# How to perform two-sample t-tests in R by inputting sample statistics rather than the raw data?

Let's say we have the statistics given below

gender mean sd n
f 1.666667 0.5773503 3
m 4.500000 0.5773503 4


How do you perform a two-sample t-test (to see if there is a significant difference between the means of men and women in some variable) using statistics like this rather than actual data?

I couldn't find anywhere on the internet how to do this. Most of the tutorials and even the manual deal with the test with the actual data set only.

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Should be homework tagged and moved to stats.stackexchange. –  DWin Jun 13 '12 at 16:21
This Wikipedia article plus the help page for R's t-distribution functions (got by ?pt) -- see especially pt() -- do have all the info you'd need to do this yourself. And you'll learn a lot about stats and R if you do that. –  Josh O'Brien Jun 13 '12 at 16:40
There are good answers here already, and indeed it's both very easy (and good practice) to write a function for this yourself; however, I'll just add that you might take a look at the tsum.test function in package BSDA, which implements a t-test (two sample; Welch or equal-variance and also one sample) from summary data you supply. It basically works like the t-test in vanilla R but on the summary info. –  Glen_b Jun 18 at 0:30

## migrated from stackoverflow.comJun 13 '12 at 17:27

This question came from our site for professional and enthusiast programmers.

You can write your own function based on what we know about the mechanics of the two-sample $t$-test. For example, this will do the job:

# m1, m2: the sample means
# s1, s2: the sample standard deviations
# n1, n2: the same sizes
# m0: the null value for the difference in means to be tested for. Default is 0.
# equal.variance: whether or not to assume equal variance. Default is FALSE.
t.test2 <- function(m1,m2,s1,s2,n1,n2,m0=0,equal.variance=FALSE)
{
if( equal.variance==FALSE )
{
se <- sqrt( (s1^2/n1) + (s2^2/n2) )
# welch-satterthwaite df
df <- ( (s1^2/n1 + s2^2/n2)^2 )/( (s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1) )
} else
{
# pooled standard deviation, scaled by the sample sizes
se <- sqrt( (1/n1 + 1/n2) * ((n1-1)*s1^2 + (n2-1)*s2^2)/(n1+n2-2) )
df <- n1+n2-2
}
t <- (m1-m2-m0)/se
dat <- c(m1-m2, se, t, 2*pt(-abs(t),df))
names(dat) <- c("Difference of means", "Std Error", "t", "p-value")
return(dat)
}
x1 = rnorm(100)
x2 = rnorm(200)
# you'll find this output agrees with that of t.test when you input x1,x2
t.test2( mean(x1), mean(x2), sd(x1), sd(x2), 100, 200)
Difference of means       Std Error               t         p-value
-0.05692268      0.12192273     -0.46687500      0.64113442

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You just calculate it by hand: $$t = \frac{(\text{mean}_f - \text{mean}_m) - \text{expected difference}}{SE} \\ ~\\ ~\\ SE = \sqrt{\frac{sd_f^2}{n_f} + \frac{sd_m^2}{n_m}} \\ ~\\ ~\\ \text{where, }~~~df = n_m + n_f - 2$$

The expected difference is probably zero.

If you want the p-value simply use the pt() function:

pt(t, df)


Thus, putting the code together:

> p = pt((((1.666667 - 4.500000) - 0)/sqrt(0.5773503/3 + 0.5773503/4)), (3 + 4 - 2))
> p
[1] 0.002272053


This assumes equal variances which is obvious because they have the same standard deviation.

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A couple things: How is this "in R"? What is the distribution of the test statistic (i.e. how do you go from this to $p$-values)? –  Macro Jun 14 '12 at 14:04
You can do the calculations based on the formula in the book (on the web page), or you can generate random data that has the properties stated (see the mvrnorm function in the MASS package) and use the regular t.test function on the simulated data.
You want the simulated data to have the exact same mean(s) and var(s) as stated in the problem. One way to do this (there are many others) is to use the mvrnorm function in the MASS package (you need to set the empirical argument to TRUE). –  Greg Snow Jun 15 '12 at 17:45
The question asks about R, but the issue can arise with any other statistical software. Stata for example has various so-called immediate commands, which allow calculations from summary statistics alone. See http://www.stata.com/manuals13/rttest.pdf for the particular case of the ttesti command, which applies here.