Computing t-tests and F ratios using summary measures in between subjects designs
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 independent 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.
For two groups which constitute a within subjects factor we use the above t statistic but we need an additional term, corr, representing the correlation between these two differences in the denominator.
(group mean 1 - mean difference 2) / Sqrt{(var1/n1) + (var2/n2) - 2 corr sd1 sd2}
2x2 interactions
If the means are differences then the t-tests can be seen to correspond 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 where both factors are within subject 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). This approach can also be used for testing an interaction involving one between and one within subjects factor comparing the within subjects group differences in the two between subjects factor groups.
In the case where both factors are between subjects you can perform the interaction test as described below (incorporated into this [attachment:bbint.xls spreadsheet]) for balanced data (ie where each of the four combinations of between subject factors occurs equally often).
- $$\sum_text{i} (n_text{i}-1) \mbox{variance}_text{i}$$ = 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. Fisher and vanBelle (1993) suggest this approach for balanced or nearly balanced data.
Finally,
Compare INT/MSE to a F statistic with 1 and n1+n2-4 degrees of freedom.
The more general unbalanced case where there are unequal numbers of observations across the four combinations of the factors uses formulae from Cohen (2002) which are incorporated into this [attachment:bbint_unb.xls spreadsheet].
References
Cohen BH (2002) Calculating a Factorial ANOVA from means and standard deviations. Understanding Statistics 1(3) 191-203. [attachment:summ_anova.pdf A PDF copy of this paper is here.]
Fisher LD and van Belle G (1993) Biostatistics: A Methodology for the Health Sciences. John Wiley and Sons, New York.
Howell DC (2002) Statistical Methods for psychologists. Fifth Edition. Duxbury Press:Belmont,CA.