Contingency: race and the death penalty

Log-linear models

Each cell count is Poisson with a log-mean built from main effects and the associations allowed by the hypothesis. For example, conditional independence of D and P given V, written [VD][VP], is

\log \mu_{dvp} = \lambda + \lambda^{D}_{d} + \lambda^{V}_{v} + \lambda^{P}_{p} + \lambda^{DV}_{dv} + \lambda^{VP}_{vp}, \qquad y_{dvp} \sim \operatorname{Poisson}(\mu_{dvp})

\begin{aligned} \log \mu_{dvp} &= \lambda + \lambda^{D}_{d} + \lambda^{V}_{v} + \lambda^{P}_{p} \\ &\quad + \lambda^{DV}_{dv} + \lambda^{VP}_{vp}, \\ y_{dvp} &\sim \operatorname{Poisson}(\mu_{dvp}) \end{aligned}

Test the hypotheses

Hypothesis (log-linear model)

[D][V][P]model.8G² = 137.93, df 4, p = 7.8 × 10⁻²⁹[DV][P]model.9G² = 8.13, df 3, p = 0.0434[DV][DP]model.10G² = 7.91, df 2, p = 0.0192[VD][VP]model.11G² = 1.88, df 2, p = 0.3903[DV][DP][VP]extensionG² = 0.70, df 1, p = 0.4025

H₀: Given the victim's race, the sentence is independent of the defendant's race. Not rejected — the model fits.

Observed counts and fitted values

Large number: observed. Small: fitted under [VD][VP]. Shading: Pearson residual (teal = more than expected, terracotta = fewer).

White victim
Defendant	Death penalty	No death penalty
White	19fit 21.2	132fit 129.8
Black	11fit 8.8	52fit 54.2

Black victim
Defendant	Death penalty	No death penalty
White	0fit 0.5	9fit 8.5
Black	6fit 5.5	97fit 97.5

Equivalent logistic regression (Q3b)

Treating the death penalty as the response for each defendant × victim group:

Proportion ~ Victim_Race

Log-linear G²: 1.8819
Logistic deviance: 1.8819

Same deviance and degrees of freedom (2): the logistic model is the log-linear model conditioned on the [DV] margin.

Analysis of deviance for [VD][VP]

Terms added sequentially, as in anova(model.11, test = "Chi").

Term added	Df	Deviance	Resid. Df	Resid. Dev	Pr(>Chi)
NULL			7	395.915
Victim_Race	1	32.456	6	363.459	1.2 × 10⁻⁸ ***
Defendant_Race	1	0.110	5	363.349	0.7396
Penalty	1	225.419	4	137.929	5.9 × 10⁻⁵¹ ***
Victim_Race:Defendant_Race	1	129.798	3	8.132	4.5 × 10⁻³⁰ ***
Victim_Race:Penalty	1	6.250	2	1.882	0.0124 *

What the 2023 analysis found

Comparing each model's residual deviance with a χ² distribution on its residual degrees of freedom:

[D][V][P]All three factors are mutually independent. G² = 137.93 on 4 df, p = 7.8 × 10⁻²⁹ — rejected.
[DV][P]The sentence is independent of both the defendant's and the victim's race. G² = 8.13 on 3 df, p = 0.0434 — rejected.
[DV][DP]Given the defendant's race, the sentence is independent of the victim's race. G² = 7.91 on 2 df, p = 0.0192 — rejected.
[VD][VP]Given the victim's race, the sentence is independent of the defendant's race. G² = 1.88 on 2 df, p = 0.3903 — not rejected.

Death-sentence rate by defendant's race

White defendants
Black defendants

Labels: rate · sentenced/defendants. Whiskers: 95% Wilson intervals; the widest is for the nine white defendants with black victims, none of whom was sentenced to death.

Data: Radelet (1981), American Sociological Review 46, 918-927.

Added in 2026

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Model [D][V][P] (All three factors are mutually independent.): residual deviance (G2): 137.9 · on 4 df, goodness-of-fit p = 7.83e-29
Model [DV][P] (The sentence is independent of both the defendant's and the victim's race.): residual deviance (G2): 8.132 · on 3 df, goodness-of-fit p = 0.0434
Model [DV][DP] (Given the defendant's race, the sentence is independent of the victim's race.): residual deviance (G2): 7.91 · on 2 df, goodness-of-fit p = 0.0192
Model [VD][VP] (Given the victim's race, the sentence is independent of the defendant's race.): residual deviance (G2): 1.882 · on 2 df, goodness-of-fit p = 0.39

Plus the model description, data source and: Goodness-of-fit p-values test each model against the saturated model; a small p-value means the model does not fit. Observational data: associations, not causal effects. Original analysis: 2023 coursework, refitted in 2026.

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Summary (JSON):
{
  "title": "Defendant's race, victim's race and the death penalty (2 x 2 x 2 table)",
  "model": "Poisson log-linear models for the cell counts; D = defendant's race, V = victim's race, P = death penalty",
  "data": "Radelet (1981): homicide cases in 20 Florida counties, 1976-77",
  "sample_size": "326 defendants",
  "quantities": [
    {
      "label": "Model [D][V][P] (All three factors are mutually independent.): residual deviance (G2)",
      "value": 137.9,
      "note": "on 4 df, goodness-of-fit p = 7.83e-29"
    },
    {
      "label": "Model [DV][P] (The sentence is independent of both the defendant's and the victim's race.): residual deviance (G2)",
      "value": 8.132,
      "note": "on 3 df, goodness-of-fit p = 0.0434"
    },
    {
      "label": "Model [DV][DP] (Given the defendant's race, the sentence is independent of the victim's race.): residual deviance (G2)",
      "value": 7.91,
      "note": "on 2 df, goodness-of-fit p = 0.0192"
    },
    {
      "label": "Model [VD][VP] (Given the victim's race, the sentence is independent of the defendant's race.): residual deviance (G2)",
      "value": 1.882,
      "note": "on 2 df, goodness-of-fit p = 0.39"
    }
  ],
  "context": [
    "Goodness-of-fit p-values test each model against the saturated model; a small p-value means the model does not fit.",
    "Observational data: associations, not causal effects.",
    "Original analysis: 2023 coursework, refitted in 2026."
  ]
}