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Functional Independent Measure and future outcomes
February 23, 2012
Technical Appendix
Regression of outcomes against FIM
The type of regression used depended on the nature of the outcome (binary
outcome, continuous measure, time to event) and whether the outcome was measured
once or repeatedly at follow-up. Outcomes measured at repeated follow-up times were
modeled using a generalized linear model with a robust variance estimator (SAS
procedure GENMOD) and a link function and error distribution dependent on the nature
of the outcome, as determined by visual inspection of the residuals in comparison with
the parametric distribution.
For residential status at discharge from rehabilitation, a one-time binary outcome
(outcome 1), we used logistic regression analysis (SAS procedure LOGISTIC). For
survival time (outcome 2), we used a Cox proportional hazards model (procedure
PHREG, SAS version 9.2, Cary, NC).
FIM scores at follow-up (outcome 3) have been treated as normal, with the
identity link function. Because residential status at follow-up (outcome 4) is binary, this
analysis used the logit link function and the binomial error distribution. Number of
hospitalizations per year (outcome 5) and number of days hospitalized per year (outcome
6) are right-skewed, non-negative values. To model these outcomes, we used the log link
function and negative binomial error distribution.
The remaining outcomes (outcomes 7-9, paid care hours/day, total care hours/day,
and work hours/week) are continuous non-negative measures, but their distributions are
not readily characterized using standard distributions because of reporting artifacts. For
example, for hours of paid care assistance per day, even numbered values are
substantially more frequent than odd numbered values, and multiples of eight are
particularly common, as is the value of 24 hours. For hours worked per week, values
divisible by 5 are far more common than other values, and a value of 40 is particularly
common. Because a substantial portion of the values for all three outcomes is zero (67%
for paid care assistance, 44% for any assistance, and 76% for paid work), this analysis
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Functional Independent Measure and future outcomes
February 23, 2012
classified these outcomes dichotomously (zero or exceeding zero) and used the logit link
function and binomial error distribution.
Imputation of missing values
We used Markov Chain Monte Carlo (MCMC) multiple imputation methods
(SAS version 9.2 procedure MI) to impute missing values for FIM measured at discharge
and for all outcomes except date of death, creating five imputed value datasets. The
multiple data sets allow for appropriate assessment of uncertainty of the parameter
estimates. Single imputation yields a standard error that is too small because it does not
take into account the fact that the true values for the missing data are unknown. The
imputed values are independent draws from the posterior predictive distribution of the
missing variables. Each of our filled-in data sets was analyzed using the regression
models described in the Methods section in the article’s main text (although the
imputation method used here is Bayesian, our analytic models are not). The reported
confidence intervals account for the within-imputation and between-imputation variance.1
The date of death may be unspecified because it is missing (death occurred but the
event was not recorded) or because it has not yet occurred. Imputation of date of death
values would introduce bias because the imputation procedure will estimate values based
on the histories of subjects who are known to have died. If at least some subjects without
a date of death value are in reality still alive, the imputation procedure will underestimate
their survival. For each regression, we created five imputed datasets.
The MCMC technique was feasible in the context of this analysis because it
makes no assumptions regarding which fields are missing for each subject. Because the
MCMC method treats all imputed variables as normally distributed, we converted
imputed binary outcomes to dichotomous values by rounding. All other imputed values
were left unchanged even if they did not fall within the valid range (e.g., non-integer
imputed FIM score values and imputed FIM scores exceeding 91 or falling below 13).
See Schafer JL. 1999. “Multiple imputation: A primer.” Statistical Methods in Medical Research. 8:315. Roderick J A Little and Donald B Rubin, Statistical Analysis with Missing Data, Second Edition, John
Wiley and Sons, Hoboken NJ, 2002.
1
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February 23, 2012
Absolute impact of FIM score change on outcomes
For non-linear regressions, the absolute impact on the outcome of a 1-point
change in the FIM score varies across subjects, depending on covariate values. For
example, the absolute incremental probability of institutionalization upon discharge from
rehabilitation ( p ) is
p 
exp    FIM  ( FIM  1)   2 x2  ...
exp    FIM  FIM   2 x2  ...
,

1  exp    FIM  ( FIM  1)   2 x2  ... 1  exp    FIM  FIM   2 x2  ...
where FIM is the FIM score reported in the NSCISC database, and x2 , x3 ... . are other
covariates. The value of p varies across subjects because FIM and x2 , x3 ... all vary
across subjects. To calculate the absolute impact of a 1-point change in the FIM score in
these cases, we computed the impact for each included subject from the NSCISC dataset
and used the average over these subjects.
Results
Appendix Table A-1 supplements manuscript Table 3 by providing the confidence
intervals for the regression coefficients. In all cases, the confidence intervals are narrow,
extending no more than 23% above and below the central estimate value in all cases. Table A-1
also compares the impact of a 1-point FIM score change estimated with and without missing
value imputation. For 7 of 9 outcomes, the difference between the two estimates is no more than
20%. For hospitalizations per year, the estimated impact computed using imputed data is 38%
smaller than the estimate computed using the original data. For hospitalized days per year, the
estimated impact computed using imputed data is 44% smaller than the estimate computed using
the original data.
Table A-1. Regression Results and Impact on Outcomes of One FIM Point Change
Regression Coefficient
Outcome
Link
Function
Mean
Lower
Conf
Limit
Upper
Conf
Limit
Institutional care following discharge
Survival(1)
Logistic
-0.040
-0.016
-0.044
-0.019
-0.036
-0.014
Technical Appendix
(2)
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Impact of 1 FIM
Point on Event(1)
Imputed Original
Data
Data
-0.34%
-0.36%
(3)
Functional Independent Measure and future outcomes
FIM score at follow-up
Institutional care
Hospitalizations per year
Days hospitalized per year
Any paid care assistance
Any care assistance
Any work for pay
Identity
Logistic
Log
Log
Logistic
Logistic
Logistic
0.76
-0.05
-0.0075
-0.0080
-0.048
-0.049
0.022
February 23, 2012
0.74
-0.05
-0.0088
-0.0098
-0.051
-0.053
0.020
0.79
-0.04
-0.0062
-0.0062
-0.044
-0.046
0.025
0.76
-0.14%
-0.0044
-0.071
-0.72%
-0.85%
0.41%
0.67
-0.16%
-0.0071
-0.13
-0.72%
-0.85%
0.50%
Notes:
(1)
See note (1) for Table 3 in main manuscript.
(2)
Survival analyzed using a Cox proportional hazards model.
(3)
Although it is not possible to compute the incremental impact on survival of a 1point FIM score change, we did compute the survival regression coefficient with
and without imputed data. The value computed without imputed data (-0.022)
modestly exceeds the value computed using the imputed data (-0.016).
Appendix Table A-2 supplements manuscript Table 4 by providing confidence intervals
for the impact of a 1-point FIM score change.
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February 23, 2012
Table A-2. Magnitude of Change Associated with a 1 Point Change in FIM Score at Discharge from Rehabilitation(1)
FIM at follow-up
Prob. of institutional care
Hospitalizations per year
Days hospitalized per year
Prob. of any paid care assistance
Prob. of any care assistance
Prob. of any work for pay
Notes: (1)
Mean
0.77
0.15%
.0045
0.070
0.73%
0.85%
0.39%
Year 1
95%
LCL
0.74
0.14%
.0037
0.054
0.68%
0.79%
0.44%
95%
UCL
0.79
0.17%
.0052
0.085
0.78%
0.91%
0.34%
Mean
0.75
0.14%
.0044
0.070
0.73%
0.86%
0.41%
Year 5
95%
LCL
0.73
0.12%
.0036
0.055
0.68%
0.80%
0.46%
95%
UCL
0.77
0.15%
.0052
0.086
0.78%
0.92%
0.36%
Mean
0.76
0.13%
.0043
0.071
0.72%
0.86%
0.43%
Year 10
95%
LCL
0.74
0.12%
.0036
0.056
0.67%
0.80%
0.48%
95%
UCL
0.79
0.14%
.0051
0.087
0.77%
0.93%
0.38%
Mean
0.76
0.12%
.0042
0.072
0.71%
0.86%
0.45%
Year 15
95%
LCL
0.74
0.10%
.0035
0.056
0.66%
0.80%
0.51%
95%
UCL
0.78
0.13%
.0050
0.088
0.76%
0.93%
0.40%
Mean
0.77
0.12%
.0042
0.074
0.71%
0.85%
0.47%
Year 20
95%
LCL
0.75
0.11%
.0035
0.058
0.66%
0.79%
0.53%
All changes were in the expected direction. See main text Table 4 for mean value signs. None of the 95% confidence intervals
included zero.
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95%
UCL
0.79
0.13%
.0050
0.090
0.76%
0.91%
0.41%