ASRB NET - Agricultural Statistics (and Computer Application / Bioinformatics) · India · Agricultural Scientists Recruitment Board (ICAR)
Original practice questions written to the published syllabus · not past papersThe general linear model in matrix form: estimability, the Gauss-Markov theorem, least squares with full and less-than-full rank, the analysis of variance as a linear model, model checking and selection, and regression diagnostics. It is the theory under every design in Phase 3.
In the exams: AICE-JRF Ag Stat Units 2 and 4 (heavy); GATE ST 'Regression Analysis' (heavy); UPSC ISS Statistics II (heavy); GATE DA regression (present)
ASRB NET / ARS Agricultural Statistics
Applied regression and linear-model questions: which model, which test, which diagnostic.
How our questions for ASRB NET spread across the units of Linear Models & Regression Analysis. Difficulty: 13 easy, 39 medium, 18 hard.
| Fixed, random and mixed effects; nested and crossed classifications | 8 | |
| Non-linear regression: estimation and inference | 8 | |
| Multiple and partial correlation | 7 | |
| Examination of residuals and model adequacy | 7 | |
| Selecting the best regression: stepwise, all-subsets, AIC and BIC | 7 | |
| Influence and leverage: Cook's distance and hat values | 6 | |
| Multicollinearity and ridge regression | 6 | |
| Weighted and generalised least squares | 6 | |
| Estimable functions and the Gauss-Markov theorem | 5 | |
| Less-than-full-rank models and generalised inverses | 5 | |
| Distribution of quadratic forms and the Fisher-Cochran theorem | 2 | |
| Testing linear hypotheses; R-squared and adjusted R-squared | 2 | |
| The general linear model and least squares in matrix form | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
In the one-way model y_ij = mu + tau_i + e_ij (written without constraints), which function is estimable?
C. tau_1 - tau_2
Contrasts among treatment effects (coefficients summing to zero) and treatment means mu + tau_i are estimable; mu, a single tau_i or a non-contrast sum like tau_1 + tau_2 are not.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
Three farms are sampled, with four fields within each farm and two soil cores per field. How many degrees of freedom do fields within farms have?
C. 9
a(b - 1) = 3 x 3 = 9. 11 = 12 - 1 ignores farms; 12 is the core-within-field df.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
What does the partial correlation r12.3 measure?
A. Correlation of 1 and 2 after removing the linear effect of 3
It is the correlation between residuals of 1 and 2 after regression on 3; the combined correlation is the multiple correlation.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
An observation has raw residual 3, the residual standard deviation is s = 2, and its leverage is 0.19. What is its internally studentised residual?
B. 1.667
r = e/(s sqrt(1 - h)) = 3/(2 x 0.9) = 1.667. e/s = 1.5 is the standardised residual ignoring leverage.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
The lag-one autocorrelation of residuals in time order is about 0.4. What is the approximate Durbin-Watson statistic?
B. 1.2
d is approximately 2(1 - rho-hat) = 2 x 0.6 = 1.2, below 2, indicating positive autocorrelation.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
Regressing predictor x_j on the other predictors gives R_j^2 = 0.9. What is its variance inflation factor?
B. 10
VIF = 1/(1 - R_j^2) = 1/0.1 = 10, usually taken as serious collinearity.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
What is the ridge trace used for?
A. Choosing k where the coefficients settle
Plotting coefficients against k shows where they settle; cross-validation is the modern alternative.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
A subset model with p = 4 parameters has RSS = 120; the full model's residual mean square is 5; n = 30. What is Mallows' Cp?
A. 2.0
Cp = RSS_p/s^2 - n + 2p = 24 - 30 + 8 = 2.0. Values near or below p indicate little bias.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
For a subset model with little bias, Mallows' Cp should be close to:
D. p, the number of parameters in the subset
E(Cp) is about p when the subset model is unbiased.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
Which plot is used to check the normality of residuals?
D. A normal quantile (Q-Q) plot
Points close to a straight line on a normal Q-Q plot indicate approximately normal residuals.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
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