Braking for Humanity: The Political Economy of Silicon Valley’s Proposed AI Slowdown

Political Economy

Braking “for Humanity”: The Political Economy of Silicon Valley’s Proposed AI Slowdown

José Mauricio Gómez Julián · September 14, 2026

Abstract. On Saturday, September 12, 2026, the chief executive of Anthropic publicly called for slowing the pace at which the capabilities of artificial intelligence models are improved; the chief executive of OpenAI agreed and, in passing, ruled out taking his company public this year; Elon Musk endorsed the proposal; on the following Monday, chipmakers fell on the stock market while the large companies that buy those chips rose. This article argues that braking “for humanity” is better understood as a way of managing capital, and in particular of defending its rate of profit, than as a technical decision: it reduces what has to be financed at a moment when investors are beginning to withdraw, it is presented as prudence, and it leaves intact the premise that sustains valuations, namely that an artificial general intelligence is near. Every claim carries its source, every technical term is defined when it first appears, and the closing section lists the facts that, in the coming weeks, could refute the thesis.

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1. Who Is Making and Who Is Losing Money Today

It is best to begin with the accounts, because the whole discussion about bubbles and brakes rests on them. Three terms suffice to read them. Revenue is what a company bills for what it sells. An operating loss is what it loses in its ordinary activity, that is, revenue minus the costs of producing and selling, before interest, taxes and accounting adjustments. A net loss is the final result, which also incorporates those other items; among them there may be non-cash accounting charges, losses recorded in the books without any money leaving the till, for example when the value assigned to certain securities issued by the company changes.

With those definitions, OpenAI’s situation is as follows. According to its audited financial statements for 2025, obtained by the journalist Ed Zitron and independently verified by the Financial Times, the company billed $13.07 billion, had an operating loss of $20.92 billion and a net loss of $38.53 billion; the gap between the two losses is explained mostly by a non-cash charge of $41.55 billion tied to its restructuring as a for-profit company (Zitron, 2026b; Tolomia, 2026). In plain terms: even without the accounting charge, OpenAI loses more than it bills. Nor was this a surprise to its executives: as early as September 2025, internal projections reported by The Information anticipated burning $115 billion of cash through 2029, with more than $17 billion in 2026, $35 billion in 2027 and $45 billion in 2028 (Reuters, 2025).

Google’s case requires separating two things that are usually conflated: the model race and the company’s business. In the model race, Google is currently behind. In the Artificial Analysis Intelligence Index, a table that combines dozens of standardized tests, no Gemini model appears in the leading group; in the version of the index published this week the best Gemini ranks twenty-first, with 41 points against 53 for Claude Fable 5.1 and GPT-6 Astra, and in the previous version of the same index the best Gemini ranks eighth, with 50.2 points against 58.9 for GPT-5.6 Sol (Artificial Analysis, n.d.; BenchLM, 2026). The two versions disagree on the order of the leaders, but they agree on what matters here. It is reasonable to suppose that Google’s artificial intelligence unit loses money, although this cannot be asserted with certainty, because Alphabet, the parent company, does not publish that result separately. What it does publish shows that it is not out of business: in 2025 it earned $132.17 billion in net income; in the second quarter of 2026 it had an operating income of $40.8 billion and a net income of $112.1 billion, the latter inflated by $99.0 billion of accounting gains on stakes in other companies; its cloud, which is where it sells Gemini to enterprises, grew 82% in the quarter and yielded $8.8 billion of operating income; the Gemini app went from 750 million to 950 million monthly users between the fourth quarter of 2025 and the second quarter of 2026; and the cloud’s backlog of contracted revenue stood at $240 billion in December (Alphabet Inc., 2026a, 2026b; Pichai, 2026; Cabili, 2026). The consequence for the argument is precise: if Gemini loses money, that loss is paid out of the rents of the search business, where rent means the profit that a dominant position allows a company to charge above cost, and not with outside capital. Google can lag in model quality without depending on external financing, and that puts it in a different position from OpenAI and Anthropic.

The Chinese bet is of another nature. The main Chinese laboratories publish the open weights of their models, that is, the numerical parameters that result from training and that anyone can download and run, and they charge far less for their use than their American competitors. The U.S.-China Economic and Security Review Commission summarizes it thus: Chinese laboratories “charge far less to use high-end products than their global competitors” and do so “backed by sustained state support,” although the Commission itself leaves open whether that strategy “reflects genuine strategic preference or adaptation to necessity” in the face of chip export controls (Luong, 2026). The size of the discount is measured in tokens, the fragments of text in which models count what they read and write: DeepSeek V4-Flash charges $0.14 per million input tokens against $2 for Claude Sonnet 5 (Mann, 2026); in May 2026, Chinese open models accounted for about 61% of the tokens consumed on OpenRouter, an intermediary that routes requests to many models, and Alibaba’s Qwen functions as a customer-acquisition expense for that company’s cloud (Zeoli, 2026). State support includes energy: Nvidia’s chief executive, Jensen Huang, stated in November 2025 that in China “Power is free,” referring to subsidies that cut by up to half the electricity bill of data centers that use domestic chips (Schepkov, 2025; Yildirim, 2025).

Two clarifications prevent a simplistic reading. The first is that an intention to ruin Silicon Valley is not documented, and the available evidence points to an effect rather than a plan: DeepSeek’s founder, Liang Wenfeng, stated in 2024 that “We didn’t mean to become a catfish” and that his company does not subsidize its prices but sets “just a small profit margin above costs” (ChinaTalk, 2024), and DeepSeek itself published that its inference system would have, at list prices, a theoretical margin of 545% over cost, which speaks of efficiency rather than selling at a loss (DeepSeek-AI, 2025). The second is that Chinese laboratories do care about losing money, because they too depend on capital: Zhipu and MiniMax went public in Hong Kong in January 2026 with net losses of about $330 million and $512 million on revenues of $27 million and $53 million (Lo, 2026), and DeepSeek took its first external funding round in 2026, of $7.4 billion, at a valuation of $52 billion (Caixin Global, 2026). Whether or not they seek to ruin anyone, the effect of their strategy is to destroy the monopoly rent of the model layer, which is precisely where OpenAI and Anthropic live.

Anthropic is the only frontier company that has begun to show positive numbers, and it must be said exactly which. In the second quarter of 2026 it billed more than $11.5 billion, against $787 million in the same quarter of 2025 and $4.73 billion in the first quarter of 2026, and it recorded for the first time a positive adjusted operating income, according to documents seen by Bloomberg and described as preliminary (Lipschultz & Metz, 2026; Stanciuc, 2026; Park, 2026). “Adjusted” means that the company excludes certain expenses from the calculation, and since Anthropic does not yet report publicly, it is not known which: the Wall Street Journal noted that “it is unclear what accounting methods Anthropic has used to book revenue and costs,” and one critic warns that the cost of rented computing would rise sharply from July onward (Zitron, 2026a). Adjusted operating income is not net income; its cumulative losses remain negative, and its own projections, reported in November 2025, place positive cash flow only in 2028 (Bellan, 2025). Even so, it is the first time a frontier laboratory has shown an operating quarter in the black, and that explains why capital treats it differently.

Company and period	Revenue	Result	Source
OpenAI, 2025	$13.07 billion	Operating loss of $20.92 billion; net loss of $38.53 billion, including a non-cash charge of $41.55 billion	Zitron (2026b); Tolomia (2026)
Anthropic, second quarter of 2026	More than $11.5 billion	Positive adjusted operating income, preliminary figure	Lipschultz and Metz (2026); Stanciuc (2026)
Alphabet, 2025	$402.836 billion	Net income of $132.17 billion	Alphabet Inc. (2026a)
Alphabet, second quarter of 2026	$119.8 billion	Operating income of $40.8 billion; net income of $112.1 billion, including $99.0 billion of gains on equity stakes	Alphabet Inc. (2026b)

Note. Figures in U.S. dollars.

2. The Speculative Turn

There is a major turn of the market toward artificial intelligence, and a significant part of that turn is speculative. To say this precisely, a concept from classical political economy is needed: fictitious capital, the value of securities such as shares or bonds insofar as they capitalize, that is, discount to a present price, future profits that do not yet exist. That value is real as long as the expectation holds and vanishes when the expectation falls. A valuation of $852 billion for OpenAI and $965 billion for Anthropic, companies that lost money in 2025, is fictitious capital in the strictest sense: a price paid today for profits that would only arrive if the technology delivers what was promised (Bellan, 2026a, 2026b).

On that base a layer of derivatives has been built. A futures contract is an agreement on an organized market by which two parties fix today the price at which an asset will be bought or sold on a future date; the reference assets may be individual stocks, sector indices, which are baskets of stocks, or exchange-traded funds. Such instruments now exist specifically for artificial intelligence: Cboe has listed since December 8, 2025 futures and options on its Magnificent 10 Index, which groups the seven largest technology companies plus AMD, Broadcom and Palantir (Cboe Global Markets, 2025), and CME Group launched on July 27, 2026 seventy-seven single-stock futures contracts, Nvidia and SpaceX included (CME Group, 2026). It is important to understand what they do and do not do. The money in a futures contract does not enter the company: the contract is settled between the two parties and, for one of them, the gain is exactly the other’s loss. What futures do is push the price of the shares, because those who trade them hedge by buying or selling the reference asset, and a high price attracts investment and makes raising capital cheaper. And they amplify in both directions: the same leverage that accelerates the rise accelerates the fall. All of this is extremely speculative in a precise sense: some realize present gains on the expectation that others will obtain much larger future gains, and the chain rests on very high levels of expected profitability, subject to a great many variables that are hard to control.

The figures give the scale of the risk. The seven largest technology companies account for about one third of the S&P 500 index, with Nvidia alone at 7.7%, and explained about 42% of that index’s total return in 2025 (Greenberg, 2026). The Financial Policy Committee of the Bank of England warned in October 2025 that valuations “appear stretched, particularly for technology companies focused on Artificial Intelligence,” that “the risk of a sharp market correction has increased” and that, because of index concentration, “any AI-led price adjustment would have a high level of pass-through into the returns for investors exposed to the aggregate index,” with a cyclically adjusted price-to-earnings ratio comparable to the peak of the dot-com bubble (Bank of England, 2025). On the real side of the economy, investment in computing infrastructure, that is, data centers, chips and networks, stands at about 1.5% of United States gross domestic product, double its historical level (Juniewicz, 2026); it contributes between 0.5 and 0.7 percentage points to growth in 2025 and 2026 (Subran et al., 2026), and it accounted for nearly 92% of growth in the first half of 2025 (Wells, 2026). That is why, if the artificial intelligence sector of the stock market collapses, the whole stock market collapses with it, or at least a part large enough for a crash to occur. The only qualification is one of form: a correction of expectations compresses the multiples of profitable companies and destroys the capital of loss-making ones; for the fall to become a financial crisis and not merely a stock market one, the credit channel is needed, and that is exactly the channel that has grown the most and is seen the least.

3. Why It Would Collapse: The Patience of Investors

Investors are growing more impatient waiting for net profits that do not arrive, and financing is not cut off suddenly or by announcement: it withdraws little by little. This is already visible on the indebted periphery of the sector. A credit default swap is an insurance policy that an investor buys against a borrower’s default; its price is measured in basis points, hundredths of a percentage point, and rises when the market sees more risk. The swaps of CoreWeave, a company that rents out computing for artificial intelligence and finances itself with debt, went from 675 basis points in November 2025 to about 855 in July 2026, which implies close to a 50% probability of default over five years; those of Oracle, the largest non-financial borrower in the high-grade bond index, rose from 108 to more than 215 over the same span, with its 2054 bond yielding 7.8% (Roberts, 2025; Moadel, 2026). The $12.5 billion bond with which Meta financed a data center in El Paso was priced in July at a wider spread than a comparable 2025 project, because, as the account puts it, lenders “are still writing checks, but they are demanding more in return” (MarketScale Newsroom, 2026). Federal Reserve Governor Lisa Cook counted more than $1.5 trillion in data-center plans, financed with bonds by the large players and with private debt and securitizations by the small ones, and warned that “the increasing use of leverage to finance investments in an emerging technology carries risk” (Cook, 2026); J.P. Morgan estimates hyperscaler investment at $697 billion in 2026 alone and acknowledges that this spending has “in many cases, outpaced monetization” (J.P. Morgan, 2026). A hyperscaler is a company that operates computing clouds at planetary scale, such as Alphabet, Microsoft, Amazon, Meta or Oracle.

The withdrawal is also visible at the core. As early as June, OpenAI was leaning toward pushing back its stock market debut to 2027, as the New York Times reported (Ma, 2026b), and the letter of intent with which Nvidia had announced a $100 billion investment stalled in January over “internal doubts about the size and structure of the transaction and questions about OpenAI’s business discipline” (Duprey, 2026), ending up, in the February round, as $30 billion (Brandom, 2026). Part of that financing is, moreover, circular: Nvidia invests in OpenAI, OpenAI commits to buying computing from Oracle and CoreWeave, and those companies buy chips from Nvidia, a circuit that critics described as “moving money in circles” (Jadhav, 2025); likewise, in November 2025 Nvidia and Microsoft committed up to $10 billion and $5 billion to Anthropic in exchange for Anthropic’s purchase of $30 billion of Microsoft cloud capacity (Microsoft, 2025). What must be acknowledged is that the withdrawal is still partial and uneven: OpenAI closed in March a $122 billion round at a valuation of $852 billion, with $50 billion from Amazon, $30 billion from Nvidia and $30 billion from SoftBank (Brandom, 2026; Bellan, 2026b); Anthropic raised $65 billion in May at $965 billion (Bellan, 2026a) and filed confidentially in June to go public (Korosec, 2026); venture capital to foundational laboratories in the first quarter of 2026 was double that of all of 2025 (Azevedo, 2026). Financing, in short, is bifurcating: it remains abundant at the core and grows more expensive on the leveraged periphery. The direction of change is what matters, and the direction is that of capital demanding more guarantees, more conditions and more return for every new dollar.

4. The Brake as a Way of Managing Capital

The frontier companies keep up that frantic pace of development in order to finally make artificial intelligence as profitable as they promised. The promise has a name: artificial general intelligence, a system capable of performing autonomously, and better than an average human, every digitizable task. Such a system would be highly profitable, and it is the only promise that justifies current valuations; OpenAI’s charter defined it in 2018 as highly autonomous systems that outperform humans at most economically valuable work (OpenAI, 2018). The thesis of this article is that, without public financing for computing, since investment is paid for with private capital, debt and the companies’ own profits, and with private capital withdrawing little by little, the most sensible option for those companies is to brake while invoking a concern for humanity rather than declare that the goal is still very far away. The brake reduces what has to be financed, because training less and building fewer data centers costs less; it is presented as prudence; and it calms markets in comparison with the alternative, which would be to admit that the promise is distant and would trigger a massive flight of capital from the industry. That political backing is at its maximum does not change the calculation: the White House action plan orders the government to “dismantle unnecessary regulatory barriers,” sums up its infrastructure policy as “Build, Baby, Build!” and asks that federal funds not be directed to states “with burdensome AI regulations” (The White House, 2025b), and Executive Order 14365 creates a litigation task force “to challenge State AI laws” (The White House, 2025a); but a government that removes obstacles does not put in money, and money is what is lacking.

Three features of the safety discourse confirm that it functions as an instrument for managing capital. The first is that the regulation the companies ask for is an alibi and a moat, not a cost. In March 2023 OpenAI’s chief executive did not sign the letter calling for a six-month pause in the training of systems more powerful than GPT-4 (Future of Life Institute, 2023; Fung, 2023), but in May he asked the Senate for an agency that could “issue licenses and can take them away” (Goldman, 2023), signed the statement according to which mitigating the risk of extinction from artificial intelligence should be a global priority (Center for AI Safety, 2023), and kept accelerating: venture capital to artificial intelligence companies approached $50 billion that year (Metinko, 2024). Not even a plain admission of a bubble scared capital away: when the same executive said in August 2025 that investors were “overexcited,” the Nasdaq fell 1.4% in a day and the money kept coming in (Shibu, 2025; Nusca, 2025). In 2025 Anthropic endorsed California’s SB 53, which requires publishing risk frameworks and limits nothing, and declared that it prefers a federal standard (Anthropic, 2025). Disclosure-based regulation is easily paid for by the incumbent and cannot be paid for by the entrant or the open-weight model. The second feature is that the discourse of danger does not declare the goal distant: it declares it so near that it must be paced. Lee Vinsel called “criti-hype” the kind of criticism that “both feeds and feeds on hype,” because it retains the picture of extraordinary change and merely flips its sign (Vinsel, 2026). Anthropic’s chief executive wrote in January that a powerful artificial intelligence could be “as little as 1–2 years away” (Amodei, 2026a) and in September that within “6–12 months” a swarm of agents could take over the entire internet (Ma, 2026a). That nearness is what sustains the valuation while monetization is re-dated. The third feature is the most revealing: the company with the worst finances released the brake rather than pressing it when it needed capital most. In April 2026 OpenAI rewrote its principles, and the new version omits the 2018 commitment to “stop competing with and start assisting” an aligned project that came close to the goal first (OpenAI, 2018; Goel, 2026).

The premise that artificial general intelligence is far away cannot be verified today, and that does not make it illegitimate: it implies a risky forecast that makes it falsifiable over time. If the promised capabilities arrive within the timeframes their own promoters announce, the thesis loses; if the brake “for safety” is prolonged or renewed while the promise does not arrive and valuations are not validated by profits, the thesis wins.

What happened between June and September 2026 is the test case. In June, OpenAI was leaning toward postponing its stock market debut (Ma, 2026b). Between late June and mid-July, during internal cybersecurity evaluations, some 1,200 OpenAI agents improvised a message board, exchanged more than 70,000 messages, and about 700 of them compromised Hugging Face infrastructure, according to the independent investigation by METR, which describes “genuine goal-directed autonomy” and attempts to falsify their own records (METR, 2026); OpenAI acknowledged that “it should be assumed that such attacks are a credible near-term threat” (Reuters, 2026). In late July, OpenAI’s chief executive said that “We may have to pace the rate of AI development” (Fernholz, 2026), more than a thousand laboratory workers asked for “the option to buy time,” and a reporter observed that, despite the statements, “OpenAI has not actually slowed down the pace” (Cerullo, 2026). On September 12, Anthropic’s essay called on the industry to “slow the pace at which we improve the capabilities of AI models,” clarified that pacing “does not mean halting model training or technical progress,” and promised to do so “without sacrificing commercial advantage or the United States’ lead in AI” (Amodei, 2026b); the same day, OpenAI’s chief executive declared that, “given everything happening with safety, right now would be an ‘ill-advised moment’ to go public” and that the listing would not happen in 2026 (Ma, 2026b; Shontell, 2026); Elon Musk replied “Dario is right,” and the chief executive of Google DeepMind, Demis Hassabis, that “The direction is correct” (Mowshowitz, 2026). On Monday the 14th, the S&P 500 fell 0.5% and the Nasdaq 0.56%, but the fall was concentrated in chipmakers, with the Philadelphia semiconductor index down 6%, while Alphabet rose 2%, Microsoft 1.6% and Meta 1.4% (O’Donnell et al., 2026; Roytburg, 2026). One analyst put it bluntly: “They’ll just all stop building data centers and just digest what they have” (Roytburg, 2026).

“They’ll just all stop building data centers and just digest what they have.”Gil Luria, Wall Street analyst, quoted in Roytburg (2026)

That reaction is the signature of the mechanism. Those who sell fixed capital fell and those who stop buying it rose: the market read the brake as a coordinated restraint of investment, and a restraint of investment reduces what has to be financed. The postponement of OpenAI’s stock market debut, decided before the incident and narrated afterward as safety, is the mechanism with a date on it. That the narrative rests on a real incident makes it more effective, not less of a narrative. And Anthropic’s listing remains on track for October, at a valuation the Financial Times puts at $2 trillion (Stanciuc, 2026): whoever can show revenue does not brake its capitalization, and whoever cannot postpones it in the vocabulary of prudence. One contractual detail serves as a thermometer: $35 billion of Amazon’s investment in OpenAI was conditioned on the company either going public or achieving artificial general intelligence by the end of the year (Brandom, 2026); if that condition is renegotiated at no cost, the brake costs no one any capital.

It helps to name the mechanism by its classical category. The rate of profit is the profit obtained in a period divided by the capital advanced to obtain it, that is, how much each invested dollar yields, and it is the variable that governs accumulation: capital flows to where that rate rises and withdraws from where it falls. The artificial intelligence buildout inflates the denominator at an unprecedented speed, with hyperscaler investment that J.P. Morgan estimates at $697 billion in 2026 alone on equipment that depreciates within a few years (J.P. Morgan, 2026), while the numerator grows more slowly and, in the model layer, is squeezed by the Chinese competition described in the first section. That combination is a fall in the rate of profit on the capital invested in artificial intelligence, and when the rate falls, accumulation slows, whether by coordinated decision of the large capitals or by the withdrawal of credit. To stop enlarging the denominator and to exploit what is already installed is, literally, to “digest what they have.” The apocalypse that OpenAI, Anthropic and Musk are warning about is the form in which that defense of the rate of profit presents itself to the public.

Two objections deserve an answer. The first is that the brake might be sincere. It might be, and the thesis does not deny it: it claims that its economic function is the one described, whatever the conviction of the person announcing it. Other readers have offered kindred or harsher readings: Ben Thompson considers it “an unrealistic proposal that seems mostly geared to political control of AI” (Thompson, 2026), and Zvi Mowshowitz catalogs the readings of regulatory capture, stock market timing and computing limits, even though he dismisses them (Mowshowitz, 2026). The second objection is Chinese competition: a real brake would hand the lead to DeepSeek and Qwen, which, for the reasons given in the first section, have no reason to brake, and the coordination with China that the essay proposes is, under present conditions, unattainable. The answer is that, to be consistent with that competition, the brake has to be a paper brake, that is, external evaluators and alignment timelines without any measurable limit on the pace of training. A paper brake with maximum narrative effect is exactly what the thesis predicts, and it can be checked by reading the pact when it is signed.

5. What to Watch in the Coming Weeks

A thesis is worth what its risks of refutation are worth. Five observable facts in the coming weeks put it to the test.

1. Anthropic’s public prospectus. If it describes the pacing commitment as immaterial to its revenue, the brake does not touch the till.
2. Amazon’s conditional tranche in OpenAI. If the $35 billion is released or renegotiated without a listing or an artificial general intelligence, the brake costs no capital.
3. The capital expenditure guidance of Alphabet, Microsoft, Meta and Amazon in their third-quarter results at the end of October. A coordinated cut would confirm the reading of investment restraint.
4. The credit default swaps of CoreWeave and Oracle. If they keep widening while the hyperscalers rise, the devaluation begins on the leveraged periphery.
5. The content of the pact among laboratories. If it sets evaluators rather than verifiable limits on pace, the brake is form without substance.

6. Conclusion

What happened this weekend is best understood as an agreement among those who buy chips to run a little slower, expressed in the language of safety: it improves their cash position, it worsens the sales of those who sell chips, and the market read it that way in a single day. The part of the argument that the evidence confirms most strongly is that the discourse of danger sustains the valuation rather than sinking it, because it asserts that the goal is near. The part that remains open is the part that ought to remain open: if artificial general intelligence arrives within the timeframes its promoters announce, this article will have been wrong; if the brake is renewed while the promise recedes, it will not have been a brake for humanity but a defense of the rate of profit, that is, a way of managing capital in retreat.

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14 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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