Canonical Correlation Analysis Using Small Number of Samples

Abstract

Canonical correlation assesses the relationship between two groups of variables. Although it has been a useful tool in a wide variety of research areas, it is not well known that weaker canonical correlations require larger sample sizes to be correctly inferred. In this article, we investigate small sample bias in canonical correlation analysis and apply the jackknife bias correction to the estimation of canonical correlations. We use bootstrap samples to obtain a better confidence interval for the jackknife canonical correlation estimator.

Keywords:

• Bootstrap
• Canonical correlation
• Jackknife
• Small sample bias

Mathematics Subject Classification:

• Primary 62H20
• Secondary 62F40

Acknowledgment

The author thanks Dr. Myunghee Cho Paik, Professor in Biostatistics at Columbia University, for suggestions that led to improvements in this article.

Notes

γ₁₀: true correlation; bias(γ₁) = γ₁ − γ₁₀; : standard estimator; : jackknife estimator; : bootstrap estimator

Pairs: number of pairs; γ₁₀: true maximum canonical correlation; ASV: asymptotic variance estimator; SJKV: standard jackknife variance estimator; 95% CR: coverage rate of 95% confidence interval

Pairs: number of pairs; γ₁₀: true maximum canonical correlation; ASV(γ₁): asymptotic variance estimator; SJKV: standard jackknife variance estimate; 95% CR: coverage rate of 95% confidence interval

Pairs: number of pairs; γ₁₀: true maximum canonical correlation; BJKV: bootstrap jackknife variance estimate; 95% CR: coverage rate of 95% confidence interval; Length: average length of confidence interval; Boot-t: using standardized pivotal distribution of bootstrap jackknife estimates; Boot-%: using distribution of bootstrap jackknife estimates

Pairs: number of pairs; γ₁₀: true maximum canonical correlation; BJKV: bootstrap jackknife variance estimate; 95% CR: coverage rate of 95% confidence interval; Length: average length of confidence interval; Boot-t: using standardized pivotal distribution of bootstrap jackknife estimates; Boot-%: using distribution of bootstrap jackknife estimates