ASRB NET Categorical Data Analysis & Generalized Linear Models: practice questions

ASRB NET - Agricultural Statistics (and Computer Application / Bioinformatics) · India · Agricultural Scientists Recruitment Board (ICAR)

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

10worked here, free

67written 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 ASRB NET asks from it

ASRB NET / ARS and research-institute interviews

Choosing the right model for germination, mortality, pest counts and adoption data; overdispersion; LD50 estimation; interpreting software output.

Where the 67 questions fall

How our questions for ASRB NET spread across the units of Categorical Data Analysis & Generalized Linear Models. Difficulty: 14 easy, 38 medium, 15 hard.

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

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

A logistic regression fails to converge, with one coefficient and its standard error growing very large. What is the likely cause and remedy?

Too few iterations: increase the iteration limit
Overdispersion: use a quasi-binomial model
Separation; use Firth's penalised likelihood
A wrong link: switch from logit to probit

Answer and working

C. Separation; use Firth's penalised likelihood

Under complete or quasi-complete separation the MLE is infinite, so estimates diverge; Firth's bias-reduced likelihood or exact methods give finite estimates. More iterations or another link do not help.

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

5 · Poisson regression for counts · medium

Trap A catches 42 thrips in 6 days; trap B catches 50 in 10 days. With log(days) as offset, what is the estimated log rate ratio A versus B?

2.000
0.336
−0.174
−0.511

Answer and working

B. 0.336

Rates are 7.0 and 5.0 per day, ratio 1.4, log 1.4 = 0.336. log(42/50) = −0.174 ignores exposure and wrongly favours trap B.

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

6 · 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

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 · Zero-inflated and hurdle models · medium

Many fields have no pest because it has not yet arrived, while infested fields have Poisson-like counts that can also be zero. Which model matches this story?

A proportional-odds model
A hurdle model
A quasi-Poisson model
Zero-inflated model

Answer and working

D. Zero-inflated model

Two sources of zeros, structural (pest absent) and sampling (present but none caught), is the zero-inflated story. A hurdle model assumes all zeros come from one binary process.

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 · 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

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