17.4.2.4 Algorithms (Repeated Measures ANOVA)

1 One-way/Two-way Repeated Measures
2 Two-way Mixed-Design

One-way/Two-way Repeated Measures

For the details of the algorithms for the one-way and two way balanced repeated measure design, please see Repeated Measures ANOVA.pdf

Two-way Mixed-Design

Multivariate Tests

Considering the model: Two-Way Mixed Design RM ANOVA with one between subject factor A and another within subject factor B.

Let $k$ be the number of levels for factor A, $p$ be the number of levels for factor B, $n_i$ be the number of subjects with ith level of factor A, $y_{ij}^{T} = (y_{ij1},...,y_{ijp})$ be observations with respect to jth subject and ith level of factor A.

Define Error matrix as: $E = \sum_{i=1}^{k}\sum_{j=1}^{n_i}(y_{ij}-\bar{y_{i.}})(y_{ij}-\bar{y_{i.}})^{T},$ and Hypothesis matrix as: $H = \sum_{i=1}^{k}n_i(\bar{y_{i.}}-\bar{y_{..}})(\bar{y_{i.}}-\bar{y_{..}})^{T},$ and Hypothesis matrix with intercept as: $H_{int} = \frac{1}{\sum_{i=1}^{k}\frac{1}{n_i}}(\sum_{i=1}^{k}\frac{y_{i.}}{n_i})(\sum_{i=1}^{k}\frac{y_{i.}}{n_i})^{T},$

where

$y_{i.} = \sum_{j=1}^{n_i}y_{ij}, y_{..} = \sum_{i=1}^{k}\sum_{j=1}^{n_i}y_{ij}, \bar{y_{i.}} = \frac{y_{i.}}{n_i}, \bar{y_{..}} = \frac{y_{..}}{N}$ and $N = \sum_{i=1}^{k}n_i.$

And the degree of freedom can be obtained by $d_E = N-k$ and $d_H = k-1$ , respectively.

Suppose the mean vectors of factor A's levels are $\mu_1,...,\mu_k$ , and we let $\bar{\mu_.} = \frac{1}{k}\sum_{i=1}^{k}\mu_i.$

Main effect of within factor B

Let the contrast matrix be

$B_{(p-1)\times p}=\begin{bmatrix} 1 & -1 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & 1 & -1 \end{bmatrix}.$

In order to test $H_0: B\bar{\mu_.}$ , we can compute the values of Wilks' Lambda, Hotelling-Lawley Trace, Pillai's Trace and Roy's Largest Root. The SS&CPs are:

$S_H = BH_{int}B^{T},S_E = BEB^{T}.$

Notes: All sum of squares are calculated based on type III.

Interaction effect of B*A

The null hypothesis is $H_0: B\mu_1 = B\mu_2 = \cdots = B\mu_k = 0.$ The SS&CPs are:

$S_H = BHB^{T},S_E = BEB^{T}.$

Mauchly's Test of Sphericity

Let design matrix be

$X = \begin{bmatrix} 1_{n_1\times 1} & 1_{n_1\times 1} & & & \\ 1_{n_2\times 1} & & 1_{n_2\times 1} & & \\ \vdots & & & \ddots & \\ 1_{n_k\times 1} & & & & 1_{n_k\times 1} \end{bmatrix}.$

The residual matrix is obtained by $R = Y - X\left( (X^TX)^{-1}XY \right).$

Let $M$ be the $(p-1)\times p$ orthogonal matrix which can be set like

$M = \begin{bmatrix} p-1 & -1 & \cdots & -1 & -1 \\ 0 & p-2 & \cdots & -1 & -1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -1 \end{bmatrix}.$

Let $T = M(R^TR)M^T.$

Let $d=p-1, \tau=\frac{2d^2+d+2}{6d}-N-r_X, \varsigma = \frac{(d+2)(d-1)(d-2)(2d^3+6d^2+3d+2)}{288d^2\tau^2}.$ Here $r_X = rank(X).$

Then Mauchly's W Statistic is

$W = \frac{det(T)}{(tr(T)/d)^d}.$

The Chi-square test value is $\chi^2 = \ln(W)\tau$ with freedom degree $df = d(d+1)/2-1.$

Greenhouse-Geisser

$\varepsilon_{gg} = \frac{tr(T)^2}{tr(T^TT)d}.$

Huynh-Feldt

$\varepsilon_{hf} = \min\left( \frac{nd\varepsilon_{gg}-2}{d(N-r_X)-d^2\varepsilon_{gg}} , 1\right).$

Lower-Bound

$\varepsilon_{lb} = \frac{1}{d}.$

Roy's Largest Root

Within and Between Test

Some basic calculations:

Sum Square of Total:

$SS_T = \sum_{i,k,j}(y_{ikj}-\bar{y_{...}})^2,$ with degree freedom $df = Np-1.$

Sum Square of Between Factor A:

$SS_A = \sum_{i,k,j}(\bar{y_{ik.}}-\bar{y_{...}})^2,$ with degree freedom $df = N-1.$

Sum Square of Within Factor B:

$SS_B = \sum_{i,k,j}(y_{ikj}-\bar{y_{ik.}})^2,$ with degree freedom $df = Np-N.$

Where

$\bar{y_{i..}} = \frac{1}{n_ip}\sum_{k,j}y_{ikj}, \bar{y_{...}} = \frac{1}{Np}\sum_{i,k,j}y_{ikj}, \bar{y_{..j}} = \frac{1}{N}\sum_{i,k}y_{ikj}, \bar{y_{i.j}} = \frac{1}{n_i}\sum_{k}y_{ikj}, \bar{y_{ik.}} = \frac{1}{p}\sum_{j}y_{ikj}, N = \sum_{i=1^{k}}n_i .$

Tests of Within-Subjects Effects

Sum Square of factor B for Within test

$SSW_B = \sum_{i,k,j}(\bar{y_{..j}}-\bar{y_{...}})^2,$ with degree freedom $df = p-1.$

Sum Square of interaction A*B for Within test

$SSW_{AB} = \sum_{i,k,j}(\bar{y_{i.j}}-\bar{y_{i..}} - \bar{y_{..j}} + \bar{y_{...}})^2,$ with degree freedom $df = (k-1)(p-1).$

Sum Square of error(factor B) for Within test

$SSW_{E} = \sum_{i,k,j}(y_{ikj}-\bar{y_{i.j}} - \bar{y_{ik.}} + \bar{y_{i..}})^2,$ with degree freedom $df = (N-k)(p-1).$

Tests of Between-Subjects Effects

Sum Square of intercept for Between test

$SSB_{int1} = \frac{p}{\sum_{i=1}^{k}\frac{1}{n_i}}\left( \sum_{i}\sum_{k}\frac{\bar{y_{ik.}}}{n_i} \right)^2 ,$ with degree freedom $df = k-1.$

Sum Square of intercept for Between test(when set intercept = 0)

$SSB_{int0} = (X_{A}^{T}X_{A})^{-1}X_{A}^{T}Y ,$ with degree freedom $df = k-1.$ Here $X_A$ is the design matrix associated to effect A, while $Y$ is a $Np \times 1$ matrix representing the indexed data.

Sum Square of between factor A for Between test

$SSB_B = \sum_{i,k,j}(\bar{y_{i..}}-\bar{y_{...}})^2,$ with degree freedom $df = k-1.$

Sum Square of error(factor A) for Between test

$SSB_E = \sum_{i,k,j}(\bar{y_{ik.}}-\bar{y_{i..}})^2,$ with degree freedom $df = N-k.$

Multiple Means Comparisons

There are various methods for multiple means comparison in Origin, and we use the ocstat_dlsm_mean_comparison() function to perform means comparisons.

Two types of multiple means comparison methods:

Single-step method. It creates simultaneous confidence intervals to show how the means differ, including Tukey-Kramer, Bonferroni, Dunn-Sidak, Fisher’s LSD and Scheffé mothods.

Stepwise method. Sequentially perform the hypothesis tests, including Holm-Bonferroni and Holm-Sidak tests