Chapter 14

Repeated Measures ANOVA

All the ANOVA stuff we have done so far has had different subjects in the various cells of the experimental design

That kind of experiment is called a between-subjects design

Sometimes, however, we run the same subjects in some or all cells of the design

Such a within-subjects (or repeated measures) design has two advantages:

• you get more data per subject
• you actually get more power because you can factor between-subject differences out of the error term

Memories of the ANOVA logic

Recall that the purpose of doing an ANOVA is to see if some difference between treatment means is sufficiently large is to be unlikely to occur by chance (less than 5% chance)

When we test that … we get an estimate of the difference we are interested in, and divide it by an estimate of variation due to chance

Specifically: Notice that this F value will increase if the difference between the means is large OR if the measurement of error is small

As you will see, repeated-measures designs allow us to reduce the error term, thereby resulting in larger Fs (more power)

An example: Within versus Between

This experiment will show the importance of the "articulatory loop" for retaining information in short-term memory

Between-Subjects Version

 Subject bla-bla Subject no bla-bla ` ` `1` `2` `3` `4` `5` `X X2` `5 25` `4 16` `6 36` `4 16` `7 49` ` ` `1` `2` `3` `4` `5` `X X2` `7 49` `6 36` `5 25` `6 36` `6 36`

Within-Subject Version

 Subject bla-bla no bla-bla ` ` `1` `2` `3` `4` `5` `X X2` `5 25` `3 9` `7 49` `2 4` `3 9` `X X2` `7 49` `4 16` `7 49` `4 16` `5 25`

Computations for Between-Subject      How is the within-subjects version different from the between-subjects version?

An assumption of the between-subjects ANOVA is that the observations in one level of the treatment are independent of those in the other level(s)

Hopefully you will notice that this assumption does not hold in our within-subjects version of the experiment

The use of the same subject in more than one level of the treatment almost always builds in a dependency because subjects who do well in one level tend to also do well in the other(s)

Can we remove this dependency? In fact we can, and when we do, there is a bonus! (the kind of thing that makes statistics geeks real happy J )

Getting Rid of the Variability Due to Subjects

The "dependency in observations" is due to some subjects doing better than others

What we are going to do to deal with this is to literally remove the variation due to subjects from the error term

For demonstration purposes only .. you can think of this as subtracting each subjects mean from all the scores they contribute

Using the data from out class:

 Subject bla-bla no bla-bla ` ` `1` `2` `3` `4` `5` `X X¢ ` `5 -1` `3 -.5` `7 0` `2 -1` `3 -1` `X X¢ ` `7 +1` `4 +.5` `7 0` `4 +1` `5 +1`

Where Within-Subjects Computations

Another way of doing what is essentially the same thing is to remove the sum of squares due to subjects from the error term.       Source Tables

Between-Subjects

 Source SS df MS F Treatment 1.60 1 1.60 1.46 Error 8.80 8 1.10 Total 10.40 9

Within-Subjects

 Source SS df MS F Subject 4 Treatment 1 Error 4 Total 9

The Advantage of Within-Subject Designs

Remember, F values are increased if the difference of interest in larger OR if the measure of variance (MSerror) gets smaller

While removing the sum of squares due to subjects does make the observations independent across levels of the treatment variable, it OFTEN reduces the MSerror, thereby resulting in increased power (larger F values)

This only occurs though if the reduction in MSerror is more than compensates for the loss in dferror .. so it is not always true

Note that you cannot remove the variance (sum of squares) due to subjects when using a between subjects design because you only have one observation per subject … thus the variance due to subjects must remain as part of the error term

Moral: Usually, it is better to use within-subject (repeated measures) designs … not only do they let you use less subjects, but they are also more powerful, statistically speaking

Assumption of Compound Symmetry

Remember when we did between-subject ANOVAs, one of the assumptions was that the variance in our various treatment groups were homogenous (i.e., roughly equivelant)

A similar but slightly more complex assumption underlies repeated measures designs

Specifically, we need to satisfy the "compound symmetry" assumption which is that in addition to the variances being equal, the covariances between pairs of variables are also equal

For this to make sense, I think we may have to do a B07 time travel to re-introduce the notion of covariance ….

Imagine any two variables such as …

 Subject Height (X) Weight (Y) 1 69 108 2 61 130 3 68 135 4 66 135 5 66 120 6 63 115 7 72 150 8 62 105 9 62 115 10 67 145 11 66 132 12 63 120 Mean 65.42 125.83 Sum(X) = 785 Sum(Y) = 1510 Sum (X2) = 51473 Sum(Y2) = 192238

Sum (XY) = 99064

The covariance of these variables is computed as: But what does it mean?

The covariance formula should look familiar to you. If all the Ys were exchanged for Xs, the covariance formula would be the variance formula

Note what this formula is doing, however, it is capturing the degree to which pairs of points systematically vary around their respective means

If paired X and Y values tend to both be above or below their means at the same time, this will lead to a high positive covariance

However, if the paired X and Y values tend to be on opposite sides of their respective means, this will lead to a high negative covariance

If there is no systematic tendencies of the sort mentioned above, the covariance will tend towards zero

The Computational Formula for Cov

Given its similarity to the variance formula, it shouldn’t surprise you that there is also a computationally more workable version of the covariance formula: For our height versus weight example then: Back to Compound Symmetry

OK, now let’s assume we ran a repeated measures study in which we were looking at practice effects on some task over 3 days

 Day 1 Day 2 Day 3 Sub 1 700 650 620 Sub 2 520 450 430 Sub 3 600 540 500 Sub 4 650 630 620 Sub 5 750 700 690 Variance 7930 9830 10970

S X = 700 + 520 + 600 + 650 + 750 = 3220
S Y = 650 + 450 + 540 + 630 + 700 = 2970
S XY = (700 * 650) + …. (750 * 700) = 1947500 The Covariance (Variance/Covariance) Matrix

These variances and covariances are often presented in a matrix such as the following:  So, the assumption of compound symmetry is simply that the variances must all be approximately equal and the covariances must all be approximately equal

The variances need not (and often do not) equal the variances though

Complicating it all

So far in this chapter, we have been dealing with only one variable that has been manipulated in a within-subject manner

However, as we saw in Chapter 13, studies usually manipulate more than one variable which raises several possibilities

2 variables

2 between subject variables .. Chapter 13
1 within - 1 between
2 within

3 variables

3 between … Chapter 13
1 within - 2 between
2 within - 1 between
3 within

Computationally, we will only focus on the 2 new "2 variable" situations

However, as was the case with 3 between subject variables, I will expect you to be able to interpret 3 variable results … we will spend time doing this as well

One Between - One Within

Imagine the following study (raw data is presented in the text, pp. 459)

Similar to Siegel’s morphine tolerance study, King (1986) was interested in conditioned tolerance to another drug … midazolam

• initially midazolam decreases motor activity
• however, tolerance develops quickly
• 3 groups … 2 got 2 injections of midazolam prior to test .. the other (the control group) got saline injections
• at test, all groups got midazolam, but one of the experimental groups received it in the same context as the had before (same group) whereas the other received it in a different context (the different group)
• motor activity measured in 6 five-minute intervals producing the following data

The Data, Steve Style

 1 2 3 4 5 6 SS 150 44 71 59 132 74 55858 88 335 270 156 160 118 230 301885 212 149 52 91 115 43 154 71796 101 Control 159 31 127 212 71 224 142532 137 159 0 35 75 71 34 38328 62 292 125 184 246 225 170 274786 207 297 187 66 96 209 74 185907 155 170 37 42 66 114 81 55946 85 214 93 97 129 123 130 1127218 131 346 175 177 192 239 140 295255 212 426 329 236 76 102 232 415417 234 359 238 183 123 183 30 268532 186 Same 272 60 82 85 101 98 111338 116 200 271 263 216 241 227 338876 236 366 291 263 144 220 180 389342 244 371 364 270 308 219 267 557151 300 497 402 294 216 284 255 687386 325 355 266 221 170 199 179 3063297 232 282 186 225 134 189 169 246983 198 317 31 85 120 131 205 182261 148 362 104 144 114 115 127 204946 161 Differ 338 132 91 77 108 169 186103 153 263 94 141 142 120 195 170475 159 138 38 16 95 39 55 34315 64 329 62 62 6 93 67 129103 103 292 139 104 184 193 122 201390 172 290 98 109 109 124 139 1355576 145 286 153 142 136 148 149 Grand 169

The Dreaded Computations

Just like when we had two between-subject variables, there are three effects of interest in the current experiment:

• The main effect of Group
• The main effect of Interval
• The Group x Interval interaction
• However, recall that we can (an do) use a different error term when testing within-subject effects than when testing between subject effects

SStotal (by the way) = 1432293

So, the first thing we must do is to decide which effects are purely between-subjects, and which have a within-subject component

For this study, Group was manipulated between-subjects, but both Interval and the Group x Interval interaction have a between subjects component (i.e., Interval)

OK, now we separately deal with our between and within-subject effects

Between-Subject Effects

We treat between subject effects like we always have. We calculate SStreat as the sum of squares of the treatment means times the relevant n, and we calculate SSerror as the sum of the variance of subjects within the group  dfgroup = k-1 = 3-1 = 2
dfw/grp = k(n-1) = 3(7) = 21

Within-Subject Effects

OK, for starters, the sums of squares for the Interval and interaction effects are calculated like we did in the 2 between-subject case  SSgrp * int = SScells - SSint - SSgrp
= 766368 - 399744 - 287472
= 79152

dfint = k-1 = 6-1 = 5
dfgrp * int = dfgrp * dfint = 2 * 5 = 10

The Within-Subject Error Term

Remember than when we are dealing with within subject effects, we use a different error term (one that does not include the variability due to subject by subject variation)

Given the computations we have done so far, we can get the rest by subtraction …

 Source SS df MS F Between 6721981 Group 287472 2 143736 7.85 Ss / Group 384726 21 18320 Within 7600952 Interval 399744 5 79949 29.85 Grp X Int 79152 10 7915 2.96 Ss / Grp * Int 2811993 1054 2678 Total 1432293 143

1obtained by adding SSgroup and SSss/group

2obtained by subtracting SSbetween from SStotal

3obtained by subtracting SSinterval and SSgrp * int from SSwithin

4obtained by subtracting dfgroup, dfss/group, dfinterval and dfgrp * int from dftotal

Critical F’s:

F(2,21) = 3.49 F(5,105) = 2.37 F(10,105) = 1.99

Conclusions from the Anova

Main Effect of Group

We can reject the null hypothesis that there was no effect of group. The F-obtained for the main effect of group was greater than the critical F suggesting the there are differences among the three group means. From looking at the means it appears that this is mostly due to the mean for the "Same" group being much higher than the other two means. Main Effect of Interval

We can also reject the null hypothesis that there was no effect of interval. The F-obtained for the main effect of interval was greater than the critical F suggesting that there are differences among the six interval means. From the means, it appears as though activity was very high in the first interval, then dropped of and stayed relatively constant.

 1 2 3 4 5 6 286 153 142 136 148 149

Interaction of Group * Interval

Finally, we can also reject the null hypothesis that the effect of interval was the same for the three groups. The F-obtained for the interaction was greater than the critical F suggesting that the effect of interval is different for the three groups. From the means, it appears as though the "Same" group stayed active longer (across more of the early intervals) than the other groups.

 1 2 3 4 5 6 Control 214 93 97 129 123 130 Same 355 266 221 170 199 179 Differ 290 98 109 109 124 139 *** Chapter 13 Flashback***

 2 mins 5 mins 10 mins phobic mean = 7 mean = 8 mean = 9 8 control mean = 5 mean = 5 mean = 5 5 6 6.5 7 6.5

 Source df SS MS F Time 2 8 4 13.79 Group 1 108 108 372.41 T x G 2 8 4 13.79 Within 42 12 0.29 Total 47 136

*** Chapter 13 Flashback***

Simple Effects for the effect of time at each level of group. Source df SS MS F T for Phob 2 16 8 27.59 T for Cont 2 0 0 0 Within 42 12 0.29 Total 47 136

So, we could describe the interaction by saying that fear increased over time for phobics, but fear did not change at all over time for the controls

*** Chapter 13 Flashback***

Simple Effects

As was the case when we had two between subject variables, we will often want to do simple-effects analyses to gain a better understanding of the interaction

Recall that there are two ways we could approach these analyses, we could ask

• At which intervals was there a significant effect or group (i.e., a difference between the groups)?, or
• For which groups was there a significant effect of interval?
• Here it makes sense to look at the interaction and consider the experimental predictions to determine which of these approaches is likely to yield the information you want

Since the predictions are focused primarily on potential differences between groups (or lack of differences), the first approach is the one we would want to take in this case

Nonetheless, we will briefly consider both situations

Simple Effects for Within-Subject Variables

We had decided that in our situations we were not interested in looking at the effect of interval separately for each group

But, if we had been, then we would have been examining the effect of a within-subject variable (interval)

For reasons that are not important, whenever you are doing simple-effects that are focused on the effect of a within-subject variable, you cannot use some general error term (like, for example SSs/grp * int)

Instead, what you do is a separate one-way, repeated measures analysis of variance for each simple effect

So, for example, if you were interested in the effect of interval for the control group, you would run a complete repeated measures ANOVA examining the interval variable but using only the data from the control group

Simple Effects for Between-Subject Variables

Step 1: Computing sums of squares for the effect of group at each interval Step 2: Mean Squareds for the group effects at each interval

Since there are three groups at each interval, there are 2 degrees of freedom for each contrast

MS = SS/df, so …

MSGrp at Int1 = 79688 / 2 = 39844.00
MSGrp at Int2 = 155125 / 2 = 77562.50
MSGrp at Int3 = 74840 / 2 = 37420.00
MSGrp at Int4 = 15472 / 2 = 7736.00
MSGrp at Int5 = 30416 / 2 = 15208.00
MSGrp at Int6 = 10888 / 2 = 5444.00
Step 3: The error term

OK, here is where we differ from the Chapter 13 way of doing things

The appropriate error term SS is the SSSs/Cell

We could calculate that by hand but it would take a lot of work

In the "trust me" category, I give you the following:

SSSs/Cell = SSSs/Group + SSSs/Grp X Int, and

dfSs/Cell = dfSs/Group + dfSs/Grp X Int

So, for our example …

SSSs/Cell = SSSs/Group + SSSs/Grp X Int

= 384726 + 281199 = 665925

dfSs/Cell = dfSs/Group + dfSs/Grp X Int

= 21 + 105 = 126

MSSs/Cell = SSSs/Cell / dfSs/Cell

= 665925 / 126 = 5285.12

Step 4: Source table depicting results

 Source df SS MS F Grp at Int1 2 79688 39844 7.54 Grp at Int2 2 155125 77562.5 14.68 Grp at Int3 2 74840 37420 7.08 Grp at Int4 2 15472 7736 1.46 Grp at Int5 2 30416 15208 2.88 Grp at Int6 2 10888 5444 1.03 Ss / Cell 126 665925 5285.12 Total 143 1432293

Fcrit(2,126) = 3.07