MANOVA

Question 1

one way ANOVA

Answer 1

• Previous lectures have looked at predicting variance (the relationship) in continuous, normally distributed dependent variable
• ANOVA is an extension of this – we are using ANALYSIS OF VARIANCE to tell us something about means
• Focus is on situations with a continuous, normally distributed dependent variable, and a nominal/categorical independent variable with 3+ different levels (e.g., a variable with 3 or more different groups)
  ANOVA is basically the same as regression – they are both LINEAR MODELS. If you understand one you understand the other

Question 2

how does ANOVA work?

Answer 2

• Pretty much just compares the amount of variability between groups (good variance; means our IV is doing something) to the amount of variability within groups (bad variance; error/noise caused by something outside of our IV)
• ANOVA protects the familywise error rate
• ANOVA is an OMNIBUS test
  
  • It tells us if there a significant effect, but does not tell us where the effect is (i.e. is there a difference between group A and B, group B and C, group A and C… etc.
  • If our ANOVA is not significant we are not justified in examining group contrasts individually.

Question 3

how does MANOVA work?

Answer 3

• Traditional MANOVA is a (between-subjects) one-way ANOVA with multiple DVs
  
  • So far, we’ve only looked at statistical tests that analyse the effect of one or more IVs on a single DV (would need to look at separate DVs using separate tests)
  • The “M” stands for multivariate, which relates to the fact that we’re now looking at multiple dependent variables at once in a single test
  • More complex variants exist for within-subjects designs, but we won’t cover them
• MANOVA assesses whether your IV has an overall effect on your DVs taken together
  
  • In our example: Does pet ownership effect mental health?
  • This is assessed using a multivariate test that produces a single p value
  • Similar to the concept of the “omnibus test” from ANOVA
  • If it is significant, then you break it down into which specific DVs the IV effects

Question 4

why do you use manova?

Answer 4

• There are two key reasons that those who like MANOVA argue for its usage
• (1) Situations where part of your research question is about whether an IV has an overall effect on all of the DVs
  
  • For example: Does pet ownership generally effect mental health?
  • Usually only happens in cases where the individuals DVs are smaller parts of a larger construct. For example: Different aspects of mental health
  • I think this is a good argument!
  • Very research question dependent though
• (2) Controlling error rates by avoiding multiple testing
  
  • Think of why we do ANOVA instead of many t-tests; more risk of a false positive
  • However, this logic is a bit weaker in my opinion
  • For ANOVA, we can ignore non-significant effects, and significant effects that we break down further, we correct for (e.g., Bonferroni)
  • For MANOVA, the multivariate test is either significant (we run all the ANOVAs we would have run anyway, without correction), or non-significant (we don’t run any follow-up ANOVAs)
    This is particularly problematic as there’s no single method for doing the multivariate test, and MANOVA can lack power in some situations, creating some real risks of false negatives

Question 5

the different multivariate tests

Answer 5

• Pillai’s Trace
  
  • Quite conservative: Lower power, but robust against violations of assumptions
  • The one we’ll use in our examples
• Wilks’ Lambda
  
  • The (apparently) most commonly used one
  • Not as robust against violations of things like homogeneity of covariances
• Roy’s Largest Root:
  
  • Quite liberal, but the least robust against violations of assumptions
    Very powerful when 1 DV is pushing the effect in the multivariate test

Question 6

mixed ANOVA vs MANOVA

Answer 6

• You’ll potentially run into many situations where your data could fit into many different statistics testing frameworks, and you need to choose which to use
• Usually, the choice will be based on your research question, as it is here
• MANOVA is usually for situations where the DVs are separate measures that fit into a larger construct, and you want to know both overall and individual effects
• Mixed ANOVA is for situations where you want to know whether the measures differ from one another (e.g., you want to know whether anxiety is lower at post-test compared to pre-test)
  The data look similar, but the tests tell us different things, making them better suited to different situations and research questions

Question 7

assumptions

Answer 7

• The standard ones:
  
  • DV should be at the scale level
  • Data should be normally distributed
• The standard between-subjects ANOVA one:
  
  • Equal Variances
  • Keep in mind that this needs to be done for each of the DVs!
• The bullet we dodged in mixed ANOVA by using linear mixed effects models:
  Equality of covariance matrices

Question 8

equality of covariance matrices

Answer 8

• A really fancy way of saying that the correlation between measurements of the within-subjects IV should be the same across the groups of the between-subjects IV
• Box’s test can be used to test this, but it’s very sensitive to potential violations
  
  • Because of this, people often do the significance test using an alpha level of 0.01 (or even 0.001) instead of 0.05
    However, for simplicity, we’ll still think about it using the standard 0.05

Question 9

normality in MANOVA

Answer 9

• Independent normality tests isn’t quite right…
• In truth, you want the multivariate residuals to follow a multivariate normal, but this test is much less straight-forward (definitely beyond the scope of this lecture)
• As long as things aren’t too non-normal, Pillai’s should be alright to use
  
  • This is essentially why I’m only using Pillai’s
  • However, non-normality is then potentially an issue for the individual ANOVAs…
  • Particularly for Emotional Intelligence…
    Remember the whole “pragmatic” thing I talked about last semester