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HW-04

# HW-04 - HOMEWORK FOR EXST 7036(Chapter 4 Student Name Tanza...

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HOMEWORK FOR EXST 7036 (Chapter 4) Student Name: Tanza Erlambang 4.1)A study used logistic regression to determine characteristics associated with Y = whether a cancer patient achieved remission (1= yes). The most important explanatory variable was a labeling index (LI) that measures proliferative activity of cells after a patient receives an injection of triated thymidine. It represents the percentage of cells that are “labeled”. Table 4.8 shows the grouped data. Sofware reports Table 4.9 for a logistic regression model using LI to predict π = P(Y=1) a. Show how software obtained π ^ = 0.068 when LI = 8 Answer: π ^ =e α+β(LI) /[1+e α+β(LI) ]=e -3.777+0.145(8) / [1+e -3.777+0.145(8) ]= 0.073/1.073= 0.068 b . Show that π ^ = 0.50 when LI= 26.0 Answer: at -α ^ ^ = 3.777/ 0.145 = 26.0 c. Show that the rate of change in π ^ is 0.009 when LI = 8 and is 0.036 when LI = 26 Answer: At LI =8, π ^ =0.068 →rate of change= β ^ π ^ (1- π ^ )= 0.145(0.068)(1-0.068)= 0.0092 At LI =26, π ^ =0.5 →rate of change= β ^ π ^ (1- π ^ )= 0.145(0.5)(1-0.5)= 0.036 d . The lower quartile and upper quartile for LI are 14 and 28. Show that π ^ increase by 0.42 from 0.15 to 0.57 between those values. Answer: LI= 14 → π ^ =e α+β(LI) /[1+e α+β(LI) ]=e -3.777+0.145(14) /[1+e -3.777+0.145(14) ]=0.1743/1.1743= 0.148 LI= 28 → π ^ =e α+β(LI) /[1+e α+β(LI) ]=e -3.777+0.145(28) /[1+e -3.777+0.145(28) ]=1.327/2.327= 0.57 The increase between upper and lower quartile= 0.57- 0.148 = 0.422 e . When LI increase by 1, show the estimate odds of remission multiply by 1.16. 1

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Answer: e β^ = e 0.145 = 1.16 4.2) Refer to the previous exercise. Using information from Table 4.9: a. Conduct a Wald test for the LI effect. Interpret Answer: Wald Statistics : (β/SE) 2 = (0.145/ 0.059) 2 = 6.04, → chi-square = z 2 = 6.04 with df=1 → see Chi-square Table: p= 0.01. Thus, HO: β= 0 is rejected→there is effect of treatment on patient. b . Construct a Wald CI for the odds ratio corresponding to a 1-unit increase in LI. Interpret Answer: Odds ratio = e β^ = e 0.145 = 1.16 CI → e β^ ± Z α/2 (SE) = 1.16 ± 1.96 (0.059), thus the CI is ( 1.044, 1.276) So, odds of remission at LI = x + 1 are estimated between 1.044 and 1.276 times the odds of remission at LI= x. c. Conduct a likelihood-ratio test for the LI effect. Interpret Answer: Likelihood-ratio statistics → See Table 4.9 → is (8.30), with df=1 → p<0.01 →there is effect of treatment on patient. d . Construct the likelihood-ratio CI for the odds ratio. Interpret Answer: From Table 4.9 (under column “Likelihood ratio 95% Conf.Limits”), we find the lower and upper limits limits (0.0425 and 0.2846). Then, exponentiating : e 0.0425 and e 02846 are (1.04, 1.33)→ Thus, the odds of remission at LI=x+1 are estimated to fall between 1.04 and 1.33 times the odds of remission at LI=x 4.9) For the horseshoe crab data, fit a logistic regression model for the probability
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HW-04 - HOMEWORK FOR EXST 7036(Chapter 4 Student Name Tanza...

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