## Agenda

• Trauma Data Example Again
• Parameter Estimation with MCSE
• Reference Priors for Binomial Regression
• O-Ring Data Presentations

## Trauma Data

Dr. Turner Osler, a trauma surgeon and former head of the Burn Unit at the University of New Mexico Trauma Center, provided data on survival of patients admitted to the University of New Mexico Trauma Center between 1991 and 1994.

• The predictor variables are injury severity score (ISS), revised trauma score (RTS), AGE, and type of injury (TI). TI is either blunt ($$TI = 0$$), e.g., the result of a car crash, or penetrating ($$TI = 1$$), e.g., gunshot or stab wounds.
• ISS is an overall index of a patient's injuries based on the approximately 1,300 injuries cataloged in the Abbreviated Injury Scale. Takes on values from 0 for a patient with no injuries to 75 for a patient with severe injuries in three or more body areas.
• RTS is an index of physiologic status derived from the Glasgow Coma Scale. It is a weighted average of a number of measurements on an incoming patient such as systolic blood pressure and respiratory rate. Takes on values from 0 for a patient with no vital signs to 7.84 for a patient with normal vital signs.

## Trauma Data

• Data available here.
# trauma <- read.table("http://www.ics.uci.edu/~wjohnson/BIDA/Ch8/trauma300.txt,
head(traumadata)
##     ID death ISS TI    RTS AGE
## 1 2979     0   1  1 7.8408  25
## 2 1167     0   9  0 7.8408  63
## 3  116     0  29  0 2.9304  32
## 4 1118     0  10  0 7.8408  61
## 5 3126     0   2  1 7.8408  43
## 6 2173     0   9  0 7.8408  27

## Trauma Data

Dr. Osler proposed a logistic regression model to estimate the probability of a patient's death using an intercept; the predictors ISS, RTS, AGE, and TI; and an interaction between AGE and TI. AGE is viewed as a surrogate for a patient's physiologic reserve, i.e., their ability to withstand trauma.

The model $\text{logit}(\theta_i) = \beta_1 + \beta_2 ISS_i + \beta_3 RTS_i + \beta_4 AGE_i + \beta_5 TI_i + \beta_6 (AGE*TI)_i$ is similar to logistic models used by trauma centers throughout the United States.

Introduce $$x_{i6} = (AGE*TI)_i$$ and correct for the mean values of all the continuous predictor variables, then the model becomes $\text{logit}(\theta_i) = \alpha + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5} + \beta_6 x_{i6}.$

## Trauma Data

x.check <- cbind(ISS-mean(ISS), RTS-mean(RTS), AGE-mean(AGE), TI, AGE*TI-mean(AGE*TI))
colnames(x.check) <- c("x2", "x3", "x4", "x5", "x6")
head(x.check)
##          x2        x3    x4 x5        x6
## [1,] -13.28  0.553934 -6.41  1 17.633333
## [2,]  -5.28  0.553934 31.59  0 -7.366667
## [3,]  14.72 -4.356466  0.59  0 -7.366667
## [4,]  -4.28  0.553934 29.59  0 -7.366667
## [5,] -12.28  0.553934 11.59  1 35.633333
## [6,]  -5.28  0.553934 -4.41  0 -7.366667
model2 <- glm(death ~ x.check, family = binomial(), x = TRUE)

## Binomial Regression Analysis

• For our Bayesian analysis we assume the same data model as the frequentist, i.e. we continue to use the logit link function.
• We assume the prior distribution of the parameters (the regression coefficients) makes them independent and identically normally distributed with mean 0 and variance $$\sigma^2 = 25$$.

## Unnormalized Posterior

• The log unnormalized posterior density calculated by the following R function.
x <- model2$x y <- model2$y
lupost <- function(beta, x, y) {
eta <- as.numeric(x %*% beta)
logp <- ifelse(eta < 0, eta - log1p(exp(eta)), - log1p(exp(- eta)))
logq <- ifelse(eta < 0, - log1p(exp(eta)), - eta - log1p(exp(- eta)))
logl <- sum(logp[y == 1]) + sum(logq[y == 0])
return(logl - sum(beta^2) / 50)
}

## Posterior Simulations

• Use mcmc package and metrop function to obtain posterior simulations.
• Following code runs the Metropolis algorithm to simulate the posterior.
library(mcmc)
set.seed(1)
beta.init <- as.numeric(coefficients(model2))
out <- metrop(lupost, beta.init, 5e4, scale=0.03, x = x, y = y)

## Diagnostics

ess(out$batch) ## [1] 228.1051 485.8939 269.3972 614.3843 225.2397 403.7840 multiESS(out$batch)
## [1] 968.6197
• Effective sample sizes are small!
• See highest correlation components 1, 3, and 5, which also have the smallest ESS.
• Try to improve mixing by using a vector for scale.

## Diagnostics

out <- metrop(out, scale=c(.1, .03, .1, .03, .1, .03), x = x, y = y)
out$accept ## [1] 0.17346 out$time
##    user  system elapsed
##   5.736   1.297   7.062

plot(ts(out$batch)) ## Diagnostics ess(out$batch)
## [1] 265.4184 789.0774 650.0411 686.0127 232.3179 322.1590