CUET-PG Categorical Data Analysis & Generalized Linear Models: practice questions

CUET-PG - Statistics · India · National Testing Agency

Original practice questions written to the published syllabus · not past papers

10worked here, free

75written to this exam in this subject

11syllabus units

What this subject covers

Models for counts, proportions and categories: contingency tables, the exponential-family GLM with its link functions, logistic and Poisson regression, log-linear models, overdispersion, ordinal outcomes and generalised linear mixed models, applied to germination, mortality, pest counts and adoption data.

In the exams: AICE-JRF Ag Stat Unit 2 'contingency tables, log linear models' (heavy); GATE DA logistic regression (present); light in UPSC ISS

Contingency tables: association, odds ratios and tests
Exponential family and the generalized linear model
Link functions, deviance and iteratively reweighted least squares
Logistic regression for binary and binomial data
Poisson regression for counts
Overdispersion: quasi-likelihood and negative binomial models
Log-linear models for multi-way tables
Ordinal and multinomial logistic models
Zero-inflated and hurdle models
Generalized linear mixed models
Model checking: residuals and goodness of fit

Where the 75 questions fall

How our questions for CUET-PG spread across the units of Categorical Data Analysis & Generalized Linear Models. Difficulty: 17 easy, 45 medium, 13 hard.

Contingency tables: association, odds ratios and tests	9
Link functions, deviance and iteratively reweighted least squares	8
Poisson regression for counts	8
Ordinal and multinomial logistic models	8
Zero-inflated and hurdle models	7
Model checking: residuals and goodness of fit	7
Exponential family and the generalized linear model	6
Overdispersion: quasi-likelihood and negative binomial models	6
Generalized linear mixed models	6
Logistic regression for binary and binomial data	5
Log-linear models for multi-way tables	5

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Contingency tables: association, odds ratios and tests · hard

For the 2 × 2 table with cells 80, 20 (treated) and 60, 40 (untreated), what is the 95% confidence interval for the odds ratio?

2.03 to 3.30
1.42 to 5.02
1.42 to 2.67
2.17 to 3.27

Answer and working

B. 1.42 to 5.02

SE(log θ̂) = √(1/80 + 1/20 + 1/60 + 1/40) = 0.323; exp(log 2.67 ± 1.96 × 0.323) = 1.42 to 5.02. Adding ±1.96 SE directly to θ̂ ignores the skewness of the odds ratio; using the variance instead of the SE gives too narrow an interval.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

2 · Exponential family and the generalized linear model · easy

What is the variance function V(μ) of the Poisson family?

μ(1 − μ)
1
μ
μ²

Answer and working

C. μ

Poisson Var(y) = μ. The normal has V = 1, the gamma μ², the binomial proportion μ(1 − μ).

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

3 · Link functions, deviance and iteratively reweighted least squares · hard

For a logit link, by roughly what factor do probit coefficients differ from logit coefficients fitted to the same data?

They are identical, since both links are symmetric
Probit coefficients are larger, about 1.6 to 1.8 times
Probit coefficients are smaller, about 1/1.6 to 1/1.8 of the logit ones
Probit coefficients are about π times the logit ones

Answer and working

C. Probit coefficients are smaller, about 1/1.6 to 1/1.8 of the logit ones

The logistic distribution has SD π/√3 ≈ 1.81 against 1 for the standard normal, so logit coefficients are larger by about 1.6-1.8. The exact ratio depends on the data.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

4 · Poisson regression for counts · medium

A Poisson GLM with log link includes an intercept. What property do the fitted values have?

Σμ̂ᵢ = Σyᵢ
μ̂ᵢ = yᵢ for every i
Σμ̂ᵢ = 0
Σ(yᵢ − μ̂ᵢ)² = 0

Answer and working

A. Σμ̂ᵢ = Σyᵢ

The likelihood equation for the intercept is Σ(yᵢ − μ̂ᵢ) = 0, so fitted and observed totals agree.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

5 · Ordinal and multinomial logistic models · medium

How can the proportional-odds assumption be checked?

By checking that category frequencies are equal
Score test, or slopes of cut-point logits
By the Hosmer-Lemeshow test on the highest category
By the deviance of the model divided by its df

Answer and working

B. Score test, or slopes of cut-point logits

Separate cumulative binary logits should give similar slopes if the assumption holds; the score test formalises this. Partial proportional-odds models relax it.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

6 · Zero-inflated and hurdle models · medium

For a zero-inflated Poisson with π = 0.3 and λ = 2, what are the mean and variance?

Mean 1.4, variance 1.4
Mean 1.4, variance 2.24
Mean 2.0, variance 2.0
Mean 2.0, variance 3.2

Answer and working

B. Mean 1.4, variance 2.24

E(Y) = (1 − π)λ = 1.4; Var(Y) = (1 − π)λ(1 + πλ) = 1.4 × 1.6 = 2.24. Zero inflation alone makes the counts overdispersed.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

7 · Zero-inflated and hurdle models · easy

In a hurdle model for pest counts, where do zeros come from?

From neither; zeros are excluded
Only from the count part
Only from the binary part
From both the binary and count parts

Answer and working

C. Only from the binary part

The count part is zero-truncated, so every zero comes from the binary zero-versus-positive model. In zero-inflated models zeros come from both sources.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

8 · Generalized linear mixed models · hard

In a Poisson GLMM with log link and a normal random intercept, how do conditional and marginal slopes compare?

They are unrelated
They are equal; only the intercept differs
Marginal slopes are larger
Marginal slopes are attenuated by √(1 + 0.346σ²)

Answer and working

B. They are equal; only the intercept differs

E(y) = exp(x′β)E(e^b) = exp(x′β + σ²/2), so only the intercept shifts by σ²/2; slopes are the same on both scales.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

9 · Generalized linear mixed models · medium

How can an observation-level random effect help in a Poisson GLMM?

It absorbs overdispersion
It removes the need for an offset
It ensures every fitted count is an integer
It converts counts to proportions

Answer and working

A. It absorbs overdispersion

A normal random effect for each observation makes the counts a Poisson-lognormal mixture, with variance exceeding the mean.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

10 · Model checking: residuals and goodness of fit · hard

How is leverage defined for a GLM?

From the weighted hat matrix W^(1/2)X(X′WX)⁻¹X′W^(1/2)
As the change in deviance when the observation is deleted
As the Pearson residual squared
From the unweighted hat matrix X(X′X)⁻¹X′

Answer and working

A. From the weighted hat matrix W^(1/2)X(X′WX)⁻¹X′W^(1/2)

IRLS weights enter the hat matrix, so leverage depends on fitted means as well as the design; Cook's distance combines leverage and residual size.

Further reading: McCullagh and Nelder, Generalized Linear Models; Agresti, Categorical Data Analysis; Dobson and Barnett, An Introduction to Generalized Linear Models

Ten here, 75 in the app. The Laisha app has every practice question we have written to CUET-PG in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.

Free for ten days, then one subscription of US$10 a month. We would rather say so here than have you find it later.

Practise the full set in Laisha →