This question has been asked and not answered before here.
I am building a model attempting to predict heroin use over time in patients based on their use of amphetamine-type substances (ATS). ATS use and heroin use are measured in the same questionnaire and therefore any time heroin use is recorded, ATS use is also recorded. There are two potential models I am considering.
Model 1
Model 1 is a three-predictor model. The predictors are:
(i) ATSUseAtBaseline
(a time-invariant categorical predictor based on days of ATS use in the 28 days previous to baseline, with three levels of ATS use, none
(0 days), low
(1-12 days' use), and high
(13-28 days' use); the none
category is the reference category)
(ii) yearsFromStart
a continuous variable indicating how many years from start of treatment the measurement was made
(iii) ATSUseAtBaseline
x yearsFromStart
interaction
This is the output from the model, a longitudinal mixed effects repeated measures regression with the above predictors as fixed factors and random slopes (i.e. yearsFromStart|participant id
. The model was fit in R using the lme()
function in the nlme
package.
Linear mixed-effects model fit by maximum likelihood
Data: workDF
AIC BIC logLik
33606.85 33672.54 -16793.42
Random effects:
Formula: ~yearsFromStart | pID
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 4.018394 (Intr)
yearsFromStart 2.992416 -0.871
Residual 4.777283
Fixed effects: heroin ~ yearsFromStart + ATSUseAtBaseline + yearsFromStart:ATSUseAtBaseline
Value Std.Error DF t-value p-value
(Intercept) 1.331320 0.1153107 3335 11.545503 0.0000
yearsFromStart -0.741715 0.2332848 1924 -3.179440 0.0015
ATSUseAtBaselinelow 2.853880 0.2878873 3335 9.913185 0.0000
ATSUseAtBaselinehigh 4.308878 0.5171080 3335 8.332647 0.0000
yearsFromStart:ATSUseAtBaselinelow -2.594455 0.5637359 1924 -4.602252 0.0000
yearsFromStart:ATSUseAtBaselinehigh -7.339846 1.8286941 1924 -4.013709 0.0001
Correlation:
(Intr) yrsFrS ATSUsAtBslnl ATSUsAtBslnh
yearsFromStart -0.410
ATSUseAtBaselinelow -0.401 0.164
ATSUseAtBaselinehigh -0.223 0.091 0.089
yearsFromStart:ATSUseAtBaselinelow 0.170 -0.414 -0.428 -0.038
yearsFromStart:ATSUseAtBaselinehigh 0.052 -0.128 -0.021 -0.327
yrsFrmStrt:ATSUsAtBslnl
yearsFromStart
ATSUseAtBaselinelow
ATSUseAtBaselinehigh
yearsFromStart:ATSUseAtBaselinelow
yearsFromStart:ATSUseAtBaselinehigh 0.053
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-2.49102026 -0.16320514 -0.16320514 -0.08182908 5.11315552
Number of Observations: 5265
Number of Groups: 3338
So far so good. All predictors are significant, with no ATS use predicting less heroin use at baseline than low ATS use or high ATS use. The rate of reduction in heroin use (in days use in the previous 28 days) each year of treatment is lowest in the group using no ATS at baseline and highest in the group with high ATS use at baseline. There are 3338 participants with 5265 observations (i.e. the overwhelming majority only have one measurement, at baseline).
Model 2
Model 2 is a four-predictor model, including the three predictors in the first model and
(iv) atsFactor
: the time-varying equivalent of ATSUseAtBaseline
, with the same three levels, none
, low
, and high
.
Here is the output
Linear mixed-effects model fit by maximum likelihood
Data: workDF
AIC BIC logLik
33560.04 33638.87 -16768.02
Random effects:
Formula: ~yearsFromStart | pID
Structure: General positive-definite, Log-Cholesky parametrization
StdDev Corr
(Intercept) 4.085715 (Intr)
yearsFromStart 2.892861 -0.89
Residual 4.716549
Fixed effects: heroin ~ yearsFromStart + ATSUseAtBaseline + atsFactor + yearsFromStart:ATSUseAtBaseline
Value Std.Error DF t-value p-value
(Intercept) 1.299861 0.1154830 3335 11.255865 0.0000
yearsFromStart -0.877436 0.2253863 1922 -3.893033 0.0001
ATSUseAtBaselinelow 0.818656 0.4252261 3335 1.925226 0.0543
ATSUseAtBaselinehigh 0.933265 0.8938787 3335 1.044062 0.2965
atsFactorlow 2.312577 0.3586254 1922 6.448447 0.0000
atsFactorhigh 3.584153 0.7797614 1922 4.596473 0.0000
yearsFromStart:ATSUseAtBaselinelow -1.346094 0.5846532 1922 -2.302381 0.0214
yearsFromStart:ATSUseAtBaselinehigh -5.079803 1.9771895 1922 -2.569204 0.0103
Correlation:
(Intr) yrsFrS ATSUsAtBslnl ATSUsAtBslnh atsFctrl atsFctrh
yearsFromStart -0.408
ATSUseAtBaselinelow -0.248 0.180
ATSUseAtBaselinehigh -0.116 0.080 0.179
atsFactorlow -0.032 -0.093 -0.735 -0.182
atsFactorhigh -0.016 -0.033 -0.173 -0.816 0.219
yearsFromStart:ATSUseAtBaselinelow 0.148 -0.415 -0.536 -0.017 0.363 -0.006
yearsFromStart:ATSUseAtBaselinehigh 0.044 -0.114 0.013 -0.499 -0.042 0.405
yrsFrmStrt:ATSUsAtBslnl
yearsFromStart
ATSUseAtBaselinelow
ATSUseAtBaselinehigh
atsFactorlow
atsFactorhigh
yearsFromStart:ATSUseAtBaselinelow
yearsFromStart:ATSUseAtBaselinehigh -0.009
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-2.52320797 -0.15744819 -0.15744819 -0.06879067 5.32278040
Number of Observations: 5265
Number of Groups: 3338
The addition of the time-varying predictor has changed things. The significant differences in intercept coefficients for the time-invariant ATS use predictor ATSUseAtBaseline
are no longer significant. Even the interaction coefficients for these time-invariant versions of the predictor are no longer as strong. The time-varying predictor is a strong predictor in this model. Low ATS use at any time is associated with an increase in heroin use of 2.31 days, and high ATS use is associated with an increase heroin use of 3.6 days!
A likelihood ratio test of the two models...
Model df AIC BIC logLik Test L.Ratio p-value
tiModel1 1 10 33606.85 33672.54 -16793.42
tvModel2 2 12 33560.04 33638.87 -16768.02 1 vs 2 50.80377 <.0001
...shows that the addition of the time-varying predictor in tvModel2
has increased the predictive power of the first model (tvModel1
) considerably.
However the problem for interpreting this model is that the values of the time-invariant predictor ATSUseAtBaseline
and the time-varying predictor atsFactor
at time = 0 are identical. The fact that these baseline measurement make up 3338/5265 = 64% of all observations makes me think that there is some serious confounding of the two predictors going on, making interpretation of either variable very tricky.
So my questions are:
1. Is it ok to use the same variable as both a time-invariant and a time-varying predictor in the same model?
Even if the answer is "No." that will help.
2. If it is ok to include both, how do I resolve the ambiguities between the time-invariant and the time-varying predictors?
ATSUseAtBaseline
and theatsFactor0
dummy which should generate a warning. @llewmills, please can you clarify and also include the actual model output for both models. $\endgroup$ATSUseAtBaseiline
is the first observation ofatsFactor
. It is compulsory for all patients being admitted to the drug and alcohol treatment service where this study was conducted to have their frequency of drug and alcohol use in the previous 28 days recorded on the first day of treatment. Follow-up measurements during treatment are far less common as they are not compulsory. $\endgroup$