ASRB Agricultural Research Service (ARS) - Agricultural Statistics scientist · India · Agricultural Scientists Recruitment Board (ICAR)
Original practice questions written to the published syllabus · not past papersAnalysing many correlated variables together: the multivariate normal and its estimation, Wishart and Hotelling's T-squared, MANOVA, and the reduction and classification tools agricultural scientists use daily (principal components, factor analysis, discriminant and cluster analysis, Mahalanobis D-squared).
In the exams: AICE-JRF Ag Stat Unit 2 (heavy: MVN, Wishart, Hotelling, MANOVA, PCA, FA, discriminant, cluster named); GATE ST 'Multivariate Analysis' (heavy); UPSC ISS Statistics II (heavy); GATE DA PCA (present)
How our questions for ASRB Agricultural Research Service spread across the units of Multivariate Analysis. Difficulty: 0 easy, 25 medium, 2 hard.
| Estimation of the mean vector and dispersion matrix | 4 | |
| Principal component analysis | 3 | |
| Discriminant analysis and classification | 3 | |
| Multidimensional scaling and correspondence analysis | 3 | |
| Hotelling's T-squared and its applications | 2 | |
| Multivariate analysis of variance (MANOVA) | 2 | |
| Factor analysis | 2 | |
| Canonical correlation analysis | 2 | |
| Cluster analysis: hierarchical and k-means | 2 | |
| Mahalanobis D-squared and genetic divergence | 2 | |
| Multivariate normal distribution: properties, marginals and conditionals | 1 | |
| Wishart distribution | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Two traits keep the same variances, but their correlation rises from 0.5 to 0.9. What happens to the generalised variance |S|?
D. It decreases
|S| = s₁²s₂²(1 − r²), which shrinks as |r| grows: the scatter ellipse flattens. The total variance tr(S) would be unchanged. |S| cannot be negative for a valid dispersion matrix.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Two seed treatments are compared on germination, vigour and root length, measured on the same seed lots. How should the multivariate comparison be made?
C. T² on within-lot differences
Paired multivariate data are analysed as differences: T² = n d̄′S_d⁻¹d̄ with (p, n − p) df after the F transformation. Treating paired data as independent ignores the lot effect.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Why is one MANOVA on five correlated quality traits preferred to five separate ANOVAs at α = 0.05?
A. It controls overall error and uses the trait correlations
Separate ANOVAs inflate the family-wise error rate and ignore correlations; MANOVA tests the vector of means jointly and can detect differences in combinations of traits. It does assume multivariate normality and equal covariance matrices.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Which method applies principal component analysis to the genotype-by-environment interaction matrix after removing main effects?
C. The AMMI model
AMMI (additive main effects and multiplicative interaction) fits additive genotype and environment effects by ANOVA and the interaction by PCA (singular value decomposition); GGE biplots apply it to genotype plus G × E.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
A factor analysis reports an estimated uniqueness of −0.05 for one variable. What is this called and what does it suggest?
C. A Heywood case
Variances cannot be negative; a zero or negative uniqueness (Heywood case) indicates an improper solution, from over-factoring, small samples or near-collinear variables.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
A first canonical correlation of 0.85 is found between soil and crop sets. Why might it overstate how much the sets share?
B. Redundancy indices, not ρ₁, measure shared variance
A high ρ₁ can reflect one strongly related pair while most variance in each set is unrelated; redundancy indices measure the proportion of one set's variance explained by the other's canonical variates.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Why are structure coefficients usually interpreted instead of raw canonical coefficients?
B. Raw weights are unstable under within-set collinearity
With multicollinearity, raw weights can have unexpected signs and large sampling variability; correlations between variables and canonical variates (structure coefficients) are more stable.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Diseased and healthy plants are to be classified from reflectance at several wavelengths plus a categorical cultivar code. Which method avoids assuming multivariate normal predictors?
A. Logistic regression
Logistic regression models P(diseased | x) directly and handles categorical predictors; LDA is more efficient when normality holds but is sensitive when it does not.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
Farm-household survey data contain many categorical variables (crop type, irrigation source, tenure, credit access). Which method maps all their categories together?
B. Multiple correspondence analysis
MCA extends correspondence analysis to several categorical variables via the indicator or Burt matrix; PCA on arbitrary category codes treats nominal codes as numbers.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
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