Statistical Techniques IIEXST7015Post-ANOVA TestsContrasts19b_Post ANOVA_Contrasts 1Overview of ANOVARecall that we are testing for differences among indicator variables. The treatments may be fixed or random.H0: µ1= µ2= µ3= ... = µkfor fixed effects.H0: σ2τ= 0 for random effects. Assume ei ∼NIDrv(0,σ2). Remember that this covers 3 separate assumptions. Also, assume no block "interactions" for the RBD. 19b_Post ANOVA_Contrasts 2Overview (continued)Every analysis can be expressed as a model with appropriate notation and subscripting. CRD : Yij= µ + τi+ εij For the moment we will be concerned only with examining for differences among the treatment levels. We will assume that we have already detected a significant difference among treatments levels with ANOVA. 19b_Post ANOVA_Contrasts 3Overview (continued)Treatments levels may be fixed or random. Determining appropriate tests depends on recognizing correctly. With random effects we are probably not interested in individual treatment levels. We are likely to be interested in the variability among the treatment levels and the distribution of the levels. With fixed effects we will probably want to compare individual levels. 19b_Post ANOVA_Contrasts 4Post ANOVA testsHaving rejected the Null hypothesis we wish to determine how the treatment levels interrelate. This is the "post-ANOVA" part of the analysis. These tests fall into two general categories. Post hoc tests (LSD, Tukey, Scheffé, Duncan's, Dunnett's, etc.)A priori tests or pre-planned comparisons (contrasts)19b_Post ANOVA_Contrasts 5Post ANOVA (continued)A priori tests are better. These are tests that the researcher plans on doing before they gather data, and if we dedicate 1 d.f. to each one we generally feel comfortable doing each at some specified level of alpha. 19b_Post ANOVA_Contrasts 6Post ANOVA (continued)However, since multiple tests do entail risks of higher experiment wide error rates, it would not be unreasonable to apply some technique, like Bonferroni's adjustment, to insure an experimentwise error rate of the desired level of alpha (α). So how might we do these "post hoc" tests? 19b_Post ANOVA_Contrasts 7Post ANOVA (continued)The simplest approach would be to do pairwise test of the treatments using something like the two-sample t-test. This tests examines the null hypothesisH0: µ1= µ2 or H0: µ1- µ2= 0, against the alternative Ha:µ1-µ2 ≠ 0, or Ha:µ1-µ2≥ 0 or Ha:µ1-µ2 ≤ 0. 19b_Post ANOVA_Contrasts 8Post-ANOVA tests The test we have seen so far are often (usually?) done with no a priori tests in mind. We do not have certain comparisons in mind before doing the experiment, we want to examine many, or all, levels of the treatments for differences from one another. The experimentwise error rate is intended to allow this (except for the LSD). 19b_Post ANOVA_Contrasts 9Post-ANOVA tests (continued)However, sometimes we do have some particular comparisons in mind when we do an experiment. When we want some lesser number of comparisons, and they are determined a priori (without looking at the data), then we can use a less stringent criteria. 19b_Post ANOVA_Contrasts 10Post-ANOVA tests (continued)We generally feel comfortable with one test per degree of freedom at some specified level of alpha (α), just as we did in regression (looking at each regression coefficient with an α level of error). This is the case with a priori contrasts. 19b_Post ANOVA_Contrasts 11ContrastsRecall out discussion of linear combinations from Multiple Regression Ai= aXi+ bYi+ cZi Var(Ai)=a*Var(Xi) + b*Var(Yi) + c*Var(Zi) +2*Covariances Var(Ai) = a2σ2Xi+ b2σ2Yi+ c2σ2Zi+2(abσXi,Yi+ acσXi,Zi+ bcσYi,Zi) If the variables are independent we can assume all covariances are zero. 19b_Post ANOVA_Contrasts 12Contrasts (continued)In multiple regression we did not assume that the regression coefficients were independent. However, in ANOVA we do consider the levels of a treatment to be independent. 19b_Post ANOVA_Contrasts 13Contrasts (continued)Suppose we want to test the mean of two groups against the mean of 3 other groups. H0: (µ1+ µ2)/2 = (µ3+ µ4+ µ5)/3 H0: (µ1+ µ2)/2 - (µ3+ µ4+ µ5)/3 = 0 H0: 1/2µ1+1/2µ2-1/3µ3-1/3µ4-1/3µ5= 0 19b_Post ANOVA_Contrasts 14Contrasts (continued)The variance of a mean is σ2/n. In ANOVA all of the σ2are equal to MSE. The n may or may not be equal. Since we do not need the covariances we can calculate the variance as a linear combination, H0: 1/2µ1+1/2µ2-1/3µ3-1/3µ4-1/3µ5= 0 Var: (1/2)2*MSE(1/n1) + (1/2)2*MSE(1/n2) + (-1/3)2*MSE(1/n3) + (-1/3)2*MSE(1/n4) +(-1/3)2*MSE(1/n5) =MSE(1/4n1+1/4n2+1/9n3+1/9n4+1/9n5) 19b_Post ANOVA_Contrasts 15Contrasts (continued)Note that we already saw this for the two sample t-test as t = (⎯Y1-⎯Y2) / √(MSE((1/n1+1/n2))). If the design is balanced we can simplify this to t = (⎯Y1-⎯Y2)/√(2MSE/n). Of course, this is still true for 2 means. We also saw another type of application for linear combinations. 19b_Post ANOVA_Contrasts 16Contrasts (continued)If you want to test an hypothesis between two or more independent estimates like,H0: µ1= 0.5µ2or µ1- 0.5µ2= 0 We note that since these are independent, the variance for this t-test will be 12Var(µ1) + 0.52Var(µ2) MSE(1/n1+0.25/n2)This is for two means. 19b_Post ANOVA_Contrasts 17Contrasts (continued)For more means we have a t-test that looks like the following. H0: (µ1+ µ2)/2 - (µ3+ µ4+ µ5)/3 = 0H0: 1/2µ1+1/2µ2-1/3µ3-1/3µ4-1/3µ5= 0 And the variance is MSE(1/4n1+1/4n2+1/9n3+1/9n4+1/9n5) 19b_Post ANOVA_Contrasts 18Contrasts (continued)The general formula, H0:m1µ1+m2µ2+m3µ3+m4µ4+m5µ5=0Linear combo: miµ3, where we will call the mithe "multipliers", estimate µiwith ⎯Yiand the variance with MSE. Test statistic: Σmi⎯Yi Variance: (MSE/n)Σm2iFor balanced data with all n equal. 19b_Post ANOVA_Contrasts 19Contrasts (continued)So all we need are the multipliers. These are often called the "contrast". I prefer these as integers. Note for the example we have examined. H0: 1/2µ1+1/2µ2-1/3µ3-1/3µ4-1/3µ5= 0 If we multiply by 6 we get, H0: 3µ1+3µ2-2µ3-2µ4-2µ5= 0 The multipliers are 3, 3, -2, -2, -2 instead of 1/2, 1/2, -1/3, -1/3, -1/3. 19b_Post ANOVA_Contrasts 20Contrasts (continued)Coming up with the multipliers. The multipliers should be a priori contrasts of interest to the investigator. As such, there are no "right or wrong" contrasts as long as