Course Analysis of Variance Topic Contrasts 1 CONTRASTS AMONG MEANS In the previous class we discussed the general formula for an ANOVA when testing the omnibus hypothesis that all population means are the same H0 1 2 3 j When we reject the H0 we conclude that the populations are not all the same When there are more than two populations however we do not know which populations differ That is some of the populations may differ and some may be equivalent When there are more than two populations we need to perform further comparisons or contrasts among the populations to determine exactly which populations differ To make our discussion of contrasts more concrete we ll continue our depression example in which we measured depression larger scores indicate greater depression following smiling therapy exercise therapy or no therapy Our bogus data are as follows No Therapy 7 7 6 Smiling Therapy 2 1 1 Exercise Therapy 3 1 2 x 6 67 s 0 58 x 1 33 s 0 58 x 2 00 s 1 00 Recall that the test of the omnibus null was significant F 2 6 45 60 p 0002 indicating that the average depression scores among the therapies were not the same At this point we would like to conduct contrasts to determine which populations differ It s possible that we formulated specific comparisons prior to looking at the data Alternatively we could develop comparisons after looking at the data e g it appears that depressions scores are smaller for persons experiencing smiling or exercise therapy than for persons who received no therapy Whether the contrasts are formulated a priori or post hoc is important and will be discussed in a later class For the moment we will discuss general procedures for testing contrasts CONTRASTS A contrast is a linear combination of population means in which the coefficients of the means sum to zero For example the null hypothesis that smiling therapy is no more effective than no therapy H 0 n s can be re expressed as a linear combination of population means H 0 n s 0 In general a contrast involves the summation of population means that are weighted by coefficients that reflect the hypothesized relation among the means c1 1 c2 2 c3 3 c j j In our depression example in which we test whether smiling therapy differs from no therapy the coefficients are 1 1 and 0 for the no therapy smiling therapy and exercise therapy conditions respectively 1 n 1 s 0 e Course Analysis of Variance Topic Contrasts 2 a A contrast can be represented more compactly as c j j and the lower case Greek letter psi j 1 a is used to represent the value of a contrast c j j j 1 Pairwise and Complex Comparisons There are two general categories of contrasts pairwise and complex Although the process and logic of testing pairwise and complex contrasts are identical the distinction is necessary when we discuss issues relevant to the testing of multiple contrasts Pairwise Contrasts Pairwise contrasts involve comparison of two populations The two populations of interest are weighted with coefficients that sum to zero and all remaining populations are weighted with a coefficient of 0 In our depression example with three populations there are a maximum of three pairwise comparisons 1 n 1 s 0 e 1 n 0 s 1 e 0 n 1 s 1 e Complex Contrasts Complex contrasts involve comparison of more than 2 populations The contrast again is a linear combination of population means weighted by coefficients that reflect the hypothesized relations among the populations In our depression example we could test whether the smiling therapy and exercise therapy are better than no therapy 2 n 1 s 1 e Notice that this complex comparison involves information from three populations If we had a fourth sample in our depression study i e persons receiving hugging therapy and wanted to compare the average of smiling and exercise with no therapy we would simply weight the hugging therapy population mean with 0 2 n 1 s 1 e 0 h As we will soon discuss the contrast is tested with an F ratio The formula of the F ratio is such that the specific values of the coefficients within a set are not important as long as they express the desired proportionate weighting of the means For example we can test whether smiling therapy and exercise therapy differ from no therapy with any of the following contrasts 1 n 5 s 5 e 2 n 1 s 1 e 4 n 2 s 2 e 1000 n 500 s 500 e The above contrasts are logically equivalent and will produce the same results However if we change the proportionate weightings within a set the meaning of the contrast changes For example the following contrasts do not test the same hypothesis as do the 4 previous contrasts 1 n 25 s 75 e 4 n 1 s 3 e The latter contrasts also compare the no therapy population with smiling and exercise therapy However they combine smiling and exercise therapy in a manner that weights exercise therapy Course Analysis of Variance Topic Contrasts 3 three times heavier than smiling therapy Whether such a weighting is meaningful depends upon the research question The important point however is that different hypotheses are tested by changing the proportionate weighting of the contrast coefficients The F Ratio as a General Formula for Testing a Contrast A contrast is tested with the general formula for an F ratio that compares the simplicity and error associated with the full and restricted models ER EF F EF df R df F df F For example the H0 and H1 for the contrast of whether no therapy is different from the average of smiling and exercise therapies 2 n 1 s 1 e are H 0 2 n s e 0 or with the abreviatio n H 0 0 H 1 2 n s e 0 or with the abreviation H 1 0 The full model which is derived from H1 is Yij j ij and the estimated error of this model EF is determined by squaring the deviations of each observation from the sample mean of the corresponding group i e the least square estimate of j n n i 1 i 1 E F ei2 Yi Y j 2 As we discussed previously degrees of freedom are generally determined by subtracting the number of independently estimated parameters from the number observations df N a Because the full model estimated the population of each mean with the mean of the sample degrees of freedom for the full model is determined as dfF N a The restricted model that is derived from H0 is also of the form Yij j ij and error is calculated by summing the squared deviations of each score from the least square estimator of j However the calculation of the least square estimator of the population mean is complicated by the restriction in the H0 that the weighted
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