Repeated Measures Analysis and MANOVA Repeated Measures Often

Repeated Measures Analysis and MANOVA

Repeated Measures • • • Often times were are interested in comparing treatments where the response of interest is measured at several points over time. For example, we might compare a drug that is supposed to reduce cholesterol to placebo where cholesterol is measured every two months over a 12 month period. This differs from a block design approach in that we are interested in comparing treatment groups not just looking at whethere has been a change over time a SINGLE group of subjects, i. e.

Repeated Measures • Hypothetical Cholesterol Study Group Initial DRUG Placebo 2 mo. 4 mo. 6 mo. 8 mo. 10 mo. 12 mo. The individual(s) who measure cholesterol levels at each follow-up are blind to which group the subjects are in. Questions Interest: This study ofwould be conducted as a randomized double-blind. 1) Is there a change in the cholesterol levels of subjects over time, i. e. is there a TIME EFFECT? (within-subjects effect) 2) Is there a TREATMENT EFFECT? (between-subjects effect) 3) Is the effect of TIME the same for both TREATMENTS? 4) (within-subjects effect) THIS IS OF PRIMARY INTEREST!

Profile Plots Illustrating the Questions of Interest TIME EFFECT ONLY Cholesterol Level (mg/dl) Cholesterol Levels for both groups decreased over TIME however the decrease appears to be the same for both treatment groups, i. e. there is NO TREATMENT effect nor a TIME*TREATMENT interaction. treatment placebo 0 2 4 6 8 TIME (months) 10 12

Profile Plots Illustrating the Questions of Interest TIME and TREATMENT EFFECT Cholesterol Level (mg/dl) Cholesterol Levels for both groups decreased over TIME and the trend over time was the same for both groups, however the decrease for those receiving the drug was larger, i. e. there is a TREATMENT EFFECT. treatment placebo 0 2 4 6 8 TIME (months) 10 12

Profile Plots Illustrating the Questions of Interest TIME*TREATMENT INTERACTION Cholesterol Level (mg/dl) Here the effect of time is NOT the same for both groups. Thus we say that there is TIME and TREATMENT interaction. treatment placebo 0 2 4 6 8 TIME (months) 10 12

Example 1: Homeopathy vs. placebo in treating pain after surgery Day of surgery Mean pain assessments by visual analogue scales (VAS) Days 1 -7 after surgery (morning and evening) Copyright © 1995 BMJ Publishing Group Ltd. Lokken, P. et al. BMJ 1995; 310: 1439 -1442

Example 2: Divalproex vs. placebo for treating bipolar depression Davis et al. “Divalproex in the treatment of bipolar depression: A placebo controlled study. ” J Affective Disorders 85 (2005) 259 -266.

Example 3: Pint of milk vs. control on bone acquisition in adolescent females Mean (w/ SE bars) percentage increases in total body bone mineral and bone density over 18 months. P-values are for the differences between groups by repeated measures analysis of variance Copyright © 1997 BMJ Publishing Group Ltd. Cadogan, J. et al. BMJ 1997; 315: 1255 -1260

Example: Two treatment groups with four measurements taken over equally spaced time intervals (e. g. , A = treatment B = placebo) id 1 2 3 4 5 6 group time 1 time 2 time 3 time 4 A A A B B B 31 24 14 38 25 30 29 28 20 34 29 28 15 20 28 30 25 16 26 32 30 34 29 34 Hypothetical data from Twisk, chapter 3, page 40, table 3. 7

Profile plots by group B A

Mean profile plots by group B A

Questions of Interest 1. Overall, are there significant differences between TIME points? From plots it looks like some differences over time, in particular times 3 and 4 look different. 2. Do the two groups differ at any time points, i. e. is there a TREATMENT effect? From plots it looks like the groups differ at baseline and there are some difference everywhere else. 3. Do the two groups differ in their responses over time, i. e is there a TIME*TREATMENT interaction? Their response profiles looks similar over time, though A and B are closer by the end.

Two Main Methods of Analyses • • Option 1: Use repeated-measures ANOVA using the “Univariate” approach (restrictive assumptions) Option 2: “Multivariate” ANOVA approach, i. e. MANOVA ( less restrictive assumptions)

Assumptions 1. 2. 3. 4. Both repeated-measures ANOVA and MANOVA assume that time intervals are equally spaced. Both methods assume response is normally distributed, but both approaches are robust against violations of normality. Repeated-measures ANOVA sphericity, or compound symmetry (see next slide) Both approaches require complete data for all subjects, i. e. no missing observations for any subjects.

Sphericity/Compound Symmetry Repeated Measures ANOVA required compound symmetry which is: (a) The variances of the response variable must be the same at each time point (b) The correlation between repeated measurements are equal, regardless of the time interval between measurements. This assumption can be tested using Mauchly’s Sphericity Test.

(a) Variances at each time point (visually) Does variance look equal across time points? Looks like most variability is at Time 1 and least at Time 4.

(b) Correlation across time points Time 1 Time 2 Time 3 Time 4 Thus for this example we Time 1 1. 00000 0. 94035 -0. 14150 0. 28445 might conclude that the Time 2 0. 94035 1. 00000 -0. 02819 0. 26921 conditions for compound Time 3 -0. 14150 -0. 02819 1. 00000 0. 27844 symmetry are met. 1. 00000 Time 4 0. 28445 0. 26921 NOT 0. 27844 Certainly does not look like we have equal correlations! Time 1 and Time 2 are highly positively correlated, but Time 1 and Time 3 are negatively correlated!

JMP and SPSS • • • Both software packages will give the results using the two approaches. Therefore you really don’t need to give much thought as to which method to use. Simply run a complete analyses and compare & contrast the results. Report MANOVA results when compound symmetry is suspect.

Hypothetical Example in JMP Treatment Group (A or B) Response measured at four equally spaced time points in separate columns (Time 1 = baseline).

Hypothetical Example in JMP IMPORTANT: Be sure to change Personality to MANOVA Put the columns of repeated measurements in the Y box Put treatment effect “Group” in the model effects box.

Hypothetical Example in JMP Check this box so JMP gives the Repeated Measure ANOVA (univariate) results as well. You can change the name to whatever, it is set to be “Time” by default. If you were doing a cross-over experiment where the repeated measures were different treatments you enter that here instead.

Hypothetical Example in JMP (MANOVA Approach) The MANOVA approach effect tests for TIME and TIME*TREATMENT interaction are boxed in the output shown. From this we see that both TIME (p =. 3287) and There is no evidence of a significant TIME*TREATMENT treatment effect =. 1408). (p =. 8932) are (pnot significant at HOWEVER, you should always look the a =. 05 level. for a Time*Treatment interaction Mean Profile Plots There does not appear to a large time effect and response varies very similarly over time for both treatments, i. e. there does not appear to be a interaction. The separation between the profiles suggests a possible treatment effect.

Hypothetical Example in JMP (Repeated Measures ANOVA) The Univariate Repeated Measures ANOVA output is also contained on the right. The Mauchly’s Sphericity Test result yields (p =. 2967) which does not suggest a problem with sphericity. However, the two methods used to adjust for a lack of sphericity, Greenhouse-Geisser and Treatment effect test for Epsilon this approach Huyn-Feldt both is the same (p Epsilon =. 1408), (which but we need to consider 1 the should if Time*Treatment sphericity isn’t interaction first!!! problem), suggest otherwise.

Hypothetical Example in JMP (Repeated Measures ANOVA) From this we see that both effects: TIME G-G e =. 486, p =. 1743 H-F e =. 885, p =. 1283 & TIME*TREATMENT G-G e = 0. 486, p =. 6954 H-F e = 0. 886, p =. 8118 are not significant at the a =. 05 level.

Summary of Findings 1. Overall, are there significant differences between TIME points? NO From MANOVA results the time effect was not significant (p =. 3287). 2. Do the two groups differ at any time points, i. e. is there a TREATMENT effect? NO From MANOVA or Repeated Measures ANOVA the treatment effect (p =. 1408) 3. Do the two groups differ in their responses over time, i. e is there a TIME*TREATMENT interaction? NO

Hypothetical Example in SPSS Subject ID, group and a numeric group (A = 1, B = 2) identifier are entered. The repeated measurements over time are entered in separate columns.

Hypothetical Example in SPSS Select Analyze > General Linear Model > Repeated Measures… You are first expected to enter a name for the within-subject factor (usually Time) and the number of levels it has (4 in this case) then click Add.

Hypothetical Example in SPSS Treatment identifier goes here. Set up Mean Profile Plots by clicking here.

Mean Profile Plots in SPSS

Hypothetical Example in SPSS (MANOVA Results and Sphericity Test) Time effect is not significant (p =. 329) & the Time*Treatment interaction is not significant either (p =. 893). Mauchly’s Sphericity Test is not significant (p =. 324)

Hypothetical Example in SPSS (Repeated Measures ANOVA) Time (G-G e, p =. 174) and Time*Treatment (G-G e, p =. 695) interaction effects are not significant. These tests are only relevant if there is a significant Time effect. They look at whether the time trend is linear, quadratic, or cubic in nature.

Hypothetical Example in SPSS There is not a significant treatment effect (p =. 141). This test result is the same regardless of approach used for the within-subject effects. Not of primary interest.

Other Ways to Test for Changes Over Time • • Compare all means relative to baseline, i. e. m 2 vs. m 1, m 3 vs. m 1, m 4 vs. m 1, etc. Compare all means to the one previous, i. e. m 2 vs. m 1, m 3 vs. m 2, m 4 vs. m 3, etc. Compare each mean to the mean of the others Compare each mean to the mean of subsequent times

Other Ways to Test for Changes Over Time (JMP) After choosing Mean option Check this box so the results of each comparison are given. Repeated Measures – this is the standard, it allows us to test for Treatment, Time, and interaction effects. Same as running Sum and Contrast, try it! Polynomial – tests for linear, quadratic, cubic time trends. Helmert – each mean vs. mean of ones following it. Mean – each mean vs. mean of other time points. Notice that when you mouse over option it tells what it will do.

Other Ways to Test for Changes Over Time (JMP ~ Mean Option) The mean at baseline (m 1) is not significantly mean of the rest (p =. 987). The nature of the difference between baseline mean and the mean of the rest does not depend on treatment received (p =. 503), i. e. no interaction. The mean at time 2 (m 2) is not significantly mean of the rest (p =. 414). The nature of the difference between mean at time 2 and the mean of the rest does not depend on treatment received

Other Ways to Test for Changes Over Time (SPSS) Deviation – each mean vs. mean of the rest Simple – each mean vs. baseline Difference – each mean vs. mean of previous. Helmert – each mean vs. mean of following time periods. Select desired contrast and then click Change button to make the change. Click here to define alternative mean comparisons to make. Repeated – each mean vs. previous mean. Polynomial – linear, quadratic, and cubic time trends.

Other Ways to Test for Changes Over Time (SPSS ~ Deviation Contrast) The mean at time 2 (m 2) is not significantly mean of the rest (p =. 414). The nature of the difference between mean at time 2 and the mean of the rest does not depend on treatment received

Replacing Missing Values • • • Often times in measuring subjects over the course of time they drop out of the study for whatever reason. A standard/simple approach is to fill missing values with the last observed value. This is called “Last Observation Carried Forward (LOCF)”. Many more complicated schemes exist!!

Example: Last Observation Carried Forward (LOCF) Subject 0 Months 1 20 2 Months 13 4 Months 6 Months 2 21 21 20 19 3 19 18 10 6 4 30 25 23 = missing value

Example: LOCF (cont’d) Last Observation Carried Forward Subject 0 Month 2 Month 4 Month 6 Month 1 20 16 2 21 21 20 19 3 19 18 10 6 4 32 25 23 16 32 16

Repeated Measures Example 2 From Daniels book with imputation for both SPSS and JMP