Lecture 10 F-tests in MLR (continued) Coefficients of DeterminationF-tests continuedRecall earlier exampleSlide 4But, Global F is part of the “summary” output so no need for the additional calculationsPartial F testANOVA tables with 3 covariatesSlide 8Interpretation of ANOVA table with >1 covariateF-tests and p-values in ANOVA tableSlide 11Slide 12ImplicationsF-testsF-tests (continued)More on the partial F testTesting more than two covariatesSlide 18Using ANOVA table resultsR: simpler approachRTesting multiple coefficients simultaneouslyContinued…Recall previous exampleHow to test the interaction terms?Approach 1Approach 2Concluding remarks r.e. F-testCoefficient of DeterminationR2Use of R2SENIC exampleMisunderstandings r.e. R2What if we remove the ‘insignificant’ X’s?R2 decreased?“Solution”Coefficients of Partial DeterminationExample: X1 = ms, X2 = INFRISKSlide 39General CaseLecture 10F-tests in MLR (continued)Coefficients of DeterminationBMTRY 701Biostatistical Methods IIF-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)0 oneleast at :0:k1210HHpRecall earlier example“Full” modelThe overall F-test tests if there is some associationiieNURSENURS EMSINFRISKLOS 2432100 oneleast at :0:k143210HH> reg1 <- lm(LOS ~ INFRISK + ms + NURSE + nurse2, data=data)> anova(reg1)Analysis of Variance TableResponse: 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 + 1.79SSE <- 276.98MSR <- SSR/4MSE <- SSE/108Fstar <- MSR/MSEFstar1 - pf(Fstar, 4, 108)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 freedomMultiple R-squared: 0.3231, Adjusted R-squared: 0.2981 F-statistic: 12.89 on 4 and 108 DF, p-value: 1.298e-08Partial 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:0or 0 :0:431430HH0:0:4140HHANOVA tables with 3 covariatesSS df MSX1SS(X1) 1 SS(X1)/1X2|X1SS(X2|X1) 1 SS(X2|X1)/1X3|X2,X1SS(X3|X2,X1) 1 SS(X3|X2,X1)/1ErrorSSE n – 4 SSE/(n-4)TotalSST n - 1ANOVA tables with 3 covariatesSS df MSRegressionSS(X1,X2,X3) 3 SSR/3X1SS(X1) 1 SS(X1)/1X2|X1SS(X2|X1) 1 SS(X2|X1)/1X3|X2,X1SS(X3|X2,X1) 1 SS(X3|X2,X1)/1ErrorSSE n – 4 SSE/(n-4)TotalSST n - 1where SS(X1,X2,X3) = SS(X1) + SS(X2|X1) + SS(X3|X2,X1)Interpretation of ANOVA table with >1 covariate> anova(reg1)Analysis of Variance TableResponse: 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 2.565SSR(INFRISK) = 116.446SSR(ms | INFRISK) = 12.897SSR(NURSE| ms, INFRISK) = 1.097SSR(nurse2| nurse, ms, INFRISK) = 1.789What are these F-tests and pvalues testing?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 < 0.00010:0:1110HHF-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 = 0.410:0:2120HHInterpretation of ANOVA table with >1 covariate> reg1a <- lm(LOS ~ ms + NURSE + nurse2 + INFRISK , data=data)> anova(reg1a)Analysis of Variance TableResponse: 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) = 36.084SSR(NURSE| ms) = 17.178SSR(nurse2| ms, NURSE) = 22.421SSR(INFRISK| ms, NURSE, nurse2 ) = 56.546Implications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 INFRISKF-testsGlobal F-testPartial F-test for ONE covariateMSEMSRpnXXSSEpXXSSRFpp),,(),,(*11MSEXXXMSRpnXXSSEXXXSSRFppppp),,|(),,(1),,|(*11111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 squaresMSEXXXXMSRpnXXSSEqpXXXXSSRFqpqpqpq),,|,(),,(),,|,(*11111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:iieNURSENURS EMSINFRISKLOS