Advanced Statistics - Biology 6030
Bowling Green State University, Fall 2017
Consider the situation where you wish to compare a series of k samples, each containing n values. You are hoping to statistically evaluate whether these samples could have all been derived from the same underlying distribution or whether this scenario is unlikely. You specifically test the Ho: µ1 = µ2 = µ3 ... = µk. To test the null hypothesis that k population means are equal, we will compare two different estimates of variance: one based on the variation of individual data points around their individual sample means [s2(within)], and the other based on an estimate of variance among sample means [s2(between)]. The logic behind this is that s2(within) is always an estimate of the true s2 (assuming that the samples have equal variance). In contrast, s2(between) is only an estimate of the true s2 if your Ho is correct. If we thus calculate the ratio of s2(between)/s2(within) then this value should be close to one under the null hypothesis. We can reject the null hypothesis if this ratio is particularly high, indicating that the variance estimate derived among the means is disproportionately large compared to that derived from the individual data points around their individual sample means.
Step-by-step: Note that when you collect small data sets from the same underlying, normal distribution, the means of these samples will all vary slightly due to chance differences in the actual values sampled. Also variances from different samples will differ from each other due to chance alone. As data points are normally distributed around their sample means with a given variance s2=S(Yi-)/n-1, so the sample means will be normally distributed around a mean of means with a given standard error SE = s/
Synonyms: Note that SSbetween is also referred to as SSmodel or SSregression. SSwithin is the same as SSresidual or SSerror.
Considering an ANOVA table, understand and develop an intuitive feeling for the derivation and meaning of all terms listed below: