Generalized linear models in R
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Generalized linear models in R
www.nr.no Generalized linear models in R Magne Aldrin, Norwegian Computing Center and the University of Oslo Sharp workshop, Copenhagen, October 2012 Linear regression Model y = β 0 + β 1 x 1 + β 2 x2 + · · · + β p x p + ε • • • • • y: response variable, continious x1, · · · xp : p explanatory variables, predictor variables, covariates β0: intercept β1, · · · βp: regressions coeffisients ε: error term, E(ε) = 0, V ar(ε) = σ 2 1 Estimation • The β’s are unknown, and must be estimated from observations • n observations of yi, x1i, . . . xpi • Estimation by Ordinary Least Squares P ? Minimize i(yi − ŷi)2, ? where ŷi = βˆ0 + βˆ1x1i + βˆ2x2i + · · · + βˆpxpi • If ε is normal distributed (Gaussian), then OLS = Maximum Likelihood (ML) 2 Generalized Linear Models - GLM • • • • • • y ∼ Distributed with mean µ and perhaps an additional parameter h(µ) = η η = β 0 + β 1 x 1 + β 2 x2 + · · · + β p x p η: linear predictor h: link fuction Estimation by maximum likelihood 3 Gaussian regression/linear regression • Continious, symmetric response • y ∼ N (µ, σ 2) • µ = η : identity link 4 Gamma regression • • • • Continous, positive, skew response y ∼ Gamma(expectation µ, parameter2) log(µ) = η : log link µ = exp(η) 5 Binary logistic regression • • • • • Binary response, take values 0 or 1, binomial with n = 1 trial P (y = 1) = p = E(y) = µ y ∼ Bin(n = 1,p) logit(p) = log(p/(1 − p)) = η : logit link p = exp(η)/(1 + exp(η)) : logistic inverse link 6 Binomial logistic regression • • • • Binomial response, n trials, take values 0, 1, . . . , n y ∼ Bin(n,p) logit(p) = log(p/(1 − p)) = η : logit link p = exp(η)/(1 + exp(η)) : logistic inverse link 7 Poisson regression • • • • Count response, take values 0, 1, 2, . . . y ∼ Poisson(λ = µ) log(µ) = η : log link µ = exp(η) 8 Ex: Ozone data - Gaussian regression • Ozone - O3 is the response variable • 9 expanatory variables (wind speed, humidity, temperature ...) • 330 observation > require(faraway) > data(ozone) > > summary(ozone) O3 vh Min. : 1.00 Min. :5320 1st Qu.: 5.00 1st Qu.:5690 Median :10.00 Median :5760 Mean :11.78 Mean :5750 3rd Qu.:17.00 3rd Qu.:5830 Max. :38.00 Max. :5950 wind Min. : 0.000 1st Qu.: 3.000 Median : 5.000 Mean : 4.848 3rd Qu.: 6.000 Max. :11.000 humidity Min. :19.00 1st Qu.:47.00 Median :64.00 Mean :58.13 3rd Qu.:73.00 Max. :93.00 9 temp Min. :25.00 1st Qu.:51.00 Median :62.00 Mean :61.75 3rd Qu.:72.00 Max. :93.00 vis Min. : 0.0 1st Qu.: 70.0 Median :120.0 Mean :124.5 3rd Qu.:150.0 Max. :350.0 ibh Min. : 111.0 1st Qu.: 877.5 Median :2112.5 Mean :2572.9 3rd Qu.:5000.0 Max. :5000.0 doy Min. : 33.0 1st Qu.:120.2 Median :205.5 Mean :209.4 3rd Qu.:301.8 Max. :390.0 dpg Min. :-69.00 1st Qu.: -9.00 Median : 24.00 Mean : 17.37 3rd Qu.: 44.75 Max. :107.00 ibt Min. :-25.0 1st Qu.:107.0 Median :167.5 Mean :161.2 3rd Qu.:214.0 Max. :332.0 10 > pdf("PAIRSozone.pdf") > pairs(ozone) > dev.off() 11 5300 5800 20 60 0 3000 200 0 300 ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ● ● ● ●● ●● ●● ●● ● ● ●● ● ●● ● ● ● ●● ●● ●● ●● ● ●● ●● ● ● ● ● ●● ●●● ● ●●● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ●●●● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ●●● ● 0 ● ●● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ●●●● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ● ●●●● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ●●●● 0 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● 0 4 8 ● ● ●● ●●● ●● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 300 ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● 70 ● ●● ● ● ● ●●●● ●● ● ● ● ●●● ● ● ●● ●● ●●●●● ● ●● ●● ● ● ● ● ● ●● ●● ● ● ● ● ●●● ● ●● ● ●● ● ● ●● ●●●●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●●● ●●●●● ●● 50 ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 30 ● ● ●● ● ●●● ● ● ● ● ● ●●●● ●●● ● ● ● ●● ● ● ● ● ● ●●● ●● ● ●●● ● ●● ● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ● ●●● ● ● ●●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ●●● ● ●●● ●●●●● ● ● ● ● ● ● ●● ●● ● 200 ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ●● ● 100 vh ● ● ● ● ● ● ● 0 ● ●●● ● ●● ●●● ●●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ●● ● −50 ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 3000 ● ● ●● ●● ●●● ●● ● ● ●● ●● ●● ●● ●● ● ● ●●● ●● ● ● ●●● ●● ●● ●● ● ● ● ● ● ● ●●●●● ●●● ● ● ● ● ● ● ● ● ●●●● ● ●● ●●● ●● ●● ●●● ●● ●●●●● 200 ● ● ● ● ● ● ● O3 50 20 60 ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●●●● ●● ● ● ●● ● ● ● ● ● ● ●● ● 0 5300 5800 200 12 > lmobj<-lm(log(O3)~vh+wind+humidity+temp+ibh+dpg+ibt+vis+doy, data=ozone) > glmobj<-glm(log(O3)~vh+wind+humidity+temp+ibh+dpg+ibt+vis+doy, family=gaussian(link=identity),data=ozone) > require(mgcv) > glmobj<-gam(log(O3)~vh+wind+humidity+temp+ibh+dpg+ibt+vis+doy, family=gaussian(link=identity),data=ozone) > summary(glmobj) Family: gaussian Link function: identity Formula: log(O3) ~ vh + wind + humidity + temp + ibh + dpg + ibt + vis + doy 13 Parametric coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.301e-01 2.684e+00 0.235 0.81453 vh -1.244e-05 4.908e-04 -0.025 0.97980 wind -8.403e-03 1.226e-02 -0.685 0.49357 humidity 5.084e-03 1.713e-03 2.968 0.00322 ** temp 2.993e-02 4.527e-03 6.611 1.60e-10 *** ibh -8.495e-05 2.687e-05 -3.161 0.00172 ** dpg 5.470e-04 1.026e-03 0.533 0.59414 ibt 4.902e-04 1.237e-03 0.396 0.69221 vis -8.395e-04 3.410e-04 -2.462 0.01434 * doy -1.021e-03 2.522e-04 -4.049 6.47e-05 *** --Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 R-sq.(adj) = 0.709 GCV score = 0.16808 Deviance explained = 71.7\% Scale est. = 0.16298 n = 330 14 If y is skew and positive, alternative 1 Take the logarithm of y, log(y) is more symmetric log(y) = η + ε y = exp (η + ε) = exp (η) · exp (ε) E(y) = exp (η) · E(exp (ε)) E(exp (ε)) = exp (σ 2/2) if ε ∼ N (0, σ 2) 15 If y is skew and positive, alternative 2 E(y) = µ log(µ) = η y ∼ Gamma(µ, parameter2) 16 Ex: Ozone data - Gamma regression > > ### GLM, O3 ~ Gamma-distributed with identity link > glmobj<-gam(O3~vh+wind+humidity+temp+ibh+dpg+ibt+vis+doy, family=Gamma(link=log),data=ozone) > summary(glmobj) Family: Gamma Link function: log Formula: O3 ~ vh + wind + humidity + temp + ibh + dpg + ibt + vis + doy 17 Parametric coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.805e-01 2.583e+00 0.147 0.882957 vh 6.323e-05 4.723e-04 0.134 0.893591 wind -8.965e-03 1.180e-02 -0.760 0.447870 humidity 5.752e-03 1.648e-03 3.490 0.000551 *** temp 2.382e-02 4.356e-03 5.469 9.15e-08 *** ibh -6.420e-05 2.586e-05 -2.483 0.013548 * dpg 9.455e-04 9.869e-04 0.958 0.338775 ibt 1.527e-03 1.191e-03 1.283 0.200561 vis -8.488e-04 3.281e-04 -2.587 0.010126 * doy -1.011e-03 2.427e-04 -4.165 4.01e-05 *** --Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 R-sq.(adj) = 0.736 GCV score = 0.15567 Deviance explained = 71.1\% Scale est. = 0.15095 n = 330 18 Ex: Precipitation - binary logistic regression • 10 years data with daily precipitation at Tryvann, Oslo • Response is P01: 1 if precipitation, 0 if not, at day t • 6 predictors ? P01.L1 - 1 if precipitation, 0 if not, at day t − 1 ? P01.L2 - 1 if precipitation, 0 if not, at day t − 2 ? P01.L3 - 1 if precipitation, 0 if not, at day t − 3 ? Prep.L1 - amount of precipitation (mm) day t − 1 ? Prep.L2 - amount of precipitation (mm) day t − 2 ? Prep.L3 - amount of precipitation (mm) day t − 3 19 R code ### Note: Linux style on directory structure used here > Tryvann.dat<-read.table("/nr/user/aldrin/Sharp/data/Tryvann.dat") > > ### load the mgcv library > require(mgcv) > > ### Logistic regression with binary precipitation (0=no, 1=yes) as r > glmobj<-gam(P01~P01.L1+P01.L2+P01.L3+Prep.L1+Prep.L2+Prep.L3, data=Tryvann.dat,family=binomial(link=logit)) > summary(glmobj) Family: binomial Link function: logit Formula: P01 ~ P01.L1 + P01.L2 + P01.L3 + Prep.L1 + Prep.L2 + Prep.L3 20 Parametric coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.774966 0.068825 -11.260 < 2e-16 *** P01.L1 1.064480 0.083536 12.743 < 2e-16 *** P01.L2 0.322184 0.085679 3.760 0.00017 *** P01.L3 0.411833 0.084189 4.892 1.00e-06 *** Prep.L1 0.032718 0.007428 4.405 1.06e-05 *** Prep.L2 -0.009255 0.006212 -1.490 0.13626 Prep.L3 -0.004704 0.006028 -0.780 0.43523 --Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 R-sq.(adj) = 0.125 UBRE score = 0.24364 Deviance explained = 9.43\% Scale est. = 1 n = 3420 21 Ex: Simulated data - Poisson regression n<-200 > x1<-rnorm(n) # N(0,1) > x2<-runif(n,-10,10) # Uniform(-10,10) > x3<-rnorm(n) > > eta<- 1 + 2*x1 - 0.2*x2^2 + 0*x3 > mu<-exp(eta) > > y<-rpois(n=n,lambda=mu) > > data.obj<-data.frame(y=y,x1=x1,x2=x2,x3=x3) > > ### load the mgcv library > require(mgcv) Loading required package: mgcv This is mgcv 1.7-6. For overview type ’help("mgcv-package")’. > 22 > ### Estimation > glmobj<-gam(y~x1+x2+x3,family=poisson(link=log),data=data.obj) > summary(glmobj) Family: poisson Link function: log Formula: y ~ x1 + x2 + x3 23 Parametric coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 0.154044 0.071537 2.153 0.0313 * x1 0.802362 0.051533 15.570 < 2e-16 *** x2 -0.038901 0.009412 -4.133 3.58e-05 *** x3 -0.107352 0.059222 -1.813 0.0699 . --Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 R-sq.(adj) = -0.00555 Deviance explained = 21.1\% UBRE score = 3.6933 Scale est. = 1 n = 200 24 Prediction for new data points > new.data.obj<-data.frame(x1=c(0,1),x2=c(0,0),x3=c(0,0)) > predict(glmobj,newdata=new.data.obj,type="response") 1 2 1.166542 2.602326 25 Model selection • Overfitting: The model has too many parameters relative to the number of observations ? large estimation uncertainty ? poor predictions • Underfitting: The model has too few parameters relative to the number of observations ? poor fit to data ? does not use all information in the data ? poor predictions • Methods for model selection ? Cross-validation - mentioned in space-time modeling course ? AIC ? BIC 26 Model selection by AIC and BIC • Want as best fit as possible with as few parameters as possible • p=number of parameters and n=number of observations • AIC = - 2 log likelihood + 2 p • BIC = - 2 log likelihood + log(n) p 27 Ex: Model selection using BIC > glmobj1<-gam(y~x1,family=poisson(link=log),data=data.obj) > glmobj2<-gam(y~x2,family=poisson(link=log),data=data.obj) > glmobj12<-gam(y~x1+x2,family=poisson(link=log),data=data.obj) > glmobj123<-gam(y~x1+x2+x3,family=poisson(link=log),data=data.obj) > > bic<-rep(NA,4) > bic[1]<-BIC(glmobj1) > bic[2]<-BIC(glmobj2) > bic[3]<-BIC(glmobj12) > bic[4]<-BIC(glmobj123) > > bic [1] 1113.910 1339.247 1104.025 1106.019 • The model with x1 and x2 has the lowest BIC value and is selected 28
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