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 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)
ICAR AICE-JRF/SRF Agricultural Statistics (Unit 2)
Multivariate normal properties and conditional distributions, Wishart, Hotelling's T² and its F transformation, MANOVA and Wilks' Λ, principal components, factor analysis, discriminant and cluster analysis: definitions, properties and one-step computations.
How our questions for ICAR AICE-JRF/SRF spread across the units of Multivariate Analysis. Difficulty: 15 easy, 38 medium, 22 hard.
| Multivariate analysis of variance (MANOVA) | 8 | |
| Mahalanobis D-squared and genetic divergence | 8 | |
| Multivariate normal distribution: properties, marginals and conditionals | 7 | |
| Wishart distribution | 7 | |
| Hotelling's T-squared and its applications | 7 | |
| Factor analysis | 7 | |
| Canonical correlation analysis | 6 | |
| Discriminant analysis and classification | 6 | |
| Principal component analysis | 5 | |
| Cluster analysis: hierarchical and k-means | 5 | |
| Multidimensional scaling and correspondence analysis | 5 | |
| Estimation of the mean vector and dispersion matrix | 4 |
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
Which condition is needed for the sample dispersion matrix S of p variables to be non-singular (with probability one, under normality)?
A. n > p
S has rank at most n − 1, so n − 1 ≥ p, i.e. n > p, is necessary; under normality it is then non-singular with probability one. In practice n should be several times p.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
W₁ ~ Wₚ(m₁, Σ) and W₂ ~ Wₚ(m₂, Σ) are independent. What is the distribution of W₁ + W₂?
D. Wₚ(m₁ + m₂, Σ)
Additivity: pooling independent Wishart matrices with the same Σ adds degrees of freedom, which is why pooled within-group SSCP matrices are Wishart.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
W ~ Wₚ(m, Σ) with m ≥ p. How is |W|/|Σ| distributed?
C. As a product of independent χ²_m, χ²_(m−1), ..., χ²_(m−p+1)
Bartlett's decomposition writes W in terms of a triangular matrix with independent chi entries on the diagonal having m, m − 1, ..., m − p + 1 df, so the determinant is their product.
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
Which statement about principal components is correct?
B. They are uncorrelated, and their variances sum to tr(Σ)
PCs are uncorrelated by construction and preserve total variance; independence needs normality. Covariance and correlation PCA differ in general. Maximising group separation is the aim of discriminant analysis.
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
Canonical correlations 0.6 and 0.3 are found between 3 soil and 2 crop variables on 50 fields. What is Bartlett's chi-square for H₀: all ρᵢ = 0, and its df?
B. 24.87 with 6 df
−[n − 1 − (p + q + 1)/2]Σ ln(1 − rᵢ²) = −46 × [ln 0.64 + ln 0.91] = 24.87, with pq = 6 df. Omitting the correction gives 26.49; using ln(1 − r) instead of ln(1 − r²) is a slip.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
What does the cophenetic correlation measure?
C. Dendrogram fidelity to original distances
Values close to 1 mean the tree represents the original dissimilarities faithfully; low values warn that the dendrogram distorts the data.
Further reading: Johnson and Wichern, Applied Multivariate Statistical Analysis; Anderson, An Introduction to Multivariate Statistical Analysis; Mardia, Kent and Bibby, Multivariate Analysis
For uncorrelated traits with standard deviations σⱼ, what does D² between two population means reduce to?
C. Σ(δⱼ/σⱼ)²
With diagonal Σ, Σ⁻¹ has entries 1/σⱼ², so D² = Σδⱼ²/σⱼ²: the sum of squared standardised differences.
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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