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Week10_Notes

# Week10_Notes - Psych 100A Week 10 Discussion Notes Dawn...

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Psych 100A Week 10 Discussion Notes Dawn Chen December 5, 2009 Two-Factor Between-Subjects ANOVA The two-factor between-subjects ANOVA is used when there are two independent variables, and each combination of the levels of the two IVs is given to (or possessed by) a different group of subjects. For example, if the first IV has 3 levels and the second IV has 2 levels (a 3 2 × design), then there are 6 separate groups of subjects (cells). Summary of Notation Used If factor A has 3 levels, factor B has 2 levels, and there are 3 scores in each combination of the levels of factors A and B , then the notation we use looks like the following: Factor A 1 A 2 A 3 A Means for Factor B 1 B 111 X 211 X 311 X Cell mean: 1 1 A B X 121 X 221 X 321 X Cell mean: 2 1 A B X 131 X 231 X 331 X Cell mean: 3 1 A B X Mean for group 1 : B 1 B X Factor B 2 B 112 X 212 X 312 X Cell mean: 1 2 A B X 122 X 222 X 322 X Cell mean: 2 2 A B X 132 X 232 X 332 X Cell mean: 3 2 A B X Mean for group 2 : B 2 B X Means for Factor A Mean for group 1 : A 1 A X Mean for group 2 : A 2 A X Mean for group 3 : A 3 A X Grand mean: G X Partitioning Variation There are now four different sources of variation in a score: The variation due to factor A , the variation due to factor B , the variation due to the interaction of factors A and B , and the variation due to random error. Therefore, the total variation is now partitioned into these four kinds of variation: total A B A B error SS SS SS SS SS × = + + + The following equation shows how to partition a score X . A X represents the mean for the A group that X belongs to, B X represents the mean for the B group that X belongs to, and AB X represents the mean for the AB cell that X belongs to. From each term in this equation, we obtain the corresponding sum of squares by calculating the term for each score, squaring it, and summing them up for all scores. ( ( ( ( total A B A B error G A G B G AB A B G AB SS SS SS SS SS X X X X X X X X X X X X × - = - + - + - - + + - dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketmidhorizcurlybracketexthorizcurlybracketexthorizcurlybracketexthorizcurlybracketextdncurlybracketright dncurlybracketlefthorizcurlybracketextdncurlybracketmidhorizcurlybracketextdncurlybracketright

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Hypothesis Testing In the two-factor between-subjects ANOVA, there are three questions we are interested in answering: (1) Is there a main effect of factor A ? (2) Is there a main effect of factor B ? (3) Is there an interaction between factors A and B (i.e., does factor A affect the dependent variable differently at different levels of factor B )? Each of these questions is answered separately by a different hypothesis test, shown in the following table for a = 3 and b = 2. Thus, we calculate three F values, one for the main effect of A
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