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Lecture 10 F-tests in MLR (continued) Coefficients of Determination

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

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Lecture 10 F-tests in MLR (continued) Coefficients of Determination

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

  8. ANOVA tables with 3 covariates 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 βk = 0 • Test whether a set of βk = 0 • Model 1: • Model 2: • Model 3:

  17. Testing more than two covariates • To test Model 1 vs. Model 3 • we are testing that β3 = 0 AND β4 = 0 • Ho: β3 = β4 = 0 vs. Ha: β3 ≠ 0 or β4 ≠ 0 • If β3 = β4 = 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 R2 • 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. R2 • The coefficient of determination is defined as SSR: Variance explained by X’s SSE:Variance left over, not explained by regression SST: Variance in Y

  31. Use of R2 • 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. R2 • A high R2 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 R2 indicates that the regression model is a ‘good fit’ • high R2 says nothing about adhering to model assumptions • standard diagnostics should still be used, even if R2 is high • R2 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. R2 decreased? • The addition of a covariate will ALWAYS increase the R2 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 R2 • 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) = R2(Y 1|2) =

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

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