assess whether the expected values of a quantitative variable within several pre-defined groups differ from each other. For example, suppose that a medical trial compares four treatments. The ANOVA F-test can be used to assess whether any of the treatments are on average superior, or inferior, to the others versus the null hypothesis that all four treatments yield the same mean response. This is an example of an "omnibus" test, meaning that a single test is performed to detect any of several possible differences. Alternatively, we could carry out pairwise tests among the treatments (for instance, in the medical trial example with four treatments we could carry out six tests among pairs of treatments). The advantage of the ANOVA F-test is that we do not need to pre-specify which treatments are to be compared, and we do not need to adjust for making multiple comparisons. The disadvantage of the ANOVA F-test is that if we reject the null hypothesis, we do not know which treatments can be said to be significantly different from the others, nor, if the F-test is performed at level α, can we state that the treatment pair with the greatest mean difference is significantly different at level α. The formula for the one-way ANOVA F-test statistic is or The "explained variance", or "between-group variability" is where denotes the sample mean in the i-th group, is the number of observations in the i-th group, denotes the overall mean of the data, and denotes the number of groups. The "unexplained variance", or "within-group variability" is where is the jth observation in the ith out of groups and is the overall sample size. This F-statistic follows the F-distribution with degrees of freedom and under the null hypothesis. The statistic will be large if the between-group variability is large relative to the within-group variability, which is unlikely to happen if the population means of the groups all have the same value. Note that when there are only two groups for the one-way ANOVA F-test, where t is the Student's statistic. Regression problems Consider two models, 1 and 2, where model 1 is 'nested' within model 2. Model 1 is the restricted model, and model 2 is the unrestricted one. That is, model 1 has p1 parameters, and model 2 has p2 parameters, where p1 < p2, and for any choice of parameters in model 1, the same regression curve can