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### MANOVA

The main idea...

Multivariate analysis of variance (MANOVA) is the multivariate analogues of univariate ANOVA test. It thus offers a very powerful method to examine the influence of factors and their interactions across groups.
Similar to ANOVA, MANOVA tests whether the assignment of objects to levels of one or more nominal explanatory variables (i.e. grouping variables) is statistically supported by response data. In contrast to ANOVA, however, this response data is contained in multiple continuous response variables rather than a single response variable (Figure 1). MANOVA is, therefore, suitable for testing the effect of different factors (e.g. experimental treatments or sampling site properties) on multiple response variables (e.g. OTU abundances).

MANOVA assesses main effects and interactions by creating artificial response axes that maximally separate the groups defined. These artificial variables are linear combinations of the original response variables. Figure 2 illustrates how a MANOVA may detect statistically significant differences that multiple ANOVAs may not. Results of MANOVA can, at times, be visualised by multiple discriminant analysis (MDA).

 Null hypothesis The (multivariate) vectors of means of two or more groups of objects are equal.

In addition to meeting certain assumptions (see below), a successful MANOVA requires a reasonably well-balanced experimental or sampling design. When performed correctly, this test has been shown to handle ecology data well (Warton & Hudson, 2004). If you are unfamiliar with these designs, please consult an introductory text on ANOVA-like statistics before applying MANOVA.

Ecological data sets are often violate the parametric assumptions of MANOVA or have more variables than objects. A non-parametric form of MANOVA (NP-MANOVA), which attempts to surmount the limitations of parametric MANOVA is also available.

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Figure 1: a) ANOVA-like methods assess if the difference between group means along a single response variable is significant by comparing the within- and between-group sums of squared differences from those means. b) MANOVA assesses the joint distributions of more than one response variable. Rather than a single mean per group, vectors of each response variable's mean are analysed. See Figure 2 for further detail.

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Figure 2: a) Examining the single-variable distributions of the three groups shown above (curves on individual axes) may prevent detection of significant differences observed when the joint distributions of both response variables are taken into account (ellipses). b) In MANOVA, linear combinations of the original response variables (grey axes) are used to create a composite variable (black axis) that maximally separates groups.

Results and interpretation

MANOVA results are very similar to those of ANOVA. A value of some test statistic (described below), the number of degrees of freedom of the test, and a p-value associated with the test statistic are displayed. Whether or not the p-value is small enough to label the test statistic significant is of primary interest. A significant rejection of the null hypothesis suggests that there is a significant difference between the vectors of means of two or more groups; however, it will not tell you which groups are significantly different from the others or which response variables account for that difference. Post hoc tests, such as a series of univariate F-tests (with corrections for multiple testing) are required to discover which variables account for this difference.

Statistics

A number of statistics can be used to test the null hypothesis of equality of mean vectors. Some MANOVA implementations will translate these statistics, either exactly or approximately, into F-statistics prior to output in order to facilitate the calculation of p-values. If there are only two groups in the data, these statistics are equivalent and the Hotelling's T test may be used. Some of these tests are more liberal (Type I errors more likely) while others are conservative (Type II errors more likely).

 Hotelling-Lawley trace A liberal test, this is the trace of the proportion of the hypothesised model effects to the error of the model. This is analogous to an F-test and is equivalent to Hotelling's T2 divided by the difference between the sample size and the number of groups. Wilk's λ (lambda) A test considered between liberal and conservative, Wilks' λ is another multivariate generalisation of the F-statistic; however, as the quotient of the model error variance and the sum of the model error and hypothesised effect variance, it has the opposite interpretation. That is, the value of Wilks' λ indicates the proportion of variation (in the response variables) not explained by the model (i.e. the "grouping" provided by the explanatory variables). This is a commonly applied statistic in MANOVA testing. Pillai's trace Pillai's Trace is often considered to be very robust, with better tolerance for non-normal data. It is, however, a conservative test. It is the quotient of the hypothesised model effects and the sum of the hypothesised model effects and model error. Roy's largest root Roy's largest root (also known as Roy's largest eigenvalue) is the proportion of the hypothesised model effects to the error of the model, but uses only the linear combination of the original variables which accounts for the most variation in the data. This linear combination has the largest eigenvalue (also known as a root) of all linear combinations generated by MANOVA. Johson and Wichern (2002) note that this statistic has most power in scenarios where one variable strongly distinguishes between one group and all of the others.

In general, Wilks' λ is used as a default test statistic. The Hotelling-Lawley trace would be used in a very controlled experimental setting where the design and data meet MANOVA's assumptions very well. Pillai's trace would be used, at the expense of power, if there are several, non-remediable assumption violations or an unbalanced experimental design. The conservatism of Roy's largest root may vary, and use of this statistic should be considered in the scenario described above.

## Key assumptions

• All observations should be independent.
• Group sizes should be approximately equal. A size ratio of 1.5 between groups is generally considered the upper-limit of permissible difference. Some corrections are available (see Warnings).
• The response variables should have a multivariate normal distribution within each group. Some deviations from multivariate normality can be tolerated by MANOVA, but results are strongly influenced by outliers.
• Linear relationships should exist between response variables and covariates. Departure from linearity will reduce the power of a MANOVA as linear combinations of response variables becomes less likely to maximise the difference between groups.
• Response variables should have (near-) equal variances across the range of explanatory variables.  Bartlett's test may  be used to evaluate if this assumption holds.
• The covariances between response variables should homogeneous. Box's test for the equality of covariance matrices may be used to verify this assumption holds.
• Variables should be homoscedastic. Levene's test may be applied to test for heteroscedasticity.
Warnings
• Differing sample sizes between groups affects the sum of squares calculations and must be corrected for. Several corrections exist such as using Type I sums of squares weighted by sample sizes or Type III sums of squares, which assumes that any imbalance is not meaningful.
• MANOVA quickly consumes degrees of freedom. The number of response factors (p) and the number of their levels (k) consume degrees of freedom after: (k-1)p.
• If there is strong correlation between response variables (multicollinearity), including all correlates in the analysis is questionable as they are likely to be redundant. A MANCOVA could be more useful.
• MANOVA is very sensitive to outliers. Type I and Type II errors may be caused by their presence. Tests for outliers in multivariate distributions should be performed to ensure they are not influencing the results. Note that each group should be screened for outliers separately. Transformations may be applied to reduce the influence of outliers.
• The design of your experiment or sampling process will affect how the significance of a MANOVA (or ANOVA) is tested. Identify whether you are using a between-group, repeated measures (longitudinal measurements, within-subjects design), between-within (i.e. between-group and repeated measures), or nested design and test significance accordingly.

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