Maximum Likelihood Estimates 4 points possible (graded) We continue with the LR-test on the HIP study. Let IT 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 ye breast cancer deaths in the control group is the product P (Yr = yt, Yo = yc) = P( Yr = yt) P (Yo = yc) . Recall the HIP mammography study data: breast cancer deaths alive total treatment 39 (0.0013) 30'961 31'000 control 63 (0.0020) 30'937 31'000 total 102 61 898 62'000 We use the binomial model for Y'r and Yc: Yr ~ Binom (31000, TT) Yo ~ Binom (31000, TC) The likelihood ratio test statistic is A (yr, yo) = -2log maxe, P (yT, yo; TT, TC) maxe, P (yT, yC; TT, TC) = -2log max,=nce0,1] P (yr, yo; T) maxty#To P (yT, yC; TT, TC) max,=me=me0,]] P (Binom (31000, 7) = yr) P (Binom (31000, 7) = yc) = -2log maxfranc P (Binom (31000, AT) = yr) P (Binom (31000, nc) = yc) P (Binom (31000, AMLE) = yr) P (Binom (31000, #MLE) = yC) = -2 log P (Binom (31000, AYLE) = yr) P (Binom (31000, AZLE) = yc) where we have used P (Binom (n, p) = y) to denote the probability that a binomial variable with parameters n, p takes value y.where we have used P (Binom (mp) = y) to denote the probability that a binomial variable with parameters mptakes value '9'. 1. Based on the observed data' Find the parameters [g-re) 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 Recall the pmf of the binomial distribution Binom (mp): P (k;n,p) = ('2') .pk _ {1 _p)nk To find the MLE for the binomial distribution, first take the logarithm of the likelihood function to make the derivative calculation easier: lawman =In[(:)-a'=-(1p)\"*1 _ =ln(k)+]np"+ln[1p)\" " Then, find 1.) where the derivative of P (p; n, k) is zero: HenceI the maximum likelihood estimate isp = . n The value air that maximizes P (Binom (31000, tr] = 39) P [Binom (31000, tr) = 63): WE : The value of fun that maximizes P (Binom (31000, IT) = 39): 3313 = The value of 770 that maximizes P (Binom {31000, arc) = 63): W = 2. What is the value of the test statistic A based on observed data? (Enter the value with a precision of 3 decimal points.)