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10 1 From t-test to one-way anovawith n 1 + n 2 2 degrees of freedom. This takes a little thought as it may appear as though^ ^we have fitted three parameters but, in fact, there are really just the two means m 1 and m 2.Our use of the treatment effects a 1 and a 2 makes it easy to move from one model to theother (by setting them both to zero) but you can easily see that^ ^ ^ì = ( m × n + m × n ) / ( n + n )1 1 2 2 1 2^ ^ ^ ^= {( ì + a 1 ) × n 1 + ( ì + a 2 ) × n 2 }/( n 1 + n 2 )and so^^a × n = a × n .1 1 2 2The difference between these two sums of squares is known as the sum of squares dueto the treatments. This measures the effect of allowing for two different means, and hasone degree of freedom. We can assess whether this exceeds the underlying level ofvariability by comparing it with RSS 1, but first we need to divide each one by its degreesof freedom to give the treatment and residual mean squares; this takes account of thedifferent number of parameters that each one represents. By dividing the treatment meansquare by the residual mean square, we obtain a statistic known as the variance ratio. Ifwe assume that the residuals follow a Normal distribution, the variance ratio will havean F distribution on 1 and (n 1 + n 2 2) degrees of freedom. (The degrees of freedom arethe degrees of freedom for the nominator that is the sum of squares due to treatments and those for the denominator that is the residual sum of squares.) The variance ratioisIt is interesting to note that, when there are only two treatments, the variance ratio is thesquare of the t-statistic. You can verify this in the example below, or prove it from thefollowing equations:and so