11-08 math

# This gives the amount of variability explained by the

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Unformatted text preview: nd the grand mean, M i " M . This difference is explainable by the treatment, because Mi represents a group of subjects who received a different type of treatment than the subjects in other groups. Therefore this second part defines SStreatment. ! 2 To calculate SSresidual, we add up ( X " M i ) for all the subjects. This gives the amount of variability that cannot be explained by the treatment, i.e. the variability that exists within the groups (this is why some people use the name SSwithin). This sum is really a sum of 2 sums, because we first have to add up ( X i " M i ) for all the subjects in each group ! separately, and then we add up the results from all the groups. 2 2 SSresidual = # ( X1 " M1 ) + # ( X 2 " M 2 ) + ... + # ( X k " M k ) =# i ! (# ( X i " M i ) 2 ) 2 (3) 2 To calculate SStreatment, we add up ( M i " M ) for all the subjects. This gives the amount of variability explained by the treatment, i.e. by differences among the groups (this is why ! some people use the name SSbetween). Because there are multiple subjects in each group, 2 m ( M i " M ) will be added ! ultiple times, once for each subject in Group i. Therefore the 2 faster way to calculate SStreatment is to calculate ( M i " M ) just once for each group, and multiply it by the number of subjects in the group, ni. Then we can simply add up ! 2 ni ! ( M i " M ) over all the groups. SStreatment ! 2 = \$ n i " ( M i # M ) (4) i Test Statistic for ANOVA. ANOVA works by testing whether the variability among the groups is greater than would be expected by chance. We do this by converting the ! treatment sum of squares to a mean square. MStreatment = SStreatment df treatment (5) It turns out that, according to the null hypothesis, the expec...
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## This document was uploaded on 02/25/2014 for the course PSYC 3101 at Colorado.

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