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Two-Factor ANOVA Outline Basic logic of a two-factor ANOVA Recognizing and interpreting main & interaction effects F-ratios How to compute & interpret a two-way ANOVA Assumptions Extension of Factorial ANOVA Factorial Designs Move beyond the one-way ANOVA to designs that have 2+ IVs The variables can have unique effects or can combine with other variables to have a combined effect Why Should We Use a Factorial Design? We can examine the influence that each factor by itself has on a behaviour, as well as the influence that combining these factors has on the behaviour Can be efficient and cost-effective Interpretation of Factorial Designs Two Kinds of Information: 1. Main effect of an IV – Effect that one IV has independently of the effect of the other IV – Design with 2 IVs, there are 2 main effects (one for each IV): Main Effect of Factor A (1st IV): Overall difference among the levels of A collapsing across the levels of B. Main Effect of Factor B (2nd IV): Overall difference among the levels of B collapsing across the levels of A. Interpretation of Factorial Designs Two Kinds of Information: 2. Interaction – Represent how independent variables work together to influence behavior – The relationship between one factor and the DV change with, or depends on, the level of the other factor that is present – The influence of changing one factor is NOT the same for each level of the other factor Two-Way ANOVA F= variance between groups variance within groups In a 2-way ANOVA, there are 3 F-ratios: 1. Main effect for Factor A 2. Main effect for Factor B 3. Interaction A x B Guidelines for the Analysis of a Factorial Design First determine whether the interaction between the independent variables is statistically significant. – If the interaction is statistically significant, identify the source of the interaction by examining the simple main effects – Main effects should be interpreted cautiously whenever an interaction is present in an experiment Then examine whether the main effects of each independent variable are statistically significant. Analysis of Main Effects When a statistically significant main effect has only 2 levels, the nature of the relationship is determined in the same manner as for the independent samples ttest When a main effect has 3 or more levels, the nature of the relationship is determined using a Tukey HSD test Effect Size Three different values of ŋ2 are computed ŋ2 for Factor A = SSA_______ SStotal – SSB - SSAxB ŋ2 for Factor B = SSB_______ SStotal – SSA - SSAxB ŋ2 for Factor AxB = SSAxB______ SStotal – SSA - SSB Effect Size – alternate formulas 2 2 2 A SS A SS A SS within B SS B SS B SS within AxB SS AxB SS AxB SS within Assumptions The observations within each sample must be independent DV is measured on an interval or ratio scale The populations from which the samples are selected have must have equal variances The populations for which the samples are selected must be normally distributed Calculating 2 Factor Between Subjects Design ANOVA by hand Influence of a specific hormone on eating behaviour IV (A): Gender – Males – Females IV (B): Drug Dose – No drug – Small dose – Large dose DV: Eating consumption over a 48-hour period The Data …. Factor B – Amount of drug No drug Factor A - Gender Male Female Small dose Large dose 1 7 3 6 7 1 1 11 1 1 4 6 1 6 4 0 0 0 3 0 2 7 0 0 5 5 0 5 0 3 Homogeneity of variance = s2 largest = s2 smallest Satisfied or violated??? Step 1: State the Hypotheses Main Effect for Factor A Main Effect for Factor B Step 1: State the Hypotheses Interaction between dosage & gender Step 2: Compute df Double Check: dftotal= dfbetween + dfwithin Step 3: Determine F-critical Use the F distribution table F Critical (df effect, df within) Using = .05 Step 4: Calculate SS SSTOTAL = 2 – G2 N Step 4: Calculate SS SSBETWEEN Tx = T2 – G2 n N Step 4: Calculate SS SSWITHIN TX = SS inside each treatment Double Check SS SSTotal SS Within Tx SS BetweenTx SS for Factor A SS A = Trow2 – G2 nrow N SS for Factor B SS B = TColumn2 – G2 nColumn N SS for Interaction SS AxB SS Between SS A SS B Step 5: Calculate MS for Factor A MSA = SSA dfA Step 5: Calculate MS for Factor B MSB = SSB dfB Step 5: Calculate MS for Interaction MSAxB = SSAxB dfAxB Step 5: Calculate MS Within Treatments MSwithin = SSwithin dfwithin Step 6: Calculate F ratios – Factor A MS A F MS within Step 6: Calculate F ratios – Factor B MS B F MS within Step 6: Calculate F ratios – Interaction MS AxB F MS within Step 7: Summary Table Source Between Tx Factor A Factor B Interaction Within Tx Total SS df MS F Extension of Factorial ANOVA 1 factor is between subject & 1 factor is within subject e.g.: pre-post-control design – All subjects are given a pre-test and a post-test – Participants divided into two groups – Experimental group vs. control group 2 x 3 mixed design Group Therapy Time Control Between-Subjects Before After 3 mos. after Within-Subjects