Homework 10 Stat 547. Problem ) Z D!
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1 Homework 0 Stat 547 Problem 74 Notaton: h s the hazard rate for the aneulod grou, h s the hazard rate for the dlod grou (a Log-rank test s erformed: H 0 : h (t = h (t Sgnfcance level α = 005 Test statstc s calculated as H : h (t h (t Z =,- * (/ = ~ N(0, ' $ ' ( $ d % ( "% " d = ( & t d q d q d d-(d/ (/(-/((-d/(-d Total , d ( #& d + #
2 56506 The above calculatons lead to the followng value for the test statstc: Z (" = = The absolute value of the test statstc s less than the crtcal value Z 0975 = 96 Thus, the null hyothess s not reected at 005 sgnfcance level Therefore, the two hazard rates are not sgnfcantly dfferent (b For detectng early dfferences the followng secal weght functon s used: W ˆ( ( t = S t The only requrement for q s that t has to be ostve Assume q = Then the test statstc becomes: Z = = = W ( t - W ( t + d, ' d & #& d # ' d $ ' $ % "% ' " * ( = Sˆ( t = ' ˆ( - S t + d ' = ' 4, d & #& d # ' d $ ' $ % "% ' " * ( The followng table facltates the comutatons t d q d q d S to bottom
3 The test statstcs s then comuted as: Z = = 8543 The absolute value of the test statstc s less than the crtcal value Z 0975 = Thus, the null hyothess s not reected at 005 sgnfcance level and, therefore, the early dfferences n hazard rates are not sgnfcant Problem 77 (a Survval functons for the three grous Untreated grou t d q S Radated grou t d q S Radated + BPA grou t d q S The grahs of all three survval functons are dslayed below: (b Parwse tests H 0 : h (t = h (t Sgnfcance level α = 005 H : h (t h (t Log-rank test s used wth the test statstc gven by
4 d +, d ( =,- Z * (/ = ~ N(0, ' $ ' ( $ d % ( "% " d = & #& ( # Untreated vs Radated grous t d q d q d d-(d/ (/(-/((-d/(-d Test statstc has value Z ( = = whch s greater than Z 0975 = 96 Thus, the hull 4797 hyothess s reected under 005 sgnfcance level, and one concludes that the hazard rates of these two grous are sgnfcantly dfferent Untreated vs Radated + BPA t d q d q d d-(d/ (/(-/((-d/(-d
5 6303 Test statstc s Z ( = = whch exceeds Z 0975 = 96 Therefore, the null 8353 hyothess s reected under 005 sgnfcance level and, thus, untreated grou death rate s sgnfcantly dfferent from the one of the radated+bpa grou 3 Radated vs Radated + BPA t d q d q d d-(d/ (/(-/((-d/(-d Test statstc s Z ( = = , ts value exceeds Z 0975 = 96 and, therefore, the null hyothess s reected under 005 sgnfcance level Therefore, the death rates for these two grous are sgnfcantly dfferent The overall concluson s that all hazard rates are arwse sgnfcantly dfferent (c Test for trends Notaton: h s hazard rate for the radated + BPA grou, h s hazard rate for the radated grou, h 3 s hazard rate for the untreated grou Therefore, the hyothess testng s set as follows: H 0 : h (t = h (t = h 3 (t Sgnfcance level α = 005 H : h (t < h (t < h 3 (t Test statstc s calculated as Z = 3 = 3 a Z (# 3 = g= g g a a " ˆ ' d $, where a =, a =, a 3 = 3 and Z ( = % d ( " = & # The tables below shows artal comutatons necessary for calculaton of Z
6 t d q d q d 3 q 3 3 d t Z Z Z 3 σ σ σ 3 σ σ 3 σ The test statstc s calculated to be Z = Ths s greater than Z 095 = 645 and, thus, the null hyothess s reected Therefore, the hazard rates are n the suggested order, e the untreated grou has the hghest death rate, the radated grou has slghtly lower rate, and the lowest death rate s ossessed by the radated+bpa grou Ths suorts the dea that the radated treatment esecally when accomaned by BPA does mrove the lfetme n rats Problem 70 ata from roblem 74 s used to erform: H 0 : h (t = h (t Sgnfcance level α = 005 H : h (t h (t (a Test usng the the Cramer-von Mses statstc Test statstc (frst verson of Cramer-von Mses statstc s comuted as follows:
7 ~ ~ [ H( t # H ( t ][ + ( t # + ( t# ] & Q = ' $ ( + (* % t "* ~ d Where H ( t = t "# s the Nelson-Aalen estmator of the cumulatve hazard functon and % ( t = d t " $ # ( ~ ~, =, Also, ( t = ( t + ( t s the estmated varance of H ( t H ( t All necessary comutatons are shown n the table below: t d q d q d H H σ σ σ (H -H *Δσ Total The test statstc s then & # Q = $ = The -value for the test statstc s 0774 (usng % 05" Table C6 from the book Thus, the null hyothess s not reected Ths certfes the fact that the two hazard rates are not sgnfcantly dfferent
8 (b Test wth the weghted dfference n the Kalan-Meer statstc W KM t d q d q d S G S G w(t W KM S A σ Total
9 The followng comutatons have to be erformed n order to calculate the test statstc: ngˆ ˆ ( t G ( t w( t =, for 0 t t n Gˆ ( t + n Gˆ, ( t ( c where " # # % G ˆ ( t = & t t &' s the Kalan-Meer estmator of the tme to censorng $ n n The test statstc s then comuted as: W = [ + " ] ( [ ˆ ( " ˆ KM t t w t S t S ( t ] The varance of W KM s n " = [ Sˆ ˆ ( t S ( t ] ˆ ˆ A ng ( t + ng ( t # ˆ = ", ˆ ( ˆ ( ˆ ( ˆ S t S t ng t G ( t = where S (t s the Kalan-Meer estmator based on the combned samle and A s comuted as: " = k= A ( t ˆ " k+ tk w( tk S ( tk The fnal test statstc used for the analyss s Z W KM = ˆ Accordng to the above table, the followng quanttes are obtaned: W KM = 8579 and σ = Thus, the test statstc s Z = = 7378 Ths s less than the crtcal value Z 0975 = 96 and, 4370 therefore, the null hyothess s not reected The two survval functons (and, thus, hazard rates are not sgnfcantly dfferent
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