Assignment 2 · Question 2 · Binomial GLM with factors

Education, gender and attitudes to women's role

In the 1974–75 US General Social Surveys, respondents were asked whether they agreed that “Women should take care of running their homes and leave the running of the country up to men.” The task: model agreement against sex and years of education, decide whether the two act additively, and find the best model that uses education's ordering.

The chosen model

For a respondent with $e$ years of education, and $F$ equal to 1 for women and 0 for men, model.5 gives each sex its own intercept and slope:

\operatorname{logit}\pi = 2.455 + 0.893\,F - 0.2635\,e - 0.0817\,e F

\begin{aligned}\operatorname{logit}\pi &= 2.455 + 0.893\,F \\ &\quad - 0.2635\,e - 0.0817\,e F\end{aligned}

Compare the candidate models

Choose a model

model.2dev 0.00 / 0 dffactor(Sex) * factor(Education)model.4dev 15.16 / 11 dffactor(Sex) + factor(Education)Parallel linesdev 25.09 / 21 dffactor(Sex) + Educationmodel.5chosendev 20.24 / 20 dffactor(Sex) + Education + Education:factor(Sex)model.6dev 20.24 / 19 dffactor(Sex) + Education + I(Education^2) + Education:factor(Sex)model.7dev 9.95 / 10 dffactor(Sex) + Education + factor(Education) + education:factor(Sex)

Separate logit slopes in education for men and women — the chosen model (Q2b, Q2c).

Share who agree, by years of education

Men (observed, area ∝ respondents)
Women (observed)
Fitted, men
Fitted, women

Fit of the selected model

Deviance: 20.244
Resid. df: 20
AIC: 136.69

Goodness of fit: P(χ²20 > 20.24) = 0.4428

Odds ratio for more education (model.5)

Extra years of education

Men

× 0.349

95% CI (0.286, 0.425)

Women

× 0.251

95% CI (0.203, 0.311)

Each extra year of schooling lowers the odds of agreeing by about 23% for men and 29% for women; the gap is the significant sex × education interaction.

Analysis of deviance

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

Term added	Df	Deviance	Resid. Df	Resid. Dev	Pr(>Chi)
NULL			23	337.903
factor(Sex)	1	0.370	22	337.533	0.5431
Education	1	312.446	21	25.087	6.4 × 10⁻⁷⁰ ***
factor(Sex):Education	1	4.843	20	20.244	0.0278 *

Coefficients

Term	Estimate	Std. error	z	p-value
(Intercept)	2.4554	0.2934	8.37	5.9 × 10⁻¹⁷ ***
factor(Sex)Female	0.8927	0.4304	2.07	0.0381 *
Education	−0.2635	0.0253	−10.43	1.9 × 10⁻²⁵ ***
factor(Sex)Female:Education	−0.0817	0.0372	−2.20	0.0280 *

The tests behind the choice

2a · model.4 vs saturated
Are sex and (nominal) education additive on the logit scale?
Deviance change
15.16 on 11 df
p
0.1753
Yes — no evidence of interaction once education is a factor.
2b · Sex × Education in model.5
With education linear, do men and women need different slopes?
Deviance change
4.843 on 1 df
p
0.0278
Yes — the slopes differ at the 5% level.
2b · Education² in model.6
Is there curvature in education?
Deviance change
0.0076 on 1 df
p
0.9304
No — the quadratic term adds nothing.
2c · model.5 vs model.7
Do nominal education effects improve on the linear trend?
Deviance change
10.297 on 10 df
p
0.4148
No — the simpler model.5 is preferred.

Part	Question	Comparison	Deviance change	p	Conclusion
2a	Are sex and (nominal) education additive on the logit scale?	model.4 vs saturated	15.16 on 11 df	0.1753	Yes — no evidence of interaction once education is a factor.
2b	With education linear, do men and women need different slopes?	Sex × Education in model.5	4.843 on 1 df	0.0278	Yes — the slopes differ at the 5% level.
2b	Is there curvature in education?	Education² in model.6	0.0076 on 1 df	0.9304	No — the quadratic term adds nothing.
2c	Do nominal education effects improve on the linear trend?	model.5 vs model.7	10.297 on 10 df	0.4148	No — the simpler model.5 is preferred.

Data: National Opinion Research Center, University of Chicago, General Social Surveys 1974-1975.

Added in 2026

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Coefficient (Intercept): 2.455 · SE 0.2934, p = 5.86e-17
Coefficient factor(Sex)Female: 0.8927 · SE 0.4304, p = 0.0381
Coefficient Education: -0.2635 · SE 0.02527, p = 1.89e-25
Coefficient factor(Sex)Female:Education: -0.08172 · SE 0.0372, p = 0.028
Odds ratio per extra year of education, men: 0.7683 · 95% CI (0.7312, 0.8074)
Odds ratio per extra year of education, women: 0.7081 · 95% CI (0.6712, 0.747)
Residual deviance: 20.24 · on 20 df, goodness-of-fit p = 0.443

Plus the model description, data source and: Observational survey data: associations, not causal effects. Original analysis: 2023 coursework, refitted in 2026.

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Message

Explain this model output.

Summary (JSON):
{
  "title": "Education, sex and agreement that women should leave running the country to men",
  "model": "Binomial GLM, logit link: factor(Sex) + Education + Education:factor(Sex) (model.5)",
  "data": "NORC General Social Surveys 1974-75, counts by years of education (6 to 17) and sex",
  "sample_size": "2678 respondents in 24 education-by-sex cells",
  "quantities": [
    {
      "label": "Coefficient (Intercept)",
      "value": 2.455,
      "note": "SE 0.2934, p = 5.86e-17"
    },
    {
      "label": "Coefficient factor(Sex)Female",
      "value": 0.8927,
      "note": "SE 0.4304, p = 0.0381"
    },
    {
      "label": "Coefficient Education",
      "value": -0.2635,
      "note": "SE 0.02527, p = 1.89e-25"
    },
    {
      "label": "Coefficient factor(Sex)Female:Education",
      "value": -0.08172,
      "note": "SE 0.0372, p = 0.028"
    },
    {
      "label": "Odds ratio per extra year of education, men",
      "value": 0.7683,
      "note": "95% CI (0.7312, 0.8074)"
    },
    {
      "label": "Odds ratio per extra year of education, women",
      "value": 0.7081,
      "note": "95% CI (0.6712, 0.747)"
    },
    {
      "label": "Residual deviance",
      "value": 20.24,
      "note": "on 20 df, goodness-of-fit p = 0.443"
    }
  ],
  "context": [
    "Observational survey data: associations, not causal effects.",
    "Original analysis: 2023 coursework, refitted in 2026."
  ]
}