01 · An observed difference does not explain its cause

A table may show that two variables change together while leaving unresolved why they do so. When the subject is crime and racial classification, that distance between observation and explanation is decisive: a territorial or administrative association does not demonstrate an individual biological predisposition.

The problem cannot be resolved by rejecting the data or accepting their interpretation without scrutiny. We must reconstruct what was counted, who was observed, how the groups were formed, and which historical and institutional processes could produce the pattern. Scientific criticism requires both rigorous measurement and clarity about the proposed mechanisms.

This version develops the pedagogical question behind the original post, which drew on Gujarati and Porter and on Ritchey. Their references are retained as background; the argument is reconstructed here through an original exposition, a reproducible exercise, and additional methodological and empirical literature. Passages from those editions are not reproduced as direct quotations. [1][2]

02 · Association, correlation, prediction, and causation

Association means that the distribution of one variable differs across values of another. Correlation usually denotes a specific measure of covariation, such as Pearson’s coefficient. Prediction involves anticipating an outcome using available information. Causation asks how an outcome would change under a specified modification or intervention. These tasks are not interchangeable.

The covariance in the numerator captures how X and Y deviate jointly from their means. The denominator removes their units through the standard deviations. The expression requires finite, positive variances. A value near zero indicates little linear association; it does not rule out a nonlinear relationship or establish independence.

In the causal sense relevant here, an association is spurious relative to an interpretation when the observed pattern does not justify the direct effect attributed to it and may arise from other processes. The term is also used for sampling coincidences, extensive searches across relationships, and shared trends. Naming the specific mechanism matters because these are not all the same problem.

The statistical relationship may be real; the causal explanation attributed to it may be mistaken.

A variable may improve a forecast because it summarizes inequalities or institutional decisions without representing a biological mechanism. A causal explanation requires assumptions beyond fitting a regression. The potential-outcomes framework helps formulate this distinction. [3]

03 · What “skin color,” “race,” and “crime” mean

Pigmentation is a phenotypic characteristic that can be measured in different ways. Socially assigned race captures how other people or institutions classify someone. Racial or ethnic self-identification records a situated personal response. Genetic ancestry estimates inheritance relationships using data and reference populations. These are not equivalent measurements. [4]

An analysis must therefore specify who classifies, with which instrument, at what time, and under which categories. It must also recognize that the meaning of those categories may vary across countries and institutions. A category recorded by police does not thereby become a measurement of ancestry.

There is no single homogeneous variable called “criminality” that allows homicide, corporate fraud, traffic offenses, and environmental crimes to be combined without qualification. Their opportunities, harms, and detection processes may differ. Selecting the behaviors to study is part of defining the problem.

04 · From event to record: how the data are constructed

An event may go unreported; a report may not lead to arrest; an arrest does not establish guilt; a conviction is an outcome of the judicial process. These transitions require analysis in their own right. Victimization surveys and police records have different coverage: the former may capture unreported events, whereas the latter record what is reported and institutionally recognized. [5]

As a teaching model, if C is the number of relevant events and q is the common probability that each event is recorded, then:

The formula assumes that each event can generate at most one record and that there are no false or duplicate records. This equality of expectations does not require independence between events; independence would be an additional assumption for a binomial distribution. Real processes may violate several of these conditions. The equation illustrates that recording depends on both occurrence and observation.

If q varies across territories, different recorded rates may reflect differences in occurrence, observation, or both. Looking only at the final total cannot determine which explanation matters more. Reporting, patrol allocation, investigation criteria, event classification, and source coverage would have to be studied.

05 · A fictional example: 84 versus 36

Construct two sets of neighborhoods, A and B, over one year. They do not represent ethnicities or countries. Their populations remain constant, and resident-years approximate exposure. Each incident is assigned to its location of occurrence and counted once. Two deprivation strata are distinguished without claiming that the label is a sufficient cause of crime.

The numerator counts events; the denominator measures exposure. 84 incidents per 1,000 resident-years does not mean that 8.4% of people committed a crime. One person might be connected with several events, several perpetrators might participate in one event, and someone committing an act might live elsewhere.

The crude rate ratio is 84/36 = 7/3 ≈ 2.333. The crude difference is 48 incidents per 1,000 resident-years. Within each stratum, however, A and B have identical rates: 100 versus 100 and 20 versus 20. Here, by construction, the aggregate difference comes from composition: A has 80% of its exposure in the higher-rate stratum, while B has 20%.

This table establishes the mathematical possibility of a compositional mechanism. It cannot estimate its importance in a real population, let alone attribute an individual predisposition to people.

06 · Comparing equivalent compositions

The overall rate is a weighted mean of stratum-specific rates. If g identifies the set and z the stratum:

r_gz is the rate within the stratum; w_gz is its exposure share. Thus r_A = 0.8 × 100 + 0.2 × 20 = 84, whereas r_B = 0.2 × 100 + 0.8 × 20 = 36. Equal rates are combined with different weights.

For a standardized descriptive comparison, assign both sets a reference exposure split equally between the strata. These common weights are an explicit choice in the exercise:

The standardized ratio is 1 and the difference is 0. Another common distribution would produce a different rate level but preserve equality because the stratum-specific rates coincide. Standardization is not automatically an intervention and does not by itself turn this comparison into a causal effect.

The example shows a difference disappearing after stratification. In a Simpson reversal, the direction of the aggregate association opposes that of the within-stratum associations. That is not what happens here. Simpson’s paper is a classic starting point for examining these aggregation problems. [6]

07 · The mathematical architecture of association

The law of total covariance separates two components of an association. For variables with finite second moments:

The first term averages the covariation of X and Y within the strata defined by Z. The second captures how their means change jointly between strata. Total covariance can be present even when every within-stratum covariance is zero.

This decomposition is algebraic. It does not identify Z as a cause or justify calling it a confounder without a causal hypothesis. Nor should the four-row table be treated as four interchangeable individual observations: their exposures differ and their units are aggregates. The exercise concerns rates; it does not estimate an individual correlation between race and behavior.

08 · Confounding, mediation, and selection: three different questions

The following diagrams are teaching models, not established maps of a society. An arrow represents a proposed causal influence; the absence of an arrow also expresses an assumption. A diagram makes these choices open to examination. [7]

Confounding: a common cause

In this model, the path X ← Z → Y may generate association without an effect of X on Y. Adjusting for an appropriate variable may block that path. Two variables being correlated with Z is not sufficient, by itself, to establish this structure.

Mediation: a pathway that is part of the effect

If the question concerns the total effect of D, adjusting for M may exclude part of the mechanism of interest. A direct effect raises another question and requires its own assumptions. This diagram does not imply that all inequalities work this way. In regressions involving racial classifications, interpreting adjustment requires attention to temporal order and to which variables might be consequences of the processes studied. [8]

Selection: studying only those who entered the records

S is a collider because two arrows point into it. Conditioning on it may open an association that was previously absent. [9] If U also affects a subsequent outcome Y, selection may distort the analysis of X and Y. Policing applications must separately model decisions to stop, record, and act afterward: this minimal diagram does not replace that work.

09 · Why a neighborhood is not a person

The ecological fallacy involves attributing a relationship observed between aggregates to individuals without information justifying that step. This does not make territorial studies useless: it delimits the conclusions they support. Robinson provided a classic formulation of the distinction between ecological and individual correlations. [10]

Suppose a neighborhood has a particular demographic composition and a high incident rate. That information does not identify perpetrators, their residence, or the circumstances of each event. It also does not automatically distinguish exposure to victimization from participation as a perpetrator. Those are different variables and social roles.

The reverse error matters too: individual data alone do not adequately describe every contextual effect. Research may require personal and territorial levels, relationships between them, and spatial dependence. A multilevel model organizes part of that structure but does not automatically remove confounding or turn coefficients into causal effects.

Moving between scales requires compatible data and defensible assumptions. Scale is not a presentation detail; it is part of the object under study.

10 · Class, racialization, and institutions: mechanisms to investigate

A historical-materialist reading asks how access to livelihoods, property, labor, urban space, and state power is organized. This essay proposes investigating how these relations may unequally distribute opportunities, exposure to harm, and possibilities for protection.

Racialization may operate through specific practices: restrictions on housing access, hiring criteria, credit barriers, or differential institutional treatment. The relationship between these practices and class position requires historical investigation. Neither reducing racism to an individual opinion nor using “class” as a variable that explains every outcome in advance is sufficient.

Criticism of a biological interpretation becomes more precise by reconstructing mechanisms and evidence rather than replacing racial essence with a destiny of poverty. A material position does not determine each person’s behavior. Offense types, opportunities to commit them, and the capacity to avoid detection may be distributed differently.

It is also necessary to study which harms become legally defined and recorded and how economic, corporate, or environmental offenses are investigated. Statistics centered on certain visible offenses do not automatically represent every harmful behavior or every legally defined infraction.

In debates such as those associated with Black Lives Matter, the relevant object includes institutions and historical experiences of racialization. Sharing distant evolutionary ancestry does not erase the differences between those contemporary social positions.

11 · What empirical research contributes and what remains open

Sampson, Morenoff, and Raudenbush (2005) studied violence using longitudinal Chicago data and individual, family, and neighborhood characteristics. They found that contextual and family-background variables statistically accounted for an important part of the differences examined. The observational design and its setting prevent turning that result into a universal law or equating a coefficient’s reduction with proof of one exclusive causal mechanism. [11]

Knox, Lowe, and Mummolo (2020) examined a different problem: when the police decision to investigate determines who appears in the data and that decision is affected by discrimination, selected records may distort estimates of subsequent treatment. Their methodological result does not license assigning one fixed magnitude of bias to all police agencies. [12]

These contributions serve different purposes. The first supplies situated evidence about social composition and context; the second formalizes a selection problem. Neither should be confused with our A/B table, which was constructed to teach the arithmetic of rates.

A residual difference after adjustment does not establish a genetic explanation: it may reflect incomplete measurement, selection, omitted causes, or misspecification. Conversely, a reduced difference does not show that the entire effect of discrimination has been identified. Each conclusion requires stating which alternative was assessed and with what evidence.

12 · Note: African ancestry and different genealogies

Cann, Stoneking, and Wilson (1987) analyzed mitochondrial DNA from 147 people and proposed a shared maternal ancestry probably located in Africa. This reconstructs one line of inheritance, not a direct date for the origin of the species or a claim that only one woman existed at that time. [13]

Subsequent nuclear-genomic findings also provided evidence of gene flow between Neanderthals and ancestors of present-day humans. A general description of human expansion as complete replacement without admixture is therefore inadequate. [14]

Distant evolutionary ancestry, a family’s genealogy, and Afro-descendant identification concern different scales and questions. The historical-social use of “Afro-descendant” may refer to experiences and ties formed within particular diasporas and societies; it does not claim to describe the whole evolutionary genealogy of humanity.

Recognizing these differences allows shared evolutionary history and the specificity of contemporary discrimination to be acknowledged together. Neither issue is resolved through a territorial correlation between population composition and incidents.

13 · A reproducible demonstration in R

The following code uses only base R functions and the stats package included in a standard installation. The data are contained in the script itself. There is no data download or random simulation. The labels A/B do not encode racial categories.

# Spurious association / asociacion espuria
# FICTIONAL data: no racial categories or empirical crime estimates.
# Datos FICTICIOS: sin categorias raciales ni estimaciones empiricas.
# Base R + stats only. Counts are incidents, not people or binary trials.

d <- data.frame(
  group = factor(c("A", "A", "B", "B"), levels = c("B", "A")),
  deprivation = factor(c("high", "low", "high", "low"),
                       levels = c("low", "high")),
  incidents = c(80, 4, 20, 16),
  exposure = c(800, 200, 200, 800)
)
stopifnot(all(d$exposure > 0), all(d$incidents >= 0))
d$rate_1000 <- 1000 * d$incidents / d$exposure
print(d)

# Crude rates / tasas crudas
crude <- aggregate(cbind(incidents, exposure) ~ group, data = d, FUN = sum)
crude$rate_1000 <- 1000 * crude$incidents / crude$exposure
print(crude)
r <- setNames(crude$rate_1000, as.character(crude$group))
cat("Crude rate ratio A/B:", unname(r["A"] / r["B"]), "\n")
cat("Crude rate difference A-B:", unname(r["A"] - r["B"]), "\n")

# Incident counts with a log-exposure offset; reference group = B.
m_crude <- glm(incidents ~ group + offset(log(exposure)),
               family = poisson(link = "log"), data = d)
m_adjusted <- glm(incidents ~ group + deprivation + offset(log(exposure)),
                  family = poisson(link = "log"), data = d)
rr_crude <- unname(exp(coef(m_crude)["groupA"]))
rr_adjusted <- unname(exp(coef(m_adjusted)["groupA"]))
rr_deprivation <- unname(exp(coef(m_adjusted)["deprivationhigh"]))
print(c(crude_A_over_B = rr_crude,
        adjusted_A_over_B = rr_adjusted,
        adjusted_high_over_low = rr_deprivation))

# Standardization to common exposure shares: low = 0.5, high = 0.5.
grid <- expand.grid(group = levels(d$group),
                    deprivation = levels(d$deprivation),
                    KEEP.OUT.ATTRS = FALSE, stringsAsFactors = FALSE)
grid$group <- factor(grid$group, levels = levels(d$group))
grid$deprivation <- factor(grid$deprivation, levels = levels(d$deprivation))
grid$exposure <- 1000
# At exposure = 1000 the predicted count is the rate per 1000.
grid$rate_1000 <- predict(m_adjusted, newdata = grid, type = "response")
weights <- c(low = 0.5, high = 0.5)
grid$weighted_rate <- grid$rate_1000 * weights[as.character(grid$deprivation)]
standardized <- aggregate(weighted_rate ~ group, data = grid, FUN = sum)
names(standardized)[2] <- "standardized_rate_1000"
print(standardized)

# Arithmetic/model checks for this constructed example only.
stopifnot(abs(unname(r["A"]) - 84) < 1e-8,
          abs(unname(r["B"]) - 36) < 1e-8,
          abs(rr_crude - 7/3) < 1e-7,
          abs(rr_adjusted - 1) < 1e-7,
          abs(rr_deprivation - 5) < 1e-7,
          all(abs(standardized$standardized_rate_1000 - 60) < 1e-6))

# Common exposure rescaling changes the intercept, not the rate ratios.
scaled <- d
scaled$exposure <- scaled$exposure * 1000
m_scaled <- glm(incidents ~ group + deprivation + offset(log(exposure)),
                family = poisson(link = "log"), data = scaled)
stopifnot(abs(unname(exp(coef(m_scaled)["groupA"])) - 1) < 1e-7,
          abs(unname(exp(coef(m_scaled)["deprivationhigh"])) - 5) < 1e-7)

# No p-values or confidence intervals are interpreted for artificial cells.
cat("All example checks passed. / Comprobaciones del ejemplo superadas.\n")
sessionInfo()

What the model represents

μ_gz is the expected count and E_gz is exposure. The term offset(log(exposure)) introduces exposure with its coefficient fixed at 1; it is not a covariate whose coefficient is estimated. I equals 1 when the condition holds and 0 otherwise. B and lower deprivation are the reference categories. Exponentiating β_A compares A/B rates at the same stratum; exponentiating β_H compares strata at the same set. [15][16]

The crude model uses only set; the adjusted model includes stratum. The Poisson family with log link is useful for displaying this count-and-exposure structure. It does not turn incidents into binary trials. A binomial model requires a success/failure outcome with defined trials; subtracting incidents from residents does not justify it.

The code standardizes through predictions for an exposure of 1,000 resident-years and common weights of 0.5/0.5. Internal checks verify the example’s values and their invariance to using a common exposure scale. Adjusted equality remains a property of the fictional dataset.

14 · How to assess a causal explanation in a real case

An investigable question must specify the population, period, outcome, and intervention. For example: how would a particular rate change under a defined housing-access policy, or how would a police decision vary if perceived racial information changed while other encounter characteristics were held constant? These questions require different designs.

1. Define the objective. Separate disparity description, prediction, and the effect of an intervention. Specify whether a total, direct, or mediated effect is of interest.
2. Document measurement. Identify origin, coverage, categories, exclusions, duplicates, and denominators. Distinguish event location from residence.
3. Represent time. Establish what precedes what. If feedback exists, use time-indexed variables; a static DAG does not represent simultaneous cycles.
4. Justify adjustment. Choose variables by their proposed causal role, not their p-value or a coefficient’s change alone.
5. Examine identification. For effects of defined interventions, discuss consistency, conditional comparability, and positivity; in interconnected neighborhoods, consider possible effects between units. [3]
6. Seek complementary designs and sources. Triangulate records, surveys, and focused studies; consider experiments, audits, or policy changes when relevant. Their assumptions must be defended in each application.
7. Quantify uncertainty and sensitivity. Distinguish sampling error from selection and measurement bias. Consider unobserved causes and alternative specifications.
8. Delimit generalization. A local estimate does not automatically transfer to other periods, countries, or offense types.

Conditional comparability assumes, for the selected objective, the absence of uncontrolled common causes; regression cannot establish it by itself. Positivity requires the compared alternatives to be possible in the relevant strata. Consistency requires the intervention and its versions to be defined sufficiently well.

From this perspective, theory guides the search for mechanisms but must also face evidence capable of correcting it. A materialist explanation becomes stronger when it specifies processes, distinguishes scales, and accepts the analysis’s identification limits.

15 · What the analysis supports

A statistical difference between groups requires explanation; it does not contain a theory of its causes. The fictional table demonstrates that composition can produce an aggregate difference even when stratum-specific rates are equal. Measurement, selection, and scale problems show why causal interpretation requires further work.

Regarding crime and skin color, the scientific question is how material conditions, social classifications, and institutions participate in defined and observed outcomes. The analysis presented supplies no evidence of a racial biological predisposition; nor does it estimate one social mechanism valid for every context.

Explaining an association requires studying both the processes that produce events and the institutions that turn them into data.

References and further reading

References 1–2 are bibliographic antecedents of the original post. The remaining sources support the methodological distinctions and identified studies; the historical-materialist interpretation and A/B example are developments of this essay. Web resources accessed November 2, 2026.

1. Gujarati, D. N., & Porter, D. C. (2010). Econometría (5.ª ed.). McGraw-Hill Interamericana. ISBN 9786071502940.
2. Ritchey, F. J. (2002). Estadística para las ciencias sociales. McGraw-Hill. ISBN 9789701031414.
3. Hernán, M. A., & Robins, J. M. (2020). Causal Inference: What If. Chapman & Hall/CRC. Read source.
4. National Human Genome Research Institute. Use of Population Descriptors in Genomics. Read source.
5. Bureau of Justice Statistics. Crime: administrative records and victimization surveys. Read source.
6. Simpson, E. H. (1951). The interpretation of interaction in contingency tables. JRSS Series B, 13(2), 238–241. DOI: 10.1111/j.2517-6161.1951.tb00088.x. Read source.
7. Greenland, S., Pearl, J., & Robins, J. M. (1999). Causal diagrams for epidemiologic research. Epidemiology, 10(1), 37–48. DOI: 10.1097/00001648-199901000-00008. Read source.
8. VanderWeele, T. J., & Robinson, W. R. (2014). On causal interpretation of race in regressions adjusting for confounding and mediating variables. Epidemiology, 25(4), 473–484. DOI: 10.1097/EDE.0000000000000105. Read source.
9. Shrier, I., & Platt, R. W. (2008). Reducing bias through directed acyclic graphs. BMC Medical Research Methodology, 8, 70. DOI: 10.1186/1471-2288-8-70. Read source.
10. Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15(3), 351–357. DOI: 10.2307/2087176. Read source.
11. Sampson, R. J., Morenoff, J. D., & Raudenbush, S. (2005). Social anatomy of racial and ethnic disparities in violence. American Journal of Public Health, 95(2), 224–232. DOI: 10.2105/AJPH.2004.037705. Read source.
12. Knox, D., Lowe, W., & Mummolo, J. (2020). Administrative records mask racially biased policing. American Political Science Review, 114(3), 619–637. DOI: 10.1017/S0003055420000039. Read source.
13. Cann, R. L., Stoneking, M., & Wilson, A. C. (1987). Mitochondrial DNA and human evolution. Nature, 325, 31–36. DOI: 10.1038/325031a0. Read source.
14. Green, R. E., et al. (2010). A draft sequence of the Neandertal genome. Science, 328(5979), 710–722. DOI: 10.1126/science.1188021. Read source.
15. R Core Team. glm: Fitting Generalized Linear Models. R statistical documentation. Read source.
16. R Core Team. family: Family Objects for Models; offset: Include an Offset in a Model Formula. R statistical documentation. Read source.