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In example 8.15, on Firth logistic regression, we mentioned alternative approaches to separation troubles. Here we demonstrate exact logistic regression. The code for this appears in the book (section 4.1.2) but we don’t show an example of it there. We’ll consider the setting of observing 100 subjects each with x=0 and x=1, observing no events when x=0, and 5 when x=1.

**SAS**

We’ll create the data as a summary, rather than for every line of data. Then we can use the “events/trials” syntax (section 4.1.1) that both `proc logistic` and `proc genmod` accept. This is another way to reduce the size of data sets (along with the `weight` option mentioned previously) but is less generally useful. The the `exact` statement in `proc logistic` will fit the exact logistic regression and generate a p-value. The `estimate` option is required to display estimated log odds ratio.

data exact;

x=0; count=0; n=100; output;

x=1; count=5; n=100; output;

run;

proc logistic data=exact;

model count/n = x;

exact x / estimate;

run;

This generates the following output:

Exact Parameter Estimates

Standard 95% Confidence

Parameter Estimate Error Limits p-Value

x 1.9414* . -0.0677 Infinity 0.0594

NOTE: * indicates a median unbiased estimate.

**R**

In R we use the `elrm()` function in the `elrm` package to approximate exact logistic regression, as described in this paper by the package’s authors. The function requires a special formula object with syntax identical to the SAS events/trials syntax. (Note that the function does not behave as expected when identical observations with trials=1 are submitted. Thus data should be collapsed into unique combinations of predictors before using the function.) In addition, it requires its data to be included in a data frame. We’ll construct the data frame in one function call to `data.frame()`.

elrmdata = data.frame(count=c(0,5), x=c(0,1), n=c(100,100))

library(elrm)

resexact = elrm(count/n ~ x, interest = ~x, iter=22000,

burnIn=2000, data=elrmdata, r=2)

summary(resexact)

producing the following result:

Call:

[[1]]

elrm(formula = count/n ~ x, interest = ~x, r = 2, iter = 22000,

dataset = elrmdata, burnIn = 2000)

Results:

estimate p-value p-value_se mc_size

x 2.0225 0.02635 0.0011 20000

95% Confidence Intervals for Parameters

lower upper

x -0.02065572 Inf

Differences between the SAS and R results most likely arise from the fact that the `elrm()` function is an approximation of the exact approach. The upper limit of infinity seen in the exact SAS analysis and approximate exact `elrm()` analysis reveals a limitation of this approach relative to the Firth approach seen in example 8.15 and the Bayesian approach we’ll examine later.

A final note: if the true Pr(Y=1|X=1) = 0.05, then the true Pr(Y=1|X=0) that results in a log odds ratio of 1.94 is about 0.0075; for a log odds ratio of 2.02, the true probability is about 0.0069.

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