Chapter 14 - Repeated Measures Designs
Chapter 14 – Repeated-Measures Designs
[As in previous chapters, there will be substantial rounding in these answers. I have attempted to make the answers fit with the correct values, rather than the exact results of the specific calculations shown here. Thus I may round cell means to two decimals, but calculation is carried out with many more decimals.]
14.1 Does taking the GRE repeatedly lead to higher scores?
a. Statistical model:
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b. Analysis:
| | |
|Subject |Test Session |
|Mean |Mean |
| | |
|1 |1 |
|566.67 |552.50 |
| | |
|2 |2 |
|450.00 |563.75 |
| | |
|3 |3 |
|616.67 |573.75 |
| | |
|4 | |
|663.33 | |
| | |
|5 | |
|436.67 | |
| | |
|6 | |
|696.67 | |
| | |
|7 | |
|503.33 | |
| | |
|8 | |
|573.33 | |
| | |
|Mean | |
|563.33 | |
| | |
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|Source |df |SS |MS |F |
|Subjects | 7 |189,666.66 | | |
|Within subj |16 | 5266.67 | | |
| Test session | 2 | 1808.33 |904.17 |3.66 ns |
| Error |14 |3458.33 |247.02 | |
|Total |23 |194,933.33 | | |
14.2 Data on first two Test Sessions in Exercise 14.1:
a. Related-sample t test:
|Subj |First |Second |Diff |
|1 |550 |570 | 20 |
|2 |440 |440 | 0 |
|3 |610 |630 | 20 |
|4 |650 |670 | 20 |
|5 |400 |460 | 60 |
|6 |700 |680 | -20 |
|7 |490 |510 | 20 |
|8 |580 |550 | -30 |
|Mean | | | 11.25 |
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[t.025(7) = +2.365]
Do not reject H0
b. Repeated-measures ANOVA:
|Source |df |SS |MS |F |
|Between subj |7 | |130,793.75 | | | |
|Within subj |8 | |3250.00 | | | |
| Test session | |1 | | 506.25 |506.25 |1.29ns |
| Error | |7 | |2743.75 |391.96 | |
|Total |15 | | 134,185.94 | | | |
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14.3 Teaching of self-care skills to severely retarded children:
|Cell means: |Phase | |
| | |Baseline |Training |Mean |
|Group: |Exp |4.80 |7.00 |5.90 |
| |Control |4.70 |6.40 |5.55 |
| |Mean |4.75 |6.70 |5.72 |
|Subject means: |S1 |S2 |S3 |S4 |S5 |
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|Source |df |SS |MS |F |
|Between Subj |19 |106.475 | | |
| Groups | 1 |1.125 |1.125 |0.19 |
| Ss w/in Grps | 18 |105.250 |5.847 | |
|Within Subj |20 |83.500 | | |
| Phase | 1 |38.025 |38.025 |15.26* |
| P x G | 1 |0.625 |0.625 |0.25 |
| P x Ss w/in Grps | 18 |44.850 |2.492 | |
|Total |39 |189.975 | | |
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There is a significant difference between baseline and training, but there are no group differences nor a group x phase interaction.
14.4 Independent t test on data in Exercise 14.3:
a. Difference scores (Training - Baseline)
|Exper. |1 |2 |
| | |Baseline |Training |Total |
|Group |Exp | 4.8 | 7.0 |5.90 |
| |Att Cont | 4.7 | 6.4 |5.55 |
| |No Att Cont | 5.1 | 4.6 |4.85 |
| |Total | 4.87 | 6.00 |5.43 |
|Subject means: |S1 |S2 |S3 |S4 |S5 |
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|Source |df |SS |MS |F |
|Between subj |29 | |159.7333 | | | |
| |Groups | |2 | |11.4333 |5.7166 |1.04 |
| |Ss w/ Grps | |27 | |148.300 |5.4926 | |
|Within subj |30 | | 95.0000 | | | |
| |Phase | |1 | |19.2667 |19.2667 |9.44* |
| |P * G | |2 | |20.6333 |10.3165 |5.06* |
| |P * Ss w/Grps | |27 | |55.1000 | 2.0407 | |
|Total | 59 | |254.733 | | |
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b. Plot:
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c. There seems to be no difference between the Experimental and Attention groups, but both show significantly more improvement than the No Attention group.
14.6 Summarization of stories by adult and children good and poor readers:
|Cell Means for Age * Readers * Items: |
| |Adults | |
|Items: |Setting |Goal |Disp |Mean |
|Good Readers |6.20 |6.0 |5.0 |5.73 |
|Poor Readers |5.40 |4.8 |2.0 |4.07 |
|Mean |5.80 |5.40 |3.50 |4.90 |
| | | | | |
| |Children | |
|Items: |Setting |Goal |Disp |Mean |
|Good Readers |5.80 |5.60 |3.00 |4.80 |
|Poor Readers |3.00 |2.40 |1.20 |2.20 |
|Mean |4.40 |4.00 |2.10 |3.50 |
|Cell Means for Age * Readers: |
| |Adults |Children |Mean |
|Good Readers |5.73 |4.80 |5.27 |
|Poor Readers |4.07 |2.20 |3.13 |
|Mean |4.90 |3.50 |4.20 |
|Cell Means for Age * Items: |
| |Adults |Children |Mean |
|Setting |5.80 |4.40 |5.10 |
|Goal |5.40 |4.00 |4.70 |
|Disposition |3.50 |2.10 |2.80 |
|Mean |4.90 |3.50 |4.20 |
|Cell Means for Reader * Items: |
| |Good Readers |Poor Readers |Mean |
|Setting | 6.00 | 4.20 |5.10 |
|Goal | 5.80 | 3.60 |4.70 |
|Disposition | 4.00 | 1.60 |2.80 |
|Mean | 5.27 | 3.13 |4.20 |
|Subject Means: |
| |Good Adult Readers: |7.00 |5.00 |5.00 |7.00 |4.67 |
| |Good Children Readers: |4.00 |6.33 |6.00 |4.33 |3.33 |
| |Poor Adult Readers: |5.33 |3.00 |4.67 |3.00 |4.33 |
| |Poor Children Readers: |2.00 |1.00 |3.33 |3.33 |1.33 |
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14.7 From Exercise 14.6:
a. Simple effect of reading ability for children:
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Because we are using only the data from Children, it would be wise not to use a pooled error term. The following is the relevant printout from SPSS for the Between-subject effect of Reader.
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b. Simple effect of items for adult good readers:
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Again, we do not want to pool error terms. The following is the relevant printout from SPSS for Adult Good readers. The difference is not significant, nor would it be for any decrease in the df if we used a correction factor.
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14.8 Within-groups covariance matrices for the data in Exercise 14.10:
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14.9 It would certainly affect the covariances because we would force a high level of covariance among items. As the number of responses classified at one level of Item went up, another item would have to go down.
14.10 Cigarette smoking quitting techniques:
a. Analysis:
|Cell Means for Group * Time * Place: |
| |Pre | |Post | |
| |Home |Work | |Home |Work |Mean |
|Taper |6.80 |6.00 | |5.80 |3.60 |5.55 |
|Immediate |7.00 |6.20 | |5.80 |4.80 |5.95 |
|Aversion |7.00 |6.20 | |4.80 |2.40 |5.10 |
|Mean |6.93 |6.13 | |5.47 |3.60 |5.53 |
|Means Group * Time: | Means Group * Place: |
| |Pre |Post |Mean |Home |Work |Mean |
|Taper |6.40 |4.70 |5.55 |6.30 |4.80 |5.55 |
|Immediate |6.60 |5.30 |5.95 |6.40 |5.50 |5.95 |
|Aversion |6.60 |3.60 |5.10 |5.90 |4.30 |5.10 |
|Mean |6.53 |4.53 |5.53 |6.20 |4.87 |5.53 |
|Time * Place: |
| |Pre |Post |Total |
|Home |6.93 |5.47 |6.20 |
|Work |6.13 |3.60 |4.87 |
|Tot |6.53 |4.53 |5.53 |
|Subject * Time: |
|Pre |
|Home |6.5 |5.0 |7.5 |7.0 |
|Between subj | 14 | 69.433 | | |
| Group | 2 | 7.233 | 3.617 | 0.70 |
| Ss w/in grp | 12 | 62.200 | 5.183 | |
|Within subj | 45 | 116.500 | | |
| Time | 1 | 60.000 | 60.000 | 109.09* |
| TxS | 14 | 14.500 | | |
| GxT | 2 | 7.900 | 3.950 | 7.18* |
| GxTw/in grps | 12 | 6.600 | 0.550 | |
| Place | 1 | 26.667 | 26.667 | 43.24* |
| PxS | 14 | 8.833 | | |
| GxP | 2 | 1.433 | 0.717 | 1.16 |
| GxPw/in grps | 12 | 7.400 | 0.617 | |
| TxP | 1 | 4.267 | 4.267 | 28.44* |
| TxPxS | 14 | 3.233 | | |
| TxPxG | 2 | 1.433 | 0.717 | 4.78* |
| TxPxSw/in grp | 12 | 1.800 | 0.150 | |
|Total | 59 | 186.933 | | |
*p < .05 [F.05(1,12) = 4.75; F.05(2,12) = 3.89]
b. There is a significant decrease in desire from Pre to Post and a significant reduction at Work relative to at Home. There is also a Time by Place interaction, with a greater Place difference after treatment. The Time by Group interaction is the real test of our hypothesis, showing that the Pre-Post difference depends on the treatment group, with the greatest difference in the Aversion condition. The 3-way interaction shows that the Time by Group interaction itself interacts with Place.
14.11 Plot of results in Exercise 14.10:
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14.12 I will look at the Group × Time interaction by looking at the simple effect of Time for each Group. For a more complex design, we should run separate analyses for each group, to avoid problems with sphericity. However, with only two levels of the repeated measures variables, we have only one off-diagonal covariance, so we don’t have a problem with constant covariances. I will test each by the same test term as was used to test the Group × Time interaction, MSGxTxSs w/in groups (The pattern of significance would not change with separate analyses. The Fs if we used separate analyses would be 44.46, 18.78, and 51.43, respectively.)
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These SSSimple Effects sum to the same total as SSTime and SSG * T:
14.45 + 8.45 + 45.00 = 67.90 = 60.00 + 7.90
Each of the methods led to a significant reduction in desire to smoke. If we then look at the effect of Group at Post:
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This would not be significant even for the maximum possible value of f ', meaning that we do not have data to allow us to recommend one method over the other. (If we had run a separate analysis on just the Posttest data, the corresponding F would have been 4.33, with p = .038.)
14.13 Analysis of data in Exercise 14.5 by BMDP:
a. Comparison with results obtained by hand in Exercise 14.5.
b. The F for Mean is a test on H0: μ = 0.
c. MSw/in Cell is the average of the cell variances.
14.14 An analysis of data in Exercise 14.6 by SPSS as if it were a factorial:
From Exercise 14.6:
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This equals the MSResidual in the SPSS printout.
14.15 Source column of summary table for 4-way ANOVA with repeated measures on A & B and independent measures on C & D.
|Source |
|Between Ss |
| |C |
| |D |
| |CD |
| |Ss w/in groups |
|Within Ss |
| |A |
| |AC |
| |AD |
| |ACD |
| |A x Ss w/in groups |
| |B |
| |BC |
| |BD |
| |BCD |
| |B x Ss w/in groups |
| |AB |
| |ABC |
| |ABD |
| |ABCD |
| |AB x Ss w/in groups |
|Total |
14.16 Analysis of Foa et al. (1991) data
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All three groups decreased over time, but the Supportive Counseling group decreased the least and the interaction was significant.
14.17 Using the mixed models procedure on data from Exercise 14.16
If we assume that sphericity is a reasonable assumption, we could run the analysis with covtype(cs). That will give us the following, and we can see that the F’s are the same as they were in our analysis above.
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However, the correlation matrix below would make us concerned about the reasonableness of a sphericity assumption. (This matrix is collapsed over groups, but reflects the separate matrices well.) Therefore we will assume an autoregressive model for our correlations.
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These F values are reasonably close, but certainly not the same.
14.18 Standard analysis with missing data:
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Notice that we have lost considerable degrees of freedom and our F for Group is no longer significant
14.19 Mixed model analysis with unequal size example.
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Notice that we have a substantial change in the F for Time, though it is still large.
14.20 Analysis of Stress data:
|Source |df |SS |MS |F |Pillai F |Prob |
|Between subj |97 |137.683 | | | | | |
| |Gender | 1 | 7.296 |7.296 |5.64* | | |
| |Role | 1 | 8.402 |8.402 |6.49* | | |
| |G * R | 1 | 0.298 |0.298 | ................
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