R2s for Regression Models

Three methods to calculate R2 for models with correlated errors, including Phylogenetic GLS, Phylogenetic Logistic Regression, Linear Mixed Models (LMMs), and Generalized Linear Mixed Models (GLMMs). See details in Ives 2018 .

This package provides three R²s for statistical models with correlated errors including classes: ‘lmerMod’ (LMM), ‘glmerMod’ (GLMM), 'gls', ‘phylolm’ (Phylogenetic GLS), and ‘binaryPGLMM/phyloglm/communityPGLMM’ (Phylogenetic Logistic Regression). Detailed technical descriptions can be found in Ives 2018.

Installation

This package can be installed with:

install.packages("rr2")
# install.packages("devtools")
devtools::install_github("arives/rr2")

Package structure

This package has three main functions: R2.resid(), R2.lik(), and R2.pred(). You can use them individually in the form of, e.g., R2.resid(mod, mod.r) where mod is the full model and mod.r is the reduced model for partial R2s. If you do not include the reduced model mod.r, then the appropriate model with just the intercept is used to give the total R². When using R2.resid and R2.pred with PGLS, you need to include the phylo object containing a phylogenetic tree, e.g., R2.resid(mod, mod.r, phy = phy).

You can calculate all three R²s at the same time with R2(mod, mod.r). You can also specify which R²(s) to calculate within this function by turning off unwanted methods, e.g., R2(mod, mod.r, resid = FALSE) or R2(mod, mod.r, pred = FALSE).

This package also has some helper functions such as inv.logit(), partialR2(), and partialR2adj().

Usage: calculating R²s for regression models

First, let’s simulate data that will be used to fit various models.

# data 
set.seed(123)
p1 <- 10; nsample <- 10; n <- p1 * nsample
d <- data.frame(x1 = rnorm(n = n), 
                x2 = rnorm(n = n), 
                u1 = rep(1:p1, each = nsample), 
                u2 = rep(1:p1, times = nsample))
d$u1 <- as.factor(d$u1); d$u2 <- as.factor(d$u2)
 
# LMM: y with random intercept
b1 <- 1; b2 <- -1; sd1 <- 1.5
d$y_re_intercept <- b1 * d$x1 + b2 * d$x2 + 
  rep(rnorm(n = p1, sd = sd1), each = nsample) +  # random intercept u1
  rep(rnorm(n = p1, sd = sd1), times = nsample) + # random intercept u2
  rnorm(n = n)
 
# LMM: y with random slope
b1 <- 0; sd1 <- 1; sd.x1 <- 2
d$y_re_slope <- b1 * d$x1 + 
  rep(rnorm(n = p1, sd = sd1), each = nsample) + # random intercept u1
  d$x1 * rep(rnorm(n = p1, sd = sd.x1), times = nsample) + # random slope u1
  rnorm(n = n)
 
# GLMM
b1 <- 1; sd1 <- 1.5
prob <- rr2::inv.logit(b1 * d$x1 + rep(rnorm(n = p1, sd = sd1), each = nsample)) 
# random intercept u1
d$y_binary <- rbinom(n = n, size = 1, prob = prob)
 
# PGLS
b1 <- 1.5; signal <- 0.7
phy <- ape::compute.brlen(ape::rtree(n = n), method = "Grafen", power = 1)
phy.x <- ape::compute.brlen(phy, method = "Grafen", power = .0001)
x_trait <- ape::rTraitCont(phy.x, model = "BM", sigma = 1)
e <- signal^0.5 * ape::rTraitCont(phy, model = "BM", sigma = 1) + (1-signal)^0.5 * rnorm(n=n)
d$x_trait <- x_trait[match(names(e), names(x_trait))]
d$y_pgls <- b1 * x_trait + e
rownames(d) <- phy$tip.label    
 
# Phylogenetic Logistic Regression
b1 <- 1.5; signal <- 2
e <- signal * ape::rTraitCont(phy, model = "BM", sigma = 1)
e <- e[match(phy$tip.label, names(e))]
d$y_phy_binary <- rbinom(n = n, size = 1, prob = rr2::inv.logit(b1 * d$x1 + e))
 
head(d)

##              x1          x2 u1 u2 y_re_intercept y_re_slope y_binary
## t31 -0.56047565 -0.71040656  1  1       3.053041 -0.2790159        1
## t37 -0.23017749  0.25688371  1  2       3.794671  1.7435372        0
## t8   1.55870831 -0.24669188  1  3       8.062178 -0.3410566        1
## t70  0.07050839 -0.34754260  1  4       3.649759  0.5076822        0
## t53  0.12928774 -0.95161857  1  5       2.526704  0.2830316        0
## t13  1.71506499 -0.04502772  1  6       7.631604 -8.5551981        0
##         x_trait     y_pgls y_phy_binary
## t31 -2.07597968 -2.8257102            0
## t37 -0.31921893 -0.6918108            0
## t8  -0.24097587 -0.3359352            1
## t70 -0.08278377 -0.6383157            0
## t53 -1.60010819 -1.3718365            0
## t13 -1.52297135 -2.0347222            1

Then, let’s fit some models and calculate their R²s.

LM

library(rr2)
z.f.lm <- lm(y_re_intercept ~ x1 + x2, data = d)
z.x.lm <- lm(y_re_intercept ~ x1, data = d)
z.0.lm <- lm(y_re_intercept ~ 1, data = d)
 
R2(mod = z.f.lm, mod.r = z.x.lm)

##    R2_lik  R2_resid   R2_pred 
## 0.2473776 0.2473776 0.2473776

partialR2(mod = z.f.lm, mod.r = z.x.lm)

## [1] 0.2473776

partialR2adj(mod = z.f.lm, mod.r = z.x.lm)

## $R2
## [1] 0.2473776
## 
## $R2.adj
## [1] 0.4982517

LMM

z.f.lmm <- lme4::lmer(y_re_intercept ~ x1 + x2 + (1 | u1) + (1 | u2), data = d, REML = F)
z.x.lmm <- lme4::lmer(y_re_intercept ~ x1 + (1 | u1) + (1 | u2), data = d, REML = F)
z.v.lmm <- lme4::lmer(y_re_intercept ~ 1 + (1 | u2), data = d, REML = F)
z.0.lmm <- lm(y_re_intercept ~ 1, data = d)
 
R2(mod = z.f.lmm, mod.r = z.x.lmm)

##    R2_lik  R2_resid   R2_pred 
## 0.5356524 0.6036311 0.6087728

R2(mod = z.f.lmm, mod.r = z.v.lmm)

##    R2_lik  R2_resid   R2_pred 
## 0.7441745 0.8373347 0.8559029

R2(mod = z.f.lmm, mod.r = z.0.lmm)

##    R2_lik  R2_resid   R2_pred 
## 0.7762978 0.8767789 0.8991618

R2(mod = z.f.lmm) # if omit mod.r, default will be the simplest model, such as z.0.lmm here.

##    R2_lik  R2_resid   R2_pred 
## 0.7762978 0.8767789 0.8991618

GLMM

z.f.glmm <- lme4::glmer(y_binary ~ x1 + (1 | u1), data = d, family = "binomial")
z.x.glmm <- lme4::glmer(y_binary ~ 1 + (1 | u1), data = d, family = "binomial")
z.v.glmm <- glm(y_binary ~ x1, data = d, family = "binomial")
 
R2(mod = z.f.glmm, mod.r = z.x.glmm)

##    R2_lik  R2_resid   R2_pred 
## 0.1170588 0.1413694 0.1373521

R2(mod = z.f.glmm, mod.r = z.v.glmm)

##    R2_lik  R2_resid   R2_pred 
## 0.1990563 0.3404476 0.3545240

R2(mod = z.f.glmm)

##    R2_lik  R2_resid   R2_pred 
## 0.2406380 0.3659939 0.3792381

# specify sigma2_d for R2.resid()
R2.resid(mod = z.f.glmm, mod.r = z.v.glmm, sigma2_d = "s2w")

## [1] 0.3404476

R2.resid(mod = z.f.glmm, mod.r = z.v.glmm, sigma2_d = "NS") 

## [1] 0.4246935

R2.resid(mod = z.f.glmm, mod.r = z.v.glmm, sigma2_d = "rNS")

## [1] 0.4553596

PGLS

z.f.pgls <- phylolm::phylolm(y_pgls ~ x_trait, phy = phy, data = d, model = "lambda")
z.v.lm <- lm(y_pgls ~ x_trait, data = d)
 
# phy is needed for phylogenetic models' R2.resid and R2.pred
R2(mod = z.f.pgls, mod.r = z.v.lm, phy = phy)

##    R2_lik  R2_resid   R2_pred 
## 0.2353912 0.3590018 0.3114035

R2(mod = z.f.pgls, phy = phy)

##    R2_lik  R2_resid   R2_pred 
## 0.8642865 0.8862266 0.8777782

# This also works for models fit with nlme::gls()
z.f.gls <- nlme::gls(y_pgls ~ x_trait, data = d, correlation = ape::corPagel(1, phy), method = "ML")
z.x.gls <- nlme::gls(y_pgls ~ 1, data = d, correlation = ape::corPagel(1, phy), method = "ML")
R2(mod = z.f.gls, mod.r = z.v.lm)

##    R2_lik  R2_resid   R2_pred 
## 0.2353912 0.3590294 0.3114048

R2(mod = z.f.gls)

##    R2_lik  R2_resid   R2_pred 
## 0.8642865 0.8862315 0.8777784

Phylogenetic Logistic Regression

Note: we modified ape::binaryPGLMM to return necessary components for rr2::R2().

z.f.plog <- rr2::binaryPGLMM(y_phy_binary ~ x1, data = d, phy = phy)
z.x.plog <- rr2::binaryPGLMM(y_phy_binary ~ 1, data = d, phy = phy)
z.v.plog <- glm(y_phy_binary ~ x1, data = d, family = "binomial")
 
# R2.lik can't be used with binaryPGLMM because it is not a ML method
R2(mod = z.f.plog, mod.r = z.x.plog)

## Models of class binaryPGLMM do not have R2.lik method.

##  R2_resid   R2_pred 
## 0.6115816 0.3344832

R2(mod = z.f.plog)

## Models of class binaryPGLMM do not have R2.lik method.

##  R2_resid   R2_pred 
## 0.8076862 0.5531285

z.f.plog2 <- phylolm::phyloglm(y_phy_binary ~ x1, data = d, start.alpha = 1, phy = phy)
z.x.plog2 <- phylolm::phyloglm(y_phy_binary ~ 1, data = d, phy = phy, 
                               start.alpha = min(20, z.f.plog2$alpha))
z.v.plog2 <- glm(y_phy_binary ~ x1, data = d, family = "binomial")
 
# R2.resid and R2.pred do not apply for phyloglm
R2(z.f.plog2, z.x.plog2) 

## Models of class phyloglm only have R2.lik method.

##    R2_lik 
## 0.3853273

# alternate
R2.lik(z.f.plog2, z.x.plog2)

## [1] 0.3853273

Contributions of predictors

We can use rr2::R2() to calculate partial R²s and compare contributions of different predictors. Here is an example using phylolm::phyloglm(). The same comparisons can be also applied to other types of models.

z.f <- phylolm::phyloglm(y_phy_binary ~ x1 + x2, data = d, start.alpha = 1, phy = phy)
z.r1 <- phylolm::phyloglm(y_phy_binary ~ x1, data = d, start.alpha = 1, phy = phy)
z.r2 <- phylolm::phyloglm(y_phy_binary ~ x2, data = d, start.alpha = 1, phy = phy)
# total R2
R2(z.f)

## Models of class phyloglm only have R2.lik method.

##    R2_lik 
## 0.4825879

# contribution of x1
R2(z.f, z.r2)

## Models of class phyloglm only have R2.lik method.

##    R2_lik 
## 0.3877523

# contribution of x2
R2(z.f, z.r1)

## Models of class phyloglm only have R2.lik method.

##      R2_lik 
## 0.004148668

It is also possible to estimate the “contribution” of correlation structrues in the model. For the above example, we can replace the phylogeny with a star phylogeny and then compare the R²s of the two models.

# see the first chunk R code for the build of phy.x, a star phylogeny
z.r3 <- phylolm::phyloglm(y_phy_binary ~ x1 + x2, data = d, start.alpha = 1, phy = phy.x)
R2(z.f, z.r3)

## Models of class phyloglm only have R2.lik method.

##    R2_lik 
## 0.3124863

Citation

Please cite the following papers if you find this package useful:

Models, LMMs, and GLMMs. Systematic Biology,
syy060.](https://doi.org/10.1093/sysbio/syy060)  

• Anthony R. Ives and Daijiang Li (2018). rr2: An R package to calculate R^2s for regression models. The Journal of Open Source Software, 3(30), 1028.

Contributing

Contributions are welcome. You can either provide comments and feedback by filing an issue on Github here or making pull requests. It may be easier if you first open an issue outlining what you will do in the pull request.

Questions about the package can also be posted as issues on Github.