1. Lecture 10 F-tests in MLR (continued) Coefficients of Determination BMTRY 701 Biostatistical Methods II

2. F-tests continued  Two kinds of F-tests  Overall F-test (or Global F-test) tests whether or not there is a regression relation between Y and the set of covariates For a regression with p covariates, the overall F-test compares F* = MSR/MSE ~ F(p, n-p-1)

3. Recall earlier example  “Full” model  The overall F-test tests if there is some association

4. > reg1 <- lm(LOS ~ INFRISK + ms + NURSE + nurse2, data=data) > anova(reg1) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) INFRISK 1 116.446 116.446 45.4043 8.115e-10 *** ms 1 12.897 12.897 5.0288 0.02697 * NURSE 1 1.097 1.097 0.4277 0.51449 nurse2 1 1.789 1.789 0.6976 0.40543 Residuals 108 276.981 2.565 --- SSR <- 116.45 + 12.90 + 1.10 + 1.79 SSE <- 276.98 MSR <- SSR/4 MSE <- SSE/108 Fstar <- MSR/MSE Fstar 1 - pf(Fstar, 4, 108)

5. But, Global F is part of the “summary” output so no need for the additional calculations > summary(reg1) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.355e+00 5.266e-01 12.068 < 2e-16 *** INFRISK 6.289e-01 1.339e-01 4.696 7.86e-06 *** ms 7.829e-01 5.211e-01 1.502 0.136 NURSE 4.136e-03 4.093e-03 1.010 0.315 nurse2 -5.676e-06 6.796e-06 -0.835 0.405 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.601 on 108 degrees of freedom Multiple R-squared: 0.3231, Adjusted R-squared: 0.2981 F-statistic: 12.89 on 4 and 108 DF, p-value: 1.298e-08

6. Partial F test  partial because it tests “part” of the model.  tests one or more covariates simultaneously  Can be done using the ANOVA table, if covariates are entered in the ‘correct’ order  Or, by comparing results from regression tables  Examples:

7. ANOVA tables with 3 covariates SSdfMS X1 SS(X1)1SS(X1)/1 X2|X1 SS(X2|X1)1SS(X2|X1)/1 X3|X2,X1 SS(X3|X2,X1)1SS(X3|X2,X1)/1 Error SSEn – 4SSE/(n-4) Total SSTn - 1

8. ANOVA tables with 3 covariates SSdfMS Regression SS(X1,X2,X3)3SSR/3 X1 SS(X1)1SS(X1)/1 X2|X1 SS(X2|X1)1SS(X2|X1)/1 X3|X2,X1 SS(X3|X2,X1)1SS(X3|X2,X1)/1 Error SSEn – 4SSE/(n-4) Total SSTn - 1 where SS(X1,X2,X3) = SS(X1) + SS(X2|X1) + SS(X3|X2,X1)

9. Interpretation of ANOVA table with >1 covariate > anova(reg1) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) INFRISK 1 116.446 116.446 45.4043 8.115e-10 *** ms 1 12.897 12.897 5.0288 0.02697 * NURSE 1 1.097 1.097 0.4277 0.51449 nurse2 1 1.789 1.789 0.6976 0.40543 Residuals 108 276.981 2.565 SSR(INFRISK) = 116.446 SSR(ms | INFRISK) = 12.897 SSR(NURSE| ms, INFRISK) = 1.097 SSR(nurse2| nurse, ms, INFRISK) = 1.789 What are these F-tests and pvalues testing?

10. F-tests and p-values in ANOVA table  They are tests for a covariate, conditional on what is above it in the table.  Example: F statistic for INFRISK tests is it adjusted for other covariates?  no  it tests INFRISK in the presence of no other covariates  p < 0.0001

11. F-tests and p-values in ANOVA table  Example: F statistic for ‘ms’ tests is it adjusted for other covariates?  yes  it tests the significance of ms, after adjusting for INFRISK  p = 0.03  Example: F-statistic for nurse2 tests significance of β4, adjusting for INFRISK, ms, NURSE. p = 0.41

12. Interpretation of ANOVA table with >1 covariate > reg1a <- lm(LOS ~ ms + NURSE + nurse2 + INFRISK, data=data) > anova(reg1a) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) ms 1 36.084 36.084 14.0699 0.0002852 *** NURSE 1 17.178 17.178 6.6980 0.0109794 * nurse2 1 22.421 22.421 8.7425 0.0038187 ** INFRISK 1 56.546 56.546 22.0481 7.857e-06 *** Residuals 108 276.981 2.565 --- SSR(ms) = 36.084 SSR(NURSE| ms) = 17.178 SSR(nurse2| ms, NURSE) = 22.421 SSR(INFRISK| ms, NURSE, nurse2 ) = 56.546

13. Implications  ANOVA table results depends on the order in which the covariates appear  If you want to use ANOVA table to test one or more covariates, they should come at the end  reg1: we can see if INFRISK is significant without any adjustments we can see if nurse2 is significant adjusting for everything else  reg1a: we can see if INFRISK is significant adjusting for everything else we can see if nurse2 is significant, adjusting for NURSE and ms, but not adjusting for INFRISK

14. F-tests  Global F-test  Partial F-test for ONE covariate

15. F-tests (continued)  Partial F-test for >1 covariate  Implications: The denominator is always the MSE from the full model The numerator can always be determined by entering the covariates in the order in which you want to test them Recall: additivity of sums of squares

16. More on the partial F test  Test whether an individual β k = 0  Test whether a set of β k = 0  Model 1:  Model 2:  Model 3:

17. Testing more than two covariates  To test Model 1 vs. Model 3 we are testing that β 3 = 0 AND β 4 = 0 Ho: β 3 = β 4 = 0 vs. Ha: β 3 ≠ 0 or β 4 ≠ 0 If β 3 = β 4 = 0, then we conclude that Model 3 is superior to Model 1 That is, if we fail to reject the null hypothesis Model 1: Model 3:

18. Interpretation of ANOVA table with >1 covariate > anova(reg1) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) INFRISK 1 116.446 116.446 45.4043 8.115e-10 *** ms 1 12.897 12.897 5.0288 0.02697 * NURSE 1 1.097 1.097 0.4277 0.51449 nurse2 1 1.789 1.789 0.6976 0.40543 Residuals 108 276.981 2.565 SSR(INFRISK) = 116.446 SSR(ms | INFRISK) = 12.897 SSR(NURSE| ms, INFRISK) = 1.097 SSR(nurse2| nurse, ms, INFRISK) = 1.789

19. Using ANOVA table results  SSR(NURSE, nurse2| INFRISK, ms) = SSR(NURSE| ms, INFRISK) + SSR(nurse2| nurse, ms, INFRISK) = 1.097+ 1.789 = 2.886  MSR = 2.886/2 = 1.443  F* = 1.443/2.565 = 0.5626 ~ F(2,108)  p-value = 0.57

20. R: simpler approach > anova(reg3) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) INFRISK 1 116.446 116.446 45.7683 6.724e-10 *** ms 1 12.897 12.897 5.0691 0.02634 * Residuals 110 279.867 2.544 --- > anova(reg1, reg3) Analysis of Variance Table Model 1: LOS ~ INFRISK + ms + NURSE + nurse2 Model 2: LOS ~ INFRISK + ms Res.Df RSS Df Sum of Sq F Pr(>F) 1 108 276.981 2 110 279.867 -2 -2.886 0.5627 0.5713

21. R > summary(reg3) Call: lm(formula = LOS ~ INFRISK + ms, data = data) Residuals: Min 1Q Median 3Q Max -2.9037 -0.8739 -0.1142 0.5965 8.5568 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.4547 0.5146 12.542 <2e-16 *** INFRISK 0.6998 0.1156 6.054 2e-08 *** ms 0.9717 0.4316 2.251 0.0263 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.595 on 110 degrees of freedom Multiple R-squared: 0.3161, Adjusted R-squared: 0.3036 F-statistic: 25.42 on 2 and 110 DF, p-value: 8.42e-10

22. Testing multiple coefficients simultaneously  Region: it is a ‘factor’ variable with 4 categories > reg4 <- lm(LOS ~ factor(REGION), data=data) > anova(reg4) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) factor(REGION) 3 103.554 34.518 12.309 5.376e-07 *** Residuals 109 305.656 2.804 ---

23. Continued… > summary(reg4) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 11.0889 0.3165 35.040 < 2e-16 *** factor(REGION)2 -1.4055 0.4333 -3.243 0.00157 ** factor(REGION)3 -1.8976 0.4194 -4.524 1.55e-05 *** factor(REGION)4 -2.9752 0.5248 -5.669 1.19e-07 *** Residual standard error: 1.675 on 109 degrees of freedom Multiple R-squared: 0.2531, Adjusted R-squared: 0.2325 F-statistic: 12.31 on 3 and 109 DF, p-value: 5.376e-07

24. Recall previous example  Interaction between REGION and MEDSCHL

25. How to test the interaction terms?  Approach 1: Fit two models  model with interactions  model without interactions Compare models using ‘anova’ command  Approach 2: fit one model find SSR for interactions, conditional on main effects calculate F-statistic calculate p-value

26. Approach 1 > reg5 <- lm(logLOS ~ factor(REGION)*ms, data=data) > reg6 <- lm(logLOS ~ factor(REGION)+ ms, data=data) > anova (reg6, reg5) Analysis of Variance Table Model 1: logLOS ~ factor(REGION) + ms Model 2: logLOS ~ factor(REGION) * ms Res.Df RSS Df Sum of Sq F Pr(>F) 1 108 2.29085 2 105 2.27831 3 0.01254 0.1926 0.9013 >

27. Approach 2 > anova(reg5) Analysis of Variance Table Response: logLOS Df Sum Sq Mean Sq F value Pr(>F) factor(REGION) 3 0.98268 0.32756 15.0961 3.077e-08 *** ms 1 0.27393 0.27393 12.6245 0.0005719 *** factor(REGION):ms 3 0.01254 0.00418 0.1926 0.9012545 Residuals 105 2.27831 0.02170 - What are degrees of freedom for the F-test?

28. Concluding remarks r.e. F-test  Global F-test: not very common, except for very small models  Partial F-test for individual covariate: not very common because it is the same as the t-test  Partial F-test for set of covariates: quite common easiest to find ANOVA table for nested models can use ANOVA table from full model to determine F- statistic

29. Coefficient of Determination  Also called R 2  Measures the variability in Y explained by the covariates.  Two questions (and think ‘sums of squares’ in ANOVA): How do we measure the variance in Y? How do we measure the variance explained by the X’s?

30. R2R2  The coefficient of determination is defined as SST: Variance in Y SSR: Variance explained by X’s SSE:Variance left over, not explained by regression

31. Use of R 2  Similar to correlation  But, not specific to just one X and Y  Partitioning of explained versus unexplained  For certain models, it can be used to determine if addition of a covariate helps ‘predict’

32. SENIC example > summary(reg1) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.355e+00 5.266e-01 12.068 < 2e-16 *** INFRISK 6.289e-01 1.339e-01 4.696 7.86e-06 *** ms 7.829e-01 5.211e-01 1.502 0.136 NURSE 4.136e-03 4.093e-03 1.010 0.315 nurse2 -5.676e-06 6.796e-06 -0.835 0.405 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.601 on 108 degrees of freedom Multiple R-squared: 0.3231, Adjusted R-squared: 0.2981 F-statistic: 12.89 on 4 and 108 DF, p-value: 1.298e-08 32% of the variance in LOS is explained by the regression model

33. Misunderstandings r.e. R 2  A high R 2 indicates that a useful prediction can be made there still may be considerable uncertainty, due to small N. recall that predictions depend on how close “X” is to the mean  A high R 2 indicates that the regression model is a ‘good fit’ high R 2 says nothing about adhering to model assumptions standard diagnostics should still be used, even if R 2 is high  R 2 near 0 indicates X and Y are not related. you can still have strong association with a lot of unexplained variance (e.g., age and cancer) for similar reasons as above, need to look at modeling X and Y may be related, but not linearly

34. What if we remove the ‘insignificant’ X’s? > reg7 <- lm(LOS ~ INFRISK, data=data) > summary(reg7) Call: lm(formula = LOS ~ INFRISK, data = data) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.3368 0.5213 12.156 < 2e-16 *** INFRISK 0.7604 0.1144 6.645 1.18e-09 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.624 on 111 degrees of freedom Multiple R-Squared: 0.2846, Adjusted R-squared: 0.2781 F-statistic: 44.15 on 1 and 111 DF, p-value: 1.177e-09

35. R 2 decreased?  The addition of a covariate will ALWAYS increase the R 2 value.  Why? there is always at least a little bit explained by the new X the only possible way to have no increase in R2 would be if the addition of the new covariate had estimated β = 0 It is ‘almost never’ true that the slope estimate is exactly.  Extreme case: perfect linear association between two covariates (e.g., age in years and age in months)

36. “Solution”  Adjusted R  Accounts for the number of covariates in the model  “Purists” do not like the adjusted R 2  The adjusted only increases with a new covariate if the new term “improves” the model more than expected by chance alone.

37. Coefficients of Partial Determination  Measures the marginal contribution of one X variable when all others are already in the model  Intuitively, how much variation in Y are we explaining, after accounting for what is already in the model?  Construction in Two Covariate case:

38. Example: X1 = ms, X2 = INFRISK > anova(reg3) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) INFRISK 1 116.446 116.446 45.7683 6.724e-10 *** ms 1 12.897 12.897 5.0691 0.02634 * Residuals 110 279.867 2.544 --- > anova(reg7) Analysis of Variance Table Response: LOS Df Sum Sq Mean Sq F value Pr(>F) INFRISK 1 116.446 116.446 44.15 1.177e-09 *** Residuals 111 292.765 2.638 ---

39. Example: X1 = ms, X2 = INFRISK SSR(X1|X2) = SSR(ms|INFRISK) = SSE(X2) = SSE(INFRISK) = R 2 (Y 1|2) =

40. General Case  Examples with 3 and 4 covariates  Can also be generalized for a set of covariates