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Introduction of Mixed effect model

http://nycdatascience.com/mixed_effect_model_supstat/index.html#1

Introduction of Mixed effect model
Learning by simulation
Supstat Inc.

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Introduction of Mixed effect model

http://nycdatascience.com/mixed_effect_model_supstat/index.html#1

Outline
· What is mixed effect model
· Fixed effect model
· Mixed effect model
- Random Intercept model
- Random Intercept and Slope Model
· General Mixed effect model
· Case study

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Introduction of Mixed effect model

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What is mixed effect model

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Introduction of Mixed effect model

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Classical normal linear model
Formation:
Yi = b0 + b1*Xi + ei
· Yi is response from suject i.
· Xi are covariates.
· b0, b1 are parameters that we want to estimate.
· ei are the random terms in the model, and are assumped to be independently and indentically
distributed from Normal(0,1). It is very important that there is no stucuture in ei and it
represents the variations that could not be controled in our studies.

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Introduction of Mixed effect model

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Violation of independence assumpation.
In many cases, responses are not independent from each other. These data usualy have some
cluster stucture.
· Repeated measures, where measurements are taken multiple times from the same sujects.
(clustered by subject)
· A survey of all the family memebers. (clustered by family)
· A survey of students from 20 classrooms in a high school. (clustered by classroom)
· Longitudial data, or known as the panel data, where several responses are collected from the
same sujects along the time. (clustered by subject)
We need new tools - Mixed effect model.

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Mixed effect model
Mixed effect model = Fixed effect + Random effect
· Fixed effects
- expected to have a systematic and predictable influence on your data.
- exhaust “the levels of a factor”.Think of sex(male/femal).
· Random effect
- expected to have a non-systematic, unpredictable, or “random” influence on your data.
- Random effects have factor levels that are drawn from a large population, but we do not
know exactly how or why they differ.

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Introduction of Mixed effect model

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Example of Fixed effects and Random effects
FIXED EFFECTS
Male or female

Individuals with repeated measures

Insecticide sprayed or not

Block within a field

Upland or lowland

Brood

One country versus another

Split plot within a plot

Wet versus dry

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RANDOM EFFECTS

Family

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Introduction of Mixed effect model

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Fixed effect model

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Introduction of Mixed effect model

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Fixed effect model
Fixed effect model is just the linear model that you maybe already know.
Yi = b0 + b1*Xi + ei
1<i<n n is number of sample
· Yi: Response Variable
· b0: fixed intercept
· b1: fixed slope
· Xi: Explanatory Variable (fixed effect)
· ei: noise (error)

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Introduction of Mixed effect model

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Data generation of fixed effect model
set.seed(1)
# genaerate x
x <- seq(1,5,length.out=100)
# generate error
noise <- rnorm(n=100,mean=0,sd=1)
b0 <- 1
b1 <- 2
# generate y
y <- b0 + b1*x + noise

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Data generation of fixed effect model
plot(y~x)

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Introduction of Mixed effect model

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Cooefficient estimation of fixed effect model
model <- lm(y~x)
summary(model)

Call:
lm(formula = y ~ x)
Residuals:
Min
1Q
-2.3401 -0.6058

Median
0.0155

3Q
0.5851

Max
2.2975

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)
1.1424
0.2491
4.59 1.3e-05 ***
x
1.9888
0.0774
25.70 < 2e-16 ***
--Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.903 on 98 degrees of freedom
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plot of fixed effect model
plot(y~x)
abline(model)

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Introduction of Mixed effect model

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Mixed effect model

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Introduction of Mixed effect model

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Random Intercept model
there are i people, and we repeat measure j times for every people. These poeple are individually
different which we don't know, so there are random effect cause by people, and there are another
random noise cause by measure for every people.
Yij = b0 + b1*Xij + bi + eij
· b0: fixed intercept
· b1: fixed slope
· Xij: fixed effect
· bi: random effect(influence intercept)
· eij: noise

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Introduction of Mixed effect model

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Data generation of Random Intercept model
b0 <- 9.9
b1 <- 2
# repeat measure times for 6 people
n <- c(13, 14, 14, 15, 12, 13)
npeople <- length(n)
set.seed(1)
# generate x(fixed effect)
x <- matrix(rep(0, length=max(n) * npeople),ncol = npeople)
for (i in 1:npeople){
x[1:n[i], i] <- runif(n[i], min = 1, max = 5)
x[1:n[i], i] <- sort(x[1:n[i], i])
}
# random effect
bi <- rnorm(npeople, mean = 0, sd = 10)

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Introduction of Mixed effect model

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Data generation of Random Intercept model
xall <- NULL
yall <- NULL
peopleall <- NULL
for (i in 1:npeople){
xall <- c(xall, x[1:n[i], i]) # combine x
# generate y
y <- rep(b0 + bi[i], length = n[i]) +
b1 * x[1:n[i],i] +
rnorm(n[i], mean = 0, sd = 2) # noise
yall <- c(yall, y) # combine y
people <- rep(i, length = n[i])
peopleall <- c(peopleall, people)
}
# final dataset
data1 <- data.frame(yall,peopleall,xall)

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Introduction of Mixed effect model

http://nycdatascience.com/mixed_effect_model_supstat/index.html#1

Cooefficient estimation of Random Intercept
model
library(nlme)
# xall is fixed effect
# bi influence intercept of model
lme1 <- lme(yall~xall,random=~1|peopleall,data=data1)
summary(lme1)

Linear mixed-effects model fit by REML
Data: data1
AIC BIC logLik
358 368
-175
Random effects:
Formula: ~1 | peopleall
(Intercept) Residual
StdDev:
7.3
1.77
Fixed effects: yall ~ xall
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Introduction of Mixed effect model

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Plot of Random Intercept model

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Introduction of Mixed effect model

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Random Intercept and slope model
Yij = b0 + (b1+si)*Xij + bi + eij
· b0: fixed intercept
· b1: fixed slope
· X: fixed effect
· bi: random effect(influence intercept)
· eij: noise
· si: random effect(influence slope)

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Introduction of Mixed effect model

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Data generation of Random Intercept and
slope model
a0 <- 9.9
a1 <- 2
n <- c(12, 13, 14, 15, 16, 13)
npeople <- length(n)
set.seed(1)
si <- rnorm(npeople, mean = 0, sd = 0.5) # random slope
x <- matrix(rep(0, length = max(n) * npeople),
ncol = npeople)
for (i in 1:npeople){
x[1:n[i], i] <- runif(n[i], min = 1,
max = 5)
x[1:n[i], i] <- sort(x[1:n[i], i])
}

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Introduction of Mixed effect model

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Data generation of Random Intercept and
slope model
bi <- rnorm(npeople, mean = 0, sd = 10) # random intercept
xall <- NULL
yall <- NULL
peopleall <- NULL
for (i in 1:npeople){
xall <- c(xall, x[1:n[i], i])
y <- rep(a0 + bi[i], length = n[i]) +
(a1 + si[i]) * x[1:n[i],i] +
rnorm(n[i], mean = 0, sd = 0.5)
yall <- c(yall, y)
people <- rep(i, length = n[i])
peopleall <- c(peopleall, people)
}
# generate final dataset
data2 <- data.frame(yall, peopleall, xall)

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Introduction of Mixed effect model

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Cooefficient estimation of Random Intercept
and slope model
# bi influence intercept and slope of model
lme2 <- lme(yall~xall,random=~1+xall|peopleall,data=data2)
print(summary(lme2))

Linear mixed-effects model fit by REML
Data: data2
AIC BIC logLik
179 194 -83.6
Random effects:
Formula: ~1 + xall | peopleall
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 11.593 (Intr)
xall
0.464 0.044
Residual
0.445

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Introduction of Mixed effect model

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Plot of Random Intercept and slope model

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Introduction of Mixed effect model

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what if we just use linear model
· complete pooling

# wrong estimation
lm1 <- lm(yall~xall,data=data2)
summary(lm1)

Call:
lm(formula = yall ~ xall, data = data2)
Residuals:
Min
1Q Median
-17.80 -6.27 -3.67

3Q
2.19

Max
24.33

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)
6.86
3.72
1.84 0.06874 .
xall
4.31
1.15
3.76 0.00032 ***
--25 of 34

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Introduction of Mixed effect model

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what if we just use linear model
· no pooling

# wrong estimation and waste too many freedom and we don't care about the exact different of pe
lm2 <- lm(yall~xall+factor(peopleall)+xall*factor(peopleall),data=data1)
summary(lm2)

Call:
lm(formula = yall ~ xall + factor(peopleall) + xall * factor(peopleall),
data = data1)
Residuals:
Min
1Q Median
-2.983 -1.194 0.054

3Q
1.092

Max
4.238

Coefficients:
(Intercept)
xall
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Estimate Std. Error t value Pr(>|t|)
18.818
1.342
14.02 < 2e-16 ***
0.929
0.413
2.25
0.028 *
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Introduction of Mixed effect model

http://nycdatascience.com/mixed_effect_model_supstat/index.html#1

General Mixed effect model

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Introduction of Mixed effect model

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Logistic Mixed effect model
Yij = exp(eta)/(1+exp(eta))
eta = b0 + b1*Xij + bi + eij
· b0: fixed intercept
· b1: fixed slope
· X: fixed effect
· bi: random effect(influence intercept)
· eij: noise

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Introduction of Mixed effect model

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Data generation of Logistic Mixed effect
model
b0 <- - 6
b1 <- 2.1
set.seed(1)
n <- c(12, 13, 14, 15, 16, 13)
npeople <- length(n)
x <- matrix(rep(0, length = max(n) * npeople),
ncol = npeople)
bi <- rnorm(npeople, mean = 0, sd = 1.5)
for (i in 1:npeople){
x[1:n[i], i] <- runif(n[i], min = 1,max = 5)
x[1:n[i], i] <- sort(x[1:n[i], i])
}

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Introduction of Mixed effect model

http://nycdatascience.com/mixed_effect_model_supstat/index.html#1

Data generation of Logistic Mixed effect
model
xall <- NULL
yall <- NULL
peopleall <- NULL
for (i in 1:npeople){
xall <- c(xall, x[1:n[i], i])
y <- NULL
for(j in 1:n[i]){
eta1 <- b0 + b1 * x[j, i] + bi[i]
y <- c(y, rbinom(n = 1, size = 1,
prob = exp(eta1)/(exp(eta1) + 1)))
}
yall <- c(yall, y)
people <- rep(i, length = n[i])
peopleall <- c(peopleall, people)
}
data3 <- data.frame(xall, peopleall,yall)

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Introduction of Mixed effect model

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Cooefficient estimation of Logistic Mixed
effect model
library(lme4)
# formula is different
lmer3 <- glmer(yall~xall+(1|peopleall),data=data3,family=binomial)
print(summary(lmer3))

Generalized linear mixed model fit by maximum likelihood ['glmerMod']
Family: binomial ( logit )
Formula: yall ~ xall + (1 | peopleall)
Data: data3
AIC
69.8

BIC
77.1

logLik deviance
-31.9
63.8

Random effects:
Groups
Name
Variance Std.Dev.
peopleall (Intercept) 3.94
1.98
Number of obs: 83, groups: peopleall, 6
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Plot of Logistic Mixed effect model

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Case study

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Introduction of mixed effect model

  • 1. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Introduction of Mixed effect model Learning by simulation Supstat Inc. 1 of 34 1/29/14, 10:51 PM
  • 2. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Outline · What is mixed effect model · Fixed effect model · Mixed effect model - Random Intercept model - Random Intercept and Slope Model · General Mixed effect model · Case study 2 of 34 1/29/14, 10:51 PM
  • 3. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 What is mixed effect model 3 of 34 1/29/14, 10:51 PM
  • 4. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Classical normal linear model Formation: Yi = b0 + b1*Xi + ei · Yi is response from suject i. · Xi are covariates. · b0, b1 are parameters that we want to estimate. · ei are the random terms in the model, and are assumped to be independently and indentically distributed from Normal(0,1). It is very important that there is no stucuture in ei and it represents the variations that could not be controled in our studies. 4 of 34 1/29/14, 10:51 PM
  • 5. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Violation of independence assumpation. In many cases, responses are not independent from each other. These data usualy have some cluster stucture. · Repeated measures, where measurements are taken multiple times from the same sujects. (clustered by subject) · A survey of all the family memebers. (clustered by family) · A survey of students from 20 classrooms in a high school. (clustered by classroom) · Longitudial data, or known as the panel data, where several responses are collected from the same sujects along the time. (clustered by subject) We need new tools - Mixed effect model. 5 of 34 1/29/14, 10:51 PM
  • 6. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Mixed effect model Mixed effect model = Fixed effect + Random effect · Fixed effects - expected to have a systematic and predictable influence on your data. - exhaust “the levels of a factor”.Think of sex(male/femal). · Random effect - expected to have a non-systematic, unpredictable, or “random” influence on your data. - Random effects have factor levels that are drawn from a large population, but we do not know exactly how or why they differ. 6 of 34 1/29/14, 10:51 PM
  • 7. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Example of Fixed effects and Random effects FIXED EFFECTS Male or female Individuals with repeated measures Insecticide sprayed or not Block within a field Upland or lowland Brood One country versus another Split plot within a plot Wet versus dry 7 of 34 RANDOM EFFECTS Family 1/29/14, 10:51 PM
  • 8. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Fixed effect model 8 of 34 1/29/14, 10:51 PM
  • 9. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Fixed effect model Fixed effect model is just the linear model that you maybe already know. Yi = b0 + b1*Xi + ei 1<i<n n is number of sample · Yi: Response Variable · b0: fixed intercept · b1: fixed slope · Xi: Explanatory Variable (fixed effect) · ei: noise (error) 9 of 34 1/29/14, 10:51 PM
  • 10. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of fixed effect model set.seed(1) # genaerate x x <- seq(1,5,length.out=100) # generate error noise <- rnorm(n=100,mean=0,sd=1) b0 <- 1 b1 <- 2 # generate y y <- b0 + b1*x + noise 10 of 34 1/29/14, 10:51 PM
  • 11. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of fixed effect model plot(y~x) 11 of 34 1/29/14, 10:51 PM
  • 12. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Cooefficient estimation of fixed effect model model <- lm(y~x) summary(model) Call: lm(formula = y ~ x) Residuals: Min 1Q -2.3401 -0.6058 Median 0.0155 3Q 0.5851 Max 2.2975 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.1424 0.2491 4.59 1.3e-05 *** x 1.9888 0.0774 25.70 < 2e-16 *** --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.903 on 98 degrees of freedom 12 of 34 1/29/14, 10:51 PM
  • 13. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 plot of fixed effect model plot(y~x) abline(model) 13 of 34 1/29/14, 10:51 PM
  • 14. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Mixed effect model 14 of 34 1/29/14, 10:51 PM
  • 15. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Random Intercept model there are i people, and we repeat measure j times for every people. These poeple are individually different which we don't know, so there are random effect cause by people, and there are another random noise cause by measure for every people. Yij = b0 + b1*Xij + bi + eij · b0: fixed intercept · b1: fixed slope · Xij: fixed effect · bi: random effect(influence intercept) · eij: noise 15 of 34 1/29/14, 10:51 PM
  • 16. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of Random Intercept model b0 <- 9.9 b1 <- 2 # repeat measure times for 6 people n <- c(13, 14, 14, 15, 12, 13) npeople <- length(n) set.seed(1) # generate x(fixed effect) x <- matrix(rep(0, length=max(n) * npeople),ncol = npeople) for (i in 1:npeople){ x[1:n[i], i] <- runif(n[i], min = 1, max = 5) x[1:n[i], i] <- sort(x[1:n[i], i]) } # random effect bi <- rnorm(npeople, mean = 0, sd = 10) 16 of 34 1/29/14, 10:51 PM
  • 17. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of Random Intercept model xall <- NULL yall <- NULL peopleall <- NULL for (i in 1:npeople){ xall <- c(xall, x[1:n[i], i]) # combine x # generate y y <- rep(b0 + bi[i], length = n[i]) + b1 * x[1:n[i],i] + rnorm(n[i], mean = 0, sd = 2) # noise yall <- c(yall, y) # combine y people <- rep(i, length = n[i]) peopleall <- c(peopleall, people) } # final dataset data1 <- data.frame(yall,peopleall,xall) 17 of 34 1/29/14, 10:51 PM
  • 18. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Cooefficient estimation of Random Intercept model library(nlme) # xall is fixed effect # bi influence intercept of model lme1 <- lme(yall~xall,random=~1|peopleall,data=data1) summary(lme1) Linear mixed-effects model fit by REML Data: data1 AIC BIC logLik 358 368 -175 Random effects: Formula: ~1 | peopleall (Intercept) Residual StdDev: 7.3 1.77 Fixed effects: yall ~ xall 18 of 34 1/29/14, 10:51 PM
  • 19. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Plot of Random Intercept model 19 of 34 1/29/14, 10:51 PM
  • 20. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Random Intercept and slope model Yij = b0 + (b1+si)*Xij + bi + eij · b0: fixed intercept · b1: fixed slope · X: fixed effect · bi: random effect(influence intercept) · eij: noise · si: random effect(influence slope) 20 of 34 1/29/14, 10:51 PM
  • 21. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of Random Intercept and slope model a0 <- 9.9 a1 <- 2 n <- c(12, 13, 14, 15, 16, 13) npeople <- length(n) set.seed(1) si <- rnorm(npeople, mean = 0, sd = 0.5) # random slope x <- matrix(rep(0, length = max(n) * npeople), ncol = npeople) for (i in 1:npeople){ x[1:n[i], i] <- runif(n[i], min = 1, max = 5) x[1:n[i], i] <- sort(x[1:n[i], i]) } 21 of 34 1/29/14, 10:51 PM
  • 22. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of Random Intercept and slope model bi <- rnorm(npeople, mean = 0, sd = 10) # random intercept xall <- NULL yall <- NULL peopleall <- NULL for (i in 1:npeople){ xall <- c(xall, x[1:n[i], i]) y <- rep(a0 + bi[i], length = n[i]) + (a1 + si[i]) * x[1:n[i],i] + rnorm(n[i], mean = 0, sd = 0.5) yall <- c(yall, y) people <- rep(i, length = n[i]) peopleall <- c(peopleall, people) } # generate final dataset data2 <- data.frame(yall, peopleall, xall) 22 of 34 1/29/14, 10:51 PM
  • 23. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Cooefficient estimation of Random Intercept and slope model # bi influence intercept and slope of model lme2 <- lme(yall~xall,random=~1+xall|peopleall,data=data2) print(summary(lme2)) Linear mixed-effects model fit by REML Data: data2 AIC BIC logLik 179 194 -83.6 Random effects: Formula: ~1 + xall | peopleall Structure: General positive-definite, Log-Cholesky parametrization StdDev Corr (Intercept) 11.593 (Intr) xall 0.464 0.044 Residual 0.445 23 of 34 1/29/14, 10:51 PM
  • 24. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Plot of Random Intercept and slope model 24 of 34 1/29/14, 10:51 PM
  • 25. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 what if we just use linear model · complete pooling # wrong estimation lm1 <- lm(yall~xall,data=data2) summary(lm1) Call: lm(formula = yall ~ xall, data = data2) Residuals: Min 1Q Median -17.80 -6.27 -3.67 3Q 2.19 Max 24.33 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.86 3.72 1.84 0.06874 . xall 4.31 1.15 3.76 0.00032 *** --25 of 34 1/29/14, 10:51 PM
  • 26. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 what if we just use linear model · no pooling # wrong estimation and waste too many freedom and we don't care about the exact different of pe lm2 <- lm(yall~xall+factor(peopleall)+xall*factor(peopleall),data=data1) summary(lm2) Call: lm(formula = yall ~ xall + factor(peopleall) + xall * factor(peopleall), data = data1) Residuals: Min 1Q Median -2.983 -1.194 0.054 3Q 1.092 Max 4.238 Coefficients: (Intercept) xall 26 of 34 Estimate Std. Error t value Pr(>|t|) 18.818 1.342 14.02 < 2e-16 *** 0.929 0.413 2.25 0.028 * 1/29/14, 10:51 PM
  • 27. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 General Mixed effect model 27 of 34 1/29/14, 10:51 PM
  • 28. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Logistic Mixed effect model Yij = exp(eta)/(1+exp(eta)) eta = b0 + b1*Xij + bi + eij · b0: fixed intercept · b1: fixed slope · X: fixed effect · bi: random effect(influence intercept) · eij: noise 28 of 34 1/29/14, 10:51 PM
  • 29. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of Logistic Mixed effect model b0 <- - 6 b1 <- 2.1 set.seed(1) n <- c(12, 13, 14, 15, 16, 13) npeople <- length(n) x <- matrix(rep(0, length = max(n) * npeople), ncol = npeople) bi <- rnorm(npeople, mean = 0, sd = 1.5) for (i in 1:npeople){ x[1:n[i], i] <- runif(n[i], min = 1,max = 5) x[1:n[i], i] <- sort(x[1:n[i], i]) } 29 of 34 1/29/14, 10:51 PM
  • 30. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Data generation of Logistic Mixed effect model xall <- NULL yall <- NULL peopleall <- NULL for (i in 1:npeople){ xall <- c(xall, x[1:n[i], i]) y <- NULL for(j in 1:n[i]){ eta1 <- b0 + b1 * x[j, i] + bi[i] y <- c(y, rbinom(n = 1, size = 1, prob = exp(eta1)/(exp(eta1) + 1))) } yall <- c(yall, y) people <- rep(i, length = n[i]) peopleall <- c(peopleall, people) } data3 <- data.frame(xall, peopleall,yall) 30 of 34 1/29/14, 10:51 PM
  • 31. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Cooefficient estimation of Logistic Mixed effect model library(lme4) # formula is different lmer3 <- glmer(yall~xall+(1|peopleall),data=data3,family=binomial) print(summary(lmer3)) Generalized linear mixed model fit by maximum likelihood ['glmerMod'] Family: binomial ( logit ) Formula: yall ~ xall + (1 | peopleall) Data: data3 AIC 69.8 BIC 77.1 logLik deviance -31.9 63.8 Random effects: Groups Name Variance Std.Dev. peopleall (Intercept) 3.94 1.98 Number of obs: 83, groups: peopleall, 6 31 of 34 1/29/14, 10:51 PM
  • 32. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Plot of Logistic Mixed effect model 32 of 34 1/29/14, 10:51 PM
  • 33. Introduction of Mixed effect model http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 Case study 33 of 34 1/29/14, 10:51 PM
  • 34. Introduction of Mixed effect model 34 of 34 http://nycdatascience.com/mixed_effect_model_supstat/index.html#1 1/29/14, 10:51 PM