CSIR-UGC NET - Mathematical Sciences (statistics option) · India · National Testing Agency for CSIR
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 CSIR-UGC NET spread across the units of Multivariate Analysis. Difficulty: 0 easy, 18 medium, 21 hard.
| Wishart distribution | 5 | |
| Canonical correlation analysis | 5 | |
| Multivariate normal distribution: properties, marginals and conditionals | 4 | |
| Factor analysis | 4 | |
| Discriminant analysis and classification | 4 | |
| Estimation of the mean vector and dispersion matrix | 3 | |
| Multivariate analysis of variance (MANOVA) | 3 | |
| Multidimensional scaling and correspondence analysis | 3 | |
| Hotelling's T-squared and its applications | 2 | |
| Principal component analysis | 2 | |
| Cluster analysis: hierarchical and k-means | 2 | |
| Mahalanobis D-squared and genetic divergence | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
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
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 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
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
With priors p₁, p₂ and costs c(1 | 2), c(2 | 1), when does the minimum-expected-cost rule assign x to group 1?
D. When f₁(x)/f₂(x) > [c(1 | 2)p₂]/[c(2 | 1)p₁]
Expected cost is minimised by comparing the density ratio with the ratio of prior-weighted costs of the two errors; with equal priors and costs the threshold is 1.
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
Which approach gives a likelihood-based choice of the number of clusters and membership probabilities?
B. Normal-mixture model-based clustering
Fitting mixtures of multivariate normals by EM allows comparison of numbers of components by BIC and gives posterior membership probabilities.
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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