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Repeated Measure Design of ANOVA AMS 572 Group 5 Outline • Jia Chen: Introduction of repeated measures ANOVA • Chewei Lu: One-way repeated measures • Wei Xi: Two-factor repeated measures • Tomoaki Sakamoto : Three-factor repeated measures • How-Chung Liu: Mixed models • Margaret Brown: Comparison • Xiao Liu: Conclusion Introduction of Repeated Measures ANOVA Jia Chen What is it ? Definition: - It is a technique used to test the equality of means. When To Use It? • It is used when all members of a random sample are measured under a number of different conditions. • As the sample is exposed to each condition in turn, the measurement of dependent variable is repeated. Introduction of One-Way Repeated Measures ANOVA Che-Wei, Lu Professor:Wei Zhu One-Way Repeated Measures ANOVA • Definition A one-way repeated measures ANOVA instead of having one score per subject, experiments are frequently conducted in which multiple score are gathered for each case. • Concept of Repeated Measures ANOVA One factor with at least two levels, levels are dependent. Dependent means that they share variability in some way. The Repeated Measures ANOVA is extended from standard ANOVA. One-Way Repeated Measures ANOVA • When to Use Measuring performance on the same variable over time – for example looking at changes in performance during training or before and after a specific treatment The same subject is measured multiple times under different conditions – for example performance when taking Drug A and performance when taking Drug B The same subjects provide measures/ratings on different characteristics – for example the desirability of red cars, green cars and blue cars Note how we could do some RM as regular between subjects designs – For example, Randomly assign to drug A or B One-Way Repeated Measures ANOVA • Source of Variance in Repeated Measures ANOVA SStotal – Deviation of each individual score from the grand mean SSb/t subjects – Deviation of subjects' individual means (across treatments) from the grand mean. – In the RM setting, this is largely uninteresting, as we can pretty much assume that ‘subjects differ’ SSw/in subjects: How Ss vary about their own mean, breaks down into: – SStreatment • As in between subjects ANOVA, is the comparison of treatment means to each other (by examining their deviations from the grand mean) • However this is now a partition of the within subjects variation – SSerror • Variability of individuals’ scores about their treatment mean One-Way Repeated Measures ANOVA • Partition of Sum of Square Repeated Measures ANOVA / / Standard ANOVA / / One-Way Repeated Measures ANOVA • Standard ANOVA Table Variation Between Within Total SS Df ( − )2 −1 ( − ) 2 =1 =1 ( − ) 2 MS F a-1 MSA= N-a MSE= −1 F= − N-1 =1 =1 • Repeated Measures ANOVA Table SS Df MS ( )2 2 − ( )2 2 − a-1 MSA= s-1 -Error ( )2 2 − − Total + ℎ N-A Between Within -Subjects F −1 N-a MSE= (−1)(−1) F= One-Way Repeated Measures ANOVA • Example: Researchers want to test a new anti-anxiety medication. They measure the anxiety of 7 participants three times: once before taking the medication, once one week after taking the medication, and once two weeks after taking medication. Anxiety is rated on a scale of 1-10,with 10 being ”high anxiety” and 1 being “low anxiety”. Are there any difference between the three condition using significant level = 0.05? Participants Before Week1 Week2 1 9 7 4 2 8 6 3 3 7 6 2 4 8 7 3 5 8 8 4 6 9 7 3 7 8 6 2 One-Way Repeated Measures ANOVA Participants Before Week1 Week2 1 9 7 4 2 8 6 3 3 7 6 2 4 8 7 3 5 8 8 4 6 9 7 3 7 8 6 2 • Define Null and Alternative Hypotheses 0 : = 1 = 2 : 0 One-Way Repeated Measures ANOVA Participants Before Week1 Week2 1 9 7 4 2 8 6 3 3 7 6 2 4 8 7 3 5 8 8 4 6 9 7 3 7 8 6 2 • Define Degrees of Freedom N=21 s=7 = − 1 = 3 − 1 = 2 ℎ = − = 21 − 3 = 18 = − 1 = 7 − 1 = 6 = ℎ - =18-6=12 = − 1 = 21 − 1 = 20 One-Way Repeated Measures ANOVA Participants Before Week1 Week2 1 9 7 4 2 8 6 3 3 7 6 2 4 8 7 3 5 8 8 4 6 9 7 3 7 8 6 2 • Analysis of Variance ( )2 2 = − , where = , = ℎ 2 ( ) ℎ = 2 − ( )2 2 = − , ℎ = ℎ One-Way Repeated Measures ANOVA • Analysis of Variance(ANOVA Table) SS Df MS F Between 98.67 2 49.34 224.27 Within 10.29 18 -Subjects 7.62 6 -Error 2.67 12 Total 108.96 20 0.22 Error=within-Subjects=10.29-7.62=2.67 Total=Beteen+Within=98.67+10.29=108.96 • Test Statistic: 0 = = ℎ 98.67 = 2.67 2 12 =244.27 One-Way Repeated Measures ANOVA SS Df MS F Between 98.67 2 49.34 224.27 Within 10.29 18 -Subjects 7.62 6 -Error 2.67 12 Total 108.96 20 0.22 • Critical Region 0 ℎ 2,12,0.05 = 3.88, ℎℎ. Now, the 0 = 224.26 > 3.88. ℎ, 0 ℎ ℎ ℎ One-Way Repeated Measures ANOVA • SAS Code In here, we don’t have CLASS statement because our data set does not have an independent variable The Before, Week1, DATA REPEAT; INPUT SUBJ BEFORE WEEK1 WEEK2; and Week2 are the time level at each DATALINES; participants. 1974 2863 3762 4873 The NOUNI(no univariate) 5884 is a request not to conduct 6973 a separate analysis for 7862 each of the three times ; variables. PROC ANOVA DATA=REPEAT; TITLE "One-Way ANOVA using the repeated Statment"; MODEL BEFORE WEEK1 WEEK2= / NOUNI; REPEATED TIME 3 (1 2 3); This indicates the labels we RUN; want to printed for each level of times One-Way Repeated Measures ANOVA • SAS Result Wei Xi Stating of the Hypothesis Within-Subjects Main Effect Between-Subjects Main Effect Between-Subjects Interaction Effect Within-Subjects By Between-Subjects Interaction Effects Two-Factor ANOVA with Repeated Measures on One Factor Hypothesis ANOVA TABLE Source DF SS MS F Factor A a-1 SSA SSA/(a-1) MSA/MSWA ～ F(a-1),n(a-1) Factor B b-1 SSB SSB/(b-1) AB Interaction (a-1)(b-1) SSAB SSAB/(a-1)(b-1) Subjects within A (n-1)a SSW A SSWA/(n-1)a Error (n-1)a(b-1) SSE SSE/((n-1)a(b-1) Total nab-1 SST ～ F(b-1),n(a-1)(b-1) MSAB/MSE ～ F(a-1)(b-1,n(a-1)(b-1) MSB/MSE Example • The shape variable is the repeated variable. This produces an ANOVA with one between-subjects factor. If you were to examine the expected mean squares for this setup, you would find that the appropriate error term for the test of calib is subject|calib. The appropriate error term for shape and shape#calib is shape#subject|calib (which is the residual error since we do not include the term in the model). SAS Code Data Q1; set pre.Q1; run; proc anova data=Q1; title' Two-way Anova with a Repeated Measure on One Factor'; class calib; model shape_1 shape_2 shape_3 shape_4 = calib/nouni; Analysis of SAS Output MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no shape Effect H = Anova SSCP Matrix for shape E = Error SSCP Matrix S=1 M=0.5 N=0 Statistic Wilks' Lambda Value 0.02529573 F Value 25.69 Num DF 3 Den DF 2 Pr > F 0.0377 Pillai's Trace 0.97470427 25.69 3 2 0.0377 Hotelling-Lawley Trace 38.53236607 25.69 3 2 0.0377 Roy's Greatest Root 38.53236607 25.69 3 2 0.0377 At α=0.05,we reject the hypothesis and conclude that there is shape Effect MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no shape*calib Effect H = Anova SSCP Matrix for shape*calib E = Error SSCP Matrix S=1 M=0.5 N=0 Statistic Wilks' Lambda Value 0.16750795 F Value 3.31 Num DF 3 Den DF 2 Pr > F 0.2404 Pillai's Trace 0.83249205 3.31 3 2 0.2404 Hotelling-Lawley Trace 4.96986607 3.31 3 2 0.2404 Roy's Greatest Root 4.96986607 3.31 3 2 0.2404 At α=0.05,we cannot reject the hypothesis and conclude that there is no shape*calib Effect Tests of Hypotheses for Between Subjects Effects Source calib Error DF 1 Anova SS 51.04166667 Mean Square 51.04166667 4 17.16666667 4.29166667 F Value 11.89 Pr > F 0.0261 Univariate Tests of Hypotheses for Within Subject Effects Adj Pr > F Source shape DF 3 Anova SS 47.45833333 Mean Square 15.81944444 F Value 12.80 Pr > F 0.0005 G-G 0.0099 H-F 0.0011 shape*calib 3 7.45833333 2.48611111 2.01 0.1662 0.2152 0.1791 Error(shape) 12 14.83333333 1.23611111 Two-Factor ANOVA with Repeated Measures on both Factors ANOVA TABLE Source DF SS MS F Subjects n-1 SSS SSS/I-1 MSS/MSE Factor A a-1 SSA SSA/(a-1) MSA/MSA*S ～ F(a-1),(n-1)(a-1) Factor B b-1 SSB SSB/(b-1) MSB/MSB*S ～ F(b-1),(n-1)(b-1) AB Interaction (a-1)(b-1) SSAB SSAB/((a-1)(b1)) MSAB/MSE ～ F(a-1)(b-1),(n-1)(a1)(b-1) A*Subjects (n-1)(a-1) SSA*S SSWA/((n-1)a) SSA*S/MSE F(a-1)(n-1),(n-1)(a-1)(b-1) B*Subjects (n-1)(b-1) SSB*S SSWB/((n-1)b) SSA*S/MSE F(n-1)(b-1),(n-1)(a-1)(b-1) SSE/((n-1)(a1)(b-1)) Error (n-a)(a-1)(b-1) SSE Total nab-1 SST Example Three subjects, each with nine accuracy scores on all combinations of the three different dials and three different periods. With subject a random factor and both dial and period fixed factors, the appropriate error term for the test of dial is the dial#subject interaction. Likewise, period#subject is the correct error term for period, and period#dial#subject (which we will drop so that it becomes residual error) is the appropriate error term for period#dial. • • • • • • • • • • • • • • SAS Code Data Q2; Input Mins1-Mins9; Datalines; 45 53 60 40 52 57 28 37 46 35 41 50 30 37 47 28 32 41 60 65 75 58 54 70 40 47 50 ; ODS RTF STYLE=BarrettsBlue; Proc anova data=Q2; Model Mins1-Mins9=/nouni; Repeated period 3, dail 3/nom; Run; ods rtf close; SAS Output Univariate Tests of Hypotheses for Within Subject Effects Source period Error(period) DF Anova SS 2 1072.666667 4 148.444444 Adj Pr > F Mean Square F Value Pr > F G - G H-F 536.333333 14.45 0.0148 0.0563 0.0394 37.111111 Greenhouse-Geisser Epsilon 0.5364 Huynh-Feldt Epsilon 0.6569 At α=0.05,we reject the hypothesis and conclude that there is period Effect Source dail Error(dail) DF 2 4 Anova SS 978.6666667 38.4444444 Mean Square 489.3333333 9.6111111 F Value Pr > F 50.91 0.0014 Adj Pr > F G-G H-F 0.0169 0.0115 Greenhouse-Geisser Epsilon 0.5227 Huynh-Feldt Epsilon 0.5952 At α=0.05,we reject the hypothesis and conclude that there is dail Effect Adj Pr > F Source period*dail Error(period*dail) DF 4 Anova SS Mean Square F Value Pr > F G - G H - F 8.66666667 2.16666667 0.30 0.8715 0.6603 0.7194 8 58.22222222 Greenhouse-Geisser Epsilon 0.2827 Huynh-Feldt Epsilon 0.4006 7.27777778 At α=0.05,we cannot reject the hypothesis and conclude the there is no period*dail Effect Three-factor Experiments with a repeated measure T. Sakamoto Example of a marketing experiment Case of this example • A company which produces some Liquid Crystal Display wants to examine the characteristics of its prototype products. Experiment The subjects who belong to a region X or Y see the Liquid Crystal Display A, B, or C. Each type of LCD is seen twice; once in the light and the other in the dark. The preferences of the LCD are measured by the subjects, on a scale from 1 to 5 (1= lowest, 5=highest). 35/87 Experimental Design and Data Three factors Type of LCD Regions to which the specimens belong In the light / In the dark Repeted measure factor : In the light / In the dark Type of LCD A X REGION Y B C subj light dark subj light dark subj light dark 1 5 4 11 4 4 21 5 5 2 4 2 12 5 6 22 5 3 3 5 4 13 3 4 23 3 3 4 3 5 14 5 4 24 4 4 5 5 3 15 4 6 25 4 3 6 4 4 16 5 5 26 3 5 7 3 5 17 4 3 27 4 3 8 4 3 18 5 3 28 5 2 9 2 5 19 5 5 29 3 4 10 5 4 20 4 4 30 4 36/87 3 SAS PROGRAM data lcd; input subj type $ region $ light dark @@; datalines; 1a542a423a544a355a53 6 a 4 4 7 a 3 5 8 a 4 3 9 a 2 5 10 a 5 4 11 b 4 4 12 b 5 6 13 b 3 4 14 b 5 4 15 b 4 6 16 b 5 5 17 b 4 3 18 b 5 3 19 b 5 5 20 b 4 4 21 c 5 5 22 c 5 3 23 c 3 3 24 c 4 4 25 c 4 3 26 c 3 5 27 c 4 3 28 c 5 2 29 c 3 4 30 c 4 3 ; run; proc anova data=lcd; title ’Three-way ANOVA with a Repeated Measure'; class type region; model light dark = type | region /nouni; repeated light_dark; means type | region; run; 37/87 OUTPUT(Part 1/4): OUTPUT(Part 2/4): OUTPUT(Part 3/4): 40/81 OUTPUT(Part 4/4): Mixed Effect Models How-Chang Liu Mixed Models • When we have a model that contains random effect as well as fixed effect, then we are dealing with a mixed model. • From the above definition, we see that mixed models must contain at least two factors. One having fixed effect and one having random effect. Why use mixed models? • When repeated measurements are made on the same statistical units, it would not be realistic to assume that these measurements are independent. • We can take this dependence into account by specifying covariance structures using a mixed model Definition • A mixed model can be represented in matrix notation by: = 0 + 1 + • is the vector of observations • 0 is the vector of fixed effects • 1 is the vector of random effects • is the vector of I.I.D. error terms • and are matrices relating 0 and 1 to Assumptions 1 ~Normal 0, G ~Normal 0, R R and G are constants We also assume that 1 and are independent • We get V = ZGZ' + R, where V is the variance of y • • • • How to estimate 0 and 1 ? If R and G are given: Using Henderson’s Mixed Model equation, we have: ′ −1 ′ −1 ′ −1 ′ −1 + −1 0 ′ −1 = ′ −1 1 So 0 = ( ′ −1 )−1 X′y And 1 = ′ −1 (y − X0 ) What if G and R are unknown? • We know that both 1 and are normally distributed, so the best approach is to use likelihood based methods • There are two methods used by SAS: • 1)Maximum likelihood (ML) • 2)Restricted/residual maximum likelihood (REML) Example Below is a table of growth measurements for 11 girls and 16 boys at ages 8, 10, 12, 14: Person 1 2 3 4 5 6 7 8 9 10 11 12 13 14 gender F F F F F F F F F F F M M M age8 age10 age12 21.0 20.0 21.5 21.0 21.5 24.0 20.5 24.0 24.5 23.5 24.5 25.0 21.5 23.0 22.5 20.0 21.0 21.0 21.5 22.5 23.0 23.0 23.0 23.5 20.0 21.0 22.0 16.5 19.0 19.0 24.5 25.0 28.0 26.0 25.0 29.0 21.5 22.5 23.0 23.0 22.5 24.0 age14 23.0 25.5 26.0 26.5 23.5 22.5 25.0 24.0 21.5 19.5 28.0 31.0 26.5 27.5 Person 15 16 17 18 19 20 21 22 23 24 25 26 27 gender M M M M M M M M M M M M M age8 age10 age12 25.5 27.5 26.5 20.0 23.5 22.5 24.5 25.5 27.0 22.0 22.0 24.5 24.0 21.5 24.5 23.0 20.5 31.0 27.5 28.0 31.0 23.0 23.0 23.5 21.5 23.5 24.0 17.0 24.5 26.0 22.5 25.5 25.5 23.0 24.5 26.0 22.0 21.5 23.5 age14 27.0 26.0 28.5 26.5 25.5 26.0 31.5 25.0 28.0 29.5 26.0 30.0 25.0 Using SAS data pr; input Person Gender $ y1 y2 y3 y4; y=y1; Age=8; output; y=y2; Age=10; output; y=y3; Age=12; output; y=y4; Age=14; output; drop y1-y4; datalines; 1 F 21.0 20.0 21.5 23.0 2 F 21.0 21.5 24.0 25.5 … ; Run; Using SAS proc mixed data=pr method=ml covtest; class Person Gender; model y = Gender Age Gender*Age / s; repeated / type=un subject=Person r; run; Results Model Information Data Set WORK.PR Dependent Variable y Covariance Structure Unstructured Subject Effect Person Estimation Method ML Residual Variance Method None Fixed Effects SE Method Model-Based Degrees of Freedom Method Between-Within As one can see, the covariance matrix is unstructured, as we are going to estimate it using the maximum likelihood method Class Level Information Class Person Gender Levels Values 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 2 FM Dimensions Covariance Parameters 10 Columns in X 6 Columns in Z 0 Subjects 27 Max Obs Per Subject 4 As one can see, we do not have a Z matrix for this model Number of Observations Number of Observations Read 108 Number of Observations Used 108 Number of Observations Not Used 0 Iteration History Iteration Evaluations -2 Log Like Criterion 0 1 478.24175986 1 2 419.47721707 0.00000152 2 1 419.47704812 0.00000000 Convergence criteria met. the convergence of the Newton-Raphson algorithm means that we have found the maximum likelihood estimates Covariance Parameter Estimates Cov Parm Subject Estimate Standard Error Z Value Pr Z UN(1,1) Person 5.1192 1.4169 3.61 0.0002 UN(2,1) Person 2.4409 0.9835 2.48 0.0131 UN(2,2) Person 3.9279 1.0824 3.63 0.0001 UN(3,1) Person 3.6105 1.2767 2.83 0.0047 UN(3,2) Person 2.7175 1.0740 2.53 0.0114 UN(3,3) Person 5.9798 1.6279 3.67 0.0001 UN(4,1) Person 2.5222 1.0649 2.37 0.0179 UN(4,2) Person 3.0624 1.0135 3.02 0.0025 UN(4,3) Person 3.8235 1.2508 3.06 0.0022 UN(4,4) Person 4.6180 1.2573 3.67 0.0001 The table lists the 10 estimated covariance parameters in order. In other words, these are the estimates for R, the variance of Solution for Fixed Effects Effect Gender Intercept Gender F Gender M Age Age*Gender F Age*Gender M Estimate Standard Error DF t Value Pr > |t| 15.8423 0.9356 25 16.93 <.0001 1.5831 1.4658 25 1.08 0.2904 0 . . . . 0.8268 0.07911 25 10.45 <.0001 -0.3504 0.1239 25 -2.83 0.0091 0 . . . . From this table, we see that the boys intercept is at 15.8423, whole the girls intercept is at 15.8423+1.5831=17.42. The estimate of the boys’ slope is at 0.827, while the girls’ slpe is at 0.827-0.350=0.477 So the girls’ starting point is higher than the girls but their growth rate is only about half of that of the boys Type 3 Tests of Fixed Effects Effect Num DF Den DF F Value Pr > F Gender 1 25 1.17 0.2904 Age 1 25 110.54 <.0001 Age*Gender 1 25 7.99 0.0091 This is probably the most important table from our results: The gender row tests the null hypothesis that girls and boys have a common intercept. As we can see we cannot reject that hypothesis The Age tests the null hypothesis that age does not affect the growth rate. As we can see, we reject the null hypothesis as the F-value is large. The Age*gender tests reveals that there is a difference in slope at the 1% significance level. Repeated Measures ANOVA vs. Independent Measures ANOVA Magarate Brown • Can we just use standard ANOVA with repeated measures data? • No, Independent Measures (standard) ANOVA assumes the data are independent. • Data from a repeated measures experiment not independent How are standard ANOVA and repeated measures ANOVA the same? • Independent measures ANOVA: an extension of the pooled variance t-test • Repeated Measures ANOVA: an extension of the paired sample t-test How are standard ANOVA and repeated measures ANOVA the same? • Independent measures ANOVA: assumes the population variances are equal (homogeneity of variance) • Repeated Measures ANOVA: sphericity assumption that the population variances of all the differences are equal How are standard ANOVA and repeated measures ANOVA the same? • Both assume Normality of the population Advantages to using Repeated Measures instead of Independent Measures • Limited number of subjects available • Prefer to limit the number of subjects • Less variability (finger tapping with caffeine example) • Can examine effects over time Drawbacks • Practice effect • Example: subjects get better at performing a task each time with “practice” • Differential transfer: “This occurs when the effects of one condition persist and affect participants’ experiences during subsequent conditions.” (format: http://www.psychmet.com/id16.html) • Example: medical treatments Resources (to be formatted) • http://www.psychmet.com/id16.html • http://www.utexas.edu/courses/schwab/sw38 8r7_spring_2007/SolvingProblemsInSPSS/Solv ing%20Repeated%20Measures%20ANOVA%2 0Problems.pdf • http://en.wikipedia.org/wiki/Repeated_measu res • http://www.mhhe.com/socscience/psycholog y/shaugh/ch07_summary.html