Ch. 12 - Experiments with More Than One Independent Variable and Factorial ANOVA

Complex designs

- Factorial design = any design w/ more than 1 IV, advantage = more closely resembles the real world because the results are due to more than 1 factor (variable) - Factorial notation = the numerical notation corresponding to a factorial design. It indicates, in brief form, the # of IVs & the # of levels of each variable, a 3 x 4 design has 2 IVs, 1 w/ 3 levels & 1 w/ 4 levels - Main effect = an effect of a single IV. A main effect describes the effect of a single variable as if there were no other variables in the study, in a study w/ 2 IVs, 2 main effects are possible-1 for each variable - Interaction effect = the effect of each IV @ the levels of other IV, interaction effects allow us to assess whether the effect of 1 variable depends on the level of the other variable. In this way, we can more closely simulate the real world, where multiple variables often interact

Results write up for interaction effect

There was a significant interaction effect, F(1,28) = 38.76,p< .05,η2 = .58. The Fobtof 38.76 exceeded theFcv= 4.20 . Therefore, the researchers rejected the null that there was no interaction between the factors. However, there is the possibility of a type I error in that the researchers rejected the null when the null was true. Participants recalled more concrete words under imagery rehearsal (M= 10 ) than under rote rehearsal (M= 5). There was no difference for recall of abstract words in the imagery(M=5) and rote (M=5) rehearsal conditions. Nearly 58% of the variability in words recalled is explained by the interaction between Word type and Rehearsal Type.

Results write up for main effect

There was a significant main effect for Word type, F(1,28) = 13.95, p< .05,η2 = .33. The Fobtof 13.95 exceeded the Fcv= 4.20 . Therefore, the researchers rejected the null that there was not difference between abstract and concrete words. However, there is the possibility of a type I error in that the researchers rejected the null when the null was true. The concrete group (M= 7.5) recalled more words compared to the abstract group (M= 5). Approximately 33% of the variability in the number of words recalled is explained by Word type.

Factorial ANOVA

•More than 1 IV •Two IV's -Two wayANOVA •Three IV's -Three wayANOVA Allows for testing an interaction effect How the influence of one IV changes the DV depending on the level of the other IV. - analysis of variance - DV = int/ratio - more IV = more participants needed making more comparisons

Mixed factorial designs

•One IV is manipulated as independent-groupsand the other is manipulated within-groups. •This design is intermediate betweenthe within-groups design andthe independent-groups design in terms of number of participants. - 2 types of IVs (b/t & w/in) = mixed model

Effect size for significant results eta squared n2

•Percentage of variability in the dependent variable that is accounted for by the Independent variable. •Ex: Percentage of variability in the number of words recalled that is explained by the independent variables. - f test = stat sig effect size - look @ SS - describe SS - proportion or % ex) 33 % of the variability in the # of words recalled is explained by rehearsal type

Main effect (marginal means)

•Test for the difference between the levels for one independent variable at a time. •There are as many main effects as there are independent variables in the Factorial design. •Main Effect F test in Factorial ANOVA - for each IV, main effect ANOVA - means of cond. regardless of IV each IV tested as main effect

Interaction Effect

•The effect of one independent variable on the dependent variable may not be the same at all levels of the other independent variable. •The effect of one independent variable may depend on the level of the other independent variable. - looks @ both IVs at the same time •EX) Does the effect of the cell phone depend on the age of the driver? Does using the cell phone affect younger drivers more than older drivers?

2 x 2 factorial design

•Using factorial designs to study manipulated variables (researcher in control of) or participant variables (ex- age, can't ran. assign.) 2 x 2, compares 4 cells - factorial design = exper. in which there are 2 or more IVs (factors)

Two-way randomized ANOVA

- Null hypothesis (H0) = the IV had no effect; the samples all represent the same pop. In a 2 way ANOVA, there are 3 null hypotheses: 1 for factor A, 1 for factor B, & 1 for the interaction of factors A & B - Alternative hypothesis (Ha) = the IV had an effect; @ least 1 of the samples rep. a diff. pop. than the others. In 1 2 way ANOVA, there are 3 alt. hyps: 1 for factor A, 1 for factor B, 1 for interaction of factors A & B - F-ratio = the ratio formed when the between-groups variance is divided by the w/in groups variance. In a 2 way ANOVA, there are 3 F-ratios: 1 for factor A, 1 for factor B, 1 for the interaction of factors A & B - Between-groups variance = an estimate of the variance of the group means from the grand mean. In a 2 way ANOVA, there are 3 types of between-groups variance: that attributable to factor A, that attributable to factor B, & that attributable to interaction of factors A & B - Within-groups variance = an estimate of the variance w/in each condition in the experiment; also known as error variance, or variance due to chance - Eta-squared (n2) = a measure of effect size-the variability in the dependent variable attributable to the independent variable. In a two-way ANOVA, eta-squared is calculated for factor A, for factor B, & for interaction of factors A & B - Tukey's post hoc test = a test conducted to determine which conditions from a variable w/ more than 2 conditions differ significantly from each other

Interaction effects

- use line graph - if interaction = lines would touch or cross parallel lines = no int. effect b/t IVs - Testing limits = a form of external validity - EX) "the drunk group was more aggressive than the placebo group, especially for heavy men" "child chess experts can remember more chess pieces than digits, but adult chess novices can remember more digits than chess pieces"

EX) of 2 x 2 factorial design

IV1: Word type (concrete; abstract) Main effect IV2: Rehearsal Type (Rote; Imagery) Main effect DV: Memory scores Word type X Rehearsal Type: Interaction effect Null hypothesis Each IV and the interaction between them has a null that states there is no difference. Factor A- No main effect Factor B - No main effect No Interaction between A X B Alternative hypothesis The alternative says there is a difference somewhere between the conditions or an interaction. - source table Main effect for word type: F (1,28) = 13.95, p ? Go to the F critical value table to determine whether the main effect is significant at the .05 level. •Fcv(1,28) = 4.20 - exceeds FCV, reject null = there is a diff. that is stat sig Main effect for rehearsal type: F (1,28) = 75.97,p ? Go to the F critical value table to determine whether the main effect is significant. •Fcv(1,28) = 4.20 - rej null = sig diff Test for interaction effect •Word Type X Rehearsal Type on Recall of Words F (1,28) = 38.76, p ? Go to the F critical value table to determine whether the main effect is significant. Fcv(1,28) = 4.20 - stat sig, look @ avgs for int & M, summarize in description to int. results

Factorial Design

More than one Independent variable (Factors) Notation- IV1X IV2 (levels x levels) Example 1 - IV1: Gender (m/f) IV2: Age (young/old) 2 X 2 factorial (design) = 4 ind. cells (unique conditions) that we compare to see interactions b/t var. Example 2-IV1: Gender (m/f) IV2: Drug Treatment 2 X 3Factorial (drug, placebo, control) - 3 lvls = 6 cells factorial analysis of variance = factorial anova