13.8 Alternative Time Metrics
Let’s go back to the simple linear growth model, this time being very explicit about the scaling and centering of the time variable. Time metric = original scores, Grade = 1, 2, 4, 6. Fitting the linear model with grade (as originally coded)
linear_grade_fit <- lme(
fixed= verb ~ 1 + grade,
random= ~ 1 + grade|id,
data=verblong,
na.action = na.exclude
)
summary(linear_grade_fit)## Linear mixed-effects model fit by REML
## Data: verblong
## AIC BIC logLik
## 5053.632 5081.844 -2520.816
##
## Random effects:
## Formula: ~1 + grade | id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 3.915535 (Intr)
## grade 1.241299 0.321
## Residual 3.581589
##
## Fixed effects: verb ~ 1 + grade
## Value Std.Error DF t-value p-value
## (Intercept) 15.15099 0.3686512 611 41.09845 0
## grade 4.67339 0.1087023 611 42.99255 0
## Correlation:
## (Intr)
## grade -0.155
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.61615651 -0.54270692 -0.02898121 0.52402086 3.16468350
##
## Number of Observations: 816
## Number of Groups: 204Important Questions to Ask
- Where should the intercept be located?
- Is there a meaningful zero-time? (choosing c1)
- time since birth, time to death, time since surgery, time since meeting the research staff
- Is there a meaningful zero-time? (choosing c1)
- What should the time units be? (choosing c2)
- years, months, minutes, grades, # of psychological events?
Using these quantities (c1 and c2) and the formula below it is possible to create alternative time metrics that may be more appropriate for the specific change process we are interested in:
\[timenew = (time - c1)/c2\]
where
- c1 is a centering constant, and
- c2 is a scaling constant.
13.8.1 Recentering time metrics
time_cG1 = Grade centered 0-point = grade 1: time_cG1 = 0, 1, 3, 5
time_cG6 = Grade centered 0-point = grade 6: time_cG6 = -5, -4, -2, 0
13.8.2 Rescaling time metric
time_cG6 = Grade centered 0-point = grade 6: time_cG6rescale = -1.0, -0.8, -0.4, 0.0
13.8.3 Remapping Time
We may also want to remap time based on some other metric. For example, if we wanted to look at the number of assessment as indexing time. Here assessment would be the number of assessments youth have been exposed to: (assessment = 1, 2, 3, 4). Note: the way the mapping is done here only works in this case with no missing data. If have missing data, need to remap in a different way.
13.8.4 Compare Growth Metrics
Let’s look at the models with different time-metrics.
13.8.4.1 Time metric = grade
linear_grade_fit <- lme(
fixed= verb ~ 1 + grade,
random= ~ 1 + grade|id,
data=verblong,
na.action = na.exclude
)
summary(linear_grade_fit)## Linear mixed-effects model fit by REML
## Data: verblong
## AIC BIC logLik
## 5053.632 5081.844 -2520.816
##
## Random effects:
## Formula: ~1 + grade | id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 3.915535 (Intr)
## grade 1.241299 0.321
## Residual 3.581589
##
## Fixed effects: verb ~ 1 + grade
## Value Std.Error DF t-value p-value
## (Intercept) 15.15099 0.3686512 611 41.09845 0
## grade 4.67339 0.1087023 611 42.99255 0
## Correlation:
## (Intr)
## grade -0.155
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.61615651 -0.54270692 -0.02898121 0.52402086 3.16468350
##
## Number of Observations: 816
## Number of Groups: 20413.8.4.2 Time metric = grade_cG1
time_cG1 = Grade centered 0-point = grade 1: time_cG1 = 0, 1, 3, 5
linear_time_cG1_fit <- lme(
fixed= verb ~ 1 + time_cG1,
random= ~ 1 + time_cG1|id,
data=verblong,
na.action = na.exclude
)
summary(linear_time_cG1_fit)## Linear mixed-effects model fit by REML
## Data: verblong
## AIC BIC logLik
## 5053.632 5081.844 -2520.816
##
## Random effects:
## Formula: ~1 + time_cG1 | id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 4.470800 (Intr)
## time_cG1 1.241299 0.558
## Residual 3.581590
##
## Fixed effects: verb ~ 1 + time_cG1
## Value Std.Error DF t-value p-value
## (Intercept) 19.82438 0.3678085 611 53.89865 0
## time_cG1 4.67339 0.1087023 611 42.99256 0
## Correlation:
## (Intr)
## time_cG1 0.14
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.61615611 -0.54270689 -0.02898113 0.52402090 3.16468325
##
## Number of Observations: 816
## Number of Groups: 20413.8.4.3 Time metric = grade_cG6
time_cG6 = Grade centered 0-point = grade 6: time_cG6 = -5, -4, -2, 0
linear_time_cG6_fit <- lme(
fixed= verb ~ 1 + time_cG6,
random= ~ 1 + time_cG6|id,
data=verblong,
na.action = na.exclude
)
summary(linear_time_cG6_fit)## Linear mixed-effects model fit by REML
## Data: verblong
## AIC BIC logLik
## 5053.632 5081.844 -2520.816
##
## Random effects:
## Formula: ~1 + time_cG6 | id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 9.460226 (Intr)
## time_cG6 1.241299 0.92
## Residual 3.581590
##
## Fixed effects: verb ~ 1 + time_cG6
## Value Std.Error DF t-value p-value
## (Intercept) 43.19133 0.6976142 611 61.91292 0
## time_cG6 4.67339 0.1087023 611 42.99256 0
## Correlation:
## (Intr)
## time_cG6 0.853
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.61615613 -0.54270686 -0.02898114 0.52402089 3.16468319
##
## Number of Observations: 816
## Number of Groups: 20413.8.4.4 Time metric = grade_cG6rescale
time_cG6 = Grade centered 0-point = grade 6: time_cG6rescale = -1.0, -0.8, -0.4, 0.0
linear_time_cG6rescale_fit <- lme(
fixed= verb ~ 1 + time_cG6rescale,
random= ~ 1 + time_cG6rescale|id,
data=verblong,
na.action = na.exclude
)
summary(linear_time_cG6rescale_fit)## Linear mixed-effects model fit by REML
## Data: verblong
## AIC BIC logLik
## 5050.413 5078.625 -2519.207
##
## Random effects:
## Formula: ~1 + time_cG6rescale | id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 9.460223 (Intr)
## time_cG6rescale 6.206483 0.92
## Residual 3.581591
##
## Fixed effects: verb ~ 1 + time_cG6rescale
## Value Std.Error DF t-value p-value
## (Intercept) 43.19133 0.6976141 611 61.91293 0
## time_cG6rescale 23.36695 0.5435110 611 42.99260 0
## Correlation:
## (Intr)
## time_cG6rescale 0.853
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.61615435 -0.54270813 -0.02898075 0.52402103 3.16468675
##
## Number of Observations: 816
## Number of Groups: 20413.8.4.5 Time metric = assessment
Assessment = 1, 2, 3, 4
linear_assessment_fit <- lme(
fixed= verb ~ 1 + assessment,
random= ~ 1 + assessment|id,
data=verblong,
na.action = na.exclude
)
summary(linear_assessment_fit)## Linear mixed-effects model fit by REML
## Data: verblong
## AIC BIC logLik
## 5134.792 5163.004 -2561.396
##
## Random effects:
## Formula: ~1 + assessment | id
## Structure: General positive-definite, Log-Cholesky parametrization
## StdDev Corr
## (Intercept) 2.960137 (Intr)
## assessment 1.892481 0.35
## Residual 3.997613
##
## Fixed effects: verb ~ 1 + assessment
## Value Std.Error DF t-value p-value
## (Intercept) 10.417770 0.4005742 611 26.00709 0
## assessment 7.968696 0.1822741 611 43.71820 0
## Correlation:
## (Intr)
## assessment -0.405
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.41214177 -0.56890698 -0.05446236 0.56436062 3.60513496
##
## Number of Observations: 816
## Number of Groups: 204