which switches attention away from the treatment , , to , , means differences in means …, etc. Note that are simple expressions in the and hence constants. ni So the orthogonal transformation has produced a set of variates with the following properties: is normally distributed with mean 0 and variance and is independent of 2, each variate , which are themselves all independent with means 0 and variance and 2, the variates are all independent of the remaining - = - 1) variates which t nt t n(t themselves are all independent with means 0 and variances 2. The of the {} can therefore be separated into three parts: logLikelihood Under the hypothesis that all treatment means are equal (that is, all , or equivalently all ), the second of these three expressions is the in the one-way ANOVA, and is Treatment SS therefore distributed as a variate with degrees of freedom. Moreover, each single t-1 component of the tests a contrast of one set of means against another, and is Treatment SS distributed as a variate with 1 degree of freedom. The final expression in the of the {} is the in the one-way logLikelihood Residual SS ANOVA and is therefore distributed as a variate with = degrees of freedom, nt-t t(n-1) It is also independent of irrespective of whether the treatment means are all equal or not. the Hence Treatment SS. The ratio of to the in the one-way ANOVA is, under the Treatment MS Residual MS hypothesis that all treatment means are equal, distributed as an F variate with t-1 numerator and denominator degrees of freedom. t(n-1) Each individual contrast component F value is distributed as an F variate with 1 numerator and denominator degrees of freedom under the assumption that that t(n-1) particular contrast of treatment means is 0. Note that the can be expressed as Residual MS
which switches attention away from the treatment means , , …, to differences in means , , etc. Note that are simple expressions in the ni and hence constants. So the orthogonal transformation has produced a set of variates with the following properties: is normally distributed with mean 0 and variance 2, and is independent of each variate , which are themselves all independent with means 0 and variance 2, and the t variates are all independent of the remaining nt - t = n(t - 1) variates which themselves are all independent with means 0 and variances 2. The logLikelihood of the {} can therefore be separated into three parts: Under the hypothesis that all treatment means are equal (that is, all , or equivalently all ), the second of these three expressions is the Treatment SS in the one-way ANOVA, and is therefore distributed as a variate with t-1 degrees of freedom. Moreover, each single component of the Treatment SS tests a contrast of one set of means against another, and is distributed as a variate with 1 degree of freedom. The final expression in the logLikelihood of the {} is the Residual SS in the one-way ANOVA and is therefore distributed as a variate with nt-t = t(n-1) degrees of freedom, irrespective of whether the treatment means are all equal or not. It is also independent of the Treatment SS. Hence The ratio of Treatment MS to the Residual MS in the one-way ANOVA is, under the hypothesis that all treatment means are equal, distributed as an F variate with t-1 numerator and t(n-1) denominator degrees of freedom. Each individual contrast component F value is distributed as an F variate with 1 numerator and t(n-1) denominator degrees of freedom under the assumption that that particular contrast of treatment means is 0. Note that the Residual MS can be expressed as