ASRB NET - Agricultural Statistics (and Computer Application / Bioinformatics) · 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)
AICE-JRF Genetics and Plant Breeding; ASRB NET
Mahalanobis D², Tocher clustering, inter- and intra-cluster distances and choice of divergent parents.
How our questions for ASRB NET spread across the units of Multivariate Analysis. Difficulty: 15 easy, 39 medium, 4 hard.
| Hotelling's T-squared and its applications | 6 | |
| Multivariate analysis of variance (MANOVA) | 6 | |
| Mahalanobis D-squared and genetic divergence | 6 | |
| Multivariate normal distribution: properties, marginals and conditionals | 5 | |
| Estimation of the mean vector and dispersion matrix | 5 | |
| Principal component analysis | 5 | |
| Factor analysis | 5 | |
| Discriminant analysis and classification | 5 | |
| Cluster analysis: hierarchical and k-means | 5 | |
| Multidimensional scaling and correspondence analysis | 5 | |
| Canonical correlation analysis | 3 | |
| Wishart distribution | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
X ~ Nₚ(μ, Σ), and A is a fixed q × p matrix. What is the distribution of AX + b?
B. N(Aμ + b, AΣA′)
Linear transformations of a multivariate normal are multivariate normal, with mean Aμ + b and dispersion AΣA′. A′ΣA has the wrong dimensions for a q-vector.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
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 varieties are compared on 3 quality traits with 12 and 10 plants. The sample Mahalanobis D² is 2.5. What is the two-sample T²?
A. 13.64
T² = [n₁n₂/(n₁ + n₂)]D² = (120/22) × 2.5 = 13.64. Its F form is [(n₁ + n₂ − p − 1)/(p(n₁ + n₂ − 2))]T² = (18/60) × 13.64 = 4.09 with (3, 18) df.
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
A correlation-matrix PCA of five traits gives eigenvalues 2.6, 1.2, 0.6, 0.4 and 0.2. What proportion of total variance do the first two components explain?
D. 76%
For a correlation matrix the eigenvalues sum to p = 5, so (2.6 + 1.2)/5 = 76%. 52% counts only the first component; 86.4% divides by the first three eigenvalues instead of the total.
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 standardised variable has loadings 0.8 and 0.3 on two orthogonal factors. What is its communality?
C. 0.73
h² = 0.8² + 0.3² = 0.64 + 0.09 = 0.73; its uniqueness is 1 − 0.73 = 0.27. Adding loadings without squaring gives 1.10.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
For two groups with a common dispersion matrix, what is Fisher's linear discriminant function coefficient vector?
B. S_p⁻¹(x̄₁ − x̄₂)
Maximising between-group to within-group variance of a′x gives a ∝ S_p⁻¹(x̄₁ − x̄₂). The first eigenvector of S_p is the first principal component, which ignores the group difference.
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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