Chapter 14- Two-Factor Anova (Independent Measures) PDF

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Chapter 14: Two-Factor ANOVA (Independent Measures)  Researchers systematically change two or more independent variables and then observe how the changes influence the dependent variable  Factorial Design- research study that involves more than one factor  Two-Factor, Independent-Measures, equal n designs o Contains two factors o Separate sample used for each treatment condition o Sample size is the same for all treatment conditions  F-Ratio o Numerator measures the actual mean differences in the data o Denominator measures the differences that would be expected if there is no treatment effect  Main Effect- the mean differences among the levels of one factor o Two-factor ANOVA allows for evaluation of main effect for each individual factor o When design of research study is represented as a matrix with one factor determining the rows and the second factor determining the columns, then the mean differences among the rows describe the main effect of one factor and the mean differences among the columns describe the main effect for the second factor o Existence of sample mean differences does not imply that the differences are statistically significant—there will always be at least some small difference o Unless hypothesis test demonstrates that the main effects are significant, must conclude that the observed mean differences are simply the result of sampling error o State hypotheses concerning main effect of factor A and then of factor B and then calculate two separate F-ratios to evaluate hypotheses  F = variance between means for factor A/variance expected if there is no treatment effect  F = variance between row means/variance expected if there is no treatment effect  F = variance between means for factor B/variance expected if there is no treatment effect  F = variance between column means variance expected if there is no treatment effect  Interaction- (between two factors) occurs whenever the mean differences between individual treatment conditions are different from what would be predicted from the overall main effects of the factors o F = variance (mean differences) not explained by main effects/variance (mean differences) expected if there is no treatment effects o H0: There is no interaction between factors A and B, the mean differences between treatment conditions are explained by the main effects of the two factors o H1: There is an interaction between factors, the mean differences between treatment conditions are not what would be predicted from the overall main effects of the two factors

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o An interaction between factors also can occur when the effect of one factor depends on the different levels of a second factor o Additionally, when the results of a two-factor study are presented in a graph, the existence of nonparallel lines indicates an interaction between the two factors Independence of Main Effects and Interactions o Two-factor ANOVA consists of three independent hypothesis tests, each evaluating specific mean differences:  The main effect of factor A—evaluates mean differences between rows  The main effect of factor B—evaluates mean differences between columns  The A x B interaction—evaluates mean differences between treatment conditions that are not predicted from the overall main effects from factors A and B  Each evaluated by F = variance between treatments/variance expected if there are no treatment effects  mean square = MS = SS/df  SS total = ∑X2 – (G2/N)  df total = N – 1  SS within = ∑SS each treatment  df within = ∑df each treatment  SS between = SS total – SS within  df between = number of cells – 1  between + within = total  SSA = ∑T row/n row – (G2/N)  dfA = number of rows – 1  SSAXB = SS(between) – SSA – SSB  dfAXB = df (between) – dfA – dfB o Because they’re independent, it is possible for data from a two-factor study to display any possible combination of significant and/or not significant main effects and interactions Effect Size o Compute 3 separate η2 o For factor A, η2 = SSA/(SSA + SS within) Assumptions o Observations within each sample must be independent o Populations from which samples are selected must be normal o Populations from which samples are selected must hva eeqaual variances...