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# Clinical Research Training Program 2021

Clinical Research Training Program 2021. ANOVA and ANCOVA. Fall 2004. www.edc.gsph.pitt.edu/faculty/dodge/clres2021.html. ANOVA vs. REGRESSION. ANOVA can be regarded as a special type of linear regressions.

## Clinical Research Training Program 2021

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### Presentation Transcript

1. Clinical Research Training Program 2021 ANOVA and ANCOVA Fall 2004 www.edc.gsph.pitt.edu/faculty/dodge/clres2021.html

2. ANOVA vs. REGRESSION • ANOVA can be regarded as a special type of linear regressions. • By using dummy coding, we can get coefficients which indicate difference in means in various groups.

3. REGRESSIONMETHODS Regression ANOVA ANCOVA/Regression

4. The null and alternative hypotheses are H0: H1: for some , where i represents the mean of population i Analysis of Variance • Hypotheses - Whether all group means are equal versus at least two group means are different

5. Analysis of Variance • Assumptions - • kindependent random samples from knormal populations with distributions N(1, 2), …, N(k, 2), respectively. • Outcome variable should be continuous. • All the populations have the same unknown variance 2 (homogeneous variance).

6. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••••••• ••••••• • •

7. Analysis of Variance • The main idea for comparing means: what matters is not how far apart the sample means are but how far apart they are relative to the variability of individual observations • ANOVA compares the variation due to specific sources within the variation among individuals who should be similar. In particular, ANOVA tests whether several populations have the same mean by comparing how far apart the sample means are with how much variation there is within the sample

8. Analysis of Variance • Between Sum of Squares (SSB) • Within Sum of Squares (SSW) • Total Sum of Squares (SST)  SST = SSB + SSW

9. MSB ~ MSW Analysis of Variance • Hypotheses: H0: H1: for some • Test statistic:

10. Analysis of Variance • Between Mean Squares (MSB) • Within Mean Squares (MSW)

11. Analysis of Variance (ANOVA Table) Summary Table

12. ANOVA vs. REGRESSION • SSR: Between Sum of Squares (SSB) • SSE: Within Sum of Squares (SSW) • SST: Total Sum of Squares (SST)  SST = SSB + SSW

13. ANOVA vs. REGRESSION • By using a reference coding, we can get similar results to ANOVA with additional information. • Additional information • Difference in means between the reference group and other groups • Difference in means between each group and overall mean (when each group has equal N).

14. REFERENCE CODING • Each variable takes on only values of 1 and 0. • 3 groups: groups 1, 2, and 3  need 2 dummy variables X2 1 if group 2 0 Otherwise X3 1 if group 3 0 Otherwise

15. REFERENCE CODING For Group 1: X2=0, X3=0: For Group 2: X2=1, X3=0: For Group 3: X2=0, X3=1:

16. REFERENCE CODING

17. REFERENCE CODING • Intercept μ is the mean of group 1 (mean ofreference group). • α2 indicates difference in mean between group 1 (reference group) and group2 • α3 indicates difference in mean between group 1 (reference group) and group3

18. . list y x x2 x3 x4 x5 • y x x2 x3 x4 x5 • 1. 5 1 0 0 0 0 • 2. 8 1 0 0 0 0 • 3. 7 1 0 0 0 0 • 4. 7 1 0 0 0 0 • 5. 10 1 0 0 0 0 • 6. 8 1 0 0 0 0 • 7. 4 2 1 0 0 0 • 8. 6 2 1 0 0 0 • 9. 6 2 1 0 0 0 • 10. 3 2 1 0 0 0 • 11. 5 2 1 0 0 0 • 12. 6 2 1 0 0 0 • 13. 6 3 0 1 0 0 • 14. 4 3 0 1 0 0 • 15. 4 3 0 1 0 0 • 16. 5 3 0 1 0 0 • 17. 4 3 0 1 0 0 • 18. 3 3 0 1 0 0 • 19. 7 4 0 0 1 0 • 20. 4 4 0 0 1 0 • 21. 6 4 0 0 1 0 • 22. 6 4 0 0 1 0 • 23. 3 4 0 0 1 0 • 24. 5 4 0 0 1 0 • 25. 9 5 0 0 0 1 • 26. 3 5 0 0 0 1 • 27. 5 5 0 0 0 1 • 28. 7 5 0 0 0 1 • 29. 7 5 0 0 0 1 • 30. 6 5 0 0 0 1

19. sort x • . by x:summarize y • _______________________________________________________________________________ • -> x = 1 • Variable | Obs Mean Std. Dev. Min Max • -------------+----------------------------------------------------- • y | 6 7.5 1.643168 5 10 • _______________________________________________________________________________ • -> x = 2 • Variable | Obs Mean Std. Dev. Min Max • -------------+----------------------------------------------------- • y | 6 5 1.264911 3 6 • _______________________________________________________________________________ • -> x = 3 • Variable | Obs Mean Std. Dev. Min Max • -------------+----------------------------------------------------- • y | 6 4.333333 1.032796 3 6 • _______________________________________________________________________________ • -> x = 4 • Variable | Obs Mean Std. Dev. Min Max • -------------+----------------------------------------------------- • y | 6 5.166667 1.47196 3 7 • _______________________________________________________________________________ • -> x = 5 • Variable | Obs Mean Std. Dev. Min Max • -------------+----------------------------------------------------- • y | 6 6.166667 2.041241 3 9

20. MeanGroup1=7.5Group2=5 α2=5- 7.5= -2.5Group3=4.3 α3=4.3-7.5= -3.2Group4=5.2 α4=5.2-7.5= -2.3Group5=6.2 α5=6.2-7.5= -1.3 • regress y x2 x3 x4 x5 • Source | SS df MS Number of obs = 30 • -------------+------------------------------ F( 4, 25) = 3.90 • Model | 36.4666667 4 9.11666667 Prob > F = 0.0136 • Residual | 58.50 25 2.34 R-squared = 0.3840 • -------------+------------------------------ Adj R-squared = 0.2854 • Total | 94.9666667 29 3.27471264 Root MSE = 1.5297 • ------------------------------------------------------------------------------ • y | Coef. Std. Err. t P>|t| [95% Conf. Interval] • -------------+---------------------------------------------------------------- • x2 | -2.5 .8831761 -2.83 0.009 -4.318935 -.6810648 • x3 | -3.166667 .8831761 -3.59 0.001 -4.985602 -1.347731 • x4 | -2.333333 .8831761 -2.64 0.014 -4.152269 -.5143981 • x5 | -1.333333 .8831761 -1.51 0.144 -3.152269 .4856019 • _cons | 7.5 .6244998 12.01 0.000 6.213819 8.786181 • ------------------------------------------------------------------------------

21. ANOVA vs. REGRESSION • SSR: Between Sum of Squares (SSB) • SSE: Within Sum of Squares (SSW) • SST: Total Sum of Squares (SST)  SST = SSB + SSW

22. ANOVA vs. REGRESSION • anova y x • Number of obs = 30 R-squared = 0.3840 • Root MSE = 1.52971 Adj R-squared = 0.2854 • Source | Partial SS df MS F Prob > F • -----------+---------------------------------------------------- • Model | 36.4666667 4 9.11666667 3.90 0.0136 • | • x | 36.4666667 4 9.11666667 3.90 0.0136 • | • Residual | 58.50 25 2.34 • -----------+---------------------------------------------------- • Total | 94.9666667 29 3.27471264

23. EFFECT CODING • 3 groups: groups 1, 2, and 3  need 2 dummy variables X2 1 if group 2 0 if group 3 -1 if group 1 X3 1 if group 3 0 if group 2 -1 if group 1

24. EFFECT CODING For Group 1: X2=-1, X3=-1: For Group 2: X2=1, X3=0: For Group 3: X2=0, X3=1:

25. EFFECT CODING

26. EFFECT CODING • Intercept μ is the unweighted average of the K group means. In the example here, K=3. If all groups have equal sample size, this is a grand mean. • α2 indicates difference between mean of group 2 and unweighted average of K group mean. • α3 indicates difference between mean of group 3 and unweighted average of K group mean.

27. list y x x2 x3 x4 x5 • y x x2 x3 x4 x5 • 1. 5 1 -1 -1 -1 -1 • 2. 8 1 -1 -1 -1 -1 • 3. 7 1 -1 -1 -1 -1 • 4. 7 1 -1 -1 -1 -1 • 5. 10 1 -1 -1 -1 -1 • 6. 8 1 -1 -1 -1 -1 • 7. 4 2 1 0 0 0 • 8. 6 2 1 0 0 0 • 9. 6 2 1 0 0 0 • 10. 3 2 1 0 0 0 • 11. 5 2 1 0 0 0 • 12. 6 2 1 0 0 0 • 13. 6 3 0 1 0 0 • 14. 4 3 0 1 0 0 • 15. 4 3 0 1 0 0 • 16. 5 3 0 1 0 0 • 17. 4 3 0 1 0 0 • 18. 3 3 0 1 0 0 • 19. 7 4 0 0 1 0 • 20. 4 4 0 0 1 0 • 21. 6 4 0 0 1 0 • 22. 6 4 0 0 1 0 • 23. 3 4 0 0 1 0 • 24. 5 4 0 0 1 0 • 25. 9 5 0 0 0 1 • 26. 3 5 0 0 0 1 • 27. 5 5 0 0 0 1 • 28. 7 5 0 0 0 1 • 29. 7 5 0 0 0 1 • 30. 6 5 0 0 0 1 • .

28. regress y x2 x3 x4 x5 • Source | SS df MS Number of obs = 30 • -------------+------------------------------ F( 4, 25) = 3.90 • Model | 36.4666667 4 9.11666667 Prob > F = 0.0136 • Residual | 58.50 25 2.34 R-squared = 0.3840 • -------------+------------------------------ Adj R-squared = 0.2854 • Total | 94.9666667 29 3.27471264 Root MSE = 1.5297 • ------------------------------------------------------------------------------ • y | Coef. Std. Err. t P>|t| [95% Conf. Interval] • -------------+---------------------------------------------------------------- • x2 | -.6333333 .5585696 -1.13 0.268 -1.783729 .5170623 • x3 | -1.3 .5585696 -2.33 0.028 -2.450396 -.1496044 • x4 | -.4666667 .5585696 -0.84 0.411 -1.617062 .683729 • x5 | .5333333 .5585696 0.95 0.349 -.6170623 1.683729 • _cons | 5.633333 .2792848 20.17 0.000 5.058136 6.208531 • ------------------------------------------------------------------------------ MeanGroup1=7.5 intercept=(7.5+5+4.3+5.2+6.2)/5=5.6 Group2=5 α2=5- 5.6= -0.6Group3=4.3 α3=4.3- 5.6= -1.3Group4=5.2 α4=5.2- 5.6= -0.4Group5=6.2 α5=6.2- 5.6= 0.6

29. Analysis of Covariance (ANACOVA) Why we need to consider control variables? • Need to produce accurate estimates of coefficients. • interaction • confounding • increase precision

30. Analysis of Covariance (ANACOVA) A Question to be answered by using ANACOVA • If each control variables have the same distributions between group A and B, what would be themean response value for group A and B?

31. Analysis of Covariance(ANACOVA) • Outcome---continuous • Covariates • Nominal (study factors of interests) • Control variables involve any level of measurements

32. Analysis of Covariance(ANACOVA) • Most importantly….. This method is applicable only when there is no interaction effect between the variable of interest with covariates. Y=β0+ β1X+ β2Z+ β3XZ+Ε See first whether H0: β3=0 is supported.

33. Analysis of Covariance(ANACOVA) • Blood pressure data example (X=age, Z=sex)

34. Analysis of Covariance(ANACOVA) Using the adjusted mean scores removes the influence of age on the comparison of mean blood pressures by considering what the mean BP in the two groups would be if both groups had the same mean age.

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