# It is computed as follows l n summationdisplay i 1 ?

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Chapter 5 / Exercise 4
Precalculus: Mathematics for Calculus
Redlin/Stewart Expert Verified
It is computed as follows.l=nsummationdisplayi=1[-λi+yiln(λi)-ln (yi!)]lS=nsummationdisplayi=1[-yi+yiln (yi)-ln (yi!)]s.t. 0 ln(0) = 0The Likelihood Ratio test statistic for testingH0:β1=· · ·=βp= 0 is the difference between the Nulland Residual Deviances, withpdegrees of freedom.Wald tests of the formH0:βj=βj0are of the forms (chi-square and Z) given below.X2W=parenleftBigˆβj-βj0parenrightBig2ˆVbraceleftBigˆβjbracerightBigRR:X2Wχ2α,1P=P(χ21X2W)zW=ˆβj-βj0ˆSEbraceleftBigˆβjbracerightBigRR:|zW| ≥zα/2P= 2P(Z≥ |zW|)For general linear tests among the regression coefficients,H0:Kprimeβ-m=0, the Wald Test is conductedas a chi-square test, withkrestrictions (number of rows inKprime).X2W=parenleftBigKprimeˆβ-mparenrightBigprimebracketleftBigKprimeˆV{ˆβ}KbracketrightBig-1parenleftBigKprimeˆβ-mparenrightBigRR:X2Wχ2α,kP=P(χ2kX2W)Confidence intervals for the regression coefficients and their corresponding risk ratios (RR) are obtainedby using the approximate normality of the estimated regression coefficients, and then exponentiating theend points for the odds ratios.(1-α)100% CI forβj:ˆβj±zα/2ˆSE{ˆβj}= (βjLjH)(1-α)100% CI forRRj:(eβjL,eβjH)In general, for more complex Risk Ratios of the following form, we can set up Tests and ConfidenceIntervals based on the following result.ˆRR= exp{(xi-xj)primeˆβ}ln{ˆRR}= (xi-xj)primeˆβˆVbraceleftBigln{ˆRR}bracerightBig= (xi-xj)primeˆV{ˆβ}(xi-xj)}= (xi-xj)primeparenleftBigXprimeˆWXparenrightBig-1(xi-xj)Tests and Confidence Intervals regarding ln{ˆRR}can be made, and Confidence Intervals for the RiskRatio can be obtained by exponentiating the endpoints.
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Chapter 5 / Exercise 4
Precalculus: Mathematics for Calculus
Redlin/Stewart Expert Verified
256CHAPTER 9.ALTERNATIVE REGRESSION MODELS9.3.2Goodness of Fit TestsWhen there is a fixed number ofndistinctxlevels, the Pearson and Likelihood-Ratio (Deviance) Goodness-of-Fit statistics given below have approximate chi-square distributions withn-pprimedegrees of freedom (seee.g. Agresti (1996 pp.89-90)).Pearson:X2P=nsummationdisplayi=1parenleftBigyi-ˆλiparenrightBig2ˆλi=nsummationdisplayi=1e2iPeiP=yi-ˆλiradicalbigˆλiDeviance:X2D=G2= 2nsummationdisplayi=1bracketleftbiggyilnyiˆλi-parenleftBigyi-ˆλiparenrightBigbracketrightbigg=nsummationdisplayi=1e2iDeiD= sgn{yi-ˆλi}bracketleftbigg2yilnyiˆλi-parenleftBigyi-ˆλiparenrightBigbracketrightbigg1/2When the independent variable(s) have many distinct level(s) or combinations, observations can begrouped based on theirXlevels in cases where there isp= 1 independent variable, or grouped based ontheir predicted means in general. The sums of events (y) and their corresponding predicted valuesparenleftBigˆλiparenrightBigareobtained for each group. Pearson residuals are obtained for each group based on the sums of their observedand predicted values. If we haveg
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