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

3. y = βx + ε β * + E ~ N(0, σ2) y1 x1 ε 1 y2 x2 ε 2 . = . . . . .=
yn Xn ε n
β *
E ~ N(0, σ2)

4. ∑ yi2 σ2 = ~ y12 + y22 + … + yn2 ^ n - 1 E ~ N(0, σ2) E ~ N(0, Cε)
covariance
Variance-covariance matrix Cε = yTy :
y1 y2 … yn x y y1y1 y1y2 … y1yn y12
y y2y1 y2y2 … y2yn y22
= =
yn yny1 yn y2 … ynyn yn2
yT y Cε
∑ yi2
σ2 = ~ y12 + y22 + … + yn2
n - 1 ^ n
variance

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

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

7. 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?