advanced.procD.lm: Procrustes ANOVA and pairwise tests for shape data, using...

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/advanced.procD.lm.r


The function quantifies the relative amount of shape variation explained by a suite of factors and covariates in a "full" model, after accounting for variation in a "reduced" model. Inputs are formulae for full and reduced models (order is not important, but it is better to list the model with the most terms first or use a geomorph data frame), plus indication if means or slopes are to be compared among groups, with appropriate formulae to define how they should be compared.


advanced.procD.lm(f1, f2, groups = NULL, slope = NULL, angle.type = c("r",
  "deg", "rad"), phy = NULL, pc.shape = FALSE, iter = 999, seed = NULL,
  print.progress = TRUE, data = NULL, ...)



A formula for a linear model, containing the response matrix (e.g., y ~ x1 + x2)


A formula for another linear model (e.g., ~ x1 + x2 + x3 + a*b). f1 and f2 should be nested.


A formula for grouping factors (e.g., ~ a, or ~ a*b). This argument should be left NULL unless one wishes to perform pairwise comparisons of different group levels. Note that this argument is used in conjunction with the argument, slope. If slope is NULL, a pairwise comparison test is performed on group least squares (LS) means. If slope is not NULL, this argument will designate the group levels to compare in terms of their slopes.


A formula with one - and only one - covariate (e.g., ~ x3). This argument must be used in conjunction with the groups argument. It will not make sense if the groups argument is left NULL. The groups argument defines the groups; the slope argument defines for which covariate group slopes are compared. Group slopes can differ in their magnitude and direction of shape change.


A value specifying whether directional differences between slopes should be represented by vector correlations (r), radians (rad) or degrees (deg).


A phylogenetic tree of class phylo - see read.tree in library ape (optional)


An argument for whether analysis should be performed on the principal component scores of shape. This is a useful option if the data are high-dimensional (many more variables than observations) but will not affect results


Number of iterations for significance testing


An optional argument for setting the seed for random permutations of the resampling procedure. If left NULL (the default), the exact same P-values will be found for repeated runs of the analysis (with the same number of iterations). If seed = "random", a random seed will be used, and P-values will vary. One can also specify an integer for specific seed values, which might be of interest for advanced users.


A logical value to indicate whether a progress bar should be printed to the screen. This is helpful for long-running analyses.


A data frame for the function environment; see If variables are transformed in formulae, they should also be transformed in the geomorph data frame. (See examples.)


Arguments passed on to (typically associated with the lm function, such as weights or offset).


The response matrix 'y' can be in the form of a two-dimensional data matrix of dimension (n x [p x k]) or a 3D array (p x k x n). It is assumed that the landmarks have previously been aligned using Generalized Procrustes Analysis (GPA) [e.g., with gpagen]. The names specified for the independent (x) variables in the formula represent one or more vectors containing continuous data or factors. It is assumed that the order of the specimens in the shape matrix matches the order of values in the independent variables. Linear model fits (using the lm function) can also be input in place of a formula. Arguments for lm can also be passed on via this function.

The function performs statistical assessment of the terms in the model using Procrustes distances among specimens, rather than explained covariance matrices among variables. With this approach, the sum-of-squared Procrustes distances are used as a measure of SS (see Goodall 1991). The SS between models is evaluated through permutation. In morphometrics this approach is known as a Procrustes ANOVA (Goodall 1991), which is equivalent to distance-based anova designs (Anderson 2001). Unlike procD.lm, this function is strictly for comparison of two nested models. (Use of procD.lm will be more suitable in most cases.) A residual randomization permutation procedure (RRPP) is utilized for reduced model residuals to evaluate the SS between models (Collyer et al. 2015). Effect-sizes (Z-scores) are computed as standard deviates of the SS or pairwise statistic sampling distributions generated, which might be more intuitive for P-values than F-values (see Collyer et al. 2015). If a phylogeny is used, the ANOVA Z-score is calculated from the sampling distributions of the F value, as the total SS will vary among permutations. For ANOVA Z-scores, a log-transformation is performed first, to assure a norammly distributed sampling distribution.

Pairwise tests are only performed if formulae are provided to compute such results. The generic functions, print, summary, and plot all work with advanced.procD.lm. The generic function, plot, produces diagnostic plots for Procrustes residuals of the linear fit.

Notes for geomorph 3.0.4 and subsequent versions

Compared to previous versions of geomorph, users might notice differences in effect sizes. Previous versions used z-scores calculated with expected values of statistics from null hypotheses (sensu Collyer et al. 2015); however Adams and Collyer (2016) showed that expected values for some statistics can vary with sample size and variable number, and recommended finding the expected value, empirically, as the mean from the set of random outcomes. Geomorph 3.0.4 and subsequent versions now center z-scores on their empirically estimated expected values and where appropriate, log-transform values to assure statistics are normally distributed. This can result in negative effect sizes, when statistics are smaller than expected compared to the avergae random outcome. For ANOVA-based functions, the option to choose among different statistics to measure effect size is now a function argument.

An optional argument for including a phylogenetic tree of class phylo is included in this function. ANOVA performed on separate PGLS models is analagous to a likelihood ratio test between models (Adams and Collyer 2017). Pairwise tests can also be performed after PGLS estimation of coefficients but users should be aware that no formal research on the statistical properties (type I error rates and statistical power) of pairwise statistics with PGLS has yet been performed. Using PGLS and analysis of pairwise statistics, therefore, assumes some risk.


Function returns an ANOVA table of statistical results for model comparison: error df (for each model), SS, MS, F ratio, Z, and Prand. A list of essentially the same components as procD.lm is also returned, and additionally LS means or slopes, pairwise differences comparisons of these, effect sizes, and P-values may also be returned. If a group formula is provided but slope formula is null, pairwise differences are Procrustes distances between least squares (LS) means for the defined groups. If a slope formula is provided, two sets of pairwise differences, plus effect sizes and P-values, are provided. The first is for differences in slope vector length (magnitude). The length of the slope vector corresponds to the amount of shape change per unit of covariate change. Large differences correspond to differences in the amount of shape change between groups. The second is for slope vector orientation differences. Differences in the direction of shape change (covariance of shape variables) can be summarized as a vector correlation or angle between vectors. See summary.advanced.procD.lm for summary options.


Michael Collyer


Collyer, M.L., D.J. Sekora, and D.C. Adams. 2015. A method for analysis of phenotypic change for phenotypes described by high-dimensional data. Heredity. 115:357-365.

Adams, D.C. and M.L. Collyer. 2016. On the comparison of the strength of morphological integration across morphometric datasets. Evolution. 70:2623-2631.

Adams, D.C. and M.L. Collyer. 2017. Multivariate comparative methods: evaluations, comparisons, and recommendations. Systematic Biology. In press.

See Also



Y.gpa<-gpagen(plethodon$land)    #GPA-alignment
gdf <-, species = plethodon$species, site = plethodon$site)

# Example of a nested model comparison (as with ANOVA with RRPP)
advanced.procD.lm(coords ~ log(Csize) + species, 
~ log(Csize)*species*site, iter=499, data = gdf)

# Example of a test of a factor interaction, plus pairwise comparisons 
advanced.procD.lm(coords ~ site*species, ~ site + species, groups = ~site*species, 
   iter=499, data = gdf)

# Example of a test of a factor interaction, plus pairwise comparisons, 
# accounting for a common allometry  
advanced.procD.lm(coords ~ Csize + site*species, 
~ log(Csize) + site + species, 
groups = ~ site*species, slope = ~log(Csize), iter = 499, data = gdf)

# Example of a test of homogeneity of slopes, plus pairwise slopes comparisons
gdf$group <- factor(paste(gdf$species, gdf$site, sep="."))
advanced.procD.lm(coords ~ log(Csize) + group, 
~ log(Csize) * group, 
groups = ~ group, 
slope = ~ log(Csize), angle.type = "deg", iter = 499, data = gdf)

# Example of partial pairwise comparisons, given greater model complexity.
# Plus, working with class advanced.procD.lm objects.
aov.pleth <- advanced.procD.lm(coords ~ log(Csize)*site*species, 
~ log(Csize) + site*species, 
groups = ~ species, slope = ~ log(Csize), angle.type = "deg", iter = 499, data = gdf)

summary(aov.pleth) # ANOVA plus pairwise tests
plot(aov.pleth) # diagnostic plots
aov.pleth$slopes # extract the slope vectors

geomorph documentation built on June 8, 2017, 9:02 a.m.

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