- 3 Likelihood Ratio Test On The Mammography Study A Bookmark This Page Exercises Due Sep 8 2021 08 59 03 In The Follow 1 (63.5 KiB) Viewed 21 times
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3. Likelihood Ratio Test on the Mammography Study A Bookmark this page Exercises due Sep 8, 2021 08:59 -03 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, Tit in the treatment group and to in the control group, are Но : TTT= 70 HA . TTT c. Parameter Spaces 2/2 points (graded) What is the parameter spaces @ and A corresponding to the Ho and HA? Recall that the breast cancer death rate at is the probability of an individual in the treatment group dying of breast cancer, and similarly for Ac. (Choose any correct answer.) @ = {(TT,TC) € (0,00) x (0,00) | TT =c} and A = {(TT,TC) € (0.00) [0,00) TT #Tc} O @ = {(TTT, c) € [0, 1] x [0, 1] TT = 7c} and A = {(TT,C) € [0, 1] x [0, 1] TT *c} @ = {TT € [0, 1]} and A = {(TT, TC) € [0, 1] x [0, 1] |TT TC} What is dim (@UBA)-dim (,)? dim ( OVA)-dim(o)- 1
Maximum Likelihood Estimates 1/4 points (graded) We continue with the LR-test on the HIP study. Let Yr 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 Yt breast cancer deaths in the treatment group and yc breast cancer deaths in the control group is the product P (YT = y, Yc = ye) = P (Yr = y) P (Yc = ye). Recall the HIP mammography study data: treatment control total breast cancer deaths 39 (0.0013) 63 (0.0020) alive total 30'961 31'000 30'937 31'000 61'898 62'000 102 We use the binomial model for Yt and Yo: YT Binom (31000, TT) Binom (31000, TC) YC The likelihood ratio test statistic is AYT,YC) -2 log max -2 log maxe, Pyt, yo; TT, TC) maxe, P (YT,YC; TT, TTC) maxy TE[0,1] P (YT,YC;) P (YT,YC; TT, TTC) max. (0,11 P (Binom (31000, 7) = yr) P (Binom (31000, 7) = yo) max. tre P (Binom (31000, TT) = yr) P (Binom (31000, Tc) = yc) P (Binom (31000, AMLE) = yr) P (Binom (31000, #MLE) = yc) -2 log P (Binom (31000, RMTE) = yr) P (Binom (31000, AULE) = yc) -2 log where we have used P (Binom (n,p) = 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 (TT, Tc) 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.
The value that maximizes P (Binom (31000, 7) = 39) P (Binom (31000, 7) = 63): ẢMLE 0.0231 X The value of at that maximizes P (Binom (31000, TTT) = 39): AMLE 0.0143 x The value of a that maximizes P (Binom (31000, 10) = 63): AMLE = 0.0023 2. What is the value of the test statistic A based on observed data? (Enter the value with a precision of 3 decimal points.) 0.0061 X Submit You have used 2 of 3 attempts Save 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