Repeated Measures Example In an exercise therapy study, sub- weightlifting programs
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Repeated Measures Example
In an exercise therapy study, subjects were assigned to one of three
weightlifting programs
(i=1) The number of repetitions of
weightlifting was increased as
subjects became stronger (RI)
(i=2) The
amount
of
weight
was
increased as subjects became
stronger (WI)
(i=3) Subjects did not participate in
weightlifting (XCont)
1
Measurements
of
strength
(Y )
were taken on days 2, 4, 6, 8, 10,
12 and 14 for each subject.
Source:
Littel,
Freund,
and
Spector
(1991) SAS System for Linear
Models
R code: RepeatedMeasures.R
2
Mixed model
Yijk = µ + αi + Sij + τk + γik + eijk
Yijk strength measurement for
prog. i, subj. j , time point k
αi
Sij
τk
γik
eijk
fixed program effect
random subject effect
fixed time effect
fixed prog.×time interaction
random error
where the random effects are all
independent and
2)
Sij ∼ N ID(0, σS
ijk ∼ N ID(0, σ2)
3
Average strength after 2k days on
the i-th program is
µik = E(Yijk )
= E(µ + αi + Sij + τk + γik + eijk )
= µ + αi + E(Sij ) + τk + γik + E(eijk )
= µ + αi + τk + γik
for i = 1, 2, 3 and k = 1, 2, . . . , 7
The variance of any single
observation is
V ar(Yijk ) = V ar(µ + αi + Sij + τk + αik + eijk )
= V ar(Sij + eijk )
= V ar(Sij ) + V ar(eijk )
2 + σ2
= σS
e
4
Correlation between observations
taken on the same subject:
Cov(Yijk , Yij`)
= Cov(µ + αi + Sij + τk + γik + eijk ,
µ + αi + Sij + τ` + γi` + eij`)
= Cov(Sij + eijk , Sij + eij`)
= Cov(Sij , Sij ) + Cov(Sij , eij`)
+Cov(eijk , Sij ) + Cov(eijk , eij`)
= V ar(Sij )
for k 6= `
2.
= σS
5
The correlation between Yijk and
Yij` is
2
σS
≡ρ
2
2
σS + σe
Observations
taken
on
different
subjects are uncorrelated.
6
For the set of observations taken
on a single subject, we have
V ar
=
Yij1
Yij3
..
Yij7
2
2
σe2 + σS
σS
···
2
2 ···
σS
σe2 + σS
..
2
σS
..
2
σS
...
2
σS
2
σS
2
σS
2 σ2 + σ2
σS
e
S
2J
= σe2I + σS
↑
J is a matrix of ones
This covariance structure is called
compound symmetry.
7
Write this model in the form
Y = Xβ + Zu + e
Y111
Y112
...
Y117
Y121
Y122
...
Y127
...
Y211
Y212
...
Y217
...
Y3,n31
Y3,n32
...
Y3,n37
=
1 1 0
1 1 0
...
1 1 0
1 1 0
1 1 0
...
1 1 0
...
1 0 1
1 0 1
...
1 0 1
...
1 0 0
1 0 0
...
...
1 0 0
0 1000000 |
0 0100000 |
...
|
0 0000001 |
0 1000000 |
0 0100000 |
...
|
0 0000001 |
|
0 1000000 |
0 0100000 |
...
|
0 0000001 |
|
1 1000000 |
1 0100000 |
...
|
1 0000001 |
21
columns
for
interaction
8
µ
α1
α2
α3
τ1
τ2
τ3
τ4
τ5
τ6
τ7
γ11
γ12
...
γ37
1
1
...
1
0
0
...
+ 0
...
...
...
0
0
...
0
0
0
...
0
1
1
...
1
...
...
...
0
0
...
0
··· 0
··· 0
...
··· 0
··· 0
··· 0
...
··· 0
...
...
...
··· 1
··· 1
...
··· 1
e111
e112
...
e117
e121
e122
...
S11
e127
S12
...
...
... + e
211
...
e212
...
S3,n3
e217
...
e3,n31
e3,n32
...
e3,n37
9
In this case:
R = V ar(e) = σe2I(7r)×(7r)
2I
G = V ar(u) = σS
r×r
where r is the number
of subjects
Σ = V ar(Y) = ZGZ T + R
is a block diagonal
matrix with one block
of the form
2J
(σe2I7×7 + σS
7×7)
for each subject
10
Mean strength at a
particular time
in a particular program
LSM EAN = µ̂ + α̂i + τ̂k + γ̂ik
= Ȳi.k =
V ar(Ȳi.k ) =
SȲ
i.k
=
v
u
u
u
u
u
u
u
t
6
1 nXi
ni j=1
Yijk
2
σe2 + σS
ni
1
1
M Serror + M SSubj
7
7
ni
↑
Cochran-Satterthwaite
degrees of freedom are 65.8
11
Program means
(averaging across time)
LSM EAN = Ȳi..
1 X
7
(τ̂k + γ̂ik )
= µ̂ + α̂i +
7 k=1
V ar(Ȳi...) =
SȲ
i..
=
2
σe2 + 7σS
7ni
v
u
u
u
u
u
u
u
t
M Ssubjects
2ni
There are n1 = 16, n2 = 21, n3 = 20
subjects in the three programs.
12
Mean strength at a
particular time point
(averaging across programs)
1 X
3
LSM EAN = µ̂ + τ̂k +
(α̂i + γ̂ik )
3 i=1
1 X
3
=
(Ȳi.k )
6=
Ȳ..k
3 i=1
because n1 = 16, n2 = 21, n3 = 20.
2
2
1 X
3 σe + σS
V ar(LSM EAN ) =
9 i=1
ni
SLSM EAN =
1
3
v
u
u
u
u
u
t
6
1
3 1
X
M Serror + M Ssubj
7
7
n
i=1 i
↑
Cochran-Satterthwaite
degrees of freedom are 65.8
13
Difference between strength
means at two time points
(averaging across programs)
τ̂k + 13 3i=1 γ̂ik − τ̂` − 13 3i=1 γ̂i`
P
P
1
1 X
3
3
X
Ȳi.k −
Ȳi.`
=
3 i=1
3 i=1
1 X
3 1 n
Xi
=
(Yijk − Yij`)
3 i=1 ni j=1
↑
subject effects
cancel out
14
Variance formula:
2σe2 3 1
9 Σi=1 ni
Standard error:
v
u
u
u
u
t
2M Serror Σ3 1 = 0.206
i=1 ni
9
↑
use degrees of freedom
for error = 324
15
Difference between strength
means for two programs
at a specific time point
(µ̂ + α̂i + τ̂ik + γ̂ik ) − (µ̂ + α̂` + τ̂k + γ̂`k )
= Ȳi.k − Ȳ`.k
1
1
2
2
V ar(Ȳi.k − Ȳ`.k ) = (σe + σS )( + )
ni n`
SȲ −Ȳ =
i.k
`.k
v
u
u
u
u
u
t
6
1
1
1
M Serror + M Ssubj +
7
7
ni
n`
↑
Use Cochran-Satterthwaite
degrees of freedom = 65.8
16
Difference between strength
means at two time points
within a particular program
(µ̂ + α̂i + τ̂k + γ̂ik ) − (µ̂ + α̂i + τ̂` + γ̂i`)
= Ȳi.k − Ȳi.`
V ar(Ȳi.k − Ȳi.`) =
ni
2σe2
1
V ar n Σj=1(Yijk − Yij`) = n
i
i
v
u
u
u
u
t
SȲ −Ȳ = M Serror n2
i.k
i.`
i
↑
use degrees of freedom
for error=324
17
Difference in strength
means for two programs
(averaging across time
points)
1 X
1 X
7
7
γ̂ik ) − (α̂` +
γ̂`k )
Ȳi.. − Ȳ`.. = (α̂i −
7 k=1
7 k=1
V ar(Ȳi.. − Ȳ`..) =
SȲ −Ȳ
=
i..
`..
(σe2
v
u
u
u
u
u
t
+
2 )(
7σS
1
7ni
M Ssubjects(
+
1
7ni
1
7n`
+
)
1
7n`
↑
54 d.f.
18
)
Specifying other covariance
matrices
We began with the model
Yijk = µ + αi + Sij + τk + γik + eijk
where
2)
Sij ∼ N ID(0, σS
eijk ∼ N ID(0, σe2)
and the {Sij } are distributed
independently of the {eijk }
19
This model was expressed in the
form
Y = Xβ + Zu + e
where
S11
..
u=
S3,20
contained the random subject effects
20
Here
2I
G = V ar(u) = σS
R = V ar(e) = σe2I
Σ = V ar(Y) = ZGZ T + R
2
= σS
=
2J
σe2I + σS
J
J
...
J
+ σe2I
...
2J
σe2I + σS
where J is a matrix of ones.
21
If you are not interested in predicting subject effects (random subject
effects are included only to introduce correlation among repeated
measures on the same subject), you
can work with an alternative expression of the same model
Y = Xβ + e∗
where
R = V ar(e∗)
=
2J
σe2I + σS
...
2J
σe2I + σS
22
Replace the mixed model
Y = Xβ + Zu + e
with the model
Y = Xβ + e∗
where
W 0 ··· 0
0 W ··· 0
∗
V ar(Y) = V ar(e ) =
............
0 0 ··· W
23
First Order Autoregressive:
AR(1)
W = σ2
1
ρ
ρ2
ρ3
ρ
1
ρ
ρ2
ρ2
ρ
1
ρ
ρ3 ρ2
ρ
1
where σ 2 ∈ (0, ∞) and ρ ∈ (−1, 1) are
unknown parameters.
24
