ICAR AICE-JRF/SRF Categorical Data Analysis & Generalized Linear Models: practice questions

ICAR AICE-JRF/SRF - Agricultural Statistics, Computer Application, Bioinformatics (PhD admission and fellowship) · India · National Testing Agency for ICAR

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

10worked here, free

65written 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

What ICAR AICE-JRF/SRF asks from it

ICAR AICE-JRF/SRF Agricultural Statistics (Unit 2)

Contingency tables, chi-square and likelihood-ratio tests, odds ratios, log-linear models and their hierarchy and df, and the logistic and probit models used in bioassay.

Where the 65 questions fall

How our questions for ICAR AICE-JRF/SRF spread across the units of Categorical Data Analysis & Generalized Linear Models. Difficulty: 8 easy, 30 medium, 27 hard.

Link functions, deviance and iteratively reweighted least squares	9
Log-linear models for multi-way tables	9
Exponential family and the generalized linear model	8
Logistic regression for binary and binomial data	6
Generalized linear mixed models	6
Model checking: residuals and goodness of fit	6
Contingency tables: association, odds ratios and tests	5
Overdispersion: quasi-likelihood and negative binomial models	5
Poisson regression for counts	4
Ordinal and multinomial logistic models	4
Zero-inflated and hurdle models	3

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 are the three components of a generalized linear model?

A prior, a likelihood and a posterior
A random component, a linear predictor and a link function
A response, an error term and a least-squares criterion
A design matrix, a hat matrix and a residual vector

Answer and working

B. A random component, a linear predictor and a link function

GLM: exponential-family response (random component), η = x′β (systematic component) and g(μ) = η (link).

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

3 · 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

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

A germination model has residual deviance 31.2 on 16 df, and dropping treatments raises it by 21.6 on 3 df. Taking the dispersion as φ̂ = 31.2/16, what is the quasi-likelihood F statistic?

7.20 on (3, 16) df
11.08 on (3, 16) df
3.69 on (3, 16) df
21.60 on (3, 16) df

Answer and working

C. 3.69 on (3, 16) df

φ̂ = 1.95; F = (21.6/3)/1.95 = 3.69. 7.20 ignores overdispersion; 11.08 forgets to divide by the 3 df.

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

5 · 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

6 · 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

7 · Overdispersion: quasi-likelihood and negative binomial models · hard

How do the variance functions of quasi-Poisson and NB2 negative binomial models differ?

Quasi-Poisson is quadratic; NB2 is linear
Both are quadratic in μ
Both are linear in μ
Linear in μ versus quadratic in μ

Answer and working

D. Linear in μ versus quadratic in μ

The shapes differ, so the models weight large counts differently; a plot of squared residuals against fitted means helps choose.

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

8 · 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

9 · Zero-inflated and hurdle models · hard

In a zero-inflated negative binomial model, which part's predictors explain the probability of a structural zero?

The log-linear model for the count mean
The offset term
The aggregation parameter k
A logistic model for π

Answer and working

D. A logistic model for π

ZINB has two linear predictors: logit π for structural zeros and log μ for the counts; the same or different covariates can enter each.

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

10 · 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

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