Lab 4 ANCOVA and Two-Way ANOVA_SR_rev

1 1. Submit answers to the Google form by 11:20a.m. on Mar. 17, 2022: https://forms.gle/mrotGVoHofCtGHpU8 LAB 4 ANCOVA and Two-Way ANOVA Review ANCOVAmodel : y ij = μ + α i + b x ij + ϵ ij . Represented using regression, the model is: y = θ 0 + θ 1 d 1 + ⋯ + θ a − 1 d a − 1 + θ k x + ϵ 1 . In the regression model, the intercept, θ 0 , costs 1 degree of freedom. The covariate, x, costs 1 degree of freedom to estimate θ k . Factor A costs a-1 degree of freedom. This is because if there are “a” groups, there will be a-1 dummy codes. Consequently, we need to estimate a-1 regression coefficients for factor A, and the degree of freedom for factor A is a-1. Finally, the degree freedom for error, ϵ , is N-1-(a-1)-1= N-a-1. For sum of squares, it is easy to calculate the sum of squares of regression residual (variance of the residual *(N-1)), which equals to the within group sum of squares. To calculate sum of squares for Factor A, the most intuitive way is to first estimate the residual sum of squares for this regression: y = γ 0 + γ 1 x + ϵ 2 . Then, SS A = SS ϵ 2 − SS ϵ 1 . This is the intuitive way to calculate sum of squares, it is mathematically equivalent to the formula you learned during the lecture. Type 3 sum of squares is also calculated in a similar way. The two-way ANOVA model is: Two way ANOVA model : y ij = μ + α + β + αβ + ε . In regression terms, suppose factor A has 3 groups and factor B has 3 groups, the model looks like: y = c 0 + c 1 da 1 + c 2 da 2 + c 3 db 1 + c 4 db 2 + c 5 da 1 db 1 + c 6 da 1 db 2 + c 7 da 2 db 1 + c 8 da 2 db 2 + ε . For degree of freedom, the intercept, c 0 , costs 1 degree of freedom. Factor A costs 3-1=2 degree of freedom to estimate c 1 and c 2 . Similarly, factor B costs 2 degree of freedom to estimate c 3 and c 4 . The interaction costs 2*2=4 degree of freedom. The error costs N-1-2-2-4 degree of freedoms. More generally, Factor A costs a-1 df, Factor B costs b-1 df, interaction costs (a-1)(b-1) df, residual costs N-1- (a-1)-(b-1)-(a-1)(b-1)=N-1-a+1-b+1-ab+a+b-1=N-ab. Sum of squares for factor A, B and interaction can be calculated in a similar way like in ANCOVA.

2 ANCOVA Exercise A researcher was interested in studying the effect of a drug on dementia patients’ memory. Thirty participants were randomly assigned to three groups: placebo, low dose, and high dose. Participants’ memory was measured using a memory test that scored from 0 to 10. The test was administered before and after the drug treatment. The data is presented in ‘ ANCOVA data.sav ’. Using what you have learned in the previous lab, conduct an ANOVA test with dose as the independent variable, and pre memory score as the dependent variable. A. Report the results. Does dose have a significant effect on pre memory score? F(2,27)=1.979m p=.158. No pre-memory score is not significant Conduct another ANOVA with dose as the independent variable, and post memory score as the dependent variable, report the results. B. Does dose have a significant effect on post memory score? What can you conclude? F(2,27)=2.416, p=.108. No, also not significant (p> .05) C. Why is this approach problematic? Because now it seems both the pre-test and post- test do not have a significant relationship with the independent variable, dosage Now, conduct an ANCOVA test by clicking Analyze -> General Linear Model -> Univariate, put post_score into the dependent variable box, put dose in the fixed factor(s) box, and put pre_score into the Covariate(s) box, click EM Means, put dose to the “Display Means for” box, check compare main effects, select LSD(none), Continue, Options, check Descriptive statistics, parameter estimates, Homogeneity tests, Continue, OK. [PASTE Levene’s Test of Equality of Error Variances here] D. Based on the Levene’s test, is the homogeneity of variance assumption satisfied?

3 No E. If not, should you be concerned? Why(not)? No, because the error variances differ between the groups Tests of Between-Subjects Effects Dependent Variable: post_score Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 31.920 a 3 10.640 3.500 .030 Intercept 76.069 1 76.069 25.020 .000 pre_score 15.076 1 15.076 4.959 .035 Dose 25.185 2 12.593 4.142 .027 Error 79.047 26 3.040 Total 683.000 30 Corrected Total 110.967 29 a. R Squared = .288 (Adjusted R Squared = .205) F. Does dose have a significant effect on post_score after controlling for pre_score? Report the statistics. yes, because p > .05. F(2,30)=4.142, p=.027 Pairwise Comparisons Dependent Variable: post_score (I) Dose (J) Dose Mean Difference (I-J) Std. Error Sig. b 95% Confidence Interval for Difference b Lower Bound Upper Bound Placebo Low Dose -1.786 * .849 .045 -3.532 -.040 High Dose -2.225 * .803 .010 -3.875 -.575 Low Dose Placebo 1.786 * .849 .045 .040 3.532 High Dose -.439 .811 .593 -2.107 1.228 High Dose Placebo 2.225 * .803 .010 .575 3.875 Low Dose .439 .811 .593 -1.228 2.107 Based on estimated marginal means

4 *. The mean difference is significant at the .05 level. b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments). G. Based on pairwise comparisons , which groups are significantly different? Placebo & Low Dose, Placebo & High Dose ANCOVA also assumes Homogeneity of regression, which suggests that the regression slope between covariate and dependent variables are the same across groups. To check this assumption, click analyze, general linear model, Univariate. Since we just conducted an ANCOVA, we can keep the rest of the settings, click Model, Build custom terms, move/drag dose and pre_score to the model box, then use the down arrow to move Dose into Build Term, click By *, select pre_score under Factors & Covariates, move pre_score into Build Term, and move them both together to the model box by clicking “Add.” This will create the interaction term . Continue, OK. (*Hint: See screenshots on the next page.)