Exercise 1 Exercise 2 Exercise 3

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Exercise 1

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Exercise 2

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Exercise 3

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3. Likelihood Ratio Test on the Mammography Study Bookmark this page Exercises due Sep 8, 2021 05:59 CST In the following problems, we apply the likelihood ratio test to the mammography study to answer the question whether the treatment changes the death rate. The corresponding hypotheses in terms of the breast cancer death rates, it in the treatment group and hic in the control group, are H : TTT= TTC HA : T + C. Parameter Spaces 0/2 points (graded) What is the parameter spaces and A corresponding to the Ho and HA? Recall that the breast cancer death rate it is the probability of an individual in the treatment group dying of breast cancer, and similarly for ac. (Choose any correct answer.) = {(TTT,TC) (0,0) x (0,0) TT = Ac} and OA = {(TIT,Ac) € (0,00) 0,0) |TT #Tc} 2. = {(1T1, Tc) € [0, 1] [0, 1] |TT = c} and OA = {(TT, Tc) € [0, 1] x [0,1] |TT #Tc} 2. = {€ [0, 1]} and © A = {(TT, c) € [0, 1] x [0, 1] (at #ac} X What is dim (O, UOA) - dim (Oo)? dim (O, UOA) - dim (Co) =

Maximum Likelihood Estimates 4 points possible (graded) We continue with the LR-test on the HIP study. Let Ys and Yc be the numbers of cancer deaths in the treatment and control groups respectively. Assuming these are Independent from each other, the probability of having y breast cancer deaths in the treatment group and we breast cancer deaths in the control group is the product P (Yr . Y = y) = P(V1 = y) P (Yc=ye) Recall the HIP mammography study data: treatment control total breast cancer deaths 39 (0.0013) 63 (0.0020) 102 alive total 30'961 31'000 30'93731'000 61'898 62'000 We use the binomial model for Yt and Yc: Yr Binom (31000, T) Yc Binom (31000, c) The likelihood ratio test statistic is A(yr, yc) maxe, Pyt, yc; TT, TTC) = -2 log maxe, P (yr, yc; it,c) = -2 log maxy=-CE|0,1 P (yr, yc;) max. ** P (yr, Yc; TT, TTC) maxxy==|0,1 P (Binom (31000, 7) = yr) P (Binom (31000, 7) = yc) = -2 log max. ** P (Binom (31000, T) = yr) P (Binom (31000,0) =YC) P (Binom (31000, AMLE) = yr) P (Binom (31000, AMLE) = y) = -2 log P (Binom (31000, AMLE) = yr) P (Binom (31000, A ALE YC) where we have used P (Binom (np) = y) to denote the probability that a binomial variable with parameters n, p takes value y 1. Based on the observed data, Find the parameters (TTT, c) that maximize the numerator and the denominator in the definition of the test statistic A. That is, find the 3 different maximum likelihood estimates (in blue ) in the expression above. Review: MLE for Binomial Distribution Show The value at that maximizes P (Binom (31000, 7) = 39) P (Binom (31000, 7) = 63): #MLE = The value of that maximizes P (Binom (31000, T) = 39) FALE The value of that maximizes P (Binom (31000,0) = 63) NALE 2. What is the value of the test statistic A based on observed data? (Enter the value with a precision of 3 decimal points)

P-value and conclusion 2 points possible (graded) Continue with the likelihood ratio test for the HIP experiment. 1. What is the p-value of the likelihood ratio test based on the observed data? (Please enter the value with a precision of 3 decimal points.) 2. Do we reject the null hypothesis with significance level 0.05? Reject Do not reject