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 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)
ICAR AICE-JRF/SRF Agricultural Statistics (Units 2 and 4)
Linear models with fixed, random and mixed effects, nested and crossed classifications, the Gauss-Markov theorem; simple, multiple and partial correlation; regression, residual examination, model adequacy and selection of the best regression.
How our questions for ICAR AICE-JRF/SRF spread across the units of Linear Models & Regression Analysis. Difficulty: 20 easy, 51 medium, 27 hard.
| The general linear model and least squares in matrix form | 10 | |
| Fixed, random and mixed effects; nested and crossed classifications | 9 | |
| Examination of residuals and model adequacy | 9 | |
| Distribution of quadratic forms and the Fisher-Cochran theorem | 8 | |
| Testing linear hypotheses; R-squared and adjusted R-squared | 8 | |
| Weighted and generalised least squares | 8 | |
| Estimable functions and the Gauss-Markov theorem | 7 | |
| Influence and leverage: Cook's distance and hat values | 7 | |
| Multicollinearity and ridge regression | 7 | |
| Non-linear regression: estimation and inference | 7 | |
| Less-than-full-rank models and generalised inverses | 6 | |
| Multiple and partial correlation | 6 | |
| Selecting the best regression: stepwise, all-subsets, AIC and BIC | 6 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
For n = 12 observations, Sxx = 40, Sxy = 60, x-bar = 5 and y-bar = 20. What is the least squares intercept?
A. 12.5
b = Sxy/Sxx = 1.5; a = y-bar - b x-bar = 20 - 1.5 x 5 = 12.5. Using Sxx/Sxy as the slope gives 16.7; adding b x-bar gives 27.5.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
When X has full column rank, what is the least squares estimator of beta in y = X beta + e?
B. (X'X)^(-1) X'y
The normal equations X'X beta-hat = X'y give beta-hat = (X'X)^(-1) X'y. The other forms have wrong order or dimension.
Further reading: Searle, Linear Models; Draper and Smith, Applied Regression Analysis
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
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
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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