Lab 5_SR_rev

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1 To receive full credit for Lab 5, submit the Google form https://forms.gle/gwfQuCm2bt8JrMjV7 by 11:20a.m. on April 7, 2022 and the Word document on e-class by Tuesday, April 12, 2022. Lab 5 Repeated Measures Repeated Measures: Model and assumptions Basic repeated measures model expressed using regression y = β 0 + ∑ i = 1 T − 1 β i ∗ time dummycode i + ∑ j = 1 N − 1 β j + T − 1 ∗ subject dummy code j + ∑ i = 1 T − 1 ∑ j = 1 N − 1 β for eachterm ∗ time dummy code i ∗ subject Degree of freedom: there are total N*T data points, intercept costs 1 df, time costs T-1 df, subject costs N-1 df, interaction costs (N-1)*(T-1)=N*T-N-T+1 df, error has 0 df. We cannot separate error terms from the interaction term. SS works the same as before. But it is important to note that even though repeated measure models look similar to regression models we learned before, they are not exactly the same. This is because the subject factor is considered as a random factor rather than a fixed factor. This influences how the F ratio is calculated for each factor. This is why a repeated measures model is considered a mixed design (it has a random factor (subject) and a fixed factor (time)). Note that SPSS uses a multivariate approach (each time point is considered a DV) to solve the repeated measure. But the univariate approach (from the lecture) is equivalent to the multivariate approach if the sphericity assumption is met. However, for now, we will not focus on the multivariate approach, which is discussed in EDPY 605. Assumptions Homogeneity of variance covariance matrix: different groups have the same population variance covariance matrix. Sphericity: The simple way to define sphericity is that the variances of the differences between any two time points are the same. This definition is not always correct, but for simplicity, we just want to state it this way. Intuitively, if we suspect change between time 1 and time 2 predicts change between time 2 and time 3, we know the assumption of sphericity is probably violated. Normality assumption: repeated measure in SPSS actually assumes that the DVs have a multivariate normal distribution, which means any linear combinations of the DVs must be normally distributed (explained in EDPY 605).

2 Exercise 1: One-way Within Subjects Design A researcher wants to examine the effect of a psychotherapy on participants’ happiness. The researcher measures participants’ happiness before the treatment, immediately after the treatment, and 1 month after the treatment. Open exercise 1.sav, click Analyze, General Linear Model, Repeated Measures, in the “Within-Subject Factor Name” box, type time, in the “number of levels” box, type 3, Add, click Define, move pre, post, and follow up to the within-subjects variables (time), click Plots, move time to the horizontal axis box, add, continue, click EM Means, move time to “Display Means for” box, check “Compare main effects”, Continue, Options, check Descriptive statistics, Estimates of effect size, continue, ok. Mauchly's Test of Sphericity a Measure: MEASURE_1 Within Subjects Effect Mauchly's W Approx. Chi- Square df Sig. Epsilon b Greenhouse- Geisser Huynh- Feldt Lower-bound time .958 .775 2 .679 .960 1.000 .500 Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. Design: Intercept Within Subjects Design: time b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. A. Interpret Mauchly’s Test of Sphericity. Has the assumption of sphericity been violated? Report the results. [Hint: Google how to do this. A correct answer must include both an interpretation and the corresponding statistics.] No, the assumption of sphericity was met, χ2 (2) = 0.775, p = .679. if not violated then you do not need to look at greenhouse-geisser B. Which website did you use to help you interpret this test? https://statistics.laerd.com/statistical-guides/sphericity-statistical-guide.php

3 Tests of Within-Subjects Effects Measure: MEASURE_1 Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared time Sphericity Assumed 9151.300 2 4575.650 143.773 .000 .883 Greenhouse-Geisser 9151.300 1.919 4768.520 143.773 .000 .883 Huynh-Feldt 9151.300 2.000 4575.650 143.773 .000 .883 Lower-bound 9151.300 1.000 9151.300 143.773 .000 .883 Error(time) Sphericity Assumed 1209.367 38 31.825 Greenhouse-Geisser 1209.367 36.463 33.167 Huynh-Feldt 1209.367 38.000 31.825 Lower-bound 1209.367 19.000 63.651 C. Does time have a significant effect on participants’ happiness? Yes, time has a significant effect on participants’ happiness, F(2, 38) = 143.733, p < .001, ηp2 = .883.

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4 3 time points, treatment is in the middle, pre, treatment (when it’s the highest, showing effect), post(higher than pre) Pairwise Comparisons Measure: MEASURE_1 (I) time (J) time Mean Difference (I-J) Std. Error Sig. b 95% Confidence Interval for Difference b Lower Bound Upper Bound 1 2 -30.250 * 1.599 .000 -33.597 -26.903 3 -15.350 * 1.913 .000 -19.354 -11.346 2 1 30.250 * 1.599 .000 26.903 33.597 3 14.900 * 1.825 .000 11.080 18.720 3 1 15.350 * 1.913 .000 11.346 19.354 2 -14.900 * 1.825 .000 -18.720 -11.080 Based on estimated marginal means *. The mean difference is significant at the .05 level. b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments). D. Based on the post hoc analysis, how do the three time points differ from each other? [Hint: Take a look at the plot and identify the corresponding values in the Pairwise Comparisons output.] There was a statistically significant difference in participants’ happiness between pre-treatment and immediately after the treatment (p < .001). The mean difference is 30.250. There was a statistically significant difference in participants’ happiness between pre-treatment and 1 month after the treatment (p < .001). The mean difference is 15.350. There was a statistically significant difference in participants’ happiness between immediately after the treatment and 1 month after the treatment (p < .001). The mean difference is 14.900. Exercise 2: Two-way Within Subjects Design This example comes from the lecture 9 notes. We examine the effect of time and type of novel on a dependent variable. [In the lecture, Seyma said she would refer to the DV as “number of books.”] Open ‘exercise 2.sav’, click Analyze, General Linear Model, Repeated Measures, write “type” in the within-subject factor name box, type 2 in the number of levels box, click add, type month in the within-subject factor name box, type 3 in the number of levels box, click add, define, move all the variables to the “within-subjects variables” box, click plots, move month to the horizontal axis box, move type to the separate lines box, add, continue, click EM Means, move type, month, and type*month to the “display means for” box, check “compare main effects”, Continue, Options, check Descriptive statistics, estimates of effect size, continue, paste, after the line ‘/EMMEANS=TABLES(type*month)’ add ‘COMPARE (type)

5 ADJ(LSD)’, run the code. Descriptive Statistics Mean Std. Deviation N Science Fiction month 1 2.4000 1.67332 5 Science Fiction month 2 3.8000 .83666 5 Science Fiction month 3 6.4000 1.14018 5 Mystery month 1 5.0000 1.87083 5 Mystery month 2 5.0000 2.23607 5 Mystery month 3 2.2000 1.64317 5 Mauchly's Test of Sphericity a Measure: MEASURE_1 Within Subjects Effect Mauchly's W Approx. Chi- Square df Sig. Epsilon b Greenhouse- Geisser Huynh- Feldt Lower-bound type 1.000 .000 0 . 1.000 1.000 1.000 month .796 .684 2 .710 .831 1.000 .500 type * month .252 4.135 2 .127 .572 .651 .500 Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. Design: Intercept Within Subjects Design: type + month + type * month b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. E. Has the sphericity assumption been satisfied? Yes because p is > .05 X 2 (2)=4.135, p=.127

6 Tests of Within-Subjects Effects Measure: MEASURE_1 Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared type Sphericity Assumed .133 1 .133 .016 .906 .004 Greenhouse- Geisser .133 1.000 .133 .016 .906 .004 Huynh-Feldt .133 1.000 .133 .016 .906 .004 Lower-bound .133 1.000 .133 .016 .906 .004 Error(type) Sphericity Assumed 33.533 4 8.383 Greenhouse- Geisser 33.533 4.000 8.383 Huynh-Feldt 33.533 4.000 8.383 Lower-bound 33.533 4.000 8.383 month Sphericity Assumed 2.867 2 1.433 2.774 .122 .410 Greenhouse- Geisser 2.867 1.661 1.726 2.774 .136 .410 Huynh-Feldt 2.867 2.000 1.433 2.774 .122 .410 Lower-bound 2.867 1.000 2.867 2.774 .171 .410 Error(month) Sphericity Assumed 4.133 8 .517 Greenhouse- Geisser 4.133 6.645 .622 Huynh-Feldt 4.133 8.000 .517 Lower-bound 4.133 4.000 1.033 type * month Sphericity Assumed 64.467 2 32.233 26.135 .000 .867 Greenhouse- Geisser 64.467 1.144 56.344 26.135 .004 .867 Huynh-Feldt 64.467 1.303 49.479 26.135 .003 .867 Lower-bound 64.467 1.000 64.467 26.135 .007 .867 Error(type*month ) Sphericity Assumed 9.867 8 1.233 Greenhouse- Geisser 9.867 4.577 2.156 Huynh-Feldt 9.867 5.212 1.893 Lower-bound 9.867 4.000 2.467

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7 F. Report the results based on the above table. There wasn’t a significant effect for either type or month, but there was an interaction effect Type*Month: (F(2)=26.135, p<.001) Type: (F(1)=.016, p=.906) Month: (F(2)=2.774,p=.122) Pairwise Comparisons Measure: MEASURE_1 month (I) type (J) type Mean Difference (I-J) Std. Error Sig. b 95% Confidence Interval for Difference b Lower Bound Upper Bound 1 1 2 -2.600 1.503 .159 -6.774 1.574 2 1 2.600 1.503 .159 -1.574 6.774 2 1 2 -1.200 1.158 .358 -4.414 2.014 2 1 1.200 1.158 .358 -2.014 4.414 3 1 2 4.200 * .860 .008 1.812 6.588 2 1 -4.200 * .860 .008 -6.588 -1.812 Based on estimated marginal means *. The mean difference is significant at the .05 level. b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments). G. Report and interpret the post hoc analysis results.

8 Month 3 was significant with Type I & J (p=.008)

9 Exercise 3: Repeated Measures with a Between Group Factor Open Exercise 3.sav, click Analyze, General Linear Model, Repeated Measures, type time in the within-subject factor name box, type 2 in the number of level box, add, define, move t1 and t2 in the within subjects variables box, move group to the between-subjects factor(s) box, click Plots, move time to the horizontal axis box, add, move time to the separate lines box, continue, click EM Means, move group, time, group*time to the Display means for box, Continue, Options, check homogeneity tests, estimates of effect size, Continue, click paste, after line “/EMMEANS=TABLES(Group*time)” add “compare (Group) ADJ(LSD)”, run the code. Box's Test of Equality of Covariance Matrices a Box's M 2.470 F .724 df1 3 df2 58320.000 Sig. .537 Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. a. Design: Intercept + Group Within Subjects Design: time H. For homogeneity of variance covariance matrix assumption, we examine the Box’ M test. It tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. Is the homogeneity of variance covariance matrix assumption violated? The test was nonsignificant (p=.537) therefore the assumption is not violated

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10 Mauchly's Test of Sphericity a Measure: MEASURE_1 Within Subjects Effect Mauchly's W Approx. Chi- Square df Sig. Epsilon b Greenhouse- Geisser Huynh- Feldt Lower-bound time 1.000 .000 0 . 1.000 1.000 1.000 Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix. a. Design: Intercept + Group Within Subjects Design: time b. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table. I. Why is the p-value not available? b/c only two measures Tests of Within-Subjects Effects Measure: MEASURE_1 Source Type III Sum of Squares df Mean Square F Sig. time Sphericity Assumed 1570.093 1 1570.093 89.014 .000 Greenhouse-Geisser 1570.093 1.000 1570.093 89.014 .000 Huynh-Feldt 1570.093 1.000 1570.093 89.014 .000 Lower-bound 1570.093 1.000 1570.093 89.014 .000 time * Group Sphericity Assumed 645.696 1 645.696 36.607 .000 Greenhouse-Geisser 645.696 1.000 645.696 36.607 .000 Huynh-Feldt 645.696 1.000 645.696 36.607 .000 Lower-bound 645.696 1.000 645.696 36.607 .000 Error(time) Sphericity Assumed 317.498 18 17.639 Greenhouse-Geisser 317.498 18.000 17.639 Huynh-Feldt 317.498 18.000 17.639 Lower-bound 317.498 18.000 17.639

11 Tests of Between-Subjects Effects Measure: MEASURE_1 Transformed Variable: Average Source Type III Sum of Squares df Mean Square F Sig. Intercept 45910.905 1 45910.905 1405.713 .000 Group 1222.719 1 1222.719 37.438 .000 Error 587.884 18 32.660 J. Report and interpret the main and interaction effects. Time has significant main effect (F(1)=89..014, p<.001) The interaction effects were significant (F(1)=36.607, p<.001)

12 Pairwise Comparisons Measure: MEASURE_1 time (I) Group (J) Group Mean Difference (I-J) Std. Error Sig. b 95% Confidence Interval for Difference b Lower Bound Upper Bound 1 .00 1.00 -3.022 2.146 .176 -7.531 1.487 1.00 .00 3.022 2.146 .176 -1.487 7.531 2 .00 1.00 -19.093 * 2.335 .000 -23.999 -14.187 1.00 .00 19.093 * 2.335 .000 14.187 23.999 Based on estimated marginal means *. The mean difference is significant at the .05 level. b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments). K. Report and interpret the post hoc analysis results. Time point 2 was significant (p<.001)

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