一般化線形モデル（GLM）
generalized linear Models
データ解析のための統計モデリング入門
久保拓也（２０１２） 岩波書店
Generalized Linear Models
• Linear Model

y i   1   2 xi   3 f i  
response variable ～ intercept + slope * explanatory variable
 lm(y~ x + f ・・・)，lm(y~x + f -1) (no intercept)
require(graphics)
## Annette Dobson (1990) "An Introduction to Generalized Linear Models".
## Page 9: Plant Weight Data.
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
group <- gl(2,10,20, labels=c("Ctl","Trt"))
weight <- c(ctl, trt)
lm.D9 <- lm(weight ~ group)
lm.D90 <- lm(weight ~ group - 1) # omitting intercept
anova(lm.D9)
summary(lm.D90)
opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0))
plot(lm.D9, las = 1)
# Residuals, Fitted, ...
Par(opar)
Generalized Linear Models
• Linear Model

y i   1   2 xi   3 f i  
response variable ～ intercept + slope * explanatory variable
 lm(y~ x + f ・・・)，lm(y~x + f -1) (no intercept)
Generalized Linear Model
 f ( y i )   1   2 xi   3 f i  
Model &Link function ～ intercept + slope * explanatory variable
 glm(y ~ x, data = d, family = poisson)
Poisson Model
（counting data of occurrence)
Poisson Model
 exp(   )
y
p( y |  ) 
  E ( y) 

y!
 λ：mean occurrence in unit time
yp ( y )
y0
Identity link  i   1   2 x i   3 f i  
Log link(canonical) log(  i )   1   2 x i   3 f i  
 i  exp(  1 ) exp(  2 x i ) exp(  3 f i ) 
正準リンク関数：最も自然なリンク関数：乗法効果）
Link function ～ intercept + slope * explanatory variable
 glm(y ~ x, data = d, family = poisson(link=“log”))
Canonical link function is set as default
Poisson Model (p49)
（counting data of occurrence)
Poisson Model for number of seeds of a plant,
regressed on plant size and nutrification (p49)
Maximize log-likelihood y
log L (  1 ,  2 ,  3 ) 
 log
i
 i exp(   i )
i
yi!
log(  i )   1   2 x i   3 f i
glm(y ~ x + f, data = d, family = poisson)
#page 42 plant data
d <- read.csv("data3a.csv")
d$y # number of seeds
d$x # plant size (hight)
d$f # nutrification (treat-control)
plot(d$x, d$y, pch =c(21, 19)[d$f])
# model p58
fit.all <- glm(y ~ x + f, data=d, family=poisson)
print(fit.all)
logLik(fit.all)
plot(d$x, d$y, pch =c(21, 19)[d$f])
xx <- seq(min(d$x), max(d$x), length =100)
lines(xx,exp(1.263 + 0.0801 * xx), lwd=2)
Poisson Model (p49)
（counting data of occurrence)
log(  i )   1   2 x i   3 f i
6
yi!
4
i
2
 log
yi
d$y
log L (  1 ,  2 ,  3 ) 
 i exp(   i )
8 10
14
Poisson Model for number of seeds of a plant,
regressed on plant size and nutrification (p49)
Maximize log-likelihood
7
#page 42 plant data
d <- read.csv("data3a.csv")
d$y # number of seeds
d$x # plant size (hight)
d$f # nutrification (treat-control)
plot(d$x, d$y, pch =c(21, 19)[d$f])
8
9
10
11
12
# model p58
d$x
fit.all <- glm(y ~ x + f, data=d, family=poisson)
print(fit.all)
logLik(fit.all)
plot(d$x, d$y, pch =c(21, 19)[d$f])
xx <- seq(min(d$x), max(d$x), length =100)
lines(xx,exp(1.263 + 0.0801 * xx), lwd=2)
Other Generalized Linear Models
(chap6 p114)
(discrete)
(continuous)
Probability
Random
numbers
generation
Family
in glm()
Standard link
function
Binomial
rbinom()
binomial
logit
Poisson
rpois()
poisson
log
Negative
Binomial
rnbinom()
(glb.nb()
function)
log
Gamma
rgamma()
gamma
log, inverse
Normal
rnorm()
gaussian
identity
Generalized Linear Models
Generalized Linear Model
f ( y i )   1   2 xi   3 f i  
glm(y ~ x, data = d, family = poisson)
Family （Modelled Probability Distribution)
binomial(link = “logit“) 2項分布（規定試行中の発生数）
gaussian(link = “identity”) 正規分布
Gamma(link = “inverse”) ガンマ分布（正のみ）
inverse.gaussian(link = “1/mu^2”) 逆ガウス分布
poisson(link = “log”) ポアソン分布（一定時間中の発生回数）
quasi(link = “identity”, variance = “constant”)正規分布（不均一）
 quasibinomial(link = “logit”) 2項分布（分散不均一）
 quasipoisson(link = “log”)
ポアソン分布（分散不均一）
Binomial Logistic Model (p118)
（occurrence number in given trials)
Binomial Model for the number of survived
plant in 8 obserbations, regressed on plant
size and nutrification (p118)
1
Maximize log-likelihood q i  logitstic ( z i ) 
N
p ( y | N , q )  
 y
 y
Ny
 q (1  q )

1  exp(  z i )
z i  log
qi
1  qi
glm(cbind(y,N-y) ~ x + f, data = d, family = binomial)
#page 117 plant data
d <- read.csv("data4a.csv")
d$N # number of trials
d$y # number of survived plant
d$x # plant size
d$f # nutrification (treat-control)
# model p122
fit.all <- glm(cbind(y, N-y) ~ x + f, data=d, family=binomi
print(fit.all)
logLik(fit.all)
Offset Term(p131)
（avoid a division calculation)
Count data for several zones having different
area, or different population
One way is define a density (occurrence in unit
area) and apply Poisson model   A exp(    x )
log L (  1 ,  2 ,  3 ) 
 log
i
 i exp(   i )
yi
yi!
i
i
1
2
i
log(  i )   1   2 x i  log Ai
glm(y ~ x, offset =log(A), data = d, family = poisson)
#page 133 plant data
d <- read.csv("data4b.csv")
d$y # number of plants in lot i
d$x # brightness at lot I
d$A # area of lot i
plot(d$A, d$y)
# model p131
fit<- glm(y ~ x, offset = log(A) , data=d, family=poisson)
print(fit)
logLik(fit)
Gamma Distribution Model (p138)
Gamma Distribution (continuous positive data)

P ( y | s, r ) 
r
s
y
s 1
exp(  ry ) 
y
s 1
y
)

s: shape parameter, r: rate parameter, theta=1/r: scale parameter
 time length before s times occurrence of random events with
occurrence rate of r. (average occurrence interval is   1 / r )
 Average : s / r  s  Variance: s / r 2  s  2
(s)
 ( s )
exp( 
s
 dgamma(y, shape, rate)
Weight of flower of a plant y (continuous, positive)
average weight   Ax  exp( a ) x  exp( a  b log x )
Loglink function of linear estimator log(  i )  a  b log
i
b
i
b
i
i
glm(y ~ log(x), data = d, family = gamma(link="log"))
xi
Gamma Distribution Model (p138)
Gamma Distribution (continuous positive data)
glm(y ~ log(x), data = d, family = gamma(link="log")
# A Gamma example, from McCullagh & Nelder (1989, pp. 300-2)
clotting <- data.frame(
u = c(5,10,15,20,30,40,60,80,100),
lot1 = c(118,58,42,35,27,25,21,19,18),
lot2 = c(69,35,26,21,18,16,13,12,12))
summary(glm(lot1 ~ log(u), data=clotting, family=Gamma))
summary(glm(lot2 ~ log(u), data=clotting, family=Gamma))
Call:glm(formula = lot1 ~ log(u), family = Gamma, data = clotting)
Deviance Residuals:
Min
1Q Median
3Q
Max
-0.04008 -0.03756 -0.02637 0.02905 0.08641
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.0165544 0.0009275 -17.85 4.28e-07 ***
log(u)
0.0153431 0.0004150 36.98 2.75e-09 ***
---