Biostatistics 360 F&t Tests and Intervals in Regression 1
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1 Bostatstcs 360 F&t Tests and Intervals n Regresson ORIGIN Model: Y = X + Corrected Sums of Squares: X X bar where: s the y ntercept of the regresson lne (translaton) s the slope of the regresson lne (scalng coeffcent) s the error factor n predcton of Y gven that t s a random varable wth N(0, ) L yy Y Y bar L xy 6 L yy L xy 70.8 Estmated Regresson Coeffcents for Y = + X: L xy b b 0.70 a Y bar b X bar a 0.73 Estmated values of Y (Y hat ): Y hat a b X Resduals: e Y hat Y ANOVA F-test/t-test for Smple Lnear Regresson and Interval Estmaton Goodness of ft of a ftted regresson lne can be tested usng the F-test for Regresson (also known as the ANOVA for Regresson). Alternatvely, and equvalently, ft of the regresson can be tested usng a t-test approach. The latter method also allows estmaton of confdence ntervals for the slope parameter. Interval estmates can also be derved for the Regresson lne tself as a mean, as well as for predcton of "new" observatons. ZAR READPRN ("c:/data/bostatstcs/zarex7.r.txt") X ZAR ^Zar Example 7. Y ZAR 3 X bar Y bar s n mean( X) X bar 0 mean( Y) Y bar Var( Y) s.799 s.638 length( Y) n 3 n Assumptons: X Y < sample estmate of < sample estmate of ZAR - Standard Lnear Regresson depends on specfyng n advance whch varable s to be consdered 'dependent' and whch 'ndependent'. Ths decson matters as changng roles for Y & X usually produces a dfferent result. - Y, Y, Y 3,..., Y n (dependent varable) s a random sample ~ N(, ). - X, X, X 3,..., X n (ndependent varable) wth each value of X matched to Y X X bar Y Y bar
2 Bostatstcs 360 F&t Tests and Intervals n Regresson Sum of Squares Decomposton of the Regresson: Once a regresson model (Y = X + ) s ftted wth data as Y hat = a + bx, one stll needs to determne how useful the regresson mght be, especally whether knowledge about the X provde nsght nto nterpretng the Y as a random varable from a Normal dstrbuton wth error. Ths s done by consderng a "partton" of total Sums of Squares n the sample of Y. SS T Y Y bar SS T L yy SS R Y hat Y bar SS R 9.3 < Total Sum of Squares < Regresson Sum of Squares SS E Y Y hat SS E These Sums of Squares tally as follows: SS T SS R SS E < Resdual (also called "Error") Sum of Squares And the rato of SS R to SS E can be used as a measure of "ft" of the data to the regresson. ANOVA Table for Lnear Regresson: SS df MS SS R Regresson: SS R 9.3 df R MS R MS R 9.3 df R Resdual: SS E SS E df E n MS E MS E df E SS T TOTAL: SS T df T n MS T MS T.638 df T ANOVA F-test for Regresson Slope: Hypotheses: H 0 : = 0 : 0 Test Statstc: < Slope of the Regresson s zero mplyng no relatonshp between X and Y < Two sded test MS R F F MS E < F s the rato of sample varances Samplng Dstrbuton: If Assumptons hold and H 0 s true, then F s ~F (,n-) Crtcal Value of the Test: 0.05 < Probablty of Type I error must be explctly set C qf n C
3 Bostatstcs 360 F&t Tests and Intervals n Regresson 3 Decson Rule: IF F > C, THEN REJECT H 0, OTHERWISE ACCEPT H 0 Probablty Value: P pf( F n ) P Prototype n R: #SIMPLE LINEAR REGRESSION #ZAR EXAMPLE 7. ZAR=read.table("c:/DATA/Bostascs/ZarEX7.R.txt") ZAR aach(zar) X=ageX Y=wnglY #CREATING LINEAR MODEL LM: LM=lm(Y~X) Yhat=fed(LM) e=resduals(lm) RESULTS=data.frame(X,Y,Yhat,e) RESULTS Analyss of Varance Table Response: Y Df Sum Sq Mean Sq F value Pr(>F) X e-0 *** Resduals Sgnf. codes: 0 *** 0.00 ** 0.0 * #ANOVA F TEST: anova(lm) The t-test Approach and Interval Estmaton for Smple Lnear Regresson: The t-test approach s exactly equvalent to the F-test above wth the statstc F = statstc t. In contrast to the F-test, however, the t-test allows for one-taled tests and for constructon of confdence ntervals of the regresson (to capture ), confdence ntervals for the regresson estmates (to capture Y hat ) and predcton ntervals for new observatons (to capture new observatons X n ). t-test for Regresson Parameter (slope): Hypotheses: H 0 : = 0 : 0 : < 0 : > 0 < Slope of the Regresson s zero mplyng no relatonshp between X and Y < Two Sded Case < One Sded Case Lower Tal) < One Sded Case (Upper Tal) Test Statstc: b t MS E t 0.07 < b s unbased pont estmate of < MSE s Mean Square Error from ANOVA table also denoted s x.y < Corrected sums of squares of X as defned n Regresson
4 Bostatstcs 360 F&t Tests and Intervals n Regresson 4 Samplng Dstrbuton of Test Statstc t : If Assumptons hold and H 0 s true, then t ~t (n-) Crtcal Values of the Test: 0.05 < Probablty of Type I error must be explctly set C qt n C.0 C qt n C.0 C C C.0 C 3 qt n C C 4 qt n C < Two sded lower Crtcal Value < Two sded upper Crtcal Value < Crtcal value used for two sded test (to smplfy) < One sded lower Crtcal Value < One sded upper Crtcal Value Decson Rules: IF t > C THEN REJECT H 0, OTHERWISE ACCEPT H 0 IF t < C 3, THEN REJECT H 0, OTHERWISE ACCEPT H 0 IF t > C 4, THEN REJECT H 0, OTHERWISE ACCEPT H 0 < Two sded case Probablty Values: P ptt n P < f t 0 P pt t n P < f t > 0 < Two sded case P pt t n P P pt t n P Confdence Interval for the Regresson Slope (): MS E MS E CI b C b C CI ( ) < Two sded case b 0.70 MS E CIL b C 3 mnus nfnty to CIL MS E CIU b C 4 CIU to nfnty
5 Bostatstcs 360 F&t Tests and Intervals n Regresson 5 t-test for Regresson Parameter (ntercept): Hypotheses: 0 0 H 0 : = 0 : 0 : 0 < 0 : 0 > 0 Test Statstc: < any value may be tested < tests for ntercept = 0 =0 here, but may be set to other values as desred < Two Sded Case < One Sded Case Lower Tal) < One Sded Case (Upper Tal) a 0 t t 4.83 X bar MS E n Samplng Dstrbuton of Test Statstc t : If Assumptons hold and H 0 s true, then t ~t (n-) Crtcal Values of the Test: 0.05 < Probablty of Type I error must be explctly set C qt n C.0 C qt n C.0 < Note degrees of freedom = (n-) C C C.0 Decson Rules: IF t > C THEN REJECT H 0, OTHERWISE ACCEPT H 0 IF t < C 3, THEN REJECT H 0, OTHERWISE ACCEPT H 0 IF t > C 4, THEN REJECT H 0, OTHERWISE ACCEPT H 0 < Two sded case Probablty Values: P pt t n P.9995 < f t 0 P pt t n P < f t > 0 < Two sded case P pt t n P P pt t n P
6 Bostatstcs 360 F&t Tests and Intervals n Regresson 6 Confdence Interval for Intercept (): X bar X bar CI a C MS E a C MS E n n CI ( ) a 0.73 < Two sded case X bar CIL a C 3 MS E n mnus nfnty to CIL X bar CIU a C 4 MS E n CIU to nfnty Prototype n R: #t TESTS FOR SLOPE & INTERCEPT: summary(lm) F t 0.07 t t 4.83 ^ Note for test of, (slope): Statstcs F = t #CONFIDENCE INTERVAL FOR SLOPE: #CALCULATED BY HAND! n=length(y) Lxx=sum((X mean(x))^) Lxx Lxy=sum((X mean(x))*(y mean(y))) Lxy b=lxy/lxx b MSE=summary(LM)$sgma^ MSE alpha=0.05 C=abs(qt( alpha/,n )) C stderr=sqrt(mse/lxx) CIL=b C*stderr CIL CIU=b+C*stderr CIU CONFIDENCE INTERVALS FOR #REGRESSION PARAMETERS #THE EASIER WAY: confnt(lm(y~x),level=0.95) > summary(lm) Call: lm(formula = Y ~ X) Resduals: Mn Q Medan 3Q Max Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) *** X e-0 *** --- Sgnf. codes: 0 *** 0.00 ** 0.0 * Resdual standard error: 0.84 on degrees of freedom Multple R-squared: , Adjusted R-squared: F-statstc: 40. on and DF, p-value: 5.67e-0 > Lxx [] 6 > Lxy [] 70.8 > MSE [] > CIL [] > CIU [] > confnt(lm(y~x),level=0.95).5 % 97.5 % (Intercept) X
7 Bostatstcs 360 F&t Tests and Intervals n Regresson 7 Confdence Interval for Regresson Estmates Y hat and New Predctons of Y: One or more values of X n must be explctly specfed to obtan a predcton CI for Y hat : n Xn X < here usng all orgnal values of X, but any X values may be specfed nstead... Confdence Interval for Regresson (CI): Xn X bar CI Y hat C MS E Y n hat C MS E n Xn X bar < Two sded case CIU Y hat C 4 MS E n Y hat n CIL Y hat C 3 MS E Xn X bar Xn X bar CI mnus nfnty to CIL CIU to nfnty Prototype n R: #CONFIDENCE INTERVALS FOR Yhat: CONF=predct(lm(Y~X),nterval="confdence",level=0.95) CN=data.frame(CONF) CN > CN ft lwr upr
8 Bostatstcs 360 F&t Tests and Intervals n Regresson 8 Predcton Interval for Regresson (PI): Xn X bar PI Y hat C MS E Y n hat C MS E n Xn X bar < Two sded case PIL Y hat C 3 MS E n PIU Y hat C 4 MS E n Xn X bar Xn X bar mnus nfnty to Y hat Prototype n R: PI #PREDICTION INTERVALS FOR Xn: PRED=predct(lm(Y~X),nterval="predcon",level=0.95) PR=data.frame(PRED) PR PIL PIU to nfnty > PR ft lwr upr > PRED=predct(lm(Y~X),nterval="predcon",level=0.95) Warnng message: In predct.lm(lm(y ~ X), nterval = "predcton", level = 0.95) : Predctons on current data refer to _future_ responses
9 Bostatstcs 360 F&t Tests and Intervals n Regresson 9 #PLOTTING INTERVALS IN R: plot(x,y) ablne(lm(y~x),col="blue") segments(x,pr$lwr,x,pr$upr,col="red") segments(x,cn$lwr,x,cn$upr,col="green") ponts(x,cn$lwr,col="green") ponts(x,cn$upr,col="green") ponts(x,pr$lwr,col="red") ponts(x,pr$upr,col="red") ponts(x,pr$ft,col="blue") Y X
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