Transcript Statistics 262: Intermediate Biostatistics Mixed models; Modeling change Within vs. Between subject effects… A significant chemical effect (time-dependent predictor) could either represent a between-subjects effect.

Statistics 262: Intermediate
Biostatistics
Mixed models; Modeling change
1
Within vs. Between subject
effects…
A significant chemical effect (time-dependent predictor) could either represent
a between-subjects effect or a within-subjects effect.
Because these data were made-up, we happened to know that there was
primarily a within-subjects effect…
Example 1: last week’s example.
Solution for Fixed Effects
CHEM: -.01283
Standard
Effect
Estimate
Error
DF
t Value
Pr > |t|
38.1287
4.1727
5
9.14
0.0003
time
-0.08163
0.3234
16
-0.25
0.8039
chem
-0.01283
0.003125
16
-4.11
0.0008
Intercept
2
Example 1..from last time…
6 patients with depression are given a drug that increases levels of a “happy
chemical” in the brain. At baseline, all 6 patients have similar levels of this
happy chemical and scores >=14 on a depression scale. Researchers measure
depression score and brain-chemical levels at three subsequent time points: at 2
months, 3 months, and 6 months post-baseline.
Here are the data in broad form:
id
time1
time2
time3
time4
chem1
chem2
chem3
chem4
1
20
18
15
20
1000
1100
1200
1300
2
22
24
18
22
1000
1000
1005
950
3
14
10
24
10
1000
1999
800
1700
4
38
34
32
34
1000
1100
1150
1100
5
25
29
25
29
1000
1000
1050
1010
6
30
28
26
14
1000
1100
1109
1500
3
Example 2…

Same as example 1, but made up to
have more between-subjects effect than
within-subjects effect.
id
1
2
3
4
5
6
time1
50
22
44
38
15
50
time2
48
24
40
34
19
58
time3
45
18
34
32
15
56
time4
40
22
40
34
19
54
chem1
chem2
chem3
chem4
200
900
500
700
1000
100
350
920
459
770
1000
110
400
805
880
950
1050
210
500
950
500
800
1010
220
4
Example 3…

Same as example 1, but made up to
have ONLY between-subjects effect
id
1
2
3
4
5
6
time1
51
47
42
31
26
19
time2
48
43
42
32
27
20
time3
45
46
41
31
26
21
time4
50
45
38
32
28
18
chem1
chem2
chem3
chem4
510
710
910
1105
1505
1750
520
720
890
1110
1505
1600
510
690
900
1109
1510
1680
525
680
905
1090
1590
1740
5
Example 2 (more between-subjects effects):
showing two subjects…
id=1: score
id=1: chem
id=2: chem
id=2: score
Example 2:
Example 2:
Example 2:
Example 2:
Example 2:
Example 2:
Example 3:all between-subjects effects
Example 3:
Example 3:
Example 3:
Example 3:
Example 3:
Results, example 1…
proc mixed data=hrp262.long2;
model score=chem time / solution;
random int/subject=id;
run;
Example 1: significant chem
effect.
Solution for Fixed Effects
CHEM: -.01283
Standard
Effect
Estimate
Error
DF
t Value
Pr > |t|
38.1287
4.1727
5
9.14
0.0003
chem
-0.01283
0.003125
16
-4.11
0.0008
time
-0.08163
0.3234
16
-0.25
0.8039
Intercept
19
Results, example 2…
proc mixed data=hrp262.long2;
model score=chem time / solution;
random int/subject=id;
run;
Example 2: significant
chem effect
CHEM: -.02644
Solution for Fixed Effects
Effect
Estimate
Standard
Error
Intercept
chem
time
52.0159
-0.02644
0.1009
4.2057
0.005403
0.2966
DF
t Value
5
16
16
12.37
-4.89
0.34
Pr > |t|
<.0001
0.0002
0.7381
20
Results, example 3…
proc mixed data=hrp262.long3;
model score=chem time / solution;
random int/subject=id;
run;
Example 3: significant
chem effect
CHEM: -.02354
The Mixed Procedure
Solution for Fixed Effects
Effect
Estimate
Intercept
chem
time
60.8810
-0.02354
-0.08592
Standard
Error
2.4237
0.002066
0.1707
DF
5
16
16
t Value
25.12
-11.39
-0.50
Pr > |t|
<.0001
<.0001
0.6217
21
All three examples give the same
result…


There’s no way to tell if change in chemical levels is
causing change in depression score (which is what
we probably care about).
What to do?...
22
Options


Examine graphs!
Evaluate baseline relationship of chemical1 and score1 using
regular linear regression



Use only baseline value of chemical (chem1) as a predictor
in GEE or Mixed and add a time*chem1 interaction to the
model to evaluate change.



Here, you will find a strong relationship only in examples 2 and 3, suggesting strong betweensubjects effects.
Drawback: you still can’t rule out within-subjects effects (could have both)
Here, you will find that chem1 main effect is significant but chem1*time is not in examples 2
and 3
Drawback: A significant time*chem1 interaction would indicate that baseline chemical levels
predict change in depression score over time, which is slightly different than saying that
change in chemical level predicts change in depression score.
Correlate change in time-dependent predictor with change in
repeated-measures outcome…


Calculate overall change or percent change in outcome and regress this on overall change or
percent change in the predictor: see chapter 8 of Twisk (2 time points only)
OR model all the changes together (vector of changes)…
23
The change model
 score2  score1 
chem2  chem1 
 score3  score2     chem3  chem2  ...
0
1



 score4  score3
chem4  chem3
24
SAS code to change data…
data hrp262.change;
set hrp262.broad;
time=0; ctime=2; cscore=time2-time1; cchem=chem2-chem1; output;
time=1; ctime=1; cscore=time3-time2; cchem=chem3-chem2; output;
time=2; ctime=3; cscore=time4-time3; cchem=chem4-chem3; output;
label cchem='change in chemical';
label cscore='change in depression score';
run;
25
id
1
2
3
4
5
6
time1
51
47
42
31
26
19
time2
48
43
42
32
27
20
time3
time4
45
46
41
31
26
21
50
45
38
32
28
18
id
So, look at
change in
depression score
as your outcome
variable.
All timedependent
predictors also
get a change
score.
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
cscore
-3
-3
5
-4
3
-1
0
-1
-3
1
-1
1
1
-1
2
1
1
-3
chem1
chem2
chem3
chem4
510
710
910
1105
1505
1750
520
720
890
1110
1505
1600
510
690
900
1109
1510
1680
525
680
905
1090
1590
1740
cchem
10
-10
15
10
-30
-10
-20
10
5
5
-1
-19
0
5
80
-150
80
60
time
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Example 1: naïve linear regression
cscore = -0.126817 - 0.011357*cchem
27
Example 2: naïve linear regression
cscore = -0.167511 - 0.012043*cchem
28
Example 3: naïve linear regression
cscore = -0.268792 - 0.004044*cchem
29
Modeling changes (mixed)…
EXAMPLE 1:
Effect
Intercept
time
cchem
EXAMPLE 2:
Solution for Fixed Effects
Standard
Estimate
Error
DF
t Value
0.3407
-0.4669
-0.01136
1.6018
1.2298
0.002336
5
10
10
Effect
Solution for Fixed Effects
Standard
Estimate
Error
DF
Intercept
time
cchem
-0.04810
-0.1151
-0.01218
Effect
EXAMPLE 3:
Intercept
time
cchem
0.21
-0.38
-4.86
Pr > |t|
0.8400
0.7122
0.0007
t Value
Pr > |t|
5
10
10
-0.03
-0.09
-1.93
0.9773
0.9269
0.0830
Solution for Fixed Effects
Standard
Estimate
Error
DF
t Value
Pr > |t|
-0.8621
0.6023
-0.00807
1.6106
1.2225
0.006324
0.9697
0.7825
0.01334
5
10
10
-0.89
0.77
-0.60
0.4147
0.4592
0.5587
30
Modeling changes (GEE)…
EXAMPLE 1:
Analysis Of GEE Parameter Estimates
Standard
95% Confidence
Parameter Estimate
Error
Limits
Intercept
time
cchem
EXAMPLE 2:
1.2757
1.2698
0.0011
-2.1596
-2.9557
-0.0136
2.8409
2.0219
-0.0091
Analysis Of GEE Parameter Estimates
Standard
95% Confidence
Parameter Estimate
Error
Limits
Intercept
time
cchem
EXAMPLE 3:
0.3407
-0.4669
-0.0114
-0.0481
-0.1151
-0.0122
1.6245
1.2030
0.0050
-3.2320
-2.4728
-0.0219
3.1358
2.2427
-0.0024
Analysis Of GEE Parameter Estimates
Standard
95% Confidence
Parameter Estimate
Error
Limits
Intercept
cchem
time
-0.7146
0.0060
0.2118
1.5997 -3.8498
0.0112 -0.0160
0.8366 -1.4280
2.4207
0.0280
1.8516
Z Pr > |Z|
0.27
-0.37
-9.93
0.7894
0.7131
<.0001
Z Pr > |Z|
-0.03
-0.10
-2.45
0.9764
0.9238
0.0143
Z Pr > |Z|
-0.45
0.53
0.25
0.6551
0.5955
0.8000
31
The change model
 score2  score1 
chem2  chem1 
 score3  score2     chem3  chem2   (tim e)  CORR  Error
0
1
2



 score4  score3
chem4  chem3
In fact, it often turns out that the changes are not correlated
within subjects (we’ve essentially already corrected for
between subject variability by using change scores).
scoreit  0  1 (chemit )  2 (t )  Errorit
e.g., just regular old linear regression, where each person contributes three “independent” observations…
32
The change model
If time intervals are unequal and unbalanced, you might choose to include
the change in time as a predictor in the model. Otherwise, you could face
confounding by the time between measurements (which might be correlated
with the time-dependent predictor)…
scoreit  0  1 (chemit )  2 (ti )  Errorit
33
SAS code
proc genmod data=hrp262.change;
class id;
model cscore=time cchem;
repeated subject=id / type=ind corrw;
run;
proc mixed data=hrp262.change;
class id;
model cscore=time cchem/ solution;
run;
Which are basically equivalent to good old linear
regression…
proc reg data=hrp262.change;
model cscore=time cchem;
run;
34
Another example…

In the runners study, does weight gain
increase BMD?
35
Weight (time-dependent
predictor)…
Solution for Fixed Effects
Standard
Effect
Estimate
Error
DF
t Value
Pr > |t|
0.8492
0.02757
150
30.80
<.0001
time
0.000274
0.000094
196
2.92
0.0039
pounds
0.001037
0.000205
196
5.05
<.0001
Intercept
Weight has a highly significant relationship with spine BMD…
Is this a between-subjects or within-subjects effect??
36
Between vs. within subjects…
Strong between-subjects effect (baseline
comparison)
Weak within-subjects effect (comparison of
changes)
Deletion of just a few subjects who gained large
amounts of weight makes the line much less steep.
Modeling changes…
data hrp262.change;
set hrp262.broad;
dxa=1; ctime=(dxaday2-dxaday1)*12/365.25; cspine=spine2-spine1; cpounds=pounds2-pounds1; output;
dxa=2; ctime=(dxaday3-dxaday2)*12/365.25; cspine=spine2-spine1; cpounds=pounds2-pounds1; output;
run;
proc mixed data=hrp262.bfitchange;
class id ;
model cspine= ctime
random int
cpounds /s;
/subject=id;
run;
It makes more sense here to model time as the interval (in months) between DXA (bone) measurements
rather than regular old time.
The change in BMD between DXAs is correlated to the length of time between DXAs (which was supposed to
be 1 year, but ended up being highly variable). It’s not unreasonable to believe that weight gain might also be
related to the amount of time that’s passed.
If, instead, dxa (clinical visit 1,2,3) or time (continuous) were put in the model, what would a significant effect
38
mean?