Factorial ANOVA Basic Concepts. Two-Way ANOVA We have two grouping variables, commonly referred to as: –Factors –Independent Variables best term if manipulated.

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Presentation transcript:

Factorial ANOVA Basic Concepts

Two-Way ANOVA We have two grouping variables, commonly referred to as: –Factors –Independent Variables best term if manipulated experimentally –Predictors –Grouping Variables –Classification Variables

We have one continuous variable, commonly referred to as the –Dependent variable best term if data collected experimentally –Criterion variable –Outcome variable –Response variable

A 2 x 2 Design Has two levels of Factor A and two levels of Factor B. This results in four combinations of level of A and level of B. Each such combination is referred to as a cell.

Your Party is Going Strong And then green aliens crash it

Data Collection You and colleagues monitor the aliens’ behavior. Half of them consume your ethanol-based punch. Half of them sample your room-mate’s barbiturate tablets. Post consumption, you blindly rate each alien’s level of intoxication, and you compute cell means.

Green Alien Party Crashers

Main Effects & Marginal Means Effect of Alcohol, ignoring Barbiturates –20 > 10 –Drinking alcohol increased intoxication. Effect of Barbiturates, ignoring Alcohol –25 > 5 –Taking Barbiturates increased intoxication.

Simple Main Effects The effect of one factor at specified level of another factor. Effect of Alcohol when Barb = none 10 – 0 = 10 Effect of Alcohol when Barb = one 30 – 20 = = 10, no interaction

Simple Main Effects Effect of Barb when Alch = none 20 – 0 = 20 Effect of Barb when Alch = one 30 – 10 = = 20, no interaction

Additive Model The effect of adding Alch and Barb is the simple sum of their separate effects. Ingest both A and B, intoxication = A + B = = 30. It does not work this way in humans!

Green Aliens Lines are parallel, no interaction Barb line higher than no-barb line, main effect of barbiturates Slope of both lines is positive, main effect of alcohol

Comparative Psychology For comparative purposes, you also observe human guests, including Pee Dee the Pirate.

Humans

Main Effects & Marginal Means Effect of Alcohol, ignoring Barbiturates –25 > 10 –Drinking alcohol increased intoxication. Effect of Barbiturates, ignoring Alcohol –30 > 5 –Taking Barbiturates increased intoxication.

Simple Main Effects Effect of Alcohol when Barb = none 10 – 0 = 10 Effect of Alcohol when Barb = one 40 – 20 =  20, there is an interaction.

Simple Main Effects Effect of Barb when Alch = none 20 – 0 = 20 Effect of Barb when Alch = one 40 – 10 =  30, there is an interaction.

Nonadditive Combination One drink makes you 10 units intoxicated. One barb makes you 20 units intoxicated Drink & Barb  Drink + Barb, that is, (+ Interaction) = 40, not 30

Monotonic Interaction The lines are not parallel, there is an interaction. Both lines have positive slope, the direction of effect of alcohol is the same at both levels of barbiturate (but effect stronger at barb = one). The interaction is monotonic.

Interpreting a Monotonic Interaction You can still interpret the main effects. Drinking alcohol will make you more intoxicated whether you took a barbiturate or not. --- but if you take a barbiturate, the alcohol will have a greater effect than it would if you had not taken a barbiturate.

Purple Aliens Arrive Later You observe them too.

Purple Alien Party Crashers

Main Effects & Marginal Means Effect of Alcohol, ignoring Barbiturates –15 = 15 –Drinking alcohol did not affect intoxication. Effect of Barbiturates, ignoring Alcohol –20 > 10 –Taking Barbiturates increased intoxication.

Simple Main Effects Effect of Alch when Barb = none 20 – 0 = 20 Effect of Alch when Barb = one 10 – 30 =  -20, there is an interaction.

Simple Main Effects Effect of Barb when Alch = none 30 – 0 = 30 Effect of Barb when Alch = one 10 – 20 =  -10, there is an interaction.

Nonmonotonic Interaction The lines are not parallel, there is an interaction. One line has positive slope, the other negative, the direction of effect of alcohol depends on whether a barbiturate was taken. The interaction is nonmonotonic.

Interpreting a Nonmonotonic Interaction Probably not a good idea to interpret the main effects. What is the effect of alcohol on purple aliens? Main effect is zero, but alcohol does have an effect. The effect depends on whether a barbiturate was taken or not.

Interpreting a Nonmonotonic Interaction How do barbiturates affect purple aliens? Main effect is it makes them more intoxicated -- but if they have been drinking alcohol, the barbiturate has the effect of reducing intoxication. Again, it does not work like this with humans!

Three-Way Factorial ANOVA Three Factors Three Main Effects – A, B, and C Three Two-Way Interactions –A x B, A x C, and B x C One Three-Way Interaction – A x B x C

Factor C = Species of Party Dude We have a Alcohol x Barbiturate x Species ANOVA The A x B interaction differs across levels of Species. Accordingly, we have a triple interaction.

Triple Interaction Green Human Purple

Hypotheses Tested in Two-Way ANOVA  1 =  2 =... =  a –That is, the mean of the criterion variable is constant across the a levels of factor A.  1 =  2 =... =  b –That is, Factor B does not affect the mean of the criterion variable. Factors A and B do not interact with one another, A and B combine additively to influence the criterion variable.