MANOVA/PERMANOVA Lecture Notes

ENVX2001 - Applied Statistical Methods: Lecture 6 - MANOVA/PERMANOVA

Outline

  • Multivariate data and extending ecology.
  • Hypothesis testing and data exploration.
  • Important techniques:
    • Cluster analysis
    • nMDS
    • Monte Carlo approaches
    • PCA
    • MANOVA/PERMANOVA

Lecture Focus

  • MANOVA
    • Uses in ecology.
    • Comparison to ANOVA.
    • Process and theory underpinning it.
    • Choosing a test statistic.
    • Corrections for multiple comparisons.
  • PERMANOVA

References

  • Quinn and Keough (2002) - Chapter 16, 18
  • Quinn and Keough (2023) - Chapter 14, 15, 16
  • Anderson, M. J. (2001). A new method for non-parametric multivariate analysis of variance. Aust. Ecol. 26:32 – 46.
  • Somerfield, P.J., et al. (2021). Analysis of similarities (ANOSIM) for 2-way layouts using a generalised ANOSIM statistic, with comparative notes on Permutational Multivariate Analysis of Variance (PERMANOVA). Aust Ecol 46:911-26.

Take-Home Messages

  • Decision-making and options.
  • Different approaches to MANOVA
    • Rationale
    • Pros and Cons
    • Interpretation of outputs
  • Parallels to PERMANOVA

MANOVA - The Questions

  • Same as ANOVA, but on linearly combined dependent variables instead of just one.
  • Examples:
    • "Why the spider crossed the web": How St Andrew’s Cross spiders invest in their webs (relative to food, habitat, etc.).
    • "Plant fitness and restoration of degraded landscapes": Responses of target species to revegetation (with respect to how "good" they are after a certain period). Use a phytometer to assess effects of landscape management.

Examples of Variables for a MANOVA

  • Spider web:
    • Stabilimenta
    • Radial threads
    • Spiral threads
    • Distance between cells
  • Prickly Parrot-pea Dillwynia sieberi (Fam. Fabacea):
    • Biomass (↑)
    • Mortality (↓)
    • Height (↑)
    • Flowers (◆)
    • Seed set (↑)

Example: Winter Flounder Growth Rates and Habitat Quality (Menge et al. 2000)

  • Benthic organisms in cores and food items in stomachs were analyzed.
  • Locations: Point Judith, Ninigret, Green Hill.
  • Variables Measured:
    • Benthic organisms: Polychaetes, Copepods, Crustaceans, Amphipods, Other, Total, No. of species.
    • Stomach contents: Stomach fullness, Polychaetes, Copepods, Crustaceans, Amphipods, Other, No. of species.
  • MANOVA tables were presented for benthic organisms in cores and stomach contents with sources, degrees of freedom, Wilks’ lambda, F, and p-values for factors like Vegetation, In/out, and Pond.

Why MANOVA?

  • Separate ANOVAs ignore relationships between dependent variables (lose information about correlations).
  • Multiple univariate tests inflate overall Type I error rate.
  • Effects along multiple dimensions, compared to single dimensions.
  • Combination of DVs gives MANOVA greater power to detect differences.
  • Requires a good theoretical basis for combining variables.
  • Avoid including variables just because they were measured.

MANOVA - The Process/Meaning

  • Multivariate extension of t-test and univariate ANOVA.
  • Extension of ANOVA - main effects and interactions are assessed on a combination of dependent variables.
  • MANOVA tests whether mean differences among groups on a combination of dependent variables are likely to occur by chance.
  • Similar to ANOVA in interpretation, test statistics, and post-hoc tests.
  • Similar theory underpins it.
  • Shortcomings?

Basic Requirements for MANOVA

  • 2 or more dependent variables (continuous or ratio).
  • 1 or more categorical independent variables (nominal or ordinal).

Kinds of Research Questions Asked by MANOVA

  • Same as ANOVA, just on the linearly combined dependent variables instead of just one (maximizes the difference between groups).
  • In factorial designs, a different linear combination of the dependent variables is created for each main effect and interaction that maximizes the group difference separately.

Questions Addressed by MANOVA

  • Are there any main effects?
    • Holding all other effects constant, is a difference among groups greater than expected by chance?
  • Which things are most important?
    • Which dependent variables are most important?
      • For significant main effects or interactions, on which individual dependent variable is there the most change (difference) “caused” by the levels of the independent variable?
      • Follow a significant MANOVA with individual ANOVAs to see the extent of effects on dependent variables.
    • Which levels of the independent variable are significantly different?
      • If there are significant main effects on independent variables with more than two levels, test which levels are different from each other.
      • If there are interactions, the interactions need to be taken apart so that the specific causes of the interaction can be uncovered.

Assumptions of MANOVA

  • Independence – observations should be statistically independent.
  • Random sampling – data should be randomly sampled from the population of interest.
  • Multivariate normality – DVs collectively have multivariate normality within groups.
  • Homogeneity of covariance matrices – same as for ANOVA plus correlations between any DVs are the same in each group.

Fundamental Considerations

  • Interpretation of MANOVA results always in the context of the research design.
  • Clever statistics and fancy graphics won’t make up for poor design.

Missing Data, Unequal Samples, Number of Subjects, and Power

  • Missing data needs to be handled in the usual ways.
  • Unequal samples cause non-orthogonality - total sums of squares is less than all of the effects and error added up.
  • Power in MANOVA depends on the relationships among the dependent variables.
    • Highly correlated dependent variables weaken the power of the analysis.
  • PERMANOVA as a nonparametric option?

Multivariate Normality

  • Assumes that the means of the various dependent variables in each cell and all linear combinations of them are normally distributed.
  • Difficult to show explicitly.
  • In practice, if individual variables are normal, then they should have MVN.
  • Key is to assess univariate normality via:
    • Plotting data to look for skewness, kurtosis, outliers, and symmetry.
    • Shapiro-Wilks test.

Homogeneity of Variance

  • Homogeneity of covariance matrices– same as for ANOVA plus correlations between any DVs is the same in each group.
  • Tests:
    • Levene’s test (for each DV).
    • Box’s test (to test the variance-covariance matrices) – Susceptible to non-normal data.

Linearity

  • MANOVA assumes linear relationships between all dependent variables.
  • Deviations from linearity reduce the power of the test because the linear combination of dependent variables does not maximize the difference between the groups.

Different Multivariate Test Criteria

  • Hotelling’s Trace
  • Wilk’s Lambda
  • Pillai’s Trace
  • Roy’s Largest Root

Considerations for Multivariate Test Criteria

  • When there are only two levels for an effect, all of the tests should be identical.
  • When there are more than two levels, the tests should be nearly identical - not always the case.
  • Wilk’s Lambda, Hotelling’s Trace, and Pillai’s trace all pool the variance from all the dimensions to create the test statistic.
  • E is error and H is hypothesized effect.
  • Roy’s largest root only uses the variance from the dimension that separates the groups most (the first canonical root, the largest “root” or difference).

Specifics of Multivariate Test Criteria

  • Wilk’s Lambda: ∣E∣∣H+E∣\frac{|E|}{|H + E|}
    • The ratio of error to effect plus error.
    • A bit conservative - middle of the road power.
  • Hotelling’s trace: T(H/E)=CT(H/E) = C
    • Sum of the eigenvalues for each variate.
    • Look up C in a table to get the F value (analogous to an F-test).
    • Very liberal test.
  • Pillai’s trace: V=HH+EV = \frac{H}{H + E}
    • Proportion of explained variation for each of the discriminant functions.
    • Very conservative.
  • Roy’s Maximum Root
    • Linear combination of the observations that maximizes the F-ratio (between group variance while minimizing the within-group variance).
    • Variable in terms of conservatism.
    • Susceptible to violations.

How to do a MANOVA in R

data(iris)
# Need to bind the variables together
Y <- cbind(iris$Sepal.Length,
 iris$Sepal.Width, iris$Petal.Length,
 iris$Petal.Width)
model <- manova(Y ~ Species, data = iris)
Summary(model)

PERMANOVA

  • Works with any distance measure that is appropriate to the data and uses permutations to make it distribution-free.
  • Theoretically:
    • Generate the distance matrix (usual choices).
    • Run the PERMANOVA using the same models as for MANOVA.
  • In R, use the package vegan called adonis.

PERMANOVA

  • Distances from centroids.
  • Pseudo F based on SS between/SS within.
  • P values by permutations.

PERMANOVA Output

  • Example output includes:
    • Source, df (Degrees of Freedom), SS (Sum of Squares), MS (Mean Square), Pseudo-F, P(perm), perms
    • Example Sources: Co, Sa, CoxSa, Res, Total.

How to do a PERMANOVA in R

#It is in the package vegan and is called ‘adonis’
adonis2(TransAntData ~Community*Sample, data = AntFactors, permutations = 999, by = “terms”)
#Code for pairwise tests given in practical notes

Choosing Between Tests

  • All fairly robust to violations of normality.
  • Wilk’s lambda is most widely used.
  • Can use Hotelling’s Trace if:
    • Manipulated (experimental) variables.
    • Very clean design with no internal validity problems.
  • Pillai’s trace is the most conservative, suitable if the design/data have problems (e.g., unbalanced, assumption violation, etc.).
  • PERMANOVA – use it for dissimilarities.

Follow-up ANOVA

  • If MANOVA is significant, undertake separate ANOVA on each of the DVs.
  • Risk of Type 1 errors.
  • Follow-ups “protected” by MANOVA may produce a non-significant result.
  • Still problematic – use corrections?

Assessing Dependent Variables

  • The overall alpha level should be controlled for considering the multiple tests.
  • The alpha levels can be divided equally, or they can be set up to give more important tests a more liberal alpha level.
  • α<em>overall=1−(1−α</em>1)(1−α<em>2)…(1−α</em>p){\alpha}<em>{overall} = 1 - (1-{\alpha}</em>1)(1-{\alpha}<em>2)…(1-{\alpha}</em>p)

Bonferroni Corrections and Multiple Comparisons

  • Experiment-wide error rates?
  • Corrections involve dividing alpha by n (or sequentially reducing n).
  • Using a p-value other than 0.05 to claim that there is a difference at alpha = 0.05.
  • E.g., for claim that there is a difference:
    • Comparison 1 - 0.05/4 = 0.0125
    • Comparison 2 - 0.05/3 = 0.017
    • Comparison 3 - 0.05/2 = 0.025
    • Comparison 4 - 0.05/1 = 0.05

MANOVA – A Summary

  • Make sure DVs are meaningful.
  • MANOVA:
    • Assumptions
    • Relationships among DVs
    • Choice of test statistic
  • Follow-up ANOVAs
  • Consider PERMANOVA as an alternative – same interpretation with fewer assumptions.