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Understanding Variance and Covariance in Statistical Inference

This article provides a mathematical description of variance and covariance in statistical inference, including their properties such as sphericity and independence. It also discusses sources of violations and how to use this knowledge for proper statistical inference.

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Understanding Variance and Covariance in Statistical Inference

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  1. n n ∑(yi - µ)2 σ2 = n - 1 ∑ yi2 σ2 = n - 1 ^ ^ Variance: (y - µ) t = σ/sqrt(n) ^ mean corrected: Degrees of freedom: n - 1

  2. y1 x1ε1 y2 x2ε2 . = . . . . . yn Xnεn β * + E ~ N(0, σ2) y = βx + ε

  3. E ~ N(0, Cε) covariance yT y Cε y1 y2 … yn x y1y1y1 y1y2 … y1yny12 y2 y2y1y2y2 … y2yn y22 . = . = . . yn yny1yny2 … ynyn yn2 n ∑ yi2 σ2 = ~ y12 + y22 + … + yn2 n - 1 ^ E ~ N(0, σ2) Variance-covariance matrix Cε= yTy : variance

  4. y12 = y22 = … = yn2 1 1 1 * k 1 1 e2 e2 e1 Cε = Cε = 1 0 0 1 4 0 0 1 e1 Properties of covariance matrix Homogeneous or identical error variance or sphericity Non-identical

  5. y1y2 = 0 0 0 0 0 y1y3= 0 0 0 0 0 . 0 0 0 0 . 0 0 0 0 ynym=0 0 0 0 0 e2 Cε = 1 3 0.5 5 e1 Non-independent Properties of covariance matrix Independent error components Ex: Temporal autocorrelation

  6. Overview • Motivation for considering variance • Mathematical description of variance: E ~N(0,Cε) • Properties of Cε: - sphericity • - independence  E ~N(0,Cε) iid • -Sources of violations: • - 1st level: - temporal autocorrelation • - unbalanced design • - unequal within-subject variance • - 2nd level: - correlated repeated measures (ex. n-back tasks) • - unequal variances between groups • How to use this knowledge to make proper statistical inference?

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