- Maximum Likelihood Estimates 4 Points Possible Graded We Continue With The Lr Test On The Hip Study Let Yt And Yc Be 1 (547.08 KiB) Viewed 81 times
Maximum Likelihood Estimates 4 points possible (graded) We continue with the LR-test on the HIP study. Let Yt 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 = yt, Yc = yc) = P (YT = yt)P (Yc = yc). 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 30'937 31'000 61'89862'000 We use the binomial model for Yī and Yc: YT Yc Binom (31000, TT) Binom (31000, 1c) The likelihood ratio test statistic is A (YT , YC) maxo, P (YT , YC; TT, 7c) -2 log maxoa P (YT , YC; IT, Tc) max.nq=nce[0,1 P (YT,YC; 7) -2 log maxrıtac P (YT , YC; TT, TC) max.xn=7c=E(0,1) P (Binom (31000, 7) = yt)P (Binom (31000, 7) = yc) -2 log P (Binom (31000, AT) = yt)P (Binom (31000, C) = yc) P (Binom (31000, îMLE) = yr) P (Binom (31000, îMLE) = yc) - 2 log P (Binom (31000, îMLE) = yr) P (Binom (31000, îMLE) = yc) max t#C 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 (TTT, ac) 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): AMLE The value of at that maximizes P (Binom (31000, TT) = 39): AMLE AT The value of ac that maximizes P (Binom (31000, 7c) = 63): AMLE = 2. What is the value of the test statistic Abased on observed data? (Enter the value with a precision of 3 decimal points.) Submit You have used 0 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