geomorph (version 2.1.2)

pairwiseD.test: Pairwise Group Comparisons

Description

Function performs pairwise comparisons among groups using the Euclidean distances among group means.

Usage

pairwiseD.test(f1, covariates = NULL, RRPP = FALSE, int.first = FALSE,
  iter = 999)

Arguments

f1

A formula for the linear model from which groups are to be compared (e.g., y~x1*x2)

covariates

A vector, matrix, or data frame with covariates (contunuous quantitative variables)

RRPP

a logical value indicating whether residual randomization should be used for significance testing

int.first

A logical value to indicate if interactions of first main effects should precede subsequent main effects

iter

Number of iterations for permutation test

Value

• Function returns a list with the following components:
• Obs.dist
  A matrix of Euclidean distances among group means or Least Squares group means
• Prob.Dist
  A matrix of pairwise significance levels based on permutation

Details

The function performs pairwise comparisons to identify shape differences among groups. The function is designed as a post-hoc test to Procrustes ANOVA, where the latter has identified significant shape variation explained by a grouping factor. The Function can handle, single-factor ANOVA, factorial ANOVA, and ANCOVA designs. As input the user provides a formula describing the linear model of how shape (y) varies as a function of a factor (a) or factorial interaction (a*b). A single covariate, matrix of covariates, or data frame of covariates can also be added. E.g., covariates = x1, covariates = cbind(x1, x2, x3,...), or covariates = data.frame(x1, x2, x3,...). The shape data (y) must be in the form of a two-dimensional data matrix of dimension (n x [p x k]), rather than a 3D array. It is assumed that the landmarks have previously been aligned using Generalized Procrustes Analysis (GPA) [e.g., with gpagen]. The function two.d.array can be used to obtain a two-dimensional data matrix from a 3D array of landmark coordinates. From the data, the Euclidean distances among group means are estimated, and used as test values. To evaluate significance of group differences, two possible resampling procedures are provided. First, if RRPP=FALSE, the rows of the matrix of shape variables are randomized relative to the design matrix. This is analogous to a 'full' randomization. Second, if RRPP=TRUE, a residual randomization permutation procedure is utilized (Collyer et al. 2014). Here, residual shape values from a reduced model are obtained, and are randomized with respect to the linear model under consideration. These are then added to predicted values from the remaining effects to obtain pseudo-values from which SS are calculated. NOTE: for single-factor designs, the two approaches are identical. However, when evaluating factorial models it has been shown that RRPP attains higher statistical power and thus has greater ability to identify patterns in data should they be present (see Anderson and terBraak 2003). Effect-sizes (Z-scores) are computed as standard deviates of the SS sampling distributions generated, which might be more intuitive for P-values than F-values (see Collyer et al. 2014). This test is essentially the same as procD.lm with post-hoc comparisons among least squares (LS) means for appropriate models. However, differences in means are calculated simultaneously with the same random permutations peformed for ANOVA, making it less so a post-hoc test and more so a simultaneous test of pairwise contrasts (see Collyer et al. 2014).

References

Anderson MJ. and C.J.F. terBraak. 2003. Permutation tests for multi-factorial analysis of variance. Journal of Statistical Computation and Simulation 73: 85-113. Collyer, M.L., D.J. Sekora, and D.C. Adams. 2014. A method for analysis of phenotypic change for phenotypes described by high-dimensional data. Heredity. 113: doi:10.1038/hdy.2014.75.

Examples

data(plethodon)
Y.gpa<-gpagen(plethodon$land)    #GPA-alignment
y<-two.d.array(Y.gpa$coords)
### Procrustes ANOVA
procD.lm(y~plethodon$species,iter=99)

### Pairwise comparisons: ANOVA design with full randomization
pairwiseD.test(y~plethodon$species*plethodon$site,iter=9)

### Pairwise comparisons: ANOVA design with residual randomization
pairwiseD.test(y~plethodon$species*plethodon$site,iter=9,RRPP=TRUE)

### Pairwise comparisons: ANCOVA design with full randomization
pairwiseD.test(y~plethodon$species*plethodon$site,covariates = data.frame(CS = Y.gpa$Csize),
             iter=9)

### Pairwise comparisons: ANCOVA design with residual randomization
pairwiseD.test(y~plethodon$species*plethodon$site,covariates = data.frame(CS = Y.gpa$Csize),
            iter=9, RRPP = TRUE)