### Mixed Linear Models, Module 12 - Repeated measures II, advanced

```Overview of this module
Overview of this module
Course 02429
Analysis of correlated data: Mixed Linear Models
1
Different view on the random effects approach
Example: Activity of rats
2
Gaussian model of spatial correlation
Example: Activity of rats
3
Other spatial correlation structures
4
Diagram of analysis
5
The semi-variogram
6
Analysing the time structure by polynomial regression
Module 12: Repeated measures II, advanced methods
Per Bruun Brockhoff
DTU Compute
Building 324 - room 220
Technical University of Denmark
2800 Lyngby – Denmark
e-mail: [email protected]
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 12
Fall 2014
1 / 24
Per Bruun Brockhoff ([email protected])
Aim of this module
Mixed Linear Models, Module 12
Fall 2014
2 / 24
Aim of this module
Aim of this module
Remember the rats data set?
Summary of experiment:
3 treatments: 1, 2, 3 (concentration)
10 cages per treatment
10 contiguous months
The response is activity (log(count) of intersections of light beam
during 57 hours)
Extend the toolbox for dealing with repeated measures
Introduce models where the covariance structure is directly specified
10.5
Improve our ability to handle long series
10.0
log(count)
See how to specify these models in R
9.5
9.0
8.5
1
2
3
4
5
6
7
8
9
10
Month
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Different view on the random effects approach
Different view on the random effects approach
Different view on the random effects approach
Activity of rats analyzed via compound symmetry model
Any mixed model can be expressed as:
The model is the same as the random effects model, but specified
directly
y ∼ N (Xβ, ZGZ0 + R),
The total covariance of all observations are described by
V = ZGZ0 + R
The ZGZ0 part is specified through the random effects of the model
The R part has so far been σ 2 I, but in this module we will put some
structure into R
For instance the structure known from the random effects model

 0
σ2
cov(yi1 , yi2 ) =
 individual
2
σindividual
+ σ2
, if individuali1 6= individuali2 and i1 6= i2
, if individuali1 = individuali2 and i1 6= i2
, if i1 = i2
This structure is known as compound symmetry
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lnc ∼ N (µ, V), where
µi = µ
+ α(treatmi ) + β(monthi ) + γ(treatmi , monthi ),
, if cagei1 6= cagei2 and i1 6= i2
 0
σd2
, if cagei1 = cagei2 and i1 6= i2
Vi1 ,i2 =
 2
2
σd + σ , if i1 = i2
lme(lnc ~ month + treatm + month:treatm,
random = ~1 | cage, data = rats)
OR directly into the R-matrix:
gls(lnc~month+treatm+month:treatm,
correlation=corCompSymm(form=~1|cage),data=rats)
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Example: Activity of rats
, if individual
i1
6= individual
i2
and i1 6= i2
, if individual
i1
= individual
i2
and i1 6= i2
ν2 + τ2
, if i1 = i2
The entire model is:
lnc
µi
∼
=
Vi1 ,i2
=
N (µ, V), where
µ + α(treatmi ) + β(monthi ) + γ(treatmi , monthi ), and


 0
, if cagei1 6= cagei2 and i1 6= i2

−(monthi1 −monthi2 )2
2
2
, if cagei1 = cagei2 and i1 6= i2
ν + τ exp
ρ2


 2
ν + τ 2 + σ2
, if i1 = i2
This model is implemented by:
lme(lnc~month+treatm+month:treatm,
random=~1|cage,
correlation=corGaus(form=~as.numeric(month)|cage,nugget=T),
data=rats)
ν2
ν2 + 0.5τ2
0
Covariance
=
Fall 2014
Rats data via spatial Gaussian correlation model
Covariance structures depending on “how far” observations are apart
are known as spatial
The following covariance structure has been proposed for repeated
measurements

Vi1 ,i2
Mixed Linear Models, Module 12
Gaussian model of spatial correlation
Gaussian model of spatial correlation
and
Implemented in R traditionally by a random effect:
Gaussian model of spatial correlation
0



−(ti1 −ti2 )2
ν 2 + τ 2 exp
2
ρ


 2
ν + τ 2 + σ2
Example: Activity of rats
0
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Distance t − t
0.83ρ
i1 12
i2
Mixed Linear Models, Module
Fall 2014
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Fall 2014
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Gaussian model of spatial correlation
Example: Activity of rats
Gaussian model of spatial correlation
Example: Activity of rats
Reduction from spatial Gaussian to random effects?
The relevant part of the R output is:
Random effects:
Formula: ~1 | cage
(Intercept) Residual
√
ˆ 2 + τˆ2 )
StdDev:
0.1404056 (= νˆ) 0.2171559 (= σ
Correlation Structure: Gaussian spatial correlation
Formula: ~as.numeric(month) | cage
Parameter estimate(s):
range
nugget
2.3863954 (= ρˆ2 ) 0.2186744 (= σ
ˆ 2 /(ˆ
σ 2 + τˆ2 ))
Number of Observations: 300
Number of Groups: 30
The spatial Gaussian model (A) is an extension of the random effects
model (B)
Use the restricted/residual likelihood ratio test
(A)
2
GA→B = 2`(B)
re − 2`re , where GA→B ∼ χ2
For the rats data we get:
Model
(A) Spatial Gaussian
(B) Random effects
Notice the R parametrization of the variance parameters
2`re
-105.3
8.6
G–value
GA→B = 113.9
df
2
P–value
PA→B < 0.0001
The spatial Gaussian structure is clearly a better description of the
covariance structure in the rats data set
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Mixed Linear Models, Module 12
Fall 2014
11 / 24
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Other spatial correlation structures
Mixed Linear Models, Module 12
Fall 2014
12 / 24
Diagram of analysis
Other spatial correlation structures
Diagram of analysis
R has several build–in correlation structures. A few examples are:
Write in R
Name
corGaus
Gaussian
corExp
corAR1
corSymm
exponential
autoregressive(1)
unstructured
Correlation term
Identify "individuals"
−(ti1 −ti2 )2
}
ρ2
−|ti1 −ti2 |
2
τ exp{
}
ρ
τ 2 ρ|i1 −i2 |
τ 2 exp{
Select covariance structure from
guided by information criteria
τi21 ,i2
Unfortunately is can be very difficult to choose — especially for
“short” individual series
Keep it simple: Numerical problems often occur with (too) complicated
structures
Graphical methods: Especially for “long” series the (semi)–variogram is
useful
Information criteria: AIC or BIC = “2` + penalty(#par)” can be used
as guideline
Try to cross–validate your main conclusion(s) by one of the “simple”
methods
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Mixed Linear Models, Module 12
Fall 2014
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Covariance parameters are tested by
likelihood ratio test
The green arrow is often omitted by
the argument that a non–significant
simplification of the mean structure
should not change the covariance
structure much
Select fixed effects
Select covariance structure
Change
model
Change
model
Test covariance parameters
Test fixed effects
Interpret results
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Fall 2014
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The semi-variogram
The semi-variogram
The semi-variogram
The semi-variogram
print(plot(Variogram(model2, form =˜as.numeric(month)|cage,
data = rats)))
●
Plotting of the empirical covariation (of residuals) versus "time"
Plot of σ 2 + γ(u), where
●
●
γ(u) = τ 2 1 − λ(u)
λ(u) = exp{−u2 /ρ2 }
●
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●
0.5
●
2
3
ν2
●
1
τ2
0.0
σ2
0
Covariance
4
Semivariogram
1.0
2
4
6
8
Distance
0
1
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2
3
4
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Time difference
The semi-variogram
The semi-variogram
The semi-variogram
The semi-variogram
model3 <- lme(lnc month + treatm + month:treatm, random = 1 |
cage, correlation = corExp(form =˜as.numeric(month) |
cage, nugget = T), data = rats)
print(plot(Variogram(model3, form =˜as.numeric(month)|cage,
data = rats)))
1.0
model4 <- lme(lnc month + treatm + month:treatm, random = 1 |
cage, correlation = corAR1(form = as.numeric(month) |
cage), data = rats)
print(plot(Variogram(model4, form =˜as.numeric(month)|cage,
data = rats)))
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1.0
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0.8
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Semivariogram
Semivariogram
0.8
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0.6
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0.4
0.2
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0.6
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0.4
0.2
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0.0
2
2
4
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8
Mixed Linear Models, Module 12
6
8
Distance
Distance
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4
Fall 2014
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Analysing the time structure by polynomial regression
Analysing the time structure by polynomial regression
Analysing the time structure by polynomial regression
Overview of this module
1
Do the factor based analysis as shown above.
2
Explorative plotting of individual and treatment average regression
lines/curves.
3
Potentially make a ”high degree” decomposition based on the simple
”split-plot” repeated measures model using lmer and lmerTest.
4
Check if a linear or quadratic regression model could be used as an
alternative to the factor based model
5
IF a regression approach seems to capture what is going on, then try
to fit the random coefficient model as an alternativ to the correlation
structure used from above - chose the best one at the end.
6
A possibility is that a factor based model is needed for the main effect
of time, whereas a quantitative model would fit the interaction effect.
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 12
Fall 2014
23 / 24
1
Different view on the random effects approach
Example: Activity of rats
2
Gaussian model of spatial correlation
Example: Activity of rats
3
Other spatial correlation structures
4
Diagram of analysis
5
The semi-variogram
6
Analysing the time structure by polynomial regression
Per Bruun Brockhoff ([email protected])
Mixed Linear Models, Module 12
Fall 2014
24 / 24
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