Functional Independent Measure and future outcomesDecember 22February 23, 20121
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 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).
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 () is
,
where FIM is the FIM score reported in the NSCISC database, and . are other covariates. The value of varies across subjects because FIM and 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 2723% 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 2016%. For hospitalizations per year, the estimated impact computed using imputed data is 385% 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 / Impact of 1 FIM Point on Event(1)Outcome / Link Function / Mean / Lower Conf Limit / Upper Conf Limit / Imputed Data / Original Data
Institutional care following discharge / Logistic / -0.040 / -0.044 / -0.036 / -0.34% / -0.36%
Survival(1) / (2) / -0.016 / -0.019 / -0.014 / (3)
FIM score at follow-up / Identity / 0.76 / 0.74 / 0.79 / 0.76 / 0.67
Institutional care / Logistic / -0.05 / -0.05 / -0.04 / -0.14% / -0.16%
Hospitalizations per year / Log / -0.0075 / -0.0088 / -0.0062 / -0.0044 / -0.0071
Days hospitalized per year / Log / -0.0080 / -0.0098 / -0.0062 / -0.071 / -0.13
Any paid care assistance / Logistic / -0.048 / -0.051 / -0.044 / -0.72% / -0.72%
Any care assistance / Logistic / -0.049 / -0.053 / -0.046 / -0.85% / -0.85%
Any work for pay / Logistic / 0.022 / 0.020 / 0.025 / 0.41% / 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 1-point FIM score change, we did compute the survival regression coefficient with and without imputed data. The value computed without imputed data (-0.023022) is similar tomodestly 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.
Technical Appendix- 1-
Functional Independent Measure and future outcomesDecember 22February 23, 20121
Table A-2. Magnitude of Change Associated with a 1 Point Change in FIM Score at Discharge from Rehabilitation(1)
Year 1 / Year 5 / Year 10 / Year 15 / Year 20Mean / 95% LCL / 95% UCL / Mean / 95% LCL / 95% UCL / Mean / 95% LCL / 95% UCL / Mean / 95% LCL / 95% UCL / Mean / 95% LCL / 95% UCL
FIM at follow-up / 0.77 / 0.74 / 0.79 / 0.75 / 0.73 / 0.77 / 0.76 / 0.74 / 0.79 / 0.76 / 0.74 / 0.78 / 0.77 / 0.75 / 0.79
Prob. of institutional care / 0.15% / 0.14% / 0.17% / 0.14% / 0.12% / 0.15% / 0.13% / 0.12% / 0.14% / 0.12% / 0.10% / 0.13% / 0.12% / 0.11% / 0.13%
Hospitalizations per year / .0045 / .0037 / .0052 / .0044 / .0036 / .0052 / .0043 / .0036 / .0051 / .0042 / .0035 / .0050 / .0042 / .0035 / .0050
Days hospitalized per year / 0.070 / 0.054 / 0.085 / 0.070 / 0.055 / 0.086 / 0.071 / 0.056 / 0.087 / 0.072 / 0.056 / 0.088 / 0.074 / 0.058 / 0.090
Prob. of any paid care assistance / 0.73% / 0.68% / 0.78% / 0.73% / 0.68% / 0.78% / 0.72% / 0.67% / 0.77% / 0.71% / 0.66% / 0.76% / 0.71% / 0.66% / 0.76%
Prob. of any care assistance / 0.85% / 0.79% / 0.91% / 0.86% / 0.80% / 0.92% / 0.86% / 0.80% / 0.93% / 0.86% / 0.80% / 0.93% / 0.85% / 0.79% / 0.91%
Prob. of any work for pay / 0.39% / 0.44% / 0.34% / 0.41% / 0.46% / 0.36% / 0.43% / 0.48% / 0.38% / 0.45% / 0.51% / 0.40% / 0.47% / 0.53% / 0.41%
Notes:(1)All changes were in the expected direction. See main text Table 4 for mean value signs. None of the 95% confidence intervals included zero.
Technical Appendix- 1-
[1] See Schafer JL. 1999. “Multiple imputation: A primer.” Statistical Methods in Medical Research. 8:3-15. Roderick J A Little and Donald B Rubin, Statistical Analysis with Missing Data, Second Edition, John Wiley and Sons, Hoboken NJ, 2002.