17.4.2.4 Algorithms (Repeated Measures ANOVA)RMANOVA-Algorithm
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 /math-8ce4b16b22b58894aa86c421e8759df3.png?v=0 "k") be the number of levels for factor A, /math-83878c91171338902e0fe0fb97a8c47a.png?v=0 "p") be the number of levels for factor B,/math-584a81dbf5bf6aa737ba43567ad6307b.png?v=0 "n_i") be the number of subjects with ith level of factor A, /math-09614e75034336faafa6e5823ad3ac4c.png?v=0 "y_{ij}^{T} = (y_{ij1},...,y_{ijp})") be observations with respect to jth subject and ith level of factor A.
Define Error matrix as:
/math-c2a24e957187868f7a5f66aea335e309.png?v=0 "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:
/math-5c05e9d7e58f8f010d395cbd393b7720.png?v=0 "H = \sum_{i=1}^{k}n_i(\bar{y_{i.}}-\bar{y_{..}})(\bar{y_{i.}}-\bar{y_{..}})^{T},")
and Hypothesis matrix with intercept as:
/math-e268b25cdd6f0588e185ac750ec02fff.png?v=0 "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
/math-6ed86e6cc990b9d1af993d4c0927715f.png?v=0 "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 /math-eb5752fb3802a9a7960258fdf838eb2d.png?v=0 "N = \sum_{i=1}^{k}n_i.")
And the degree of freedom can be obtained by /math-fb02db9e63440cd2e05d80f42d461504.png?v=0 "d_E = N-k") and /math-74801925a7dfba9046ad170e97859025.png?v=0 "d_H = k-1") , respectively.
Suppose the mean vectors of factor A's levels are /math-3de7e1b2c368a355b1e2304c7eea22a0.png?v=0 "\mu_1,...,\mu_k") , and we let /math-bdd78b6a720ed92a3892028772a4577f.png?v=0 "\bar{\mu_.} = \frac{1}{k}\sum_{i=1}^{k}\mu_i.")
Main effect of within factor B
Let the contrast matrix be
/math-b8927dc7ffec56da7d89517bdf88c99f.png?v=0 "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 /math-182b16b132638a9d5f73df5455c7e79f.png?v=0 "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:
/math-0a7eeb38a8d6559fc2d585c56ef2a178.png?v=0 "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 /math-d5a9ebbbdd2249c50f21c02d80b0ee99.png?v=0 "H_0: B\mu_1 = B\mu_2 = \cdots = B\mu_k = 0.") The SS&CPs are:
/math-8740d5daf3633ac0aa8055fd00c487c7.png?v=0 "S_H = BHB^{T},S_E = BEB^{T}.")
Mauchly's Test of Sphericity
Let design matrix be
/math-511b707f131edaf39ee2a6a958a35e85.png?v=0 "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 /math-1e08c5818b30602c926a294072259b93.png?v=0 "R = Y - X\left( (X^TX)^{-1}XY \right).")
Let /math-69691c7bdcc3ce6d5d8a1361f22d04ac.png?v=0 "M") be the /math-e3ceee9ec583c7faf39dd069ddeb4f3f.png?v=0 "(p-1)\times p") orthogonal matrix which can be set like
/math-10621239e770e8df473420e90d18816d.png?v=0 "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 /math-d25d17d844562411064a9f23b26a6175.png?v=0 "T = M(R^TR)M^T.")
Let /math-c36f976ce9ed7b7cff6af4cb5ff5c753.png?v=0 "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 /math-9861855a82345bcd4eaa0d3e47cf6ea0.png?v=0 "r_X = rank(X).")
Then Mauchly's W Statistic is
/math-dfa34554beee8930d717da684a7aa28c.png?v=0 "W = \frac{det(T)}{(tr(T)/d)^d}.")
The Chi-square test value is /math-6d1c809f7dfe136ca756ba72be767636.png?v=0 "\chi^2 = \ln(W)\tau") with freedom degree /math-a767a173a88aa034d131ec4b1c4678b2.png?v=0 "df = d(d+1)/2-1.")
/math-6cb8aa6f3d4995e2a9949f4118384329.png?v=0 "\varepsilon_{gg} = \frac{tr(T)^2}{tr(T^TT)d}.")
/math-2308f07b4e36581d5399e8e055c3adc7.png?v=0 "\varepsilon_{hf} = \min\left( \frac{nd\varepsilon_{gg}-2}{d(N-r_X)-d^2\varepsilon_{gg}} , 1\right).")
/math-e04e1cc8d5e7de346b48a15ada4e180e.png?v=0 "\varepsilon_{lb} = \frac{1}{d}.")
Within and Between Test
Some basic calculations:
/math-d77c0dc584970f5ddf3f8c76ddddfb58.png?v=0 "SS_T = \sum_{i,k,j}(y_{ikj}-\bar{y_{...}})^2,") with degree freedom /math-d8e1c07214fd6285d4206fb683352c15.png?v=0 "df = Np-1.")
- Sum Square of Between Factor A:
/math-5ce6c446b06d95d1235b2bab6c3d0ee1.png?v=0 "SS_A = \sum_{i,k,j}(\bar{y_{ik.}}-\bar{y_{...}})^2,") with degree freedom /math-b54b9bba2b59c6585a5b50cc1ace25ae.png?v=0 "df = N-1.")
- Sum Square of Within Factor B:
/math-3d7f70012333abcbb9b3f0a197c08b7d.png?v=0 "SS_B = \sum_{i,k,j}(y_{ikj}-\bar{y_{ik.}})^2,") with degree freedom /math-3066706bcb6ae6279cbd20354905ccb5.png?v=0 "df = Np-N.")
Where
/math-e38f8fe6d7ffb147886aed2f6b95877d.png?v=0 "\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
/math-a0399b25e8a990662496b013d6e78c4e.png?v=0 "SSW_B = \sum_{i,k,j}(\bar{y_{..j}}-\bar{y_{...}})^2,") with degree freedom /math-c0e52469ec501fd48c935d53a33e1831.png?v=0 "df = p-1.")
- Sum Square of interaction A*B for Within test
/math-1076b0939d92ec5d378d55d9831cf350.png?v=0 "SSW_{AB} = \sum_{i,k,j}(\bar{y_{i.j}}-\bar{y_{i..}} - \bar{y_{..j}} + \bar{y_{...}})^2,") with degree freedom /math-4dbf7d16253ce4aedf3d5a1ceb25d1b3.png?v=0 "df = (k-1)(p-1).")
- Sum Square of error(factor B) for Within test
/math-11007c96d117ab07ad0e07c096256209.png?v=0 "SSW_{E} = \sum_{i,k,j}(y_{ikj}-\bar{y_{i.j}} - \bar{y_{ik.}} + \bar{y_{i..}})^2,") with degree freedom /math-af69b65318cb407302efc4b495c4d44c.png?v=0 "df = (N-k)(p-1).")
Tests of Between-Subjects Effects
- Sum Square of intercept for Between test
/math-6443281de8e5073438892356febe1f0a.png?v=0 "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 /math-607e0e920ccf75242a73abeb509266d0.png?v=0 "df = k-1.")
- Sum Square of intercept for Between test(when set intercept = 0)
/math-ad771ad1001b53a2e0ba9067e8b72e41.png?v=0 "SSB_{int0} = (X_{A}^{T}X_{A})^{-1}X_{A}^{T}Y ,") with degree freedom Here /math-8933fce11a2d04e1400e749b44dce205.png?v=0 "X_A") is the design matrix associated to effect A, while /math-57cec4137b614c87cb4e24a3d003a3e0.png?v=0 "Y") is a /math-e4d2a34ee9561bb27bcfa53efe63c16b.png?v=0 "Np \times 1") matrix representing the indexed data.
- Sum Square of between factor A for Between test
/math-2a9c76b5d2ca68bfaefe2782cc3d588b.png?v=0 "SSB_B = \sum_{i,k,j}(\bar{y_{i..}}-\bar{y_{...}})^2,") with degree freedom
- Sum Square of error(factor A) for Between test
/math-ba0afaef0639b15f981b9c5920d0a2e7.png?v=0 "SSB_E = \sum_{i,k,j}(\bar{y_{ik.}}-\bar{y_{i..}})^2,") with degree freedom /math-ce7d8d33214b606d9702fb021bb9470a.png?v=0 "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
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