and 11.5% of the total. The denominator must be stated: using 235 describes prevalence in the full sample; using 161 describes the composition of those who actually received E6.

IV. GENERAL DESIGN OF THE INFERENTIAL METHODOLOGY

IV.1. Information of interest

The analytical design sought to extract or construct information on age; internet access; geographic location; nationality; barriers to virtual education; relation to social wealth; changes in diet, hydration, sleep, and physical activity; variation in income; sex; and household economic sufficiency. The intention was to study jointly dimensions that, although separated in the questionnaire for operational reasons, are materially related.

Transforming some responses into dichotomous variables is useful for binary models, but every dichotomization sacrifices information. Binary variables are therefore treated as analytical constructions rather than ontological substitutes for the continuous or multinomial phenomena from which they arise.

IV.2. Measurement mechanism and operationalization

1. Age: SD2, retained quantitatively whenever possible; for some models adult25 = 1 is also constructed if the person is 25 years of age or older.
2. Nationality: SD3; Costa Rican/non-Costa Rican is distinguished and DK/NA is treated as missing.
3. Sex: SD1.
4. Fixed internet: TC1.
5. Rural/urban location: the design intended to construct it from SD13 and INEC’s geographic classification. That classification was not completed during the exercise and therefore does not enter the models actually estimated.
6. Observed property-capital: AE6, according to the substantive definition given above. 1 if the source exists —increased, decreased, or remained unchanged—; 0 if the person states that it is not a source; NA if there is no response.
7. Occupational position/business ownership: TE1 is retained as a separate variable. Employer, independent producer, and wage worker are not collapsed into a single indicator of “capital.”
8. Income contraction: may be defined as the existence of at least one decrease in AE3-AE9 or a decrease in the income of another household member in AE10. AE11 should not be mechanically included in this indicator because it measures current sufficiency, not temporal change.
9. Monthly economic stress: AE11, retained ordinally or dichotomized only when the objective requires it.
10. Barriers to virtual education: E6 is analyzed among applicable cases. It cannot simply be converted into “access/no access” for all 235 people, because absence generated by a skip has a different meaning from a negative response.
11. Diet: FN6-FN15 require differentiated substantive rules by food type. A decrease is not necessarily negative, and an increase is not necessarily positive.
12. Hydration: FN2.
13. Sleep: FN17 and the level of hours in FN16, where relevant.
14. Physical activity: FN18.

IV.3. Dichotomous working variables

To reproduce the core models, the following variables are defined:

• VD1: nationality, 1 = non-Costa Rican, 0 = Costa Rican.
• VD2: observed presence of property-capital income according to AE6.
• VD4: sex, 1 = woman, 0 = man.
• VD7: 1 = age 25 or older, 0 = younger than 25.
• VD8: income contraction under the explicit rule defined in IV.2.

The rurality variable remains a design proposal rather than an estimated variable. The barrier to virtual education is used descriptively unless the analytical universe is restricted to cases for which E6 applies.

IV.4. Proposed combined analyses

The first model studies the observed presence of property-capital income as a function of sex, age, and nationality:

\[ VD2_i = f(VD4_i,VD7_i,VD1_i). \]

Rurality is not included because it was not constructed reproducibly.

A second analysis could study income contraction in relation to sex, age, nationality, and barriers to virtuality, but only after precisely defining the E6 universe and the contraction variable. It is therefore retained as an analytical extension rather than being confused with the model actually estimated in Section IX.

IV.5. Correlation and causality

Correlation can describe covariation; it does not by itself transform an association into a causal mechanism. In social research, causal interpretation requires substantive theory, temporality, design, and identification assumptions that are not automatically contained in a coefficient.

Ritchey uses examples involving poverty, racial composition, and crime to show how an aggregate association may reflect institutional history, segregation, economic exclusion, and omitted variables. The methodological argument is not to replace one correlation with another monocausal explanation, but to remember that the coefficient does not contain the entire causal structure of the phenomenon.

The same caution applies to contemporary racial categories. Findings on the African origin of modern human lineages and later evidence of admixture with archaic populations do not turn social categories of race into discrete biological units capable of replacing historical and causal analysis. Regression quantifies relations under a model; causality requires an additional theory of how the phenomenon is produced.

V. FOUNDATIONS OF SAMPLE SURVEYS

V.1. Mathematical statistics, estimation, and surveys

Cochran emphasizes that survey theory historically adopted simple and robust procedures because a single survey may contain many attributes with different distributions. This does not imply a separation between sampling theory and regression: ratio and regression estimators use auxiliary information to improve precision.

It is useful, however, to distinguish two objects:

\[ \text{regression estimator in sampling} \neq \text{regression model fitted to survey variables}. \]

The former uses known auxiliary information to improve estimation of a population mean or total; the latter represents a conditional relationship among variables and may serve inferential or predictive purposes. The fact that both use the word “regression” does not make their objectives identical.

V.2. Sample size and the actual role of the Central Limit Theorem

A reader could quite reasonably believe that using p=0.5 in a sample-size formula is justified because, as n increases, the original responses “become normal,” eventually distribute themselves symmetrically, and therefore half must fall in each category. That is not the case.

For a Bernoulli variable X_i taking values 0 and 1, the sample proportion is:

\[ \hat p=\frac1n\sum_{i=1}^nX_i, \]

with:

\[ E(\hat p)=p, \qquad \operatorname{Var}(\hat p)=\frac{p(1-p)}n \]

under the ideal independent scheme. In sampling without replacement from a finite population there is also a finite-population correction:

\[ \operatorname{Var}(\hat p) =\frac{p(1-p)}n\frac{N-n}{N-1}. \]

The Central Limit Theorem enters elsewhere. Under appropriate conditions, it allows the sampling distribution of the standardized proportion to be approximated by a normal distribution:

\[ \frac{\hat p-p}{\sqrt{p(1-p)/n}} \approx N(0,1). \]

This is a statement about the distribution of a statistic across repeated samples, not a claim that the original distribution of responses itself becomes normal. Nor does it imply that p tends to 0.5.

The reason for using p=0.5 when the population proportion is unknown is algebraic. Bernoulli variance depends on:

\[ f(p)=p(1-p)=p-p^2. \]

Its derivative is:

\[ f'(p)=1-2p, \]

and the maximum occurs at:

\[ p=\frac12, \qquad p(1-p)=\frac14. \]

Therefore:

\[ p(1-p)\le 0.25. \]

Choosing p=0.5 maximizes the variance and yields the most conservative sample size for a prespecified margin of error. For a large population, or one of unknown size:

\[ n_0=\frac{z_{\alpha/2}^2p(1-p)}{d^2}. \]

With z=1.96, p=0.5, and d=0.0639:

\[ n_0\approx235.2. \]

This gives the scale of the 235 interviews. The result makes sense as an idealized calculation of sampling precision for a proportion under SRS. By itself, it does not demonstrate that the actual design has total error of ±6.39%.

If sample size approached the size of a finite population, the factor (N-n)/(N-1) would tend to zero. The reason is elementary: when nearly the entire population is observed, progressively less uncertainty remains because of sampling. The original responses do not undergo a “normalization.”

V.3. Nonresponse and the Birnbaum-Sirken formula

Birnbaum and Sirken, as discussed by Cochran, studied bias due to nonavailability. The expression discussed in the original paper can be written, using the notation employed there, as:

\[ n=\frac{t_\alpha^2}{4d(d-W_2)W_1}-1. \]

The condition built into the formula itself shows that there is no admissible n when W_2>d. If substitution of a nonresponse rate larger than the tolerated error changes the sign of the denominator and produces n<0, the result is not a physically negative magnitude that should be repaired by taking its absolute value. It indicates that the calculation has left the domain in which the formula can satisfy the stated guarantee.

An analogy with the absolute value of an economic elasticity does not repair this violation of conditions. The correct conclusion is simply: the formula does not provide an admissible solution for that combination of nonresponse and tolerated error.

This point is also consistent with Cochran’s warning: substantial amounts of nonresponse are not mechanically neutralized by increasing sample size. If nonrespondents systematically differ from respondents, bias may persist even with a large n.

V.4. Types of nonresponse and call dispositions

Cochran distinguishes problems such as noncoverage, temporary absence, inability to respond, and persistent refusal. In a modern telephone survey it is also useful to distinguish completed interview, partial interview, refusal, noncontact, nonresidential number, ineligible case, and unknown eligibility. Only with such final dispositions can a standardized response rate be constructed.

The ENAVIRPA fieldwork manual distinguished busy number, no answer, inactive, completed, pending, not completed, incomplete, refusal, business, and duplicate. The raw log is useful for documenting the process, but a standard rate requires the final disposition of each unit and explicit denominator rules.

V.5. Margin of error and total survey error

Under an ideal SRS, n=235, p=0.5, and 95% confidence give:

\[ 1.96\sqrt{\frac{0.25}{235}}\approx0.0639. \]

That is, ±6.39 percentage points. This number refers to idealized sampling error. It does not automatically incorporate coverage, differential nonresponse, measurement, processing, telephone selection, weighting, or the additional uncertainty of subsequent models.

V.6. Expansion factor and weighting

The base weight of a probabilistically selected unit is the inverse of its inclusion probability:

\[ w_i=\frac1{\pi_i}. \]

In a multistage design, if a geographic unit h, a household j within h, and a person i within the household have conditional probabilities f_h, f_(j|h), and f_(i|jh), the joint probability is:

\[ f_{ijh}=f_hf_{(j|h)}f_{(i|jh)}, \]

and the basic weight is:

\[ W_{ijh}=\frac1{f_{ijh}} =\frac1{f_h}\frac1{f_{(j|h)}}\frac1{f_{(i|jh)}}. \]

Similarity between some sample and population proportions does not prove that weighting is unnecessary. The decision depends on inclusion probabilities and subsequent adjustments for nonresponse and calibration. ENAVIRPA did not reconstruct a complete weighting system; therefore its estimates are not presented as though every observation had been demonstrated to carry the same population weight.

VI. GENERALIZED LINEAR MODELS

The generalized linear model is one of the clearest examples of how a statistical theory can develop by preserving an earlier structure while simultaneously extending its domain of application. The classical linear model does not disappear: it remains contained as a special case within a more general construction. The essential idea is to preserve a linear systematic component,

\[ \eta_i=x_i^T\beta, \]

while abandoning the requirement that every response variable be Gaussian and that its mean coincide directly with the linear predictor. The conditional mean,

\[ \mu_i=E(Y_i\mid x_i), \]

is connected to the predictor through a link function:

\[ g(\mu_i)=\eta_i. \]

In this way, Gaussian linear regression, logistic regression, Poisson count models, and certain gamma specifications can be studied within a common framework.

VI.1. Preliminary concepts: variables, scales, and predictors

A traditional classification of measurement scales distinguishes nominal, ordinal, interval, and ratio variables. The distinction between discrete and continuous, however, belongs to a different axis. Number of children, for example, is discrete but has a meaningful zero and interpretable ratios; it can therefore be regarded as a discrete variable on a ratio scale. It is not appropriate to turn “discrete/continuous” into additional levels of the nominal-ordinal-interval-ratio hierarchy.

The word covariate also deserves clarification. Some applied traditions used it in a restricted sense for continuous predictors. In modern regression theory it is clearer to use it broadly: a covariate or predictor may be continuous, discrete, binary, or categorical through appropriate coding. This is different from covariance, which denotes a measure of association between variables.

The word factor likewise depends on context. In experimental design it may denote a controlled or explanatory categorical variable; in factor analysis it denotes a latent construct. Neither meaning should be inferred solely from the name assigned to a column by a software package.

These clarifications matter because the model is not determined by the computer format of a variable, but by the mathematical and probabilistic structure of the phenomenon one wishes to represent.

VI.2. Genetic connection with linear regression

Gujarati and Porter recall that the term regression originated in Galton’s work on the heights of parents and children. Its contemporary meaning is much broader: the aim is to represent how the distribution of a response —and very often its conditional mean— changes when one or more predictors change.

In the simple linear model:

\[ E(Y_i\mid x_i)=\alpha+\beta x_i. \]

In matrix form:

\[ \mu=X\beta. \]

The fitted line is not merely a geometric object superimposed on points. Its coordinates represent observed magnitudes, and its parameters acquire statistical meaning from a probabilistic specification. This distinction allows us to separate geometric fitting from a statistical model, even when both share mathematical tools.

VI.2.1. Regression and causality

Regression does not automatically transform an association into causality. A coefficient can quantify a conditional relationship under a set of assumptions without identifying, by itself, the mechanism that produces it. Causal interpretation requires substantive theory, temporal structure, design, interventions, or additional identification assumptions.

Ritchey uses examples involving poverty, racial composition, and crime to show that an aggregate correlation may condense institutional history, segregation, economic exclusion, selection, and omitted variables. The methodological point is not to replace a bivariate association with another monocausal explanation; it is to deny that the coefficient contains, by itself, the totality of the causal process.

The same caution applies to contemporary racial categories. Genetic evidence concerning a recent African origin of modern human lineages and later admixture with archaic populations does not transform social racial categories into discrete biological units that can replace historical analysis. Regression measures relations under a representation; causal explanation requires reconstructing how those relations are produced.

VI.3. Regression and least squares are not synonyms

Three objects should be distinguished: model, estimation criterion, and numerical algorithm. In the classical Gaussian model:

\[ Y=X\beta+\varepsilon, \qquad \varepsilon\sim N(0,\sigma^2I), \]

maximum-likelihood estimation of \(\beta\) coincides with minimizing:

\[ S_2(\beta)=\sum_{i=1}^n(y_i-x_i^T\beta)^2 =\|y-X\beta\|_2^2. \]

This is the source of the close historical relation between linear regression and least squares. But regression can use other criteria —absolute deviations, robust losses, penalization— and least squares can also be used in problems that do not seek to construct a complete probabilistic model. The coincidence in the Gaussian case does not justify identifying the two concepts.

VI.4. Geometric digression: norms, metrics, convexity, and discrepancy

Geometry helps explain why least squares has such a natural form. If

\[ e=y-X\beta, \]

OLS minimizes \(\|e\|_2^2\). In a finite-dimensional vector space, for \(p\ge1\),

\[ \|x\|_p=\left(\sum_i|x_i|^p\right)^{1/p} \]

is a norm and induces the metric

\[ d(x,y)=\|x-y\|_p. \]

The logical relation should not, however, be reversed: not every metric is induced by a norm, and a topology does not by itself determine a unique distance. Topological structure determines which sets are open and which notions of continuity are preserved; metric structure adds further information about distance.

Convexity should likewise not be confused with subadditivity and positive homogeneity. A function \(F\) is convex when

\[ F(\lambda x+(1-\lambda)y) \le \lambda F(x)+(1-\lambda)F(y), \qquad 0\le\lambda\le1. \]

Norms are convex and satisfy additional properties, but those properties are not the general definition of convexity.

McCullagh and Nelder use discrepancy in a broader sense than metric distance. In least squares, discrepancy has an immediate Euclidean interpretation. In a GLM, by contrast, a central measure is the deviance:

\[ D=2\{\ell(\text{saturated model})-\ell(\text{fitted model})\}. \]

Deviance arises from log-likelihoods and need not satisfy the axioms of a metric. The geometric intuition can be preserved —searching within a class for a representation that disagrees as little as possible with the data under a criterion— but the criterion need not be an \(L_p\) norm.

VI.5. Exponential families

Nelder and Wedderburn identified a class of distributions broad enough to contain several models of interest and structured enough to support a common theory. A standard form of an exponential dispersion family is:

\[ f(y_i;\theta_i,\phi) = \exp\left\{ \frac{y_i\theta_i-b(\theta_i)}{a(\phi)} +c(y_i,\phi) \right\}. \]

Here \(\theta_i\) is the natural or canonical parameter and \(\phi\) the dispersion parameter. Under the usual regularity conditions:

\[ E(Y_i\mid x_i)=b'(\theta_i)=\mu_i, \]

\[ \operatorname{Var}(Y_i\mid x_i)=a(\phi)b''(\theta_i). \]

In standard GLM notation this is summarized as:

\[ \operatorname{Var}(Y_i\mid x_i)=\phi V(\mu_i), \]

apart from known weight factors introduced by particular parameterizations. The function \(V(\mu)\) is the variance function. This expression displays an essential generalization: in the homoskedastic Gaussian model variance may be constant, whereas in Bernoulli and Poisson models it depends systematically on the mean.

For example:

VI.6. The three components of a GLM

VI.6.1. Stochastic component

The response belongs to an exponential family. A standard GLM generally works with conditionally independent responses:

\[ Y_i\mid x_i\sim\mathcal F(\mu_i,\phi). \]

This does not require identical distributions in the sense of sharing the same mean, because

\[ \mu_i=g^{-1}(x_i^T\beta) \]

may vary from one observation to another.

That statement should be distinguished from a different historical observation by McCullagh and Nelder. In discussing general principles of discrepancy, the authors state that observations should be independent or at least interchangeable in some sense that permits impartial treatment. That formulation belongs to their conceptual framework and is not equivalent to requiring equality of all conditional means in a regression.

VI.6.2. Systematic component

The predictors generate:

\[ \eta_i = \beta_0+\beta_1x_{i1}+\cdots+\beta_px_{ip} = x_i^T\beta. \]

The relevant linearity is linearity in the parameters. Thus,

\[ \eta=\beta_0+\beta_1x+\beta_2x^2 \]

is nonlinear in \(x\) but remains linear in \(\beta_0,\beta_1,\beta_2\).

VI.6.3. Link function

The mean and the predictor are connected through:

\[ g(\mu_i)=\eta_i, \qquad \mu_i=g^{-1}(\eta_i). \]

The link makes it possible to model the mean on a scale appropriate to its domain. A probability must remain between 0 and 1; a Poisson mean must be positive. The transformation allows the linear predictor to operate on a more convenient scale without violating these restrictions.

VI.7. Identity link and canonical link

The identity link is:

\[ g(\mu)=\mu, \]

and only in that case:

\[ \mu=\eta. \]

A canonical link means something different: \(g\) is chosen so that the linear predictor coincides with the natural parameter of the family:

\[ \boxed{\eta_i=\theta_i}. \]

Thus identity is the canonical link for the Gaussian, logit for the binomial, and log for the Poisson. For Gamma, the canonical link is inverse, with the sign convention depending on the parameterization.

The canonical link has algebraic advantages, but it is not compulsory. A binomial response may use probit instead of logit if the theoretical or empirical construction justifies it.

VI.8. Maximum likelihood

Parameters are normally estimated by maximizing:

\[ \ell(\beta)=\sum_{i=1}^n\ell_i(\beta). \]

In general the score equations,

\[ U(\beta)=\frac{\partial\ell}{\partial\beta}=0, \]

do not admit a closed-form algebraic solution. This leads to iterative procedures.

VI.9. IRLS: distinguishing two algorithms that share a name

There is a family of algorithms called iteratively reweighted least squares used for \(L_p\) losses, approximations to least absolute deviations, and robust regression. In that setting weights such as

\[ w_i\propto|e_i|^{p-2} \]

may appear. The formulation discussed by Burrus belongs to this context. It is a legitimate IRLS algorithm, but it does not explain the IRLS ordinarily used to fit GLMs.

In GLMs, IRLS arises from applying Fisher scoring to the log-likelihood; with canonical links there is an especially close relationship with Newton-Raphson. The conceptual sequence is:

\[ \text{exponential family} \rightarrow \text{log-likelihood} \rightarrow \text{score} \rightarrow \text{Fisher information} \rightarrow \text{Fisher scoring} \rightarrow \text{IRLS}. \]

VI.9.1. Score and Fisher information

Let

\[ \eta_i=x_i^T\beta, \qquad \mu_i=g^{-1}(\eta_i). \]

The score can be written, omitting parameterization details that do not alter the central idea, as:

\[ U(\beta) = X^T \operatorname{diag}\left( \frac{d\mu_i/d\eta_i}{\phi V(\mu_i)} \right) (y-\mu). \]

The expected information takes the form:

\[ \mathcal I(\beta)=X^TWX, \]

with working weights:

\[ \boxed{ w_i = \frac1\phi \frac{(d\mu_i/d\eta_i)^2}{V(\mu_i)} }. \]

When \(1/\phi\) is a common factor it can be omitted when updating \(\beta\).

VI.9.2. Working response

At iteration \(t\):

\[ \eta_i^{(t)}=x_i^T\beta^{(t)}, \qquad \mu_i^{(t)}=g^{-1}(\eta_i^{(t)}). \]

The first-order linear approximation leads to the working response:

\[ \boxed{ z_i^{(t)} = \eta_i^{(t)} + g'(\mu_i^{(t)})(y_i-\mu_i^{(t)}) }. \]

Since \(g'(\mu)=d\eta/d\mu\), it may also be written:

\[ z_i^{(t)} = \eta_i^{(t)} + \frac{y_i-\mu_i^{(t)}}{d\mu_i/d\eta_i}. \]

VI.9.3. Weighted step

The update is obtained by solving:

\[ \boxed{ \beta^{(t+1)} = (X^TW^{(t)}X)^{-1}X^TW^{(t)}z^{(t)} }. \]

Hence the name of the algorithm: at each iteration a weighted least-squares problem is solved, but the weights and the working variable change because they depend on the estimated mean at the current iteration.

The procedure can be summarized as follows:

1. choose admissible initial values;
2. compute \(\eta^{(t)}\);
3. obtain \(\mu^{(t)}=g^{-1}(\eta^{(t)})\);
4. construct \(W^{(t)}\);
5. construct \(z^{(t)}\);
6. solve the weighted problem;
7. repeat until a convergence criterion is satisfied.

Three cases show the unity of the method:

The Gaussian case reveals the connection with OLS: because the weights are constant, the iteration reduces to the ordinary linear problem.

VI.10. What does not explain IRLS convergence

It may be tempting to think that reweighting “brings” observations closer together, reduces predictor variance, and that the Central Limit Theorem makes the algorithm converge toward a normal distribution. That does not happen. The weights depend on conditional variance and the derivative of the link; they do not physically move the values of \(X\) or force \(\operatorname{Var}(X)\to0\).

The CLT appears elsewhere in the theory: under regularity conditions it contributes to the asymptotic normal approximation of maximum-likelihood estimators. That is an inferential property of sequences of estimators, not the numerical mechanism of IRLS.

Nor is it sufficient to write

\[ \beta^{(t+1)}=F(\beta^{(t)}) \]

to invoke Banach’s fixed-point theorem. That theorem requires proving that \(F\) is a contraction on an appropriate complete space. The mere existence of an iteration does not prove contractivity. In practice, GLMs may exhibit convergence problems due to poor numerical conditioning, collinearity, separation, boundary solutions, or problematic specifications.

VI.11. Sufficient statistics under a canonical link

When

\[ \theta_i=\eta_i=x_i^T\beta, \]

the part of the log-likelihood jointly linking \(y\) and \(\beta\) contains:

\[ \frac1{a(\phi)}\beta^TX^Ty. \]

For fixed dispersion and under the usual conditions of an exponential family, the relevant dependence of the sample on \(\beta\) appears through \(X^Ty\). This explains the sufficient-statistic structure emphasized in classical GLM theory. In canonical log-linear models with factors, the score equations may additionally imply equality between certain observed and fitted margins; this is a property of those specifications, not a universal rule for every link.

The conceptual unity can finally be summarized as:

\[ \boxed{ \text{probability family} + \text{linear predictor} + \text{link function} }. \]

The generalization does not consist in erasing differences among distributions, but in identifying which structure can remain while their particular determinations change systematically.

VII. PROBIT AND LOGIT MODELS

Probit and logit models arise when the response is binary and the immediate object is to model a conditional probability:

\[ p_i=P(Y_i=1\mid x_i). \]

The problem with the linear probability model is that \(x_i^T\beta\) can take any real value, whereas a probability must remain in \([0,1]\). Probit and logit links solve this incompatibility by transforming the probability onto an unrestricted scale.

VII.1. Probit model

The probit uses the standard normal distribution function:

\[ \Phi^{-1}(p_i)=x_i^T\beta, \]

so that:

\[ p_i=\Phi(x_i^T\beta). \]

The observed response remains Bernoulli; the normal distribution appears in the link function or, equivalently, in a latent-variable interpretation in which an unobserved normal variable crosses a threshold and generates the 0/1 response. It should therefore not be claimed that the observed binary variable is normal.

One may imagine:

\[ Y_i^*=x_i^T\beta+u_i, \qquad u_i\sim N(0,1), \]

\[ Y_i=1\quad\text{if}\quad Y_i^*>0, \]

which yields precisely \(P(Y_i=1\mid x_i)=\Phi(x_i^T\beta)\).

VII.2. Logit model

The logit uses:

\[ \operatorname{logit}(p_i) = \log\left(\frac{p_i}{1-p_i}\right) = x_i^T\beta. \]

The odds satisfy:

\[ \frac{p}{1-p}\in(0,\infty), \]

whereas their logarithm ranges over the entire real line. Inverting the link:

\[ p_i = \frac{1}{1+\exp(-x_i^T\beta)}. \]

The general logistic function can be written:

\[ f(x)=\frac{L}{1+e^{-k(x-x_0)}}, \]

with asymptotes 0 and \(L\) and central symmetry around \((x_0,L/2)\). In ordinary binary regression one takes \(L=1\).

VII.2.1. Interpretation of coefficients

In logit, a coefficient \(\beta_j\) represents the change in log-odds associated with a one-unit increase in \(x_j\), holding the other covariates constant. Therefore:

\[ e^{\beta_j} \]

is the multiplicative odds ratio associated with that change. An odds ratio greater than 1 implies increased odds; less than 1, reduced odds; equal to 1, no multiplicative change.

This should not be confused with a constant marginal effect on probability. Since

\[ p(x)=\Lambda(x^T\beta), \]

the marginal effect of a continuous covariate is:

\[ \frac{\partial p}{\partial x_j} = \beta_jp(1-p), \]

so it depends on the individual’s position on the logistic curve.

VII.3. Probit and logit: similarities and differences

Both links are sigmoidal, monotonic, and yield admissible probabilities. Their results are often qualitatively similar in many applications; their coefficients, however, are not on the same scale. Logit has a particularly convenient interpretation through odds ratios; probit is naturally connected to a normal latent-variable formulation.

The choice should not be reduced to habit. It may depend on disciplinary tradition, interpretability, substantive assumptions, fit comparison, or computational convenience.

VII.4. Inference

Maximum likelihood yields \(\hat\beta\). Under regularity conditions, the inverse information matrix approximates the covariance matrix of the estimators. An individual Wald test uses:

\[ z_j=\frac{\hat\beta_j}{SE(\hat\beta_j)}. \]

Likelihood-ratio tests compare nested models through:

\[ LR=2\{\ell(\hat\beta_1)-\ell(\hat\beta_0)\}. \]

The AIC criterion is:

\[ AIC=-2\ell(\hat\beta)+2k. \]

These quantities answer different questions. An individual p-value does not summarize predictive ability; an AIC does not prove causality; and a model that classifies reasonably well does not automatically make all of its coefficients well-identified parameters.

VII.5. Separation and unstable estimates

In binary regression, complete separation may occur when a combination of predictors perfectly classifies the cases, or quasi-complete separation when separation occurs except for some overlaps. Under those conditions, ordinary maximum likelihood may drive some coefficients toward very large magnitudes; Wald standard errors grow and conventional inference ceases to be reliable.

Thus enormous coefficients accompanied by enormous standard errors should not simply be interpreted as “very large but nonsignificant effects”: they may be a geometric symptom of separation or extremely sparse cells. Possible responses include reviewing coding, simplifying categories when substantively defensible, obtaining more observations, penalization, or bias-reduction methods such as Firth’s correction.

VIII. STATISTICAL LEARNING THEORY

VIII.1. General definition

Following James, Witten, Hastie, and Tibshirani, an elementary representation of supervised learning is:

\[ Y=f(X)+\varepsilon. \]

The problem is to estimate the unknown function \(f\) from the data. In some problems the main interest is predicting new observations; in others, understanding which variables are associated with the response and how; in many cases both goals coexist.

The expression learning from data does not mean dispensing with theory. Every choice of variables, representation, functional class, loss, and validation procedure contains decisions about which regularities are regarded as possible and relevant.

Historically, statistics developed a tradition particularly concerned with probability models, estimation, uncertainty, and inference, whereas machine learning emerged under a strong influence from computer science, algorithms, and predictive performance. This difference helps explain cultural emphases, but it does not establish an essential boundary. Statistics predicts, and machine learning uses probability and inference.

Nor is it correct to say that machine-learning methods “make few assumptions” in an absolute sense. Many shift assumptions from an explicit parametric form toward an inductive bias: hypothesis class, architecture, loss function, regularization, invariances, and training rules.

A hypothesis space or class can be written:

\[ \mathcal H=\{h:X\rightarrow Y\}. \]

It is not the same object as the \(H_0\) and \(H_1\) hypotheses of Neyman-Pearson. There is a very broad logical kinship —delimiting possibilities— but the mathematical objects and inferential problems are different.

VIII.2. Learning paradigms

In an elementary classification:

• supervised learning: pairs \((x_i,y_i)\) are observed and the aim is to learn a relation that predicts \(y\) from \(x\);
• unsupervised learning: \(x_i\) are observed without an explicit supervising response, and patterns or representations are sought;
• reinforcement learning: an agent interacts with an environment through states, actions, and rewards.

Online learning belongs to another axis. It describes a procedure that updates the model as observations arrive. There can be supervised online learning, unsupervised online learning, and so forth.

In unsupervised learning, clustering is only one possibility. Others include dimensionality reduction, density estimation, anomaly detection, association rules, and representation learning.

VIII.3. Binary classification

If the positive class is the event of interest, there are four possibilities:

• true positive (TP): actual positive, predicted positive;
• false positive (FP): actual negative, predicted positive;
• true negative (TN): actual negative, predicted negative;
• false negative (FN): actual positive, predicted negative.

The confusion matrix organizes these cases:

The matrix may be computed on training, validation, test, or external data. When it is computed on the same cases used to estimate the model, it describes in-sample performance.

VIII.4. Classification metrics

Accuracy is:

\[ Accuracy=\frac{TP+TN}{TP+TN+FP+FN}. \]

The error rate is its complement:

\[ Error=\frac{FP+FN}{N}=1-Accuracy. \]

Sensitivity or recall is:

\[ Sensitivity=\frac{TP}{TP+FN}, \]

and measures the proportion of actual positives detected.

Specificity is:

\[ Specificity=\frac{TN}{TN+FP}. \]

Classification precision, also called positive predictive value, is:

\[ Precision=\frac{TP}{TP+FP}. \]

Negative predictive value is:

\[ NPV=\frac{TN}{TN+FN}. \]

A terminological ambiguity should be avoided: classification precision is not metrological precision understood as low dispersion among repeated measurements. They are different concepts that share a word.

Accuracy can be misleading with imbalanced classes. If 90% of cases belong to the negative class, a classifier that always answers “negative” obtains 90% accuracy and yet has zero sensitivity. Every evaluation should therefore be compared with a baseline and with metrics consistent with the scientific cost of errors.

VIII.5. Training error and generalization error

For a classifier \(\hat f\), the observed error on the training data is:

\[ \widehat{Err}_{train} = \frac1n\sum_{i=1}^nI(y_i\ne\hat y_i). \]

But the predictive objective is not to memorize already observed data. What matters is performance on a new observation \((X_0,Y_0)\):

\[ Err_{test} = E\left[I\{Y_0\ne\hat f(X_0)\}\right]. \]

The difference is fundamental. A highly flexible model may fit the training data extraordinarily well and fail on new observations. This is overfitting. Test sets, cross-validation, or other procedures are used to estimate generalization while preventing the algorithm from being evaluated only on cases it already used to learn.

VIII.6. Probabilities, thresholds, and classification

A probabilistic model such as logit produces \(\hat p_i\), not logically necessary classes. To classify, a threshold \(c\) must be chosen:

\[ \hat Y_i=I\{\hat p_i>c\}. \]

Changing \(c\) changes the confusion matrix. Lowering the threshold usually detects more positives and raises sensitivity, but it may also produce more false positives and reduce specificity. Thus 0.5 is not a sacred value, and 0.25 cannot be adopted without explanation either. The choice should follow costs, prevalence, the study objective, or a rule defined on validation data, not the value that produces the most favorable appearance on the training sample.

ROC curves examine sensitivity against the false-positive rate across all thresholds. In problems with a rare positive class, precision-recall curves may be especially informative because they display directly the tradeoff between positive detection and the purity of positive predictions.

VIII.7. GLMs and supervised learning

GLMs are historically statistical models. Nevertheless, when fitted to \((X,Y)\) pairs to predict new responses they also function as supervised-learning procedures. Bayesian treatment is not what creates this status: a frequentist, Bayesian, or penalized GLM may be used predictively.

Logistic regression is the clearest example. It models:

\[ \log\left( \frac{P(Y=1\mid X)}{1-P(Y=1\mid X)} \right) =X\beta. \]

By its structure, it is a probabilistic regression model. But its probabilities can legitimately be converted into classes through a threshold. There is no contradiction:

\[ \boxed{ \text{logistic regression = probabilistic regression model} } \]

and simultaneously:

\[ \boxed{ \text{it can be used as a supervised classifier} }. \]

The historical origin of the method explains its construction, but does not exhaust all of its later scientific uses.

VIII.8. Inference and prediction are not two different models

Different questions can be asked of the same logit. From an inferential perspective: how much does the conditional association change with each covariate? How uncertain is the coefficient? Which model is compatible with the data? From a predictive perspective: what probability does it assign to an unseen case? Does it discriminate better than a baseline? What errors does it make out of sample?

These questions require different diagnostics, but they do not automatically turn logit into two algorithms. The same fit can be studied for both inferential and predictive properties. This is precisely why an individual coefficient may be imprecise while a combination of predictors still has some predictive utility; or, conversely, a model may contain statistically detectable associations yet discriminate new cases poorly.

The ENAVIRPA evaluation that follows deliberately keeps the two questions separate.

IX. REPRODUCIBLE INFERENTIAL RESULTS

IX.1. Analytical database and coding audit

The ENAVIRPA2021csv.csv database contains 235 rows and 161 variables. AE1 contains 56 “SÍ” and 179 “NO”. In AE2A-AE2D, the 179 cases excluded by the AE1 skip appear as NA, confirming that they must be classified as not applicable rather than DK/NA.

AE3-AE9 already appear in the database as textual labels. The visible discrepancy between the questionnaire codes 4-3-2-1 and the 1-2-3-4 order used in some presentation tables therefore ceases to affect reproducibility: categories are processed by name.

IX.2. Main variable: property-capital income

AE6 contains:

• 192 cases “No es una fuente de ingresos/apoyo”;
• 21 “Sin cambios”;
• 13 “Disminuyó”;
• 5 “Aumentó”;
• 4 “No responde”.

The main binary variable is therefore defined with 39 positive cases, 192 negative cases, and four missing cases. After also removing four missing ages and one DK/NA nationality, the complete model sample consists of 227 observations: 39 positive and 188 negative.

IX.3. Correlations among the model’s binary variables

With capital=1 for presence of the AE6 source, mujer=1, adulto25=1, and extranjero=1, the Pearson correlation matrix —equivalent to the phi coefficient when both variables are binary— is:

The associations are small in magnitude. This anticipates that a model using these three predictors will have little ability to explain variation in the AE6 indicator within this sample.

IX.4. Logit model

The estimated model is:

\[ \operatorname{logit}[P(VD2_i=1)] =\beta_0+\beta_1Woman_i+\beta_2Adult25_i+\beta_3Foreign_i. \]

The model contains 227 observations, a log-likelihood of -103.51, a null log-likelihood of -104.13, McFadden’s pseudo-R² of 0.0060, and a joint likelihood-ratio test with p=0.740. None of the three predictors provides individual statistical evidence of an association different from zero at conventional levels in this specification.

Unlike the operationalization based on TE1 business ownership, the AE6 indicator does not produce explosive coefficients or standard errors measured in thousands of units. This difference confirms that construct specification and cell distribution can radically alter the numerical behavior of a model.

IX.5. Probit model

The probit leads to the same substantive conclusion: sex, dichotomized age, and nationality together have very little explanatory ability for the observed property-capital income indicator in AE6. Logit and probit use different link scales, so their coefficients should not be compared one by one in magnitude.

IX.6. Rurality and other planned predictors

Rurality appeared in the conceptual design but was not constructed during the available period. It would not be methodologically correct to include it by pretending that a textual locality is already a rural/urban classification. INEC classifies geographic units using specific criteria; until that classification is applied reproducibly to SD13, the variable must remain outside the model.

Similarly, E6 does not provide a universal variable of “access to virtual education”: it was asked only of 161 cases according to questionnaire flow. It can be analyzed as a barrier among applicable cases or incorporated into a model restricted to that universe, but it cannot be imputed as “yes” or “no” to the 74 cases omitted by the skip.

X. PREDICTIVE EVALUATION: WHAT IT SHOWS AND WHAT IT DOES NOT SHOW

The same logistic regression can be studied from two perspectives. In inference, we ask what the coefficients and their uncertainty indicate about a modeled relationship. In prediction, we ask how well the estimated probabilities generalize to observations the algorithm did not see during fitting. These are not necessarily two different models: they can be two evaluations of the same model.

A reader might believe that it is enough to fit the logit to the 235 observations and then compute a confusion matrix on those same observations. That answers only: “how well does the model reproduce the data on which it was estimated?” It does not answer: “how well will it predict new cases?” This is the difference between in-sample performance and generalization.

Under the AE6 operationalization used here, the fitted logit probabilities range approximately from 0.129 to 0.208. With the threshold c=0.25, no observation exceeds the cutoff. The resulting classifier predicts “no property-capital income” for all 227 people:

The 82.8% accuracy is not a predictive victory: it coincides with the proportion of the majority class —82.8%— and is obtained at the cost of zero sensitivity. The model identifies no positive cases at this threshold.

This illustrates why accuracy should not be evaluated in isolation with imbalanced classes. An algorithm that always predicts the majority class can achieve high accuracy and nevertheless be useless for detecting the minority class.

The threshold is not a universal mathematical constant either. Choosing 0.25, 0.5, or another value changes the tradeoff between sensitivity and specificity. It should be defined by a cost function, a substantive need, or a validation procedure, not by post-hoc inspection of the same sample.

Finally, five-fold stratified cross-validation on these three predictors yields, as a reference, an approximate ROC AUC of 0.485 and an average precision close to 0.167. These results are compatible with virtually no discriminative ability in this specification. The pedagogical objective is not to make “classical statistics” or “machine learning” win, but to show that a predictive claim requires out-of-sample evaluation.

XI. LIMITATIONS

XI.1. Collective character of the design

The modules were designed by different members of the course and assembled for a common survey. There was no single theoretical framework determining every question from the outset. This limits later constructions that require combining items originally written for different objectives.

XI.2. Number of interviews

Three hundred interviews were planned and 235 were completed. The difference reduces precision and, above all, limits subgroup analyses and models with many categories. The issue is not that 235 is intrinsically “small”; adequacy depends on prevalence, the number of parameters, and cell structure.

XI.3. Design, coverage, and weighting

The telephone frame excludes persons outside operational coverage and may produce unequal inclusion. Complete design weights and nonresponse/calibration adjustments were not reconstructed. Results are therefore not presented as fully weighted national estimates.

XI.4. Nonresponse

The aggregate call-log data do not permit calculation of a complete AAPOR response rate. In addition, nonresponse and noncontact may be associated with socioeconomic conditions, so merely increasing n does not guarantee absence of bias.

XI.5. Rurality

The rural/urban variable was conceived but not constructed. INEC assigns degrees of urbanization to specific geographic units; the work needed to link SD13 to that classification was not completed.

XI.6. Skip patterns and analysis universes

AE2, TE6, and E6 show why NA does not have a single meaning. It may mean not applicable because of a skip, nonresponse, or missing data. The correct denominator depends on the universe of each question. Treating every missing value as DK/NA changes percentages and may bias derived variables.

XI.7. Dichotomous variables

Dichotomizing age, diet, income, or economic stress simplifies modeling but sacrifices gradients. Whenever possible, original variables should be retained and dichotomies used only when justified by the model or substantive question.

XI.8. Capital as an observed variable

AE6 is an indicator of property-capital income, not an asset census. It can distinguish observed presence/absence of that source during the period, but it cannot legally verify every asset owned by a person or detect capital with no observed flow. This is an observational limitation.

XI.9. Income and poverty

SD5 records bands of gross monthly household income, not continuous equivalized disposable income. It therefore cannot directly implement the 60%-of-median-equivalized-disposable-income methodology used by the European/UNECE at-risk-of-poverty indicator.

XI.10. Causality

The study is observational and cross-sectional. Regression associations do not, by themselves, identify causal effects of sex, age, nationality, or other variables.

XI.11. Predictive evaluation

Training performance is not a measure of generalization. Any predictive claim should rely on independent test data or appropriate cross-validation and should consider class imbalance and threshold selection.

XII. CONCLUSIONS

1. Government support. Fifty-six people, 23.8% of the sample, reported having received support. Among them, 85.7% reported financial resources and 41.1% food. Because AE2 allows multiple responses, those percentages are calculated over eligible people and may sum to more than 100%.

2. Labor income. Income or earnings from paid work decreased for 37.0% of the full sample, increased for 3.8%, and remained unchanged for 38.3%.

3. Household income. In AE10, 44.3% reported a decrease in the income of another household member. In AE11, 34.5% indicated that family income was insufficient to live on and 47.2% that it was just enough; 17.0% stated that it was enough and allowed them to save.

4. Observed property-capital. For 81.7% of respondents, AE6 was not a source of income/support. In the specific sense used in this analysis, this indicates an observed absence of income derived from property-capital within the categories asked. It does not mean that an exhaustive legal inspection of assets was carried out.

5. Own business and independent producer. TE1 shows 33 people with an own business and no employees and 11 with an own business employing other people. Those positions should not be collapsed into one another or automatically confused with AE6.

6. Civil organizations. For 91.1% of the sample, support from NGOs, foundations, or nonprofit organizations was not a source of support in AE9.

7. International remittances. AE5 was an effective source for 28 people —17 with no change and 11 with a decrease— plus five nonresponses; 202 reported that it was not a source. Among the 28 people with a substantive response confirming the source, 39.3% reported a decrease.

8. Work. A total of 44.7% worked for an employer. TE6 was applicable to only 38 people; within that universe, 42.1% reported more working hours.

9. Health and habits. Hydration increased for 37.0%. Hours of sleep remained the same for 57.0%, decreased for 26.8%, and increased for 14.9%. Physical activity increased for 26.0% and decreased for 24.3%.

10. Virtual education. Among the 161 applicable E6 cases, 6.8% reported lack of money and 16.8% lack of time. Relative to all 235 interviews, these correspond to 4.7% and 11.5%, respectively.

11. Sample size. The value p=0.5 is used because it maximizes p(1-p) and yields a conservative calculation, not because the CLT forces responses to split 50/50. The ±6.39% is a reference sampling error under ideal SRS, not a guarantee of ENAVIRPA’s total error.

12. Nonresponse. The 30.0546% figure computed from busy and unanswered calls does not by itself constitute a standardized response rate. The Birnbaum-Sirken formula cannot be repaired by taking an absolute value when its conditions produce an inadmissible n.

13. Inference on capital. Under the AE6 operationalization, logit and probit provide no evidence of a substantive association between observed property-capital income and sex, age ≥25, or nationality in the analytical sample of 227 cases.

14. Prediction. With the same predictors, the model has little discriminative ability. A 0.25 threshold classifies every case as negative and obtains 82.8% accuracy only because that is the majority class. Out-of-sample evaluation is indispensable.

15. Poverty. No “UNECE poverty rate” is presented by taking 60% of mean gross household income. The relative standard requires the median of equivalized disposable income; SD5 does not contain that variable in the required form.

16. Scope. ENAVIRPA 2021 is especially valuable as an exercise in applied methodology: it shows that design, skips, coding, nonresponse, operationalization, and evaluation criteria determine what can be claimed from data. Technical correction does not diminish the empirical value of the survey; it more precisely delimits its informational content.

XIII. APPENDIX A. TOPOLOGICAL DISTANCES, METRIC DISTANCES, AND COLLECTIVE BEHAVIOR

In graph theory, a graph is a pair G=(V,E). An isomorphism between G_1 and G_2 is a bijection f:V(G_1)→V(G_2) preserving adjacency:

\[ \{u,v\}\in E(G_1) \Longleftrightarrow \{f(u),f(v)\}\in E(G_2). \]

Preservation of adjacency implies preservation of graph distance, defined as the minimum number of edges in a path between two vertices. It does not imply preservation of Euclidean distances in a particular drawing of the graph.

Three ideas must be kept separate: graph isomorphism; topological invariance under homeomorphisms in general topology; and “topological distance” in studies of collective behavior. A homeomorphism does not in general preserve metric distances —that is the role of an isometry—. It is therefore incorrect to define a topological distance simply as a distance that remains invariant under perturbations.

In Ballerini and colleagues’ work on starling flocks, “topological distance” means neighbor rank: first neighbor, second, third, and so on. The empirical evidence indicated interaction with approximately six or seven neighbors, relatively independently of metric density. The conceptual distinction is:

• metric rule: a physical radius is kept fixed and the number of neighbors inside it changes;
• topological rule: approximately the number/rank of neighbors is kept fixed and the physical radius containing them changes.

The anisotropy of the angular distribution of neighbors made it possible to estimate up to what rank interaction persisted; for an isotropic distribution, the expected value of the relevant factor is 1/3. Later studies of jackdaws also showed that the regime can change with context, with topological interactions appearing during transit and metric interactions in other situations. The methodological lesson is material: the relevant structure must be determined empirically and may change with the concrete conditions of the system.

XIV. APPENDIX B. INCOME DISTRIBUTION AND THE POVERTY CRITERION

SD5 recorded bands of total monthly household income. The database contains:

The problem is not that poverty cannot be studied with a survey, but that the available variable must be matched with a definition corresponding to what it actually measures. A reader might believe that it is enough to take 60% of national mean income and compare that amount with gross monthly household income. That is not the relative criterion used by Eurostat/UNECE.

The usual at-risk-of-poverty measure uses 60% of the median equivalized disposable income. There are three important differences:

1. median, not mean;
2. disposable income, not unadjusted gross income;
3. equivalized, that is, adjusted for household size and composition.

SD5 records intervals of total household income and a nonresponse category. Without sufficient information to reconstruct individual equivalized disposable income and its population median, it is not methodologically legitimate to call a percentage obtained by applying 60% of mean gross income a “UNECE poverty rate.”

It is possible to describe how many households in the sample fall within each band, and AE11 may be used as an indicator of subjective income sufficiency. A future measurement compatible with the relative standard could also be designed by collecting disposable income, household composition, and the elements required for equivalization.

XV. R CODE FOR REPRODUCTION

The following code is designed to run with ENAVIRPA2021csv.csv in the same directory as the analysis file. The database is encoded in Windows-1252. Variable names and categorical values remain in Spanish because they are the literal values stored in the original database.

datos <- read.csv(
  "ENAVIRPA2021csv.csv",
  fileEncoding = "Windows-1252",
  stringsAsFactors = FALSE,
  check.names = FALSE
)

stopifnot(nrow(datos) == 235)
stopifnot(all(c("AE1","AE2A","AE2B","AE2C","AE2D",
                "AE3","AE4","AE5","AE6","AE7","AE8","AE9",
                "AE10","AE11") %in% names(datos)))

XV.1. Verification of AE1-AE2 and skip patterns

table(datos$AE1, useNA = "ifany")
table(datos$AE2A, useNA = "ifany")
table(datos$AE2B, useNA = "ifany")
table(datos$AE2C, useNA = "ifany")
table(datos$AE2D, useNA = "ifany")

# There must be 179 NAs in AE2A-AE2D because AE1=NO skips to AE3.
stopifnot(sum(is.na(datos$AE2A)) == 179)
stopifnot(sum(is.na(datos$AE2B)) == 179)
stopifnot(sum(is.na(datos$AE2C)) == 179)
stopifnot(sum(is.na(datos$AE2D)) == 179)

XV.2. AE3-AE9 tables using labels rather than numerical codes

vars_ae <- paste0("AE", 3:9)
lapply(datos[vars_ae], table, useNA = "ifany")

XV.3. Construction of the observed property-capital indicator

datos$capital <- ifelse(
  datos$AE6 == "No es una fuente de ingresos/apoyo", 0,
  ifelse(
    datos$AE6 %in% c("Disminuyó", "Aumentó", "Sin cambios"), 1, NA
  )
)

table(datos$capital, useNA = "ifany")

XV.4. Covariates and model sample

datos$edad <- suppressWarnings(as.numeric(datos$SD2))
datos$adulto25 <- ifelse(is.na(datos$edad), NA, as.integer(datos$edad >= 25))
datos$mujer <- ifelse(datos$SD1 == "Mujer", 1,
                      ifelse(datos$SD1 == "Hombre", 0, NA))
datos$extranjero <- ifelse(datos$SD3 == "No", 1,
                           ifelse(datos$SD3 == "Sí", 0, NA))

modelo <- na.omit(datos[c("capital","mujer","adulto25","extranjero")])
stopifnot(nrow(modelo) == 227)
table(modelo$capital)
cor(modelo)

XV.5. Logit and probit

fit_logit <- glm(
  capital ~ mujer + adulto25 + extranjero,
  family = binomial(link = "logit"),
  data = modelo
)

fit_probit <- glm(
  capital ~ mujer + adulto25 + extranjero,
  family = binomial(link = "probit"),
  data = modelo
)

summary(fit_logit)
summary(fit_probit)

# Odds ratios and approximate Wald confidence intervals for the logit
b <- coef(fit_logit)
se <- sqrt(diag(vcov(fit_logit)))
OR <- exp(b)
IC_OR <- cbind(exp(b - 1.96 * se), exp(b + 1.96 * se))
cbind(Coeficiente = b, EE = se, OR = OR, IC_OR)

XV.6. In-sample evaluation at threshold 0.25

p_hat <- predict(fit_logit, type = "response")
y_hat <- ifelse(p_hat > 0.25, 1, 0)
y <- modelo$capital

TP <- sum(y == 1 & y_hat == 1)
TN <- sum(y == 0 & y_hat == 0)
FP <- sum(y == 0 & y_hat == 1)
FN <- sum(y == 1 & y_hat == 0)

accuracy <- (TP + TN) / length(y)
sensitivity <- TP / (TP + FN)
specificity <- TN / (TN + FP)
precision <- if ((TP + FP) == 0) NA else TP / (TP + FP)
npv <- TN / (TN + FN)

c(TP=TP,TN=TN,FP=FP,FN=FN,
  accuracy=accuracy,sensitivity=sensitivity,
  specificity=specificity,precision=precision,npv=npv)

# Majority-class baseline
mean(y == 0)
range(p_hat)

XV.7. Five-fold stratified cross-validation

set.seed(2021)
K <- 5
fold <- rep(NA_integer_, nrow(modelo))

# Simple stratified assignment by class
for (cl in sort(unique(modelo$capital))) {
  idx <- which(modelo$capital == cl)
  fold[idx] <- sample(rep(1:K, length.out = length(idx)))
}

p_cv <- rep(NA_real_, nrow(modelo))
for (k in 1:K) {
  train <- modelo[fold != k, ]
  test  <- modelo[fold == k, ]
  fit_k <- glm(capital ~ mujer + adulto25 + extranjero,
               family = binomial(link="logit"), data=train)
  p_cv[fold == k] <- predict(fit_k, newdata=test, type="response")
}

# AUC function without additional packages (Mann-Whitney / ranks)
auc_rank <- function(y, p) {
  pos <- y == 1; neg <- y == 0
  r <- rank(p, ties.method="average")
  (sum(r[pos]) - sum(seq_len(sum(pos)))) / (sum(pos) * sum(neg))
}

auc_cv <- auc_rank(modelo$capital, p_cv)
auc_cv

# Confusion matrix at the same threshold, only as an example.
y_cv <- ifelse(p_cv > 0.25, 1, 0)
table(Real=modelo$capital, Predicho=y_cv)

XV.8. Income contraction without mixing it with AE11 sufficiency

vars_ingreso <- paste0("AE", 3:9)
disminucion_fuente <- apply(
  datos[vars_ingreso], 1,
  function(x) any(x == "Disminuyó", na.rm = TRUE)
)

datos$contraccion_ingreso <- as.integer(
  disminucion_fuente | datos$AE10 == "Ingreso disminuido"
)

# AE11 is kept separate as an indicator of sufficiency/economic stress.
table(datos$contraccion_ingreso, useNA="ifany")
table(datos$AE11, useNA="ifany")

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Birnbaum, Z. W., & Sirken, M. G. (1950). Bias due to non-availability in sampling surveys. Journal of the American Statistical Association, 45(249), 98-111.

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