Today we'll show how to build a Monte Carlo experiment to compare these approaches. Suppose we have 100 observations with x=0 and 100 with x=1, and suppose that the Pr(Y=1|X=0) = 0.001, while the Pr(Y=1|X=1) = 0.05. Thus the true odds ratio is (0.05/0.95)/(0.001/0.999) = 52.8 and the log odds ratio we want to find is 3.96. But we will rarely observe any y=1 when x=0. Which of these approaches is most likely to give us acceptable results?

Note that in all of the MCMC analyses we use only 6000 iterations, which is likely too few to trust in practice.

The code is long enough here that we annotate within rather than write much text around the code.

**SAS**

All the SAS procedures used accept the events/trials syntax (section 4.1.1), so we'll generate example data sets as two observations of binomial random variates with the probabilities noted above. We also make extensive use of the ODS system to suppress all printed output (section A.7.1) and to save desired pieces of output as SAS data sets. The latter usage requires first using the

`ods trace on/listing`statement to find the name of the output before saving it. Finally, we use the

`by`statement (section A.6.2) to replicate the analysis for each simulated data set.

data rlog;

do trial = 1 to 100;

/* each "trial" is a simulated data set with two observations

containing the observed number of events with x=0 or x=1 */

x=0; events = ranbin(0,100,.001); n=100; output;

x=1; events = ranbin(0,100,.05); n=100; output;

end;

run;

ods select none; /* omit _all_ printed output */

ods output parameterestimates=glm; /* save the estimated betas */

proc logist data = rlog;

by trial;

model events / n=x; /* ordinary logistic regression */

run;

ods output parameterestimates=firth; /* save the estimated betas */

/* note the output data set has the same name

as in the uncorrected glm */

proc logist data = rlog;

by trial;

model events / n = x / firth; /* do the firth bias correction */

run;

ods output exactparmest=exact;

/* the exact estimates have a different name under ODS */

proc logist data=rlog;

by trial;

model events / n = x;

exact x / estimate; /* do the exact estimation */

run;

data prior;

input _type_ $ Intercept x;

datalines;

Var 25 25

Mean 0 0

;

run;

ods output postsummaries=mcmc;

proc genmod data = rlog;

by trial;

model events / n = x / dist=bin;

bayes nbi=1000 nmc=6000

coeffprior=normal(input=prior) diagnostics=none

statistics=summary;

/* do the Bayes regression, using the prior made in the

previous data step */

run;

Now I have four data sets with parameter estimates in them. I could use them separately, but I'd like to merge them together. I can do this with the

`merge`statement (section 1.5.7) in a

`data`step. I also need to drop the lines with the estimated intercepts and rename the variables that hold the parameter estimates. The latter is necessary because the names are duplicated across the output data sets and desirable in that it allows names that are meaningful. In any event, I can use the

`where`and

`rename`data set options to include these modifications as I do the merge. I'll also add the number of events when x=0 and when x=1, which requires merging in the original data twice.

data lregsep;

merge

glm (where = (variable = "x") rename = (estimate = glm))

firth (where = (variable = "x") rename = (estimate = firth))

exact (rename = (estimate = exact))

mcmc (where = (parameter = "x") rename = (mean=mcmc))

rlog (where = (x = 1) rename = (events = events1))

rlog (where = (x = 0) rename = (events = events0));

by trial;

run;

ods select all; /* now I want to see the output! */

/* check to make sure the output dataset looks right */

proc print data = lregsep (obs = 5) ;

var trial glm firth exact mcmc;

run;

/* what do the estimates look like? */

proc means data=lregsep;

var glm firth exact mcmc;

run;

With the following output.

Obs trial glm firth exact mcmc

1 1 12.7866 2.7803 2.3186 3.9635

2 2 12.8287 3.1494 2.7223 4.0304

3 3 10.7192 1.6296 0.8885 2.5613

4 4 11.7458 2.2378 1.6906 3.3409

5 5 10.7192 1.6296 0.8885 2.5115

Variable Mean Std Dev

----------------------------------------

glm 10.6971252 3.4362801

firth 2.2666700 0.5716097

exact 1.8237047 0.5646224

mcmc 3.1388274 0.9620103

----------------------------------------

The ordinary logistic estimates are entirely implausible, while the three alternate approaches are more acceptable. The MCMC result has the least bias, but it's unclear to what degree this is a happy coincidence between the odds ratio and the prior precision. The Firth approach appears to be less biased than the exact logistic regression

**R**

The R version is roughly analogous to the SAS version. The notable differences are that 1) I want the "weights" version of the data (see example 8.15) for the

`glm()`and

`logistf()`functions and need the events/trials syntax for the

`elrm()`function and the expanded (one row per observation) version for the

`MCMClogit()`funtion. The

`sapply()`function (section B.5.3) serves a similar function to the

`by`statement in SAS. Finally, rather than spelunking through the

`ods trace`output to find the parameter estimates, I used the

`str()`function (section 1.3.2) to figure out where they are stored in the output objects and indexes (rather than data set options) to pull out the one estimate I need.

# make sure the needed packages are present

require(logistf)

require(elrm)

require(MCMCpack)

# the runlogist() function generates a dataset and runs each analysis

# the parameter "trial" keeps track of which time we're calling runlogist()

runlogist = function(trial) {

# the result vector will hold the estimates temporarily

result = matrix(0,4)

# generate the number of events once

events.0 =rbinom(1,100, .001) # for x = 0

events.1 = rbinom(1,100, .05) # for x = 1

# following for glm and logistf "weights" format

xw = c(0,0,1,1)

yw = c(0,1,0,1)

ww = c(100 - events.0, events.0, 100 - events.1,events.1)

# run the glm and logistf, grab the estimates, and stick

# them into the results vector

result[1] =

glm(yw ~ xw, weights=ww, binomial)$coefficients[2]

result[2] = logistf(yw ~ xw, weights=ww)$coefficients[2]

# elrm() needs a data frame in the events/trials syntax

elrmdata = data.frame(events=c(events.0,events.1), x =c(0,1),

trials = c(100,100))

# run it and grab the estimate

result[3]=elrm(events/trials ~ x, interest = ~ x, iter = 6000,

burnIn = 1000, data = elrmdata, r = 2)$coeffs

# MCMClogit() needs expanded data

x = c(rep(0,100), rep(1,100))

y = c(rep(0,100-events.0), rep(1,events.0),

rep(0, 100-events.1), rep(1, events.1))

# run it and grab the mean of the MCMC posteriors

result[4] = summary(MCMClogit(y~as.factor(x), burnin=1000,

mcmc=6000, b0=0, B0=.04,

seed = list(c(781306, 78632467, 364981736, 6545634, 7654654,

4584),trial)))$statistics[2,1]

# send back the four estimates, plus the number of events

# when x=0 and x=1

return(c(trial, events.0, events.1, result))

}

Note the construction of the

`seed=`option to the

`MCMClogit()`function. This allows a different seed in every call without actually using sequential seeds.

Now we're ready to call the function repeatedly. We'll do that with the

`sapply()`function, but we need to nest that inside a

`t()`function call to get the estimates to appear as columns rather than rows, and we'll also make it a data frame in the same command. Note that the parameters we change within the

`sapply()`function are merely a list of trial numbers. Finally, we'll add descriptive names for the columns with the

`names()`function (section 1.3.4).

res2 = as.data.frame(t(sapply(1:10, runlogist)))

names(res2) <- c("trial","events.0","events.1", "glm",

"firth", "exact-ish", "MCMC")

head(res2)

mean(res2[,4:7], na.rm=TRUE)

trial events.0 events.1 glm firth exact-ish MCMC

1 1 0 6 18.559624 2.6265073 2.6269087 3.643560

2 2 1 3 1.119021 0.8676031 1.1822296 1.036173

3 3 0 5 18.366720 2.4489268 2.1308186 3.555314

4 4 0 5 18.366720 2.4489268 2.0452446 3.513743

5 5 0 2 17.419339 1.6295391 0.9021854 2.629160

6 6 0 9 17.997524 3.0382577 2.1573979 4.017105

glm firth exact-ish MCMC

17.333356 2.278344 1.813203 3.268243

The results are notably similar to SAS, except for the unacceptable

`glm()`results.

In most Monte Carlo experimental settings, one would also be interested in examining the confidence limits for the parameter estimates. Notes and code for doing this can be found here. In a later entry we'll consider plots for the results generated above. As a final note, there are few combinations of event numbers with any mass worth considering. One could compute the probability of each of these and the associated parameter estimates, deriving a more analytic answer to the question. However, this would be difficult to replicate for arbitrary event probabilities and Ns, and very awkward for continuous covariates, while the above approach could be extended with trivial ease.

## 3 comments:

From your text... "Suppose we have 100 observations with x=0 and 100 with x=1, and suppose that the Pr(Y=1|X=0) = 0.001, while the Pr(Y=1|X=0) = 0.05."

Should one of those read Pr(Y=1|X=1)?

Yup. The second one. Thanks! (I swear I spotted this & would have been sure I fixed it.)

Since P(Y=1|X=0) = 0.001, we have P(Y=0|X=0) = 0.999. On the other hand, P(Y=1|X=1) = 0.05, so P(Y=0|X=1) = 0.95. As a result, we would see Y=0 with a very high chance when x=0 or 1. Why does this situation imply a quasi separation problem?

Next, why log[(0.05/0.95)/(0.001/0.999)] = 4 is the theoretical coefficient of x in the model?

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