Ecology Exam Review

SIMPER (Similarity Percentages Analysis)

• Ratio is average / standard deviation.
• A and B represent the average abundance in group A and group B, respectively.
• A zero score indicates absence in Group I (presumably Group A) but presence in Group B.
• A score of 16.8 represents the average count of the species.
• Percentage is calculated by dividing the average for each species by the average dissimilarity which is cumulative.
• Typically, the top 70% or 90% of species are considered, but the selection should be justified.
• SIMPER identifies species that contribute most to the dissimilarity between treatments.
• High A and B differences suggest a significant contribution to dissimilarity.
• The cumulative summation represents the cumulative contribution of each species to the groups.
• Command lines are in the vegan package in R.

NMDS, ANOSIM, PERMANOVA Workflow

• Samples are collected.
• A similarity or dissimilarity matrix (e.g., Bray-Curtis) is created.
• Data may be transformed (referencing previous lectures).
• Non-metric multidimensional scaling (NMDS) is used for ordination.
• Differences are tested using ANOSIM or PERMANOVA.
• SIMPER identifies which groups are different.

Advantages of NMDS

• Robust ordination technique.
• Few assumptions.
• ANOSIM and PERMANOVA are suitable for hypothesis testing with ordinations.
• CLARC1993 is an effective approach for species abundance data.
• ANOSIM requires at least four replicates and struggles with complex designs.
• PERMANOVA will be covered next.

Introduction to MANOVA and PERMANOVA

• MANOVA (multivariate analysis of variance) is a metric technique similar to principal components analysis, ANOVA, and regression.
• PERMANOVA is a versatile technique suitable as an alternative to ANOSIM.
• MANOVA is used in ecology, agriculture, and other fields.
• Comparison to ANOVA.
• Covers the process, theory, and test statistics.
• Multiple comparisons and corrections are discussed.
• PERMANOVA is also addressed.
• Marni Anderson developed PERMANOVA and initially called it nonparametric MANOVA, later changing the name to emphasize permutations.

Biological Examples of MANOVA

• Spider web analysis: Measuring radial threads, spiral threads, and cell distances to create a multivariate dataset and compare different sites.
• Prickly parrot pea plant study: Measuring biomass, mortality, height, flower, and seed set in plants from different landscapes (replanted vs. remnant sites) to compare treatments.

Flounder Diet Example

• Researchers compared flounder diets across different areas.
• Benthic cores were sampled to assess available food items.
• Stomach contents of flounders were analyzed.
• ANOVA on core samples showed no significant differences between sites.
• Wilks' Lambda is introduced as a key statistic.
• ANOVA on stomach contents revealed significant differences between ponds.
• This suggests flounders selectively consume different foods based on the environment.
• MANOVA is used to analyze all dietary items simultaneously.

Why Use MANOVA Instead of Separate ANOVAs?

• Separate ANOVAs lose information about the relationships between dependent variables.
• MANOVA reduces inflated Type I errors (false positives).
• MANOVA maximizes differences between samples relative to within-sample variation.
• MANOVA combines dependent variables to increase power to detect differences.
• Variables should be logically related to the hypothesis and treatment.

MANOVA as an Extension of t-tests and ANOVAs

• MANOVA is a multivariate extension of t-tests and univariate ANOVAs.
• It tests for mean differences between dependent variables, considering chance alone.
• It includes main effects and interactions.
• Post hoc tests can be used.

Shortcomings of MANOVA

• Variables should not be highly correlated.
• Has tight assumptions.
• PERMANOVA can overcome these limitations.

Evaluating MANOVA Output

• Consider variation, experimental design, and independent variables.
• Evaluate someone else’s ANOVA output.

Basic Requirements for MANOVA

• Two or more continuous dependent variables (interval or ratio scales).
• One or more categorical independent variables (nominal or ordinal scales).
• Similar questions to ANOVAs but applied to linearly combined dependent variables.

Differences Between MANOVA and Principal Components Analysis (PCA)

• PCA maximizes variation within the dataset.
• MANOVA maximizes differences between predefined groups.
• PCA does not have predefined groups.
• MANOVA tests differences between linear combinations of variables across treatments.

Interpreting MANOVA Results

• Examine main effects.
• Analyze significant interactions by holding one variable constant.
• Use ANOVAs to determine which dependent variables are most important after finding a significant overall difference in MANOVA.
• Use post hoc tests on variables with significant differences and more than two levels.
• Handle interactions as in ANOVAs.

MANOVA Assumptions

• Observations should be statistically independent (no pseudo-replication, random sampling).
• Multivariate normality: Variables collectively have multivariate normality within groups.
  Homogeneity of covariance matrices: All variables have homogeneity of variance, and correlations between variables are the same across groups.
• Interpretation should always be in the context of the research question, avoiding bad designs.

Problems with MANOVA

• Missing variables: Can require removing the variable for that sample.
• Unequal samples: Can cause issues with total sums of squares.
• Reduced power: Correlated variables reduce power.

Multivariate Normality

• Means of dependent variables and their linear combinations are normally distributed.
• Difficult to show explicitly.
• In practice, individual variables being normal suggests overall multivariate normality.
• Assess skewness, kurtosis, outliers, and symmetry for individual variables.
• Use tests like Shapiro-Wilk.

Homogeneity of Variance

• All variables should have homogeneity of variance.
• Equal correlation between variables.
• Use Levene's test or plots for individual variables.
• Box's test in MANOVA tests variance-covariance matrices but is sensitive to non-normal data.
• Linear relationships: Variables should follow linear relationships.
• Multiple test statistics are used to calculate the F value.

MANOVA Test Statistics

• Hotelling's Trace
• Wilks' Lambda
• Pillai's Trace
• Roy's Largest Root

Choosing a Test Statistic

• Wilks' Lambda: The most commonly used because it is a compromise test, not too conservative or too liberal.
• With only two levels, test results are identical.
• Wilks' Lambda calculates error over the effect plus the error.

Other Test Statistics

• Hotelling’s test: Hypothesis over the error and is the most liberal test.
• Pillai's trace: Hypothesis, the effect over the effect plus the error, conservative.
• Roy's Largest Root: It uses only the first difference.

MANOVA in R

• Combine the data into a single model using a simple model which statistically test the species between the lot.

Guidelines for Choosing a Test Statistic

• If there are unequal sample sizes, deviations from normality, or deviations from homogeneity of covariance matrices, use Pillai's Trace.
• If there is a very clean experimental design with no deviations and equal sample sizes, use Hotelling's Trace.
• Wilks' Lambda is generally used in most cases without deviations.

PERMANOVA

• In cases where MANOVA assumptions cannot be met, particularly with species data, PERMANOVA can be used.
• PERMANOVA can use various distance metrics (e.g., Bray-Curtis, clear dent, genetic differences).
• It is non-parametric and permutational.
• In vegan package, PERMANOVA is called Adonis.
• A MANOVA-type statistic is calculated using permutations to determine the p-value

PERMANOVA Calculations

• It gives you the same sorts of statistics as you get from a manova table.
• Calculates a pseudo F because it doesn't follow the F distribution.
• Sampling nine ninety nine is generally what we use so we get 0.03.
• Calculates the F ratio and swaps out all the data.

PERMANOVA vs. MANOVA and ANOSIM

• PERMANOVA can be slightly more conservative than MANOVA; if data suits MANOVA, it should be used for a higher likelihood of a significant test.
• Both Anosim and Permanova are recommended.

Guidelines for Choosing a Test (Revisited)

• For data meeting assumptions, Wilkes Lambda is a good default.
• Pillai's Trace handles unbalanced designs and deviations.
• For data violating assumptions, PERMANOVA is preferred.

Follow-up Analyses

• If a MANOVA shows a significant difference, follow up with ANOVAs.
• ANOVAs are protected by MANOVA, so they're only performed on significant results.

Correcting for Type I Errors

• Sequential Bonferroni test (Home test):
• Controls the over Alpha level, dividing 0.05 by four, is the new significance level.
• Rank comparison and divid 0.05 by four.

Summary of MANOVA and PERMANOVA

• Ensure dependent variables are meaningful.
• For MANOVA, verify linearity, multivariate normality, and homogeneity of covariance matrices.
• Choose the statistic (Wilkes Lambda, Pillai's Trace).
• PERMANOVA is suitable when assumptions are violated and you don't have anything that fits near those assumptions.