General Form of Linear Contrast


 Linear Contrasts 
 Week 4 Prof. Patrick Bennett Omnibus vs. Focussed F tests • A significant omnibus F test supports a very general hypothesis not all means are equal not all group effects are zero • Significant F doesn’t tell us how group means differ • Generality of omnibus F often comes at cost of reduced power Bennett, PJ PSY710 Chapter 3 Ȳ u is simply the mean of the group means; di↵erences in the size of the groups (if they exist) are ignored, and so Ȳ u is said to be the unweighted mean of the group means. ↵̂ j is simply the di↵erence between the mean of group j and Ȳ u . For the reduced model, setting the one free parameter, μ, to the grand average, Ȳ , minimizes the sum of squared residuals. 3.3.1 F formula Next, we need to derive a quantitative measure of the relative goodness-of-fit of the two models. We denote the sum of squared residuals for the best-fitting full and reduced models as E F and E R , respectively. Associated with E F and E R are degrees-of-freedom df F = N a and df R = N 1, respectively, where N is the total number of observations and a is the number of groups. Note that df R df F = a 1 is the di↵erence between the number of parameters estimated in the full model (3 ↵’s and 1 intercept) and the reduced model (1 intercept). The formula for computing the di↵erence between the two models is F = (E R E F )/(df R df F ) E F /df F (13) Equation 13 can be used to compare all nested linear models. All tests in ANOVA, analysis of covariance, and multiple regression can be computed using this formula. 3.3.2 Null Hypothesis Testing Finally, we are in a position to evaluate the hypothesis of no di↵erence between the goodness-of-fit of the full and reduced models. Note that this comparison is equivalent to evaluating the hypothesis that all of the groups have the same mean; or (equivalently) that all ↵ j ’s are zero. More formally, we are comparing the hypotheses H0 : ↵1 = ↵2 = · · · = ↵a = 0 H1 : ↵ j 6= 0 The null hypothesis is that all of the e↵ects are zero, and therefore that all group means are equal. The alternative hypothesis is that at least one e↵ect is not zero, and therefore that not all group means are equal. When the residuals, e ij , are distributed as independent, normal random variables, with mean of zero and a constant variance, then F in Equation 13 follows a F distribution with (df R df F ) and df F degrees of freedom in the numerator and denominator, repsectively (Figure 1). Under the null hypothesis, therefore, large values of F should be relatively rare (Figure 2). Using 4 Omnibus vs. Focussed F tests Bennett, PJ PSYCH 710 Chapter 4