1 / 40

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

muniya
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related