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


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

Header = T - means data file does have header titles


  • 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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