CSIR-UGC NET - Mathematical Sciences (statistics option) · India · National Testing Agency for CSIR
Original practice questions written to the published syllabus · not past papersModels 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
How our questions for CSIR-UGC NET spread across the units of Categorical Data Analysis & Generalized Linear Models. Difficulty: 1 easy, 17 medium, 24 hard.
| Exponential family and the generalized linear model | 6 | |
| Log-linear models for multi-way tables | 5 | |
| Generalized linear mixed models | 5 | |
| Link functions, deviance and iteratively reweighted least squares | 4 | |
| Poisson regression for counts | 4 | |
| Overdispersion: quasi-likelihood and negative binomial models | 4 | |
| Ordinal and multinomial logistic models | 4 | |
| Zero-inflated and hurdle models | 3 | |
| Model checking: residuals and goodness of fit | 3 | |
| Contingency tables: association, odds ratios and tests | 2 | |
| Logistic regression for binary and binomial data | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What is the variance function V(μ) of the Poisson family?
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
For a logit link, by roughly what factor do probit coefficients differ from logit coefficients fitted to the same data?
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
A Poisson GLM with log link includes an intercept. What property do the fitted values have?
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
How do the variance functions of quasi-Poisson and NB2 negative binomial models differ?
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
How can the proportional-odds assumption be checked?
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
For a zero-inflated Poisson with π = 0.3 and λ = 2, what are the mean and variance?
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
In a zero-inflated negative binomial model, which part's predictors explain the probability of a structural zero?
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
In a Poisson GLMM with log link and a normal random intercept, how do conditional and marginal slopes compare?
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
How is leverage defined for a GLM?
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
For a Poisson GLM with the canonical log link, which statistic is sufficient for β?
D. X′y
With θ = η = Xβ the log-likelihood is β′X′y − Σexp(xᵢ′β) − Σlog yᵢ!, so X′y is sufficient.
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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