AMS577. Repeated Measures ANOVA: The Univariate and the Multivariate Analysis Approaches

Transcription

1 AMS577. Repeated Meaure ANOVA: The Univariate and the Multivariate Analyi Approache 1. One-way Repeated Meaure ANOVA One-way (one-factor) repeated-meaure ANOVA i an extenion of the matched-pair t-tet to deign with more column of correlated obervation. Aume that the data ued in the computing example for betweenubject ANOVA repreented performance core of the ame 4 repondent under three different tak condition: Tak 1 Tak 2 Tak 3 Mean P P P P Mean In an analyi of Tak effect, any variance between ubject (variation in the right-hand column) i not of interet. Thi variance imply reflect that participant differ overall (e.g. P1 eem to perform le well than P4), but i independent of effect of the different tak condition on people performance. Between-ubject variance thu i removed before the effect of the repeated-meaure factor are teted. Thi could be achieved by ubtracting each core from that peron mean core acro the three tak, yielding:between-ubject variance thu i removed before the effect of the repeated-meaure factor are teted. Thi could be achieved by ubtracting each core from that peron mean core acro the three tak, yielding: Tak 1 Tak 2 Tak 3 Mean P P P P Mean

2 Thee new core do not reflect difference between ubject any more, wherea the difference between tak condition are till reflected in the column mean. The ANOVA for repeated meaurement achieve thi by firt partitioning the um of quare into a between-ubject component (i.e. variation between the row mean of the firt table) and a within-ubject component (all remaining variation). The within-ubject component i further ubdivided in variation between treatment (i.e. between the column mean in the econd table) and error variation (i.e. within the column of the econd table). SS total SS between-ubject SS within-ubject SS between-treatment SS error [The between-ubject component can alo be further ubdivided if there are between-ubject factor to be conidered; but thi will not concern u for the moment.] The degree of freedom are divided into the ame component: df total = kn 1 df between- = n 1 df within- = n(k 1) df between-treatment = k 1 df error = (n 1)(k 1) Mean quare are derived a uual by dividing um of quare with their aociated df. 2

3 The F-ratio of MS between-treatment and MS error thu derived i ued to tet the effect of the within-ubject factor againt the null hypothei that all pairwie difference between treatment are zero. Note that both MS error and MS between-treatment in thi model may contain variation that i due to a treatment x ubject interaction, which itelf i not tetable (why?). So the fact that different people may react differently to variou treatment cannot be eparated from chance variation or treatment main effect. The tet of treatment effect i not affected by thi problem becaue the potential interaction term i hidden in both the numerator and the denominator of the F-ratio. Thi would not be the cae for a tet of the between-ubject effect (for which a conervative bia would be introduced if an interaction were in fact preent). But note that we would not normally tet the between-ubject variation for ignificance in thi kind of deign. (If ignificant, it would only tell u that people are different - and didn t we know thi all along?) 3

4 Change in Brain Functional Level Brain Functional Level Example: Suppoe there are k region of interet (ROI ) and n ubject. Each ubject wa canned on baeline (oda) a well a after drinking alcohol. Our main hypothei i whether the change between baeline and alcohol i homogeneou among the ROI. That i H0 : 1 2 k, where j i the effect of alcohol on the jth ROI, j 1,, k. Profile Plot Illutrating the Quetion of Interet Tet for Equal Change in Different ROI (That i, whether the two erie are parallel.) Alcohol Baeline k Brain Region of Interet (ROI) Figure 1. Hypothei in term of the original data Profile Plot Illutrating the Quetion of Interet Tet for Equal Change in Different ROI (That i, whether the two erie are parallel.) Difference k Brain Region of Interet (ROI) Figure 2. Hypothei in term of the paired difference 4

5 The Univariate Analyi Approach For ubject i, let Y ij denote the paired difference between baeline and alcohol for the jth ROI, then the (univariate) repeated meaure ANOVA model i: Yij j Si ij, where j i the (fixed) effect of ROI j, S i i the (random) effect of ubject i, ij i the random error independent of S i. With normality aumption, we have: Let Y Y, Y,, Y ' i i1 i2 ik, and are independent to each other., we have, i 1,, n, where 1, 2,, k and ' with and 2 2. Thi particular tructure of the variance covariance matrix i called compound ymmetry. For each ubject, it aume that the variance of the k ROI are equal 2 and the correlation between each ROI pair i contant which may not be realitic., The univariate approach to one-way repeated meaure ANOVA i equivalent to a two-way mixed effect ANOVA for a randomized block deign with ubject a the block and ROI a the treatment. The degree of freedom for the ANOVA F-tet of equal treatment effect i k 1 and n1k 1 repectively. That i,. We will reject the null hypothei at the ignificance level if F0 F k1, n1 k1. 5

6 The Multivariate Analyi Approach Alternatively, we can ue the multivariate approach where no tructure, other than the uual ymmetry and non-negative definite propertie, i impoed on the variance covariance matrix in, i 1,, n. Certainly we have more parameter In thi model than the univariate repeated meaure ANOVA model. The tet tatitic i 2 ' ' 1 T0 n n 1 CY CQC CY where Y n Y, Q Y Y Y Y ', and i1 i n i C i i Under the null hypothei,. Recall that if, then 2 Therefore the Hotelling Tk 1, n 1tatitic ha the following relationhip with the F tatitic: We will reject the null hypothei at the ignificance level if F0 F k1, nk1 (upper tail percentile). When to ue what approach? There are more parameter to be etimated in the multivariate approach than in the univariate approach. Thu, if the aumption for univariate analyi i atified, one hould ue the univariate approach becaue it i more powerful. Huynh and Feldt (1970) give a weaker requirement for the validity of the univariate ANOVA F-tet. It i referred to a the Type H Condition. A tet for thi condition i called the Machly Sphericity Tet. In SAS, thi tet i requeted by the PrintE option in the repeated tatement. 6

7 Example 1. One-way Repeated Meaure ANOVA (n=4, k=4), Paired Difference in Brain Functional Level Subject ROI 1 ROI 2 ROI 3 ROI SAS Program: One-way Repeated Meaure Analyi of Variance data repeatm; input ROI1-ROI4; dataline; ; proc anova data=repeatm; title 'one-way repeated meaure ANOVA'; model ROI1-ROI4 = /nouni; repeated ROI 4 ( )/printe; run; (Note: SAS Proc GLM and Proc Mixed can alo be ued for the repeated meaure ANOVA /MANOVA analye.) 7

8 SAS Output: One-way Repeated Meaure Analyi of Variance 1. Etimated Error Variance-Covariance Matrix ROI_1 ROI_2 ROI_3 ROI_ ROI_ ROI_ Tet for Type H Condition --- Mauchly' Sphericity Tet (Note: p-value for the tet i big, o we can ue the univariate approach) Variable DF Criterion Chi-Square Pr > ChiSq Orthogonal Component Multivariate Analyi Approach --- Manova Tet Criteria and Exact F Statitic for the Hypothei of no drug Effect Statitic Value F Value Num DF Den DF Pr > F Wilk' Lambda Pillai' Trace Hotelling-Lawley Trace Roy' Greatet Root Univariate Analyi Approach --- Univariate Tet of Hypothee for Within Subject Effect Adj Pr > F Source DF Anova SS Mean Square F Value Pr > F G - G H - F ROI Error(ROI) Greenhoue-Geier Epilon Huynh-Feldt Epilon Interpretation Note that the multivariate F-tet ha value of 36.33, degree of freedom of 3 and 1, and the p-value i While the univariate F-tet ha value of 11.38, with degree of freedom of 3 and 9, and the p-value i In thi cae, ince the aumption for the univariate approach i atified, we ue the univariate approach which i more powerful (maller p-value). 8

9 2. Two-way Repeated Meaure ANOVA However, thing may not alway be o eay. For example, intead of comparing whether the change between two condition (baeline and alcohol) are contant acro everal brain region of interet (ROI) a we had introduced previouly, now, our ituation i: We have only one region of interet for each ubject. We have two condition: heavy alcohol, light alcohol We monitor the brain functional level in time. Example 2. Two-way Repeated Meaure ANOVA (n=4, k1=2, k2=4 ) with repeated meaure on both factor Heavy Alcohol Subject Time 1 Time 2 Time 3 Time Light Alcohol The following 2-factor model, however, ha repeated meaure on only one factor (time) Example 3: Two treatment group with four meaurement taken over equally paced time interval (e.g., A = treatment B = placebo) id group time1 time2 time3 time4 1 A A A B B B Hypothetical data from Twik, chapter 3, page 40, table 3.7 9

10 Choleterol Level (mg/dl) Profile Plot Illutrating the Quetion of Interet TIME EFFECT ONLY treatment placebo TIME (month) 10

11 Choleterol Level (mg/dl) Choleterol Level (mg/dl) Profile Plot Illutrating the Quetion of Interet TREATMENT EFFECT ONLY treatment placebo TIME (month) Profile Plot Illutrating the Quetion of Interet TIME and TREATMENT EFFECTS ONLY (No TIME*TREATMENT INTERACTION) treatment placebo TIME (month) 11

12 Choleterol Level (mg/dl) Profile Plot Illutrating the Quetion of Interet TIME*TREATMENT INTERACTION treatment placebo TIME (month) Homework: Now, for each 2-factor model (Example 2 and 3), how do we formulate the model, hypothee, and conduct the tet/analye in the univariate and multivariate approache repectively? SAS for Mixed Model, Second Edition 12