# Phylogenetic Generalised Linear Mixed Model for Community Data This function performs Generalized Linear Mixed Models for binary and continuous phylogenetic data, estimating regression coefficients with approximate standard errors. It is modeled after `lmer` but is more general by allowing correlation structure within random effects; these correlations can be phylogenetic among species, or any other correlation structure, such as geographical correlations among sites. It is, however, much more specific than `lmer` in that it can only analyze a subset of1 the types of model designed handled by `lmer`. It is also much slower than `lmer` and requires users to specify correlation structures as covariance matrices. `communityPGLMM` can analyze models in Ives and Helmus (2011). It can also analyze bipartite phylogenetic data, such as that analyzed in Rafferty and Ives (2011), by giving sites phylogenetic correlations. ```r communityPGLMM( formula, data = list(), family = "gaussian", sp = NULL, site = NULL, random.effects = list(), REML = TRUE, s2.init = NULL, B.init = NULL, reltol = 10^-6, maxit = 500, tol.pql = 10^-6, maxit.pql = 200, verbose = FALSE ) communityPGLMM.gaussian( formula, data = list(), family = "gaussian", sp = NULL, site = NULL, random.effects = list(), REML = TRUE, s2.init = NULL, B.init = NULL, reltol = 10^-8, maxit = 500, verbose = FALSE ) communityPGLMM.binary( formula, data = list(), family = "binomial", sp = NULL, site = NULL, random.effects = list(), REML = TRUE, s2.init = 0.25, B.init = NULL, reltol = 10^-5, maxit = 40, tol.pql = 10^-6, maxit.pql = 200, verbose = FALSE ) communityPGLMM.binary.LRT(x, re.number = 0, ...) communityPGLMM.matrix.structure( formula, data = list(), family = "binomial", sp = NULL, site = NULL, random.effects = list(), ss = 1 ) ## S3 method for class 'communityPGLMM' summary(object, digits = max(3, getOption("digits") - 3), ...) ## S3 method for class 'communityPGLMM' plot(x, digits = max(3, getOption("digits") - 3), ...) communityPGLMM.predicted.values(x, show.plot = TRUE, ...) ``` ## Arguments - `formula`: a two-sided linear formula object describing the fixed-effects of the model; for example, `Y ~ X`. - `data`: a `data.frame` containing the variables named in formula. The data frame should have long format with factors specifying species and sites. `communityPGLMM` will reorder rows of the data frame so that species are nested within sites. Please note that calling `as.data.frame.comparative.comm` will return your `comparative.comm` object into this format for you. - `family`: either `gaussian` for a Linear Mixed Model, or `binomial` for binary dependent data. - `sp`: a `factor` variable that identifies species - `site`: a `factor` variable that identifies sites - `random.effects`: a `list` that contains, for non-nested random effects, lists of triplets of the form `list(X, group = group, covar = V)`. This is modeled after the `lmer` formula syntax `(X | group)` where `X` is a variable and `group` is a grouping factor. Note that `group` should be either your `sp` or `site` variable specified in `sp` and `site`. The additional term `V` is a covariance matrix of rank equal to the number of levels of group that specifies the covariances among groups in the random effect `X`. For nested variable random effects, `random.effects` contains lists of quadruplets of the form `list(X, group1 = group1, covar = V, group2 = group2)` where `group1` is nested within `group2`. - `REML`: whether REML or ML is used for model fitting. For the generalized linear mixed model for binary data, these don't have standard interpretations, and there is no log likelihood function that can be used in likelihood ratio tests. - `s2.init`: an array of initial estimates of s2 for each random effect that scales the variance. If s2.init is not provided for `family="gaussian"`, these are estimated using in a clunky way using `lm` assuming no phylogenetic signal. A better approach is to run `link[lme4:lmer]{lmer}` and use the output random effects for `s2.init`. If `s2.init` is not provided for `family="binomial"`, these are set to 0.25. - `B.init`: initial estimates of $B$, a matrix containing regression coefficients in the model for the fixed effects. This matrix must have `dim(B.init)=c(p+1,1)`, where `p` is the number of predictor (independent) variables; the first element of `B` corresponds to the intercept, and the remaining elements correspond in order to the predictor (independent) variables in the formula. If `B.init` is not provided, these are estimated using in a clunky way using `lm` or `glm` assuming no phylogenetic signal. A better approach is to run `lmer` and use the output fixed effects for `B.init`. - `reltol`: a control parameter dictating the relative tolerance for convergence in the optimization; see `optim`. - `maxit`: a control parameter dictating the maximum number of iterations in the optimization; see `optim`. - `tol.pql`: a control parameter dictating the tolerance for convergence in the PQL estimates of the mean components of the binomial GLMM. - `maxit.pql`: a control parameter dictating the maximum number of iterations in the PQL estimates of the mean components of the binomial GLMM. - `verbose`: if `TRUE`, the model deviance and running estimates of `s2` and `B` are plotted each iteration during optimization. - `x`: `communityPGLMM` object - `re.number`: which `random.effect` in `x` to be tested - `...`: additional arguments to summary and plotting functions (currently ignored) - `ss`: which of the `random.effects` to produce - `object`: communityPGLMM object to be summarised - `digits`: minimal number of significant digits for printing, as in `print.default` - `show.plot`: if `TRUE` (default), display plot ## Returns an object of class `communityPGLMM` - **formula**: the formula for fixed effects - **data**: the dataset - **family**: either `gaussian` or `binomial` depending on the model fit - **random.effects**: the list of random effects - **B**: estimates of the regression coefficients - **B.se**: approximate standard errors of the fixed effects regression coefficients - **B.cov**: approximate covariance matrix for the fixed effects regression coefficients - **B.zscore**: approximate Z scores for the fixed effects regression coefficients - **B.pvalue**: approximate tests for the fixed effects regression coefficients being different from zero - **ss**: random effects' standard deviations for the covariance matrix $sigma^2 V$ for each random effect in order. For the linear mixed model, the residual variance is listed last - **s2r**: random effects variances for non-nested random effects - **s2n**: random effects variances for nested random effects - **s2resid**: for linear mixed models, the residual vairance - **logLIK**: for linear mixed models, the log-likelihood for either the restricted likelihood (`REML=TRUE`) or the overall likelihood (`REML=FALSE`). This is set to NULL for generalised linear mixed models - **AIC**: for linear mixed models, the AIC for either the restricted likelihood (`REML=TRUE`) or the overall likelihood (`REML=FALSE`). This is set to NULL for generalised linear mixed models - **BIC**: for linear mixed models, the BIC for either the restricted likelihood (`REML=TRUE`) or the overall likelihood (`REML=FALSE`). This is set to NULL for generalised linear mixed models - **REML**: whether or not REML is used (`TRUE` or `FALSE`) - **s2.init**: the user-provided initial estimates of `s2` - **B.init**: the user-provided initial estimates of `B` - **Y**: the response (dependent) variable returned in matrix form - **X**: the predictor (independent) variables returned in matrix form (including 1s in the first column) - **H**: the residuals. For the generalized linear mixed model, these are the predicted residuals in the $logit -1$ space - **iV**: the inverse of the covariance matrix for the entire system (of dimension (nsp*nsite) by (nsp*nsite)) - **mu**: predicted mean values for the generalized linear mixed model. Set to NULL for linear mixed models - **sp, sp**: matrices used to construct the nested design matrix - **Zt**: the design matrix for random effects - **St**: diagonal matrix that maps the random effects variances onto the design matrix - **convcode**: the convergence code provided by `optim` - **niter**: number of iterations performed by `optim` ## Details The vignette 'pez-pglmm-overview' gives a gentle introduction to using PGLMMS. For linear mixed models (`family = 'gaussian'`), the function estimates parameters for the model of the form, for example, $$ Y = \beta_0 + \beta_1x + b_0 + b_1xy = beta_0 + beta_1x + b_0 + b_1x $$ $$ b_0 ~ Gaussian(0, \sigma_0^2I_{sp})b_0 ~ Gaussian(0, sigma_0^2I_(sp)) $$ $$ b_1 ~ Gaussian(0, \sigma_0^2V_{sp})b_0 ~ Gaussian(0, sigma_0^2V_(sp)) $$ $$ \eta ~ Gaussian(0,\sigma^2)e ~ Gaussian(0,sigma^2) $$ where $beta_0$ and $beta_1$ are fixed effects, and $V_(sp)$ is a variance-covariance matrix derived from a phylogeny (typically under the assumption of Brownian motion evolution). Here, the variation in the mean (intercept) for each species is given by the random effect $b_0$ that is assumed to be independent among species. Variation in species' responses to predictor variable $x$ is given by a random effect $b_0$ that is assumed to depend on the phylogenetic relatedness among species given by $V_(sp_$; if species are closely related, their specific responses to $x$ will be similar. This particular model would be specified as `re.1 <- list(1, sp = dat$sp, covar = diag(nspp))` `re.2 <- list(dat$X, sp = dat$sp, covar = Vsp)` `z <- communityPGLMM(Y ~ X, data = data, family = "gaussian",random.effects = list(re.1, re.2))` The covariance matrix covar is standardized to have its determinant equal to 1. This in effect standardizes the interpretation of the scalar $sigma^2$. Although mathematically this is not required, it is a very good idea to standardize the predictor (independent) variables to have mean 0 and variance 1. This will make the function more robust and improve the interpretation of the regression coefficients. For categorical (factor) predictor variables, you will need to construct 0-1 dummy variables, and these should not be standardized (for obvious reasons). For binary generalized linear mixed models (`family = 'binomial'`), the function estimates parameters for the model of the form, for example, $$ y = \beta_0 + \beta_1x + b_0 + b_1xy = beta_0 + beta_1x + b_0 + b_1x $$ $$ Y = logit^{-1}(y)Y = logit^(-1)(y) $$ $$ b_0 ~ Gaussian(0, \sigma_0^2I_{sp})b_0 ~ Gaussian(0, sigma_0^2I_(sp)) $$ $$ b_1 ~ Gaussian(0, \sigma_0^2V_{sp})b_0 ~ Gaussian(0, sigma_0^2V_(sp)) $$ where $beta_0$ and $beta_1$ are fixed effects, and $V_(sp)$ is a variance-covariance matrix derived from a phylogeny (typically under the assumption of Brownian motion evolution). `z <- communityPGLMM(Y ~ X, data = data, family = 'binomial',random.effects = list(re.1, re.2))` As with the linear mixed model, it is a very good idea to standardize the predictor (independent) variables to have mean 0 and variance 1. This will make the function more robust and improve the interpretation of the regression coefficients. For categorical (factor) predictor variables, you will need to construct 0-1 dummy variables, and these should not be standardized (for obvious reasons). ## Note These function **do not** use a `comparative.comm` object, but you can use `as.data.frame.comparative.comm` to create a `data.frame` for use with these functions. The power of this method comes from deciding your own parameters parameters to be determined (the data for regression, the random effects, etc.), and it is our hope that this interface gives you more flexibility in model selection/fitting. ## Examples ```r ## Structure of examples: # First, a (brief) description of model types, and how they are specified # - these are *not* to be run 'as-is'; they show how models should be organised # Second, a run-through of how to simulate, and then analyse, data # - these *are* to be run 'as-is'; they show how to format and work with data ## Not run: ######################################################### #First section; brief summary of models and their use#### ######################################################### ## Model structures from Ives & Helmus (2011) # dat = data set for regression (note: *not* an comparative.comm object) # nspp = number of species # nsite = number of sites # Vphy = phylogenetic covariance matrix for species # Vrepul = inverse of Vphy representing phylogenetic repulsion # Model 1 (Eq. 1) re.site <- list(1, site = dat$site, covar = diag(nsite)) re.sp.site <- list(1, sp = dat$sp, covar = Vphy, site = dat$site) # note: nested z <- communityPGLMM(freq ~ sp, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.site, re.sp.site), REML = TRUE, verbose = TRUE, s2.init=.1) # Model 2 (Eq. 2) re.site <- list(1, site = dat$site, covar = diag(nsite)) re.slope <- list(X, sp = dat$sp, covar = diag(nspp)) re.slopephy <- list(X, sp = dat$sp, covar = Vphy) z <- communityPGLMM(freq ~ sp + X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.site, re.slope, re.slopephy), REML = TRUE, verbose = TRUE, s2.init=.1) # Model 3 (Eq. 3) re.site <- list(1, site = dat$site, covar = diag(nsite)) re.sp.site <- list(1, sp = dat$sp, covar = Vrepul, site = dat$site) # note: nested z <- communityPGLMM(freq ~ sp*X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.site, re.sp.site), REML = TRUE, verbose = TRUE, s2.init=.1) ## Model structure from Rafferty & Ives (2013) (Eq. 3) # dat = data set # npp = number of pollinators (sp) # nsite = number of plants (site) # VphyPol = phylogenetic covariance matrix for pollinators # VphyPlt = phylogenetic covariance matrix for plants re.a <- list(1, sp = dat$sp, covar = diag(nspp)) re.b <- list(1, sp = dat$sp, covar = VphyPol) re.c <- list(1, sp = dat$sp, covar = VphyPol, dat$site) re.d <- list(1, site = dat$site, covar = diag(nsite)) re.f <- list(1, site = dat$site, covar = VphyPlt) re.g <- list(1, site = dat$site, covar = VphyPlt, dat$sp) #term h isn't possible in this implementation, but can be done with available matlab code z <- communityPGLMM(freq ~ sp*X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.a, re.b, re.c, re.d, re.f, re.g), REML = TRUE, verbose = TRUE, s2.init=.1) ## End(Not run) ######################################################### #Second section; detailed simulation and analysis ####### #NOTE: this section is explained and annotated in ####### # detail in the vignette 'pez-pglmm-overview'####### # run 'vignette('pez-pglmm-overview') to read####### ######################################################### # Generate simulated data for nspp species and nsite sites nspp <- 15 nsite <- 10 # residual variance (set to zero for binary data) sd.resid <- 0 # fixed effects beta0 <- 0 beta1 <- 0 # magnitude of random effects sd.B0 <- 1 sd.B1 <- 1 # whether or not to include phylogenetic signal in B0 and B1 signal.B0 <- TRUE signal.B1 <- TRUE # simulate a phylogenetic tree phy <- rtree(n = nspp) phy <- compute.brlen(phy, method = "Grafen", power = 0.5) # standardize the phylogenetic covariance matrix to have determinant 1 Vphy <- vcv(phy) Vphy <- Vphy/(det(Vphy)^(1/nspp)) # Generate environmental site variable X <- matrix(1:nsite, nrow = 1, ncol = nsite) X <- (X - mean(X))/sd(X) # Perform a Cholesky decomposition of Vphy. This is used to # generate phylogenetic signal: a vector of independent normal random # variables, when multiplied by the transpose of the Cholesky # deposition of Vphy will have covariance matrix equal to Vphy. iD <- t(chol(Vphy)) # Set up species-specific regression coefficients as random effects if (signal.B0 == TRUE) { b0 <- beta0 + iD %*% rnorm(nspp, sd = sd.B0) } else { b0 <- beta0 + rnorm(nspp, sd = sd.B0) } if (signal.B1 == TRUE) { b1 <- beta1 + iD %*% rnorm(nspp, sd = sd.B1) } else { b1 <- beta1 + rnorm(nspp, sd = sd.B1) } # Simulate species abundances among sites to give matrix Y that # contains species in rows and sites in columns y <- rep(b0, each=nsite) y <- y + rep(b1, each=nsite) * rep(X, nspp) y <- y + rnorm(nspp*nsite) #add some random 'error' Y <- rbinom(length(y), size=1, prob=exp(y)/(1+exp(y))) y <- matrix(outer(b0, array(1, dim = c(1, nsite))), nrow = nspp, ncol = nsite) + matrix(outer(b1, X), nrow = nspp, ncol = nsite) e <- rnorm(nspp * nsite, sd = sd.resid) y <- y + matrix(e, nrow = nspp, ncol = nsite) y <- matrix(y, nrow = nspp * nsite, ncol = 1) Y <- rbinom(n = length(y), size = 1, prob = exp(y)/(1 + exp(y))) Y <- matrix(Y, nrow = nspp, ncol = nsite) # name the simulated species 1:nspp and sites 1:nsites rownames(Y) <- 1:nspp colnames(Y) <- 1:nsite par(mfrow = c(3, 1), las = 1, mar = c(2, 4, 2, 2) - 0.1) matplot(t(X), type = "l", ylab = "X", main = "X among sites") hist(b0, xlab = "b0", main = "b0 among species") hist(b1, xlab = "b1", main = "b1 among species") #Plot out; you get essentially this from plot(your.pglmm.model) image(t(Y), ylab = "species", xlab = "sites", main = "abundance", col=c("black","white")) # Transform data matrices into "long" form, and generate a data frame YY <- matrix(Y, nrow = nspp * nsite, ncol = 1) XX <- matrix(kronecker(X, matrix(1, nrow = nspp, ncol = 1)), nrow = nspp * nsite, ncol = 1) site <- matrix(kronecker(1:nsite, matrix(1, nrow = nspp, ncol = 1)), nrow = nspp * nsite, ncol = 1) sp <- matrix(kronecker(matrix(1, nrow = nsite, ncol = 1), 1:nspp), nrow = nspp * nsite, ncol = 1) dat <- data.frame(Y = YY, X = XX, site = as.factor(site), sp = as.factor(sp)) # Format input and perform communityPGLMM() # set up random effects # random intercept with species independent re.1 <- list(1, sp = dat$sp, covar = diag(nspp)) # random intercept with species showing phylogenetic covariances re.2 <- list(1, sp = dat$sp, covar = Vphy) # random slope with species independent re.3 <- list(dat$X, sp = dat$sp, covar = diag(nspp)) # random slope with species showing phylogenetic covariances re.4 <- list(dat$X, sp = dat$sp, covar = Vphy) # random effect for site re.site <- list(1, site = dat$site, covar = diag(nsite)) simple <- communityPGLMM(Y ~ X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.site), REML=TRUE, verbose=FALSE) # The rest of these tests are not run to save CRAN server time; # - please take a look at them because they're *very* useful! ## Not run: z.binary <- communityPGLMM(Y ~ X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.1, re.2, re.3, re.4), REML = TRUE, verbose = FALSE) # output results z.binary plot(z.binary) # test statistical significance of the phylogenetic random effect # on species slopes using a likelihood ratio test communityPGLMM.binary.LRT(z.binary, re.number = 4)$Pr # extract the predicted values of Y communityPGLMM.predicted.values(z.binary, show.plot = TRUE) # examine the structure of the overall covariance matrix communityPGLMM.matrix.structure(Y ~ X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.1, re.2, re.3, re.4)) # look at the structure of re.1 communityPGLMM.matrix.structure(Y ~ X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.1)) # compare results to glmer() when the model contains no # phylogenetic covariance among species; the results should be # similar. communityPGLMM(Y ~ X, data = dat, family = "binomial", sp = dat$sp, site = dat$site, random.effects = list(re.1, re.3), REML = FALSE, verbose = FALSE) # lmer if(require(lme4)){ summary(glmer(Y ~ X + (1 | sp) + (0 + X | sp), data=dat, family = "binomial")) # compare results to lmer() when the model contains no phylogenetic # covariance among species; the results should be similar. communityPGLMM(Y ~ X, data = dat, family = "gaussian", sp = dat$sp, site = dat$site, random.effects = list(re.1, re.3), REML = FALSE, verbose = FALSE) # lmer summary(lmer(Y ~ X + (1 | sp) + (0 + X | sp), data=dat, REML = FALSE)) } ## End(Not run) ``` ## References Ives, A. R. and M. R. Helmus. 2011. Generalized linear mixed models for phylogenetic analyses of community structure. Ecological Monographs 81:511-525. Rafferty, N. E., and A. R. Ives. 2013. Phylogenetic trait-based analyses of ecological networks. Ecology 94:2321-2333. ## Author(s) Anthony R. Ives, cosmetic changes by Will Pearse

pglmm function