What is two factor analysis of variance?
- builds on the single factor ANOVA
- looks at the effect of two categoircal variables, we can look at the effect that each level has on the numerical variable, as well as the interaction between the categorical variables
How can two factor anova be visualized?
as a two dimensional grid. one factor shown as columns, one as rows
What is the factor?
the categorical variables are called factors, and the different types within each factor are called levels
What is a cell?
the intersection of two levels is called a cell. within each cell, there will typically be several sampling units and numericalvariable is measured on each sampling unit
What 3 questions does two factor ANOVAs answer?
- main effects A are questions about the differences among the levels of factor A averaging across the levels of factor B. These are comparisons among full columns (or full rows)
- Main effects B are questions about the differences among the levels of factor B averaging across the levels of factor A. these are comparisons among full rows (or full columns)
- interactions are questions about the differences among the levels of one factor within each level of the other factor. these are cell by cell comparisons
What is the most valuable part of a two factor ANOVA?
can evaluate whether there is an interaction between the factors
What is an interactions?
a deviation from the assumption that the levels of each factor simply add together. this is called additivity
what is additivity?
Additivity is when the response to the combination of two levels is simply the sum of the two.
What is the best way to visualize interactions?
- interactions plot
- use the same information as a box plot, but present the data in a way that showcases the relationship between the categorical variables
How are interaction plots set up?
- the y axis shows the numerical variable in the same way as box plots
- the x axis shows the levels one categorical variable
- lines are used to connect cells across the x-axis according to the levels of the other categorical variable
If the categorical variables are additive, how does the interaction plot appear?
the lines on an interaction plot will be parallel
if the categorical variables are not additive, how will the interaction plot appear?
the lines of the plot are not parallel
what are the fout steps a two factr anova tests follow for the hypothesis test?
- define the null and alternative hypothesis
- establish the null distribution
- conduct the statistical test
- draw scientific conclusions
Null and alternative hypothesis for main effects A?
null: the means are the same across all levels of factor A
alternative: the means are different somewhere.
Null and alternative hypothesis for main effects B?
null: the means are the same across all levels
alternative: means are different somewhere
Null and alternative hypothesis for interaction?
null: the deviation of each cell relative to additivity is zero
alternative: there is a deviation
What is the null distribution for each of the three F-tests?
- Main Effects A: the null distribution is the sampling distribution from repeatdely sampling a statistical population where the means are the same across all levels of factor A
- main effects B: the null distribution is the sampling distribution from repeatedly sampling a statistical population where the means are the same across all levels of factor B
- interactions: the null distribution is the sampling distribution from repeatedly sampling a statistical population where the cell means are additive
4 Sources of variation for the F-tests in a two factor anova?
- Group Variation Factor A (MSa): the variation among the means of the levels of factor A.
- Group Variation Factor B (MSb): the variation among the means of the levels of factor B
- AB interactions (MSab): the amount of variation attributable to the deviation from additivity
- Residual Variation (MSe): the variation among sampling units within a cell
What are the 3 F-tests built arround different ratios of the sources of variation?
- Main effects a: the F-score is F=MSa/MSe
- Main effects B: The F-score is F=MSb/MSe
- Interactions: The F-score is F=MSab/MSe
what is the null and alternative hypothesis for all f tests?
null: F</= 1
alternative: F>1
How are the scientific conclusions for two factor anova test done?
- Step 1: Evaluate the interaction
- the scientific conclusions are either: reject the null hypothesis and conclude there is evidence that there is at least one cell that deviates from additivity, or fail to reject it
- if the conclusion is to reject the null hypothesis, then the main effects should not be evaluated
- Step 2: evaluation of main effects
* if the interaction is not sig, then the factors are considered additive and it is app. to look at the main effects for each
* reject the null hypothesis and conclude there is evidence that the means of at least two levels are different in the factor
* fail to reject the null hypothesis and conclude there is no evidence that the means among levels are different in the factor
