Presentation on theme: "Introduction to Repeated Measures. MANOVA Revisited MANOVA is a general purpose multivariate analytical tool which lets us look at treatment effects on."— Presentation transcript:
MANOVA Revisited MANOVA is a general purpose multivariate analytical tool which lets us look at treatment effects on a whole set of DVs As soon as we got a significant treatment effect, we tried to “unpack” the multivariate DV to see where the effect was
MANOVA Repeated Measures ANOVA Put differently, we didn’t have any specialness of an ordering among DVs Sometimes we take multiple measurements, and we’re interested in systematic variation from one measurement taken on a person to another Repeated measures is a multivariate procedure cause we have more than one DV
Repeated Measures ANOVA We are interested in how a DV changes or is different over a period of time in the same participants
When to use RM ANOVA Longitudinal Studies Experiments
Why are we talking about ANOVA? When our analysis focuses on a single measure assessed at different occasions it is a REPEATED MEASURE ANOVA When our analysis focuses on multiple measures assessed at different occasions it is a DOUBLY MULTIVARIATE REPEATED MEASURES ANALYSIS
Between- and Within-Subjects Factor Between-Subjects variable/factor –Your typical IV from MANOVA –Different participants in each level of the IV Within-Subjects variable/factor –This is a new IV –Each participant is represented/tested at each level of the Within-Subject factor –TIME
Data are means and standard deviations Y Dependent variable Repeated measure exptal control Group Between-subjects factor Different subjects on each level Period of treatment Within-subjects factor Same subjects on each level y1y2y3 Trial or Time
Between- and Within-Subjects Factor In Repeated Measures ANOVA we are interested in both BS and WS effects We are also keenly interested in the interaction between BS and WS –Give mah an example
RMANOVA Repeated measures ANOVA has powerful advantages –completely removes within-subjects variance, a radical “blocking” approach –It allows us, in the case of temporal ordering, to see performance trends, like the lasting residual effects of a treatment –It requires far fewer subjects for equivalent statistical power
Repeated Measures ANOVA The assumptions of the repeated measures ANOVA are not that different from what we have already talked about –independence of observations –multivariate normality There are, however, new assumptions –sphericity
Sphericity The variances for all pairs of repeated measures must be equal –violations of this rule will positively bias the F statistic More precisely, the sphericity assumption is that variances in the differences between conditions is equal If your WS has 2 levels then you don’t need to worry about sphericity
Sphericity Example: Longitudinal study assessment 3 times every 30 days variance of (Start – Month1) = variance of (Month1 – Month2) = variance of (Start – Month 2) = Violations of sphericity will positively bias the F statistic
Univariate and Multivariate Estimation It turns out there are two ways to do effect estimation One is a classic ANOVA approach. This has benefits of fitting nicely into our conceptual understanding of ANOVA, but it also has these extra assumptions, like sphericity
Univariate and Multivariate Estimation But if you take a close look at the Repeated Measures ANOVA, you suddenly realize it has multiple dependent variables. That helps us understand that the RMANOVA could be construed as a MANOVA, with multivariate effect estimation (Wilk’s, Pillai’s, etc.) The only difference from a MANOVA is that we are also interested in formal statistical differences between dependent variables, and how those differences interact with the IVs Assumptions are relaxed with the multivariate approach to RMANOVA
Univariate and Multivariate Estimation It gets a little confusing here....because we’re not talking about univariate ESTIMATION versus multivariate ESTIMATION...this is a “behind the scenes” component that is not so relevant to how we actually run the analysis
Univariate Estimation Since each subject now contributes multiple observations, it is possible to quantify the variance in the DVs that is attributable to the subject. Remember, our goal is always to minimize residual (unaccounted for) variance in the DVs. Thus, by accounting for the subject-related variance we can substantially boost power of the design, by deflating the F-statistic denominator (MS error ) on the tests we care about
RMANOVA Design: Univariate Estimation SS T Total variance in the DV SS Between Total variance between subjects SS Within Total variance within subjects SS RES Within-subjects Error SS M Effect of experiment
RMANOVA Design: Multivariate Let’s consider a simple design SubjectTime1 Time2 Time3 d t1-t2 d t1-t3 d t2-t3 1 710 12 3 5 2 2 5 4 7 -1 2 3 3 6 8 10 2 4 2.......................................……………………………….. n 3 7 3 4 0 -3 In the multivariate case for repeated measures, the test statistic for k repeated measures is formed from the (k-1) [where k = # of occasions] difference variables and their variances and covariances
Univariate or Multivariate? If your WS factor only has 2 levels the approaches give the same answer! If sphericity holds, then the univariate approach is more powerful. When sphericity is violated, the situation is more complex Maxwell & Delaney (1990) “All other things being equal, the multivariate test is relatively less powerful than the univariate approach as n decreases...As a general rule, the multivariate approach should probably not be used if n is less than a + 10” (a=# levels of the repeated measures factor).
Univariate or Multivariate? If you can use the univariate output, you may have more power to reject the null hypothesis in favor of the alternative hypothesis. However, the univariate approach is appropriate only when the sphericity assumption is not violated.
Univariate or Multivariate? If the sphericity assumption is violated, then in most situations you are better off staying with the multivariate output. –Must then check homogeneity of V-C If sphercity is violated and your sample size is low then use an adjustment (Greenhouse- Geisser [conservative] or Huynh-Feldt [liberal])
Univariate or Multivariate? SPSS and SAS both give you the results of a RMANOVA using the –Univariate approach –Multivariate approach You don’t have to do anything except decide which approach you want to use
Effects RMANOVA gives you 2 different kinds of effects Within-Subjects effects Between-Subjects effects Interaction between the two
Within-Subjects Effects This is the “true” repeated measures effect Is there a mean difference between measurement occasions within my participants?
Between-Subjects Effects These are the effects on IV’s that examine differences between different kinds of participants All our effects from MANOVA are between- subjects effects The IV itself is called a between-subjects factor
Mixed Effects Mixed effects are another named for the interaction between a within-subjects factor and a between-subjects factor Does the within-subjects effect differ by some between-subjects factor
EXAMPLE Lets say Eric Kail does an intervention to improve the collegiality of his fellow IO students He uses a pretest—intervention—posttest design The DV is a subjective measure of collegiality Eric had a hypothesis that this intervention might work differently depending on the participants GPA (high and low)
Within-Subjects RMANOVA A within-subjects repeated measures ANOVA is used to determine if there are mean differences among the different time points There is no between-subjects effect so we aren’t worried about anything BUT the WS effect The within-subjects effect is an OMNIBUS test We must do follow-up tests to determine which time points differ from one another
Example 10 participants enrolled in a weight loss program They got weighed when thy first enrolled and then each month for 2 months Did the participants experience significant weight loss? And if so when?
Write Up In order to determine if there was significant weight loss over the three occasions a repeated measures analysis of variance was conducted. Results indicated a significant within- subjects effect [F(1.29, 11.65) = 8.77, p <.05, η 2 =.49] indicating a significant mean difference in weight among the three occasions. As can be seen in Figure 1, the mean weight at month 2 and 3 was significantly lower relative to month 1 [F(1, 9) = 12.73, p <.05, η 2 =.58]. There was additional significant weight loss from month 2 to month 3 [F(1,9) = 5.38, p <.05, η 2 =.49.
Within and between-subject factors When you have both WS and BS factors then you are going to be interested in the interaction! IV = intgrp (4 levels) DV = speed at pretest and posttest
RMANOVA: Univariate estimation of between-subjects effects Tests of Between-Subjects Effects Measure: speed Transformed Variable: Average 2099.9801 349.858.000.123 1169.1073389.70264.925.000.072 15011.94825016.002 Source Intercept intgrp Error Type III Sum of SquaresdfMean SquareFSig. Partial Eta Squared
Pairwise Comparisons Measure: speed -.110.1391.000-.477.257 1.456*.138.0001.0921.819 -.201.138.881-.5188.8.131.521.000-.257.477 1.565*.138.0001.2001.931 -.091.1391.000-.459.276 -1.456*.138.000-1.819-1.092 -1.565*.138.000-1.931-1.200 -1.656*.138.000-2.021-1.2184.108.40.2061-.165.567.091.1391.000-.276.459 1.656*.138.0001.2922.021 (J) Intervention group Reasoning Speed Control Memory Speed Control Memory Reasoning Control Memory Reasoning Speed (I) Intervention group Memory Reasoning Speed Control Mean Difference (I-J)Std. ErrorSig. a Lower BoundUpper Bound 95% Confidence Interval for Difference a Based on estimated marginal means The mean difference is significant at the.05 level.*. Adjustment for multiple comparisons: Bonferroni. a. This is the difference between the levels of the IV collapsed across BOTH measures of speed (pre and post)
The only intgrp difference is speed versus all others, and that is only at posttest—exactly what we would expect /EMMEANS = TABLES(intgrp*prepost) COMPARE(intgrp) ADJ(BONFERRONI)
RMANOVA: What does it look like? I am missing something. What is it?
Practice IV = group ( 2 = training and 1 – control) DV = Letter series –Letser (pretest) and letser2 (posttest) Are the BS and WS effects More importantly is there an interaction? –If there is an interaction than you need to decompose it!