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