binary response (from {0, 1}) covariates (predictors)

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for i=1:M Z = rand_nort(designX * beta, ones(size(designX * beta)), left, right); Zs=[Zs Z]; sigma=inv(designX’ * designX); betaMLE = inv(designX’ * designX)* designX’ * Z; beta = rand_MVN(1, betaMLE, sigma)’; betas=[betas beta]; piz = [piz .5*(1+erf(designX * beta/sqrt(2)))]; end
1.2
Slice Sampler.
Slice sampler generates pairs (X, Y ) that fill in the area under (possibly unnormalized) target density in a uniform fashion. Then the X -projection is a sample from the density. The slice sampler works as follows: Suppose the target density f (x) is known up to normalizing constant, i.e., we know g (x) such that f (x) ∝ g (x).
ቤተ መጻሕፍቲ ባይዱ
Responses Yi are indicators of a particular attribute (event, property, outcome). In such situations the researcher is interested in modeling p = P (Y = 1|X ), i.e., the probability that the attribute is present, given the vector of covariates, X . Assume that Zi = Xi β + i , i = 1, . . . ,
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This example and data have been provided by Matthew Wiggins, BioE Graduate Research Assistant at GaTech
ISyE8843A, Brani Vidakovic
Handout 11
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1.1
Various Models and Related MCMC Stuff.
Latent Variable Binary Regression.
In the binary responses the observations are of the form binary response (from {0, 1}) Y1 Y2 ... Yn covariates (predictors) X1 = (X1,1 , X1,2 , . . . , X1,p ) X2 = (X2,1 , X2,2 , . . . , X2,p ) ... Xn = (Xn,1 , Xn,2 , . . . , Xn,p )
i
> 0) = P ( i > −Xi β ) = 1 − F (−Xi β ).
If the error distribution is symmetric about zero (as usually assumed), pi = 1 − F (−Xi β ) = F (Xi β ), is equivalent to probit, logit, and related models. However, the formulation that assumes latent variable Zi is allowing Gibbs sampling scheme (eg., Chib and Albert 1993) and Johnson and Albert (1999)). Successive sampling from full conditionals, (i) [β |Z, Y ] and (ii) [Z |β, Y ]. Assume that F is normal distribution and that the above model is probit. Then the distribution for β given Z is simply the multivariate normal distribution from the least square theory, [β |Z, Y ] ∼ MVN p ((X X )−1 X Z, (X X )−1 ), where X is design matrix with rows X1 , . . . , Xn . Next, we find the conditional distribution for each component of Z . If Yi and β are given, Zi is truncated at 0 normal distribution with mean Xi β . Recall the connection P (Yi = 1) = P (Zi > 0). Truncation is to the left, if Yi = 1 and to the right if Yi = 0. The conditional for Zi given Yi and β is [Zi |β, Yi ] ∼ T N (Xi β, 1, −∞, 0), T N (Xi β, 1, 0, ∞), if Yi = 0 if Yi = 1,
i iid
∼ F,
is a multivariate regression model in which Zi ’s are not observable, but the indicators Yi = 1(Zi > 0) are. Then, pi = P (Yi = 1) = P (Zi > 0) = P (Xi β +
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−µ) where T N (µ, σ 2 , a, b) is truncated normal with density proportional to exp{− (z2 }1(a ≤ z ≤ b). σ2 These two conditional distributions are now defining an easy Gibbs sampler. Part of the matlab file albertmc3.m is given below. It requires two m-files from BAYES L AB: (i) rand nort.m that simulates truncated normal T N distribution, and (ii) rand MVN.m that simulates multivariate normal distribution.
Arrithmia Example.1 Patients who undergo Coronary Artery Bypass Graft Surgery (CABG) have an approximate 19-40% chance of developing atrial fibrillation (AF). AF can lead to blood clots forming causing greater in-hospital mortality, strokes, and longer hospital stays. While this can be prevented with drugs, it is very expensive and sometimes dangerous if not warranted. Ideally, several risk factors which would indicate an increased risk of developing AF in this population could save lives and money by indicating which patients need pharmacological intervention. Researchers began collecting data from CABG patients during their hospital stay such as demographics like age and sex, as well as heart rate, cholesterol, operation time, etc. Then, the researchers recorded which patients developed AF during their hospital stay. Researchers now want to find those pieces of data which indicate high risk of AF. In the past, indicators like age, hypertension, and body surface area (BSA) have been good indicators, though these alone have not produced a satisfactory solution. The data set looks has 81 records and the first 3 and the last 3 are provided: ARR 1 1 1 0 0 0 INT 1 1 1 ... 1 1 1 AGE 68 75 69 71 73 55 ACCT 64 111 63 51 68 44 CBT 126 167 94 83 122 78 ICUT 20.25 13.5 12 16.75 17.83 11.5 AHR 85 50 74 102 78.2 94.8 LVEF 81 75 62 59 50 40