**SAS and R**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Apologies for the long and unannounced break– the longest since we started blogging, three and a half years ago. I was writing a 2-day course for SAS users to learn R. Contact me if you’re interested. And Nick and I are beginning work on the second edition of our book– look for it in the fall. Please let us know if you have ideas about what we omitted last time or would otherwise like to see added. In the mean time, we’ll keep blogging, though likely at a reduced rate.

Today: what can you say about the probability of an event if the observed number of events is 0? It turns out that the upper 95% CI for the probability is 3.69/N. There’s a sweet little paper with some rationale for this, but it’s in my other office. And I couldn’t recall the precise value– so I used SAS and R to demonstrate it to myself.

**R**

The R code is remarkably concise. After generating some Ns, we write a little function to perform the test and extract the (exact) upper 95% confidence limit. This is facilitated by the “…” notation, which passes along unused arguments to functions. Then we use `apply()` to call the new function for each N, passing the numerator 0 each time. Note that `apply()` needs a matrix argument, so the simple vector of Ns is converted to a matrix before use. [The `sapply()` function will accept a vector input, but took about 8 times as long to run.] Finally, we plot the upper limit * N against N. showing the asymptote. A log scaled x-axis is useful here, and is achieved with the `log='x'` option. (Section 5.3.12.) the result is shown above.

bin.m = seq(10, 10000, by=5)

mybt = function(...) { binom.test(...)$conf.int[2] }

uci = apply(as.matrix(bin.m), 1, mybt, x=0)

plot(y=bin.m * uci, x=bin.m, ylim=c(0,4), type="l",

lwd=5, col="red", cex=5, log='x',

ylab="Exact upper CI", xlab="Sample size",

main="Upper CI when there are 0 cases observed")

abline(h=3.69)

**SAS**

In SAS, the data, really just the N and a numerator of 0, are generated in a `data` step. The CI are found using the `binomial` option in the `proc freq tables` statement and saved using the `output` statement. Note that the `weight` statement is used here to avoid having a row for each Bernoulli trial.

data binm;

do n = 10 to 10000 by 5;

x=0;

output;

end;

run;

ods select none;

proc freq data=binm;

by n;

weight n;

tables x / binomial;

output out=bp binomial;

run;

ods select all;

To calculate the upper limit*N, another `data` step is needed– note that in this setting SAS will only produce the lower limit against the probability that all observations share the same value, thus the subtraction from 1 shown below. The log scale x-axis is obtained with the `logbase` option to the `axis` statement. (Section 5.3.12.) The result is shown below.

data uci;

set bp;

limit = (1-xl_bin) * n;

run;

axis1 order = (0 to 4 by 1);

axis2 logbase=10 logstyle=expand;

symbol1 i = j v = none c = red w=5 l=1;

proc gplot data=uci;

plot limit * n / vref=3.69 vaxis=axis1 haxis=axis2;

label n="Sample size" limit="Exact upper CI";

run;

quit;

It’s clear that the upper 95% limit on the number of successes asymptotes to about 3.69. Thus the upper limit on the binomial probability p is 3.69/N.

**An unrelated note about aggregators:**We love aggregators! Aggregators collect blogs that have similar coverage for the convenience of readers, and for blog authors they offer a way to reach new audiences. SAS and R is aggregated by R-bloggers, PROC-X, and statsblogs with our permission, and by at least 2 other aggregating services which have never contacted us. If you read this on an aggregator that does not credit the blogs it incorporates, please come visit us at SAS and R. We answer comments there and offer direct subscriptions if you like our content. In addition, no one is allowed to profit by this work under our license; if you see advertisements on this page, the aggregator is violating the terms by which we publish our work.

**leave a comment**for the author, please follow the link and comment on their blog:

**SAS and R**.

R-bloggers.com offers

**daily e-mail updates**about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.

Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.