17.4.2.4 Algorithms (Repeated Measures ANOVA)

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