The t-test
The t-test is a statistical test of whether two sample means (averages) or proportions are equal. It was invented by William Sealy Gosset, who wrote under the pseudonym “student” to avoid detection by his employer (the Guinness Brewing Company). Guinness prohibited publications by employees, because another employee had divulged trade secrets in writing.
There are also one-sample versions of a the t-test, to tell if a sample has a mean equal to some fixed value, but these are relatively little used.
When to use a t-test
A t-test can be used to compare two means or proportions. The t-test is appropriate when all you want to do is to compare means, and when its assumptions are met (see below). In addition, a t-test is only appropriate when the mean is an appropriate when the means (or proportions) are good measures. See my earlier article for guidance on when to use the mean.
Matched and unmatched t-tests
There are two forms of the t-test. In the unmatched t-test, or independent t-test, it is assumed that the two samples are independent. In non-technical language, two samples are independent when knowing something about one does not affect what we know abou the other. For example, the average heights of men and women, drawn randomly from a population, are independent, since knowing the height of a particular man tells us nothing about the height of any particular woman. In a matched t-test, the two sample are not independent; for example, the heights of husbands and wives are not independent, since taller men may be married to taller women. More obviously, the length of people’s right and left feet are dependent, because knowing the size of a right foot tells us a lot about the size of the left foot.
Assumptions of the t-test
As noted above, the independent samples t-test assumes the two samples are independent. In addition, both forms of the t-test assume that the variances of the two populations are equal. There are good ways to adjust for unequal variances, provided that the sample sizes of the two
samples are approximately equal. However, if the variances are very different and the sample sizes are different, then the t-test is not a good measure. In addition, as noted above, the t-test only makes sense when the mean makes sense.
If not the t-test, then what?
If the t-test is not appropriate, then one alternative is a nonparametric test, such as Wilcoxon’s test. Another alternative is a permutation test, or a bootstrap. In my opinion, all three alternatives ought to be used more often.
The t-test in SAS
Suppose one wishes to test if men are heavier than women in a given population. If you sample 5 men and 5 women at random, you might get something like this:
Men: 140 180 188 210 190
Women: 120 190 129 120 130
You could read that into SAS® using
data ttest;
input sex $ weight @@;
datalines;
M 140 F 120 M 180 F 190 M 188 F 129 M 210 F 120 M 190 F 130
;
run;
and then run a t-test by using
proc ttest data = ttest;
class sex;
var weight;
run;
The t-test in R
In R, one could read the same data in using
sex <- c(rep(‘M’, 5), rep(‘F’, 5))
weight <- c(140, 180, 188, 210, 190, 120, 190, 129, 120, 130)
and then run a t-test using
t.test(weight~sex)
The output looks like
Welch Two Sample t-test
data: weight by sex
t = -2.4982, df = 7.851, p-value = 0.03758
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-84.364290 -3.235710sample estimates:
mean in group F mean in group M
137.8 181.6
Which is terser than the SAS output, but says essentially the same thing. However, by default, R uses the Welch t-test, which does not assume the variances are equal. To get the test with the assumption, you would use
t.test(weight~sex, var.equal = T)
Very nice and interesting post! I didn’t know the anecdote about the Student name.
Thanks!
Thank you, very useful.
Nice Post. You probably don’t drink Guinness.
Good old Gossett. I have had Guiness in Ireland – it’s so much better than here in the USA that I no longer drink it here
You referenced an earlier article on when to use means. Which article were you talking about specifically? I’m not sure using the mean is appropriate and I’m quite sure my situation requires something other than a t-test, but it’s been long enough that I’m not sure what I should be using.
Interesting about Guiness as I’ve heard that from people I know who have traveled to Ireland…something in the shipping that changes the flavor or something along those lines.
Hi Tom
I was talking about this article
thanks
Peter
Excellent! Thanks Peter!