![Prediction intervals
> simhack <- function(params) {
glmnew <- glm1
glmnew$coefficients <- params
## simulates on PROBABILITY scale
simulate(glmnew)[[1]]
1.0
}
> set.seed(101)
0.8 > params <- MASS::mvrnorm(1000,mu=coef(glm1),
Sigma=vcov(glm1))
q
> sims <- apply(params,1,simhack)
Killed/Initial
0.6 q q
> qmat <- t(apply(sims,1,quantile,
q q q
q
q
c(0.5,0.025,0.975)))
q q
0.4 q q q
q
q
q
q
q
q
q
q
(Constructing the simulated
0.2 q
q
values at Initial densities from
0.0 1 to 100 is a bit more work —
0 20 40 60 80 100
ideally all simulate methods
Initial
would have newdata and
newparam arguments . . . )
Ben Bolker (McMaster University) GLMs in R 7 January 2011 17 / 25](https://image.slidesharecdn.com/gtaglm-110105112229-phpapp02/85/GLMs-and-extensions-in-R-17-320.jpg)
GLMs and extensions in R
- 1. Generalized linear models, and extensions, in R Ben Bolker Departments of Mathematics & Statistics and Biology, McMaster University 7 January 2011 Ben Bolker (McMaster University) GLMs in R 7 January 2011 1 / 25
- 2. 1 Introduction 2 Example 3 Challenges, tricks, extensions 4 (Extended examples) Ben Bolker (McMaster University) GLMs in R 7 January 2011 2 / 25
- 3. What are generalized linear models? Modeling framework to solve two common statistical problems: Non-normal data Non-linearity (continuous predictors) . . . superset of, and often confused with, “general” linear models (i.e. ANOVA/ANCOVA/regression: SAS PROC GLM) Ben Bolker (McMaster University) GLMs in R 7 January 2011 3 / 25
- 4. GLMs: technical details Constraints: Distributions from exponential family (Normal, Poisson, binomial, Gamma, inverse Gaussian) Invertible nonlinearities, i.e. there exists a link function that would make the relationship linear (log, logit, probit, inverse, square root, “cauchit” . . . ) , Efficient, stable algorithm: iteratively re-weighted least squares (IRLS) / Fisher scoring) standard methods (methods(class="glm")): coef, summary, plot, predict, residuals, vcov, profile, update, confint, simulate, anova, add1/drop1, logLik, AIC, . . . logistic and Poisson regression probably make up 99% of GLMs . . . Ben Bolker (McMaster University) GLMs in R 7 January 2011 4 / 25
- 5. Google scholar scraping logistic+regression q 580000 Poisson+regression q 39300 generalized+linear+model q 28700 binomial+regression q 13500 104 104.5 105 105.5 106 Ghits Ben Bolker (McMaster University) GLMs in R 7 January 2011 5 / 25
- 6. Example: reed frog predation data 1.0 Vonesh and Bolker (2005): 0.8 q > library(emdbook) Fraction killed 0.6 q q > data(ReedfrogFuncresp) q q q > glm1 <- glm(Killed/Initial~ q q 0.4 q q q q Initial, q q weight=Initial, 0.2 q family=binomial, q data=ReedfrogFuncresp) 0.0 20 40 60 80 100 Initial density Ben Bolker (McMaster University) GLMs in R 7 January 2011 6 / 25
- 7. Summary > summary(glm1) Call: glm(formula = Killed/Initial ~ Initial, family = binomial, data = ReedfrogFuncresp, weights = Initial) Deviance Residuals: Min 1Q Median 3Q Max -4.4132 -0.7275 0.4347 1.0120 1.8172 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.094563 0.188952 -0.50 0.61675 Initial -0.008416 0.002697 -3.12 0.00181 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 47.518 on 15 degrees of freedom Residual deviance: 37.717 on 14 degrees of freedom AIC: 98.639 Number of Fisher Scoring iterations: 4 Ben Bolker (McMaster University) GLMs in R 7 January 2011 7 / 25
- 8. Diagnostics Residuals vs Fitted Normal Q−Q 20 2 q13 5q 13q 16q q q q q 10 q q q q qq q diagnostics inherit Std. deviance resid. 0 q qq q q 0 Residuals q q q q qq q q q from plot.lm −10 −2 q overdispersion: −20 −4 residual deviance −30 q11 q11 −0.8 −0.6 −0.4 Predicted values −0.2 −2 −1 0 Theoretical Quantiles 1 2 ≈ χ2 n−p Scale−Location Residuals vs Leverage 11q (Venables and Ripley, q13 2 16q q 2002, p. 209): 1 5 q 0.5 q q q16 q q q13 sum(residuals(glm1, Std. deviance resid. q 4 q Std. Pearson resid. 0 q q q q q type="pearson")^2) 3 q 0.5 q q q q q 1 −2 =34.3: 2 q q q q q q 1 q p 0.05 −4 q11 Cook's distance 0 −0.8 −0.6 −0.4 −0.2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Predicted values Leverage Ben Bolker (McMaster University) GLMs in R 7 January 2011 8 / 25
- 9. Inference Coefficients: may be hard to communicate (reflect differences on the scale of linear predictor, e.g. logit/log-odds differences) Wald statistics: beware the Hauck-Donner effect (Venables and Ripley, 2002, p. 198). Wald CI of slope: stats:::confint.lm(glm1) (-0.0142,-0.0026) Likelihood ratio test, via anova: > anova(glm1,test="Chisq") ## OR > glm0 <- update(glm1, . ~ -Initial) > anova(glm1,glm0,test="Chisq") Likelihood profiles (via MASS::profile.glm), profile confidence intervals: MASS:::confint.glm(glm1) (-0.0137,-0.0031) Ben Bolker (McMaster University) GLMs in R 7 January 2011 9 / 25
- 10. Estimation issues Convergence difficulties, especially with non-standard links: set starting values, center/scale variables (?) Complete separation: brglm, logistf, arm (bayesglm) Big data: biglm (bigglm) Many predictors (penalized regression): glmnet, glmpath, penalized (Machine learning task view) Ben Bolker (McMaster University) GLMs in R 7 January 2011 10 / 25
- 11. Tricks (within GLM framework) non-standard link functions: fitting hyperbolic models of predator attack rates (Michaelis-Menten) via binomial/inverse link (http://emdbolker.wikidot.com/voneshglm) exponential survivorship models via binomial/log link (Strong et al., 1999; Tiwari et al., 2006) Gaussian family with log link: fit exponential growth models with constant variance subtleties with Gamma GLMs and dispersion parameter: V&R MASS online complements, Paul Johnson’s notes offsets: variation in sampling area/intensity (e.g. strict proportionality) Ben Bolker (McMaster University) GLMs in R 7 January 2011 11 / 25
- 12. Overdispersion Quasilikelihood models: > glmQ <- update(glm1,family="quasibinomial") > anova(glmQ,test="F") ˆ (φ = 2.45). No likelihood: qAIC requires some contortions extended GLMs negative binomial: MASS (glm.nb) beta-binomial: aod (betabin) gnlm (gnlr) VGAM (vglm) bbmle (mle2) GLMMs: lognormal-Poisson, logit-normal-binomial robust estimation (lmtest, sandwich): > coeftest(glm1,vcov=sandwich) See also the vignette for the pscl package. Ben Bolker (McMaster University) GLMs in R 7 January 2011 12 / 25
- 13. Extensions Generalized additive models (Wood, 2006): mgcv, gamlss Zero-inflated/altered/hurdle models: pscl, VGAM Beta regression: betareg Generalized regression models: bbmle, VGAM, gnlm Random effects (generalized linear mixed models): lme4 and other packages (http://glmm.wikidot.com/faq) Ben Bolker (McMaster University) GLMs in R 7 January 2011 13 / 25
- 14. References Strong, D.R., Whipple, A.V., et al., 1999. Ecology, 80:2750–2761. Tiwari, M., Bjorndal, K.A., et al., 2006. Marine Ecological Progress Series, 326:283–293. Venables, W. and Ripley, B.D., 2002. Modern Applied Statistics with S. Springer, New York, 4th edition. Vonesh, J.R. and Bolker, B.M., 2005. Ecology, 86(6):1580–1591. Wood, S.N., 2006. Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC. Ben Bolker (McMaster University) GLMs in R 7 January 2011 14 / 25
- 15. Basic ggplot code > qplot(Initial,Killed/Initial,data=ReedfrogFuncresp)+ geom_smooth(method=glm,family=binomial, aes(weight=Initial,group=NA)) Ben Bolker (McMaster University) GLMs in R 7 January 2011 15 / 25
- 16. Confidence intervals on # killed, by hand > pframe <- data.frame(Initial=1:100) > pp <- predict(glm1,newdata=pframe,se.fit=TRUE) > pmat <- with(pp,plogis(cbind(fit, fit-1.96*se.fit, fit+1.96*se.fit))) > par(bty="l",las=1) > with(ReedfrogFuncresp,plot(Initial,Killed/Initial, xlim=c(0,100),ylim=c(0,1), pch=16)) > matlines(pframe$Initial,pmat,lty=c(1,2,2),col=1,type="l") Ben Bolker (McMaster University) GLMs in R 7 January 2011 16 / 25
- 17. Prediction intervals > simhack <- function(params) { glmnew <- glm1 glmnew$coefficients <- params ## simulates on PROBABILITY scale simulate(glmnew)[[1]] 1.0 } > set.seed(101) 0.8 > params <- MASS::mvrnorm(1000,mu=coef(glm1), Sigma=vcov(glm1)) q > sims <- apply(params,1,simhack) Killed/Initial 0.6 q q > qmat <- t(apply(sims,1,quantile, q q q q q c(0.5,0.025,0.975))) q q 0.4 q q q q q q q q q q q (Constructing the simulated 0.2 q q values at Initial densities from 0.0 1 to 100 is a bit more work — 0 20 40 60 80 100 ideally all simulate methods Initial would have newdata and newparam arguments . . . ) Ben Bolker (McMaster University) GLMs in R 7 January 2011 17 / 25
- 18. Alternative display (display, coefplot from arm package) −0.015 −0.010 −0.005 0.000 Initial q > display(glm1) glm(formula = Killed/Initial ~ Initial, family = binomial, data = Re weights = Initial) coef.est coef.se (Intercept) -0.09 0.19 Initial -0.01 0.00 --- n = 16, k = 2 residual deviance = 37.7, null deviance = 47.5 (difference = 9.8) Ben Bolker (McMaster University) GLMs in R 7 January 2011 18 / 25
- 19. Beta-binomial with aod > library(aod) > glmBB1 <- betabin(cbind(Killed, Initial-Killed)~Initial, random=~1, data=ReedfrogFuncresp) Ben Bolker (McMaster University) GLMs in R 7 January 2011 19 / 25
- 20. Beta-binomial with bbmle > library(bbmle) > glmBB3 <- mle2(Killed~dbetabinom(prob=plogis(logitp), theta=exp(logtheta),size=Initial), parameters=list(logitp~Initial), data=ReedfrogFuncresp, start=list(logitp=0,logtheta=0)) Ben Bolker (McMaster University) GLMs in R 7 January 2011 20 / 25
- 21. Beta-binomial with VGAM > library(VGAM) > glmBB4 <- vglm(cbind(Killed,Initial-Killed)~Initial, betabinomial, data=ReedfrogFuncresp) > coef(glmBB4,matrix=TRUE) Ben Bolker (McMaster University) GLMs in R 7 January 2011 21 / 25
- 22. Beta-binomial with gnlm > library(gnlm) > attach(ReedfrogFuncresp) ## no data= argument! > glmBB2 <- gnlr(cbind(Killed,Initial-Killed), dist="beta binomial", pmu=c(0,0),pshape=0, mu=function(p,linear) plogis(linear), linear=~Initial) > detach(ReedfrogFuncresp) > detach("package:gnlm") > detach("package:rmutil") Ben Bolker (McMaster University) GLMs in R 7 January 2011 22 / 25
- 23. Logit-normal-Poisson with lme4 > library(lme4) > ReedfrogFuncresp$ID <- 1:nrow(ReedfrogFuncresp) > glmLNP <- glmer(cbind(Killed,Initial-Killed)~Initial+(1|ID), family=binomial, data=ReedfrogFuncresp) > summary(glmLNP) Ben Bolker (McMaster University) GLMs in R 7 January 2011 23 / 25
- 24. Alternate link functions for reed frog data 1.0 0.8 Fraction killed q 0.6 q q q q q q q q 0.4 q q q q q 0.2 q q 0.0 20 40 60 80 100 Initial density Ben Bolker (McMaster University) GLMs in R 7 January 2011 24 / 25
- 25. Comparing overdispersion estimates LN−binomial q beta−binomial q sandwich q model q−binom Wald q binomial profile q binomial Wald q −0.015 −0.010 −0.005 0.000 initial density effect Ben Bolker (McMaster University) GLMs in R 7 January 2011 25 / 25