# Two-Way ANOVA

• Created by: rosieevie
• Created on: 09-01-18 13:31

## Two-Way ANOVA

Use ANOVA whenever you want to compare averages:

• One-way ANOVA tests effect of one factor on response
• Two-way ANOVA tests effects of 2 simultaneous factors on response

Degrees of freedom = no. bits information had  - no. bits information needed

With a two-way ANOVA - divide samples in each treatment into sub-samples representing different level of second factor

• Difference if 2 variables don't response same way - lines not parallel to each other7

Graph will have one response on y-axis and two factors, one on x-axis and one on graph itself

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## Data Tables

Useful to assist in ANOVA by splitting factor levels and showing data points and samples

Factorial design - all cells represent all treatment combinations

Balanced design - each level of one factor is measured against each level of other factor

Gives 3 F values:

• Effect of one factor
• Effect of other factor
• Effect of interactions between two factors

The values are obtained for 5 sources of variation - n scores themselves, the a cell means, the sample mean (`Y), the row means (`R), the c column means (`C) and the single global mean ()

Response = Total explained **: ∑(`Y - Ḡ)2 + Unexplained **: ∑(`Y -`Y)2

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## Degrees of Freedom

• 1st factor d.f. = factor levels - 1
• 2nd factor d.f. = factor levels - 1
• Interacton d.f. = product of two sets of degrees of freedom (1st factor levels - 1) x (2nd factor levels - 1)
• Error d.f. = ((total sample size - factor levels and interaction d.f.)-1) = N - a (number of observations minus the sample means)
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## Two-Way ANOVA in R

Output contains 3 values of F-statistic and signficance of them

3 columns - 1 for each factor and 1 for response

Assumptions:

• Homogenous variances - use Bartlett test to check
• Normality of residuals - use Shapiro-Wilk to test
• Random sampling
• Independence of data points

Always read the R output table from the bottom up - interaction first

• Only write summaries of each finding, don't copy and paste the table
• F must be greater than P to obtain a significant difference
• P value must be below 0.05 to rejct N0
• In general, if interaction is significant, both factor effects must be signficant
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## Interpreting Interaction Plots

Interaction plots illustrate way in which response variable depends on two factors

• Plot response against one of independent effects then plot on the graph the sample means for each level of other indpendent effect
• Means plotted w/out error bars
• Assume each mean only have small residual variation above and below it

8 different types of outcome are possible:

• If lines are parallel to each other - no significant effect betwen two factors
• If values between the x-axis factor are different for each sample, there is a significant effect
• If lines between the plotted factor are difference for each sample, there is a signficant effect

Draw graphs and explain for posters

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## Other Two-Way ANOVA Tests

If two-factored design has no replication within each cell - not possible to look for interaction effects = negligible effects

Therefore, use Latin square - used in situations where single main effect is tested but in presence of 2nd 'nuisance' effect:

• Lay out sampling areas in structured pattern, rather than random allocation
• Each 4 levels of factor sampled against each 4 levels of other factor = orthogonal design
• Test model - Response = Factor + Block
• Response is tested against main factor and blocking variable with error mean square provided by unexplained interaction Factor:Block
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