Computing t-tests using summary measures
It is sometimes handy to be able to work out t-tests 'by hand'. For example you may not have access to the raw data (e.g. for comparing your data with published norms in a test manual). Don't forget that when performing parametric tests we are assuming within group normality and equal group variances. For two groups with given means, variances and sample sizes:
(group mean 1 - group mean 2) / Sqrt{(var1/n1) + (var2/n2)}
will have a t distribution on n1-n2-2 degrees of freedom if the group means are equal.
2x2 interactions
If the means are differences then the above corresponds to a test of a two-way 2x2 interaction where one of the factors is within subjects. A t-test on differences is equivalent to an interaction test in an ANOVA for a 2x2 interaction (see [attachment:int.doc here for examples where at least one of the factors is within subjects).]
Suppose you wish to test for an interaction between two within subject factors then you could do a paired t-test on one of the two possible sets of differences with an additional term, corr, representing the correlation between these two differences.
(mean difference 1 - mean difference 2) / Sqrt{(var1/n1) + (var2/n2) - 2 corr sd1 sd2}
which again will have a t distribution on n1-n2-2 degrees of freedom if the mean differences are equal. The above are also useful for incorporating partially complete cases if you can assume that the missing data would not influence the observed complete case means, sds and correlations (ie missing completely at random).
In the case where both factors are between subjects you can perform the interaction test as follows which is also used in this [attachment:bbint.xls spreadsheet]:
- $$\sum_text{i} (n_text{i}-1) \mbox{variance}_text{i}$$ = MSE
$$\sum_text{ij}text{2} n (\bar{AB}_text{ij} - \bar{A}_text{i} - \bar{B}_text{j} + \mbox{overall mean})text{2}$$ = INT
where \bar{AB} are the four means for the i-jth combination of factors A and B which each have means $$\bar{A}_text{i}$$ and $$\bar{B}_text{j}$$ and n is the number in that combination. If there are different numbers of i,j combinations then the harmonic mean is used for n.
Finally,
Compare INT/MSE to a F statistic with 1 and n1+n2-4 degrees of freedom.
Reference
Howell DC (2002) Statistical Methods for psychologists. Fifth Edition. Duxbury Press:Belmont,CA.