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1 Archives of Physical Medicine and Rehabilitation journal homepage: Archives of Physical Medicine and Rehabilitation 2013;94: SPECIAL COMMUNICATION Descriptive Modeling of Longitudinal Outcome Measures in Traumatic Brain Injury: A National Institute on Disability and Rehabilitation Research Traumatic Brain Injury Model Systems Study Christopher R. Pretz, PhD, a,b Allan J. Kozlowski, PhD, c,d Kristen Dams-O Connor, PhD, e Scott Kreider, MS, a,b Jeffery P. Cuthbert, MPH, MS, a,b John D. Corrigan, PhD, f Allen W. Heinemann, PhD, c,g Gale Whiteneck, PhD a,b From a Craig Hospital and the b Traumatic Brain Injury National Statistical and Data Center, Englewood, CO; c Center for Rehabilitation Outcomes Research, Rehabilitation Institute of Chicago, Chicago, IL; d Center for Healthcare Studies and the g Department of Physical Medicine and Rehabilitation, Feinberg Medical School, Northwestern University, Chicago, IL; e Mount Sinai School of Medicine, New York, NY; and f Department of Physical Medicine and Rehabilitation, The Wexner Medical Center at Ohio State University, Columbus, OH. Abstract Establishing accurate mathematical models of outcome measures is essential in understanding change throughout the rehabilitation process. The goal of this study is to identify the best-fitting descriptive models for a set of commonly adopted outcome measures found within the Traumatic Brain Injury Model Systems National Database where the modeling is based on data submission through 2011 and the complete range of recorded time points since injury for each individual, where time points range from admission to rehabilitation to 20 years postinjury. The statistical methodology and the application of the methodology contained herein may be used to assist researchers and clinicians in (1) modeling the outcome measures considered, (2) modeling various portions of these outcomes by stratification and/or truncating time periods, (3) modeling longitudinal outcome measures not considered, and (4) establishing models as a necessary precursor in conducting individual growth curve analysis. Archives of Physical Medicine and Rehabilitation 2013;94: ª 2013 by the American Congress of Rehabilitation Medicine The Traumatic Brain Injury Model Systems National Database (TBIMS NDB) was established in 1987 with the purpose of improving care and treatment outcomes for individuals with traumatic brain injury (TBI). 1 A goal of the TBIMS National Data and Statistical Center (TBIMS NDSC), the curator of the TBIMS NDB, is to advance the field of medical rehabilitation by using statistically sophisticated methods to understand outcomes related to TBI. This goal is highly applicable in the study of various Supported by the Traumatic Brain Injury Model Systems National Data and Statistical Center, the National Institute on Disability and Rehabilitation Research (NIDRR) (grant no. H133A110006); the NIDRR through the Rehabilitation Research and Training Center on Improving Measurement of Medical Rehabilitation Outcomes (grant no. H133B090024); and Traumatic Brain Injury Model System Centers grants from the NIDRR to Ohio State University (grant no. H133A070029) and Mount Sinai Medical Center (grant no. H133A070033). No commercial party having a direct financial interest in the results of the research supporting this article has or will confer a benefit on the authors or on any organization with which the authors are associated. injury-related outcomes over time. By the end of 2011, the TBIMS NDB had acquired data for 11,058 persons with TBI whose outcomes were assessed at rehabilitation admission and discharge, 1, 2, 5, and every subsequent 5 years postinjury. Patterns of recovery vary depending on the individuals included (eg, mild/moderate/severe TBI) and the outcome of interest. For instance, some outcomes steadily increase or decrease in value while others increase or decrease rapidly and level off, exhibiting a ceiling or floor effect, respectively. 2 Modeling patterns of recovery can range from the relatively straightforward to the very complex. The most commonly used longitudinal analytic strategies in rehabilitation research are cross-sectional analyses and comparisons of pretreatment and posttreatment scores. These strategies are limited in describing temporal change because they neither explicitly model timedthat is, the outcome is not a function of timednor do they account for correlations /13/$36 - see front matter ª 2013 by the American Congress of Rehabilitation Medicine

2 580 C.R. Pretz et al between measures from the same individual or unequal variances of the outcome over time, resulting in potentially misleading results. 3 A powerful statistical tool better suited to understanding the complexity of change over time is individual growth curve (IGC) analysis. IGC analysis has advantages that include understanding change at the individual, subgroup, and group level and the ability to explain the nature of change through the use of covariates. Detailed discussions of the utility of IGC analysis in rehabilitation research can be found elsewhere. 4-6 A crucial step before performing an IGC analysis is the accurate characterization of the relationship between time and the outcome of interest. Although the process of conducting IGC analyses is reserved for future studies (no IGC analyses are performed in this article), if the nature of the change is not specified correctly, information about patient recovery may be lost and IGC analysis may fail to adequately explain factors related to recovery. Therefore, the purpose of this article is to demonstrate how various rehabilitation outcomes can be expressed as a function of time so that the model used for conducting an IGC analysis is properly identified. The process of identifying the best-fitting model should not be confused with identifying models that demonstrate meaningful change (either statistical or clinical) in outcome. In fact, in terms of identifying the best-fitting model, whether the model identifies meaningful change or not is inconsequential because the best-fitting model is simply used as a basis for comparisons. 3 This article is intended to serve 3 main purposes: 1. To determine the mathematical functions that best describe the longitudinal nature of the following outcomes in the TBIMS NDB: Cognitive and Motor FIM (previously known as the FIM), Disability Rating Scale (DRS), Supervision Rating Scale (SRS), Satisfaction With Life Scale (SWLS), Glasgow Outcome ScaleeExtended (GOS-E), and weeks of employment per year, across the full timeline of available data for each person in the TBIMS NDB. 2. To report the growth parameter estimates for the optimal model of each outcome and use these estimates to describe outcome trajectories. 3. To describe how the modeling strategies discussed in this article can be applied to future IGC analysis. Overview of modeling techniques This section provides an introduction to commonly used modeling techniques, beginning with simple techniques and proceeding to the more complex. A number of texts devoted to longitudinal data analysis provide detailed accounts of these methods. 7-9 List of abbreviations: AIC Akaike information criterion DRS Disability Rating Scale GOS Glasgow Outcome Scale GOS-E Glasgow Outcome ScaleeExtended IGC individual growth curve IRC instantaneous rate of change SRS Supervision Rating Scale SWLS Satisfaction With Life Scale TBI traumatic brain injury TBIMS Traumatic Brain Injury Model Systems TBIMS NDB TBIMS National Database TBIMS NDSC TBIMS National Data and Statistical Center Linear trends A linear trend is defined as a pattern of data that can be represented by a straight line. In essence, linear trends relate time to outcome by 2 parameters (called growth or change parameters): an intercept and slope. The intercept b 0 represents the average baseline (ie, initial observation) score, while the slope b 1 represents the average constant rate of change expressed as a change in outcome based on a single increment of time (1s, 1h, 1d, etc). The slope is constant because the change in outcome remains the same for equal intervals of time. When the rate of change is positive, this implies that as time increases, outcomes increase; if the rate of change is negative, as time increases, outcomes decrease. In its most basic form, the equation relating time to the outcome (ie, linear change ) is as follows: byz b b 0 þ b b 1 where by corresponds to the estimate of the average outcome at a particular time point; b b 0 and b b 1 are the estimates of the intercept and slope, respectively; and represents the time point of interest. Quadratic trends Quadratic trends resemble parabolas and are described by 3 parameters; the additional parameter helps account for a rate of change that differs at each time point. A rate of change that is time dependent (ie, for each time point the rate of change is different) is called an instantaneous rate of change (IRC). In describing quadratic trends, b 0 remains the average baseline measure, but the IRC is a combination of b 1 and the additional parameter b 2, where the average IRC is expressed by b 1 þ 2b 2 ðþ. When used in conjunction with a graphical representation of the outcome, a helpful tool in understanding the nature of a quadratic trend is an IRC plot of IRC versus time. Many quadratic trends display a peak or trough, commonly referred to as a local minimum or maximum, respectively. The time at which the peak or trough occurs is found at the intersection of the IRC function and time axis (ie, abscissa). IRC plots replace the need to use equations to calculate local minimum and maximum values. Since IRC plots provide the rate of change at a given time point, they are useful in identifying when the outcome changes rapidly, gradually, or not at all. As the IRC tends toward zero, change becomes more gradual, and when the IRC equals zero, no change occurs. As the absolute value of the IRC increases, so does the rate of change. Likewise, if the IRC is positive, change is increasing; if the IRC is negative, change is decreasing. IRC plots are helpful in understanding the various ways in which outcomes change, and in comparing different trends based on stratification of outcome in an ensuing IGC analysis. An example of an IRC plot for a quadratic trend is provided later in this article (see SWLS IRC plot). The following equation is used to calculate the estimated average value of the outcome by at a given time point: byz b b 0 þ b b 1 þ b b2 2 For brevity, we refer to b 0 as the estimate of the average outcome at baseline, b 1 as an estimate of linear change, and b 2 as an estimate of quadratic change.

3 Modeling longitudinal TBI outcome measures 581 Cubic trends Cubic trends display a temporal pattern where the outcome increases, decreases, and then increases again (or vice versa). Similar to quadratic trends, the intercept b 0 is the average outcome at baseline, and the rate of change is time dependent. Because of the increased complexity of cubic change, calculation of the IRC includes yet another parameter, b 3, resulting in b 1 þ 2b 2 ðþ ti þ 3b 3 ðþ 2 ti. As deduced from its rising and falling trajectory, a cubic trend peaks and troughs. Used in conjunction with a graph of the outcome, times at which the outcome peaks (local maximum) and troughs (local minimum) can be determined from an IRC plot, where these times occur at the intersection of the IRC function and time axis. Like it does for a quadratic trend, the IRC plot identifies times when changes in outcome are rapid or less extreme as well as the direction of change. However, different from quadratic change, cubic change contains an inflection point or the moment when the outcome transitions from decreasing to increasing (or vice versa). In cubic trends, the graph of the IRC function is a parabola. Depending on whether the parabola opens down or up, the time associated with the inflection point is located at either the maximum or minimum of the parabola. Examples of IRC plots for cubic trends are also displayed later (see the GOS-E, SRS, and weeks of employment per year IRC plots). The equation used to calculate the estimate of the average value of the outcome for a cubic trend is as follows: byz b b 0 þ b b 1 þ b b2 2þ b b3 3 where the additional term b b 3 is the estimate of cubic change. Figure 1 shows examples of linear, quadratic, and cubic trends. It shows that change over time is constant for a linear trend, and in this case, is increasing, while rates of change for the quadratic and cubic trends are time dependent. The negative exponential Several outcomes exhibit abrupt change and then level off, exhibiting ceiling or floor effects. A ceiling effect is described by a rapid increase in outcome followed by a plateau, while a floor effect is described by a rapid decrease and ensuing plateau (fig 2). Such trends are ideally fit with a negative exponential model. The shape of the negative exponential is governed by 3 parameters: a pseudointercept (b 0 ), rate (b 1 ), and asymptote (a 0 ) (see fig 2). The asymptote (dashed line) indicates the location of the plateau or average stability point. The pseudointercept provides the location of the average starting point (ie, average baseline value). It is called a pseudointercept because unlike in previously described models, the negative exponential never traverses the ordinate (y axis). Although not considered a true rate of change, the final parameter, b 1, is the rate in which the asymptote is approached. If the rate increases, the portion of the trajectory that connects the pseudointercept to the asymptote becomes more vertical. Conversely, as the rate decreases, the same portion of the trajectory assumes a more gradual change. Related directly to the rate is the estimate of the time in which average maximum recovery is achieved. The formula for calculating the estimated time before reaching the asymptote is as follows: d ln jða Estimated Before Asymptote Z 0 b 0 Þj b 1 where d represents the distance between the function and the asymptote. The choice of the value of d is dependent on the units of measurement expressed by the outcome. If the value of d is too small, the time point will be overestimated, while if it is too large, the time point will be underestimated. The estimated time before asymptote can be used as a proxy for recovery time. There is no limit to how complicated modeling can become. As the complexity of methods used to describe various trajectories increases, so does the complexity of interpretation. In choosing the best model for the data, parsimony and interpretability should be considered along with model fit. It is important to incorporate theoretic considerations in applying modeling techniques, when possible. Methods Participants All participants are individuals with TBI who provide informed consent or consent by legal proxy to be enrolled in the TBIMS NDB. The TBIMS is a multicenter, longitudinal study of TBI funded by the U.S. Department of Education, National Institute on Disability and Rehabilitation Research. 10 Participants have sustained either penetrating or nonpenetrating TBI with at least 1 of the following characteristics: Glasgow Coma Scale score <13 on emergency admission (not attributable to intubation, sedation, or intoxication); loss of consciousness >30 minutes (not attributable to sedation or intoxication); posttraumatic amnesia >24 hours; or trauma-related intracranial abnormality on neuroimaging. All TBIMS enrollees are aged 16 years, receive medical care in a TBIMS-affiliated trauma center within 72 hours of injury, and are transferred to an affiliated inpatient TBI rehabilitation program. A standard assessment protocol is followed during acute care and inpatient rehabilitation, and during a series of follow-up assessments (1, 2, and 5y postinjury and every 5y thereafter). We treat time as a continuous variable with years as the unit. At least 3 time points were required for an individual to be included in the analysis to allow for the identification of curvilinear trends, as 2 points would force one s trajectory to be linear. Outcomes The FIM is an 18-item measure of burden of care associated with physical and cognitive functioning; each item is scored using a rating scale that ranges from 1 (Total Assistance) to 7 (Complete Independence). 11 We used the 13-item FIM Motor and 5-item FIM Cognitive subscales; scores range from 13 to 91 and 5 to 35, respectively. FIM data in the TBIMS cover the entire timeline from rehabilitation admission to most recent assessment, which for some individuals is up to 20 years postinjury. The FIM instrument is subject to ceiling effects with very high or very low functioning persons with TBI, and is less responsive to measuring change than measures of activities of daily living The DRS is a measure of functioning across several domains ranging from eye opening to employability. The DRS is used to track outcomes across a wide span of recovery by assessing a wide range of functions. Scores range from 0 (no disability) to 29 (extreme vegetative state). DRS data in the TBIMS cover the entire timeline from rehabilitation admission to latest assessment (up to 20-y follow-up). Although the DRS measures early recovery quite well, it is less sensitive to measuring mild deficits

4 582 C.R. Pretz et al Fig 1 Examples of linear, quadratic, and cubic change. and may not reflect subtle but important changes in long-term recovery. 15 The SRS is a structured interview used to rate the level of postmorbid supervision from caregivers on a 13-point ordinal scale. Supervision needs range from independent to full-time direct supervision. 15 Ratings are based on the level of supervision the patient actually receives and may not accurately reflect true supervision needs, which may prove a limitation in studies interested in supervision needs over time. SRS data in the TBIMS begin at 1 year postinjury and extend to 20-year follow-up. The GOS-E is a global measure of functioning that extends the original 5-point Glasgow Outcome Scale (GOS) to measure functioning across the spectrum of mild to severe TBI. A trained rater conducts a structured interview of the participant or proxy, or both. 16 The GOS-E distinguishes 8 ordinal categories: dead, vegetative state, lower severe disability, upper severe disability, lower moderate disability, upper moderate disability, lower good recovery, and upper good recovery. The expansion of categories in the GOS-E minimizes some limitations of the GOS, such as ceiling effects and limited sensitivity to detecting changes in functioning. 17 The GOS-E is commonly used to measure outcomes in the first months or year after injury and may not be sensitive to functional change many years postinjury. Given the current project s focus on change, individuals categorized as dead were removed from these analyses. Note that for the GOS-E we analyzed the data in 2 different ways. First we analyzed the data removing individuals with values of 1 (dead) at any time. Then the data for those individuals who died were retained (up until their time of death) in the analysis. In either approach, the conclusions remained the same, a cubic function best describes the GOS-E. GOS-E data begin at 1 year postinjury and extend to 10-year follow-up because the revised 8-category version was adopted July 1, The SWLS is a global measure of life satisfaction consisting of 5 self-report items that are rated on a 7-point Likert scale ranging from strongly disagree to strongly agree; total scores range from 5 to SWLS data in the TBIMS begin at 1 year postinjury and extend to 20-year follow-up. Pavot and Diener 19 reported that mean SWLS values for nonclinical populations tend to average above the theoretic midpoint of the scale; however, skewness was not pronounced and was not associated with a ceiling effect. Weeks of paid competitive employment per year reflects the number of weeks used in the year leading up to the interview. Information is provided by the most reliable source (participant or proxy) and includes the number of weeks worked in paid employment postinjury, including both legal and illegal competitive work. Data values range from 0 to 52 weeks. For this variable, inclusion was limited to those participants with GOS-E scores ranging from lower moderate disability to upper good recovery at the 1-year postinjury follow-up interview, indicating that the participant was functioning at least at a level requisite for performing sheltered work postinjury. Because the GOS-E variable was not included in the TBIMS NDB until 1998 and is subject to the above inclusion criteria, data are limited to those collected up to 10 years postinjury. 20 Although these data provide Fig 2 The negative exponential: ceiling and floor effect.

5 Modeling longitudinal TBI outcome measures 583 Fig 3 Average trajectory and IRC plot for the GOS-E, SWLS, SRS, and weeks of paid competitive employment per year.

6 584 C.R. Pretz et al a good measure of weeks of employment, they may not be exact because the data are not reconciled with quantitative employment data such as Social Security disability income or federal tax documentation. Furthermore, the variable inclusion criteria applied for these analyses preclude inclusion of people with TBI whose GOS-E score rises above or falls beneath the level requisite for performing sheltered work at 1 year postinjury, which may underinclude or overinclude the true number of individuals who are capable of performing some manner of work at each time point. Table 1 provides the total number of available cases (top number) for each outcome at each time point along with the number of cases used in the analysis for each outcome at each time point (bottom number). Note that not all time points are reported, and recall that individuals must have at least 3 assessments to be included in the analysis. Table 2 provides the total number of cases used in each analysis. Centers adhere to guidelines posted on the TBIMS NDSC website. 20 NDSC policy dictates that centers make every possible attempt to conduct an interview. The data center has established missing data targets in which no variable has more than 10% missing if the interview was conducted. Reasons for omissions include not collecting data because of death, incarceration, refusals, withdrawals, and loss to follow-up. Page 20 of the TBIMS presentation, found on the website provides the attrition rate for each follow-up year. 20 Analyses We expected that the negative exponential would adequately account for rehabilitation outcomes that exhibit floor or ceiling effects and that other outcomes would display linear, quadratic, or cubic changes. We used SAS version 9.3 a PROC MIXED procedure to model linear, quadratic, and cubic change, and PROC NLMIXED to model the negative exponential. We evaluated the adequacy of the models by examining the Akaike information criterion (AIC). Burnham and Anderson 21 indicate models with AIC values that differ by less than 2 are equivalent, while models with AIC values Table 1 Number of available cases per outcome per select year Outcome 0 1y 5y 10y 15y 20y FIM Cognitive 10, FIM Motor 10, DRS 10, GOS-E N/A SWLS N/A SRS N/A Employment Week N/A N/A N/A NOTE. In each table cell, the top number indicates the number of available cases for each outcome at each time point, and the bottom number indicates the number of cases used in the analysis for each outcome at each time point. Abbreviation: N/A, not applicable. that differ by more than 10 show separation. We selected models with the smallest AIC because they fit the data best. We also examined variance-covariance matrices to evaluate model fit: autoregressive, autoregressive moving average, variance components, compound symmetry, and spatial. These covariance matrices model a slightly different aspect of the correlation between measurements from the same individual over time and allow for refined assessment of model fit. 22 Results Table 2 contains the AIC, the optimal variance-covariance matrix for each combination of modeling technique and outcome, and the number of individuals in the analyses. The right-hand column delineates the best-fitting model for each outcome. The negative exponential was the best fit for 3 of the 7 outcomes (FIM Motor, FIM Cognitive, and DRS). The SWLS was best modeled by a quadratic trend, while the SRS, GOS-E, and weeks of employment per year were best modeled by cubic trends. Table 3 displays the change parameter estimates and respective levels of significance for the outcomes not modeled by the negative exponential. Each parameter estimate differs significantly from zero, meaning each contributes to its respective model and therefore each estimate is retained. The P values for each parameter estimate indicate a high level of significance, suggesting parameter estimates would likely remain significant in the presence of smaller sample sizes or the addition of new cases from the same population. Since measures on these outcomes were not obtained at rehabilitation admission and discharge, the estimate of the intercept is not meaningful. We used the equations described previously in the overview of modeling techniques section to find the average outcome at 1 year. For example, the average SRS score at year 1 is estimated to be 3:165 0:2675ð1Þþ0:0284ð1 2 Þ0:0009ð1 3 Þ,or The same equation can be used to calculate the average SRS scores at other time points. The time points used in estimating average outcome should fall within the range of temporal markers to avoid extrapolation beyond the range of data. The trajectories for the outcomes in table 3 are displayed in figure 3. The graph of each trajectory is accompanied by its respective IRC plot. We provide a description of each outcome along with a graph of each trajectory and corresponding IRC plot below. Glasgow Outcome ScaleeExtended The GOS-E displays a cubic trend. By inspection, the GOS-E begins with a 1-year follow-up average of 5.7 and increases until reaching a peak near 6 years (indicated by the IRC plot), where the peak value of the GOS-E is slightly above 6. After reaching the peak, the GOS-E declines and reaches the inflection point at around 10 years. Beyond the inflection point, the GOS-E troughs at 14 years at an average of 5.9. Then, scores increase until a maximum average of 6.3 is achieved at 20 years. As seen in the IRC plot, the most rapid increase in GOS-E scores occurs between years 19 and 20, while the rate of change in the GOS-E is at its most gradual between 9 and 11 years. Satisfaction With Life Scale The trend for the SWLS is quadratic. From figure 4 we see that the average SWLS score at year 1 is Satisfaction with life increases until reaching a peak at 10 years. From this point it

7 Modeling longitudinal TBI outcome measures 585 Table 2 AIC values for each outcome measure based on modeling approach Model Outcome Linear Change Quadratic Change Cubic Change Negative Exponential Lowest AIC GOS-E (nz3949) 44,508 (CS) 44,462 (CS) 44,426 (CS) 51,591 (VC) Cubic change SWLS (nz3121) 71,553 (CS) 71,544 (CS) 71,555 (CS) 75,459 (VC) Quadratic change SRS (nz4042) 60,673 (CS) 60,605 (CS) 60,570 (CS) 68,111 (VC) Cubic change Weeks employment per year 55,178 (CS) 54,983 (CS) 54,763 (SP) 56,776 (VC) Cubic change (nz1983) FIM Cognitive (nz9157) 289,723 (CS) 284,270 (CS) 278,643 (CS) 275,438 (VC) Negative exponential FIM Motor (nz8995) 367,414 (CS) 361,708 (CS) 356,140 (CS) 345,427 (VC) Negative exponential DRS (nz9101) 257,869 (CS) 252,083 (CS) 246,685 (CS) 243,845 (VC) Negative exponential Abbreviations: AIC, Akaike information criterion; CS, compound symmetry; SP, spatial; VC, variance components. declines, and as indicated by the IRC plot, the rate of decline is fastest at between 19 and 20 years. At 20 years, the average SWLS score is 20.8, falling below the initial year 1 measure. Supervision Rating Scale Figure 3 shows that the average SRS score is slightly above 2.9 at 1 year after admission to rehabilitation, after which scores decline. The IRC graph shows that the steepest decline in SRS scores is between years 1 and 2, while the decline becomes less extreme in approaching 6.7 years. At 6.7 years, SRS scores reach a minimum of 2.4. After 6.7 years, the scores increase until the average score peaks at 15.5 years, just below 2.7. Then scores decrease, reaching a minimum of 2.4 at 20 years. The inflection point for the SRS occurs at year 11 with a corresponding score of 2.5. Weeks of employment per year At 1 year postinjury, the average number of weeks worked per year is From years 1 to 3.5, the number of weeks worked increases to a maximum average of 27. The IRC plot in figure 3 indicates the most rapid increase in weeks worked per year occurs between years 1 and 2. From years 3.5 to 8, weeks of employment declines when a minimum average of 16.5 weeks occurs. The inflection point is at 5.9 years (or just above 21 weeks of employment per year). Then weeks of employment per year rises, until at 10 years, the average number of weeks worked per year is Cognitive and Motor FIM Table 4 shows the estimates of the change parameters for the outcome measures that are modeled by the negative exponential and the number of days until the average point of stability (asymptote) is approached, a proxy for the average time to recovery. Statistical significance of the parameter estimates is high, indicating that these results would likely hold if a smaller sample size was used. Figure 4 illustrates that the FIM Cognitive score at admission is 16.6, and the FIM Motor score is During rehabilitation both measures increase rapidly, and then the FIM Cognitive score stabilizes at an average of 30.2, while the FIM Motor score stabilizes at an average of The time to average maximum recovery for the FIM Motor score is 37 days, much shorter than the 131 days for the FIM Cognitive score, evidence that motor recovery occurs sooner. Disability Rating Scale In contrast to the FIM Cognitive and Motor scores, the DRS exhibits a floor effect, meaning the point of stability is lower than the average baseline score, since a high score on the DRS means high disability. The trajectory of the DRS is displayed in figure 5. Table 4 indicates that the average DRS score at baseline is 12.4, while scores begin to level off 55 days after admission at an average score of 3.4. Discussion The goal of this article is to demonstrate the way in which various rehabilitation outcomes can be modeled and to set the stage for more extensive investigations of these outcomes through IGC analysis. In accomplishing this goal we determined which models best describe temporal change for common TBI outcome measures. Based on clinical experience, we expected models Table 3 Outcome Estimates of change parameters for linear, quadratic, or cubic change Change Parameter Estimates Intercept (Estimate of b 0 ) Linear Change (Estimate of b 1 ) Quadratic Change (Estimate of b 2 ) Cubic Change (Estimate of b 3 ) GOS-E * * * * SWLS 21.05* * * N/A SRS 3.165* * * * Weeks employment per year 2.562* * 3.922* * Abbreviation: N/A, not applicable. * P<.001.

8 586 C.R. Pretz et al Fig 4 Average trajectory for the FIM Cognitive and Motor. describing gradual change would fit the SWLS, SRS, GOS-E, and weeks of employment per year best, as the initial measure for these outcomes is not appropriate until after the rehabilitation process begins. In contrast, we expected the FIM Cognitive and FIM Motor scores and the DRS (recall that lower numbers indicate an improvement in status) to improve quickly, then level off; the negative exponential fit these outcomes best. Had we excluded admission and discharge time points from the FIM Cognitive, FIM Motor, and DRS data, a linear, quadratic, or cubic model would likely fit best. For some outcome measures, change is not clinically significant. This can be seen in figure 3, as the greatest average change in SRS is half a point. Although witnessing meaningful change may be interesting in its own right, the presence of change is neither essential nor a necessary component in assessing the utility of these results. What is important is identification of an appropriate model that can be used as a basis for IGC analysis, where introduction of covariates may indeed reveal meaningful changes in outcome. When modeling longitudinal outcomes, it is assumed that every individual s response pattern (a plot of the individual s scores on an outcome measure over time) will not conform to the average trajectory. For instance, consider figure 6, which contains the average trajectory (bold black line) and a random sample of 30 individual response patterns (dashed black lines) for the Cognitive FIM. There is considerable variability in individual response patterns, not unlike that which surrounds a typical least squares regression line. Consequently, a trajectory is an aggregate response or representation of the overall trend assumed by the sample. Although we did not consider individual variation in this study, accounting for individual variability is an integral aspect of IGC analysis. The modeling techniques discussed thus far can be applied to outcome measures in a variety of ways. For instance, time can be truncated such that only a portion of the data is considered, allowing the researcher to concentrate on a particular segment of the trajectory. As an example, one may wish to investigate longer Table 4 Estimates of change parameters for the negative exponential Change Parameter Estimates Outcome Pseudointercept Asymptote Rate FIM Cognitive 16.59* 30.16* 13.64* 131 FIM Motor 36.65* 80.74* 63.78* 37 DRS 12.35* 3.35* 29.33* 55 * P<.001. Days to Asymptote Fig 5 Average trajectory for the DRS.

9 Modeling longitudinal TBI outcome measures 587 term patterns of functional recovery. The researcher could determine the best-fitting model for observations recorded 5 years and later, ignoring measures taken at admission, discharge, and 1-year follow-up. Likewise, the same process could be applied if a researcher is interested in other segments of the recovery continuum such as the first 3 FIM measurements. The ability to model different periods of an outcome signifies the versatility of this approach. The shapes of the trajectories themselves may promote further inquiry. For instance, trends for the GOS-E and weeks of employment per year each turn upward during the latter part of data collection, indicating improvement. To explain this observation, one may hypothesize that the increase during the later years results from survivor bias. That is, those with greater function tend to survive longer, increasing the average of the outcome along with the trend. This hypothesis could be investigated by applying IGC analysis to determine whether those who live longer (or simply compare those who are still alive with those who have died) tend to retain both higher average outcomes and faster rates of increasing change during later follow-up periods. The discussion thus far has illustrated not only the versatility of modeling techniques discussed in this article, but also their importance in setting the stage for IGC analysis. In fact, studies are presently underway in which IGC analysis is being applied to patient and injury characteristics and the change parameters of the FIM Cognitive and Motor scores. The variability across individuals with regard to baseline scores, rate of change, and point of stability is being examined through introduction of covariates, while potential relationships between change parameters and the covariates themselves are being explored. Similarly, studies are in the planning stages for the other outcomes presented in this article as well some not mentioned; in each study, a robust set of hypotheses will be explored. Study limitations Although the modeling techniques used in this study can be applied to multiple outcomes, the best-fitting models are pertinent to future studies if an identical dataset is utilized. In this study, we only model outcomes for the complete timeline of available data in the TBIMS NDB; it is incumbent on the reader to understand how modeling procedures would be applied to different periods or groups defined by other inclusion/exclusion criteria. That is, in outcomes where rapid improvement is expected because of spontaneous recovery and participation in rehabilitation (ie, the FIM), a negative exponential will most likely fit the data best when the data available capture this period of early recovery. However, if the data are based on a truncated timeline, such as 10 to 20 years postinjury, a different modeling procedure (perhaps a linear trend) would be more suitable. For example, a study of FIM data designed to examine age-related decline by considering only outcomes collected after inpatient rehabilitation has been completed may find that linear change describes the data best. Datasets are also likely to change if subgroups are investigated or after additional participants are introduced from new sources. In addition, it is also possible (although not likely) that the trajectories described in this article might change when transitioning from strictly modeling the outcome to conducting a more involved IGC analysis. Consequently, modifications to the original data or increasing the complexity of the model may require appropriate modeling adaptations. In addition to the limitations discussed above, we highlight the following: 1. Because at least 3 temporal measures must be taken on each individual, sample size is inevitably reduced. 2. The TBIMS NDB is a dynamic dataset, meaning that outcome measures and patients can enter and exit the database as time advances; thus, samples analyzed across outcomes likely include samples of different individuals and may be influenced by the periods in which variables have been collected. 3. Variables have inherent limitations, mentioned in their respective descriptions. For instance, the authors recognize that when conducting IGC analysis, outcome measures are ideally continuous. In the same vein we also recognize that many outcome measures in the social sciences and rehabilitation medicine alike are quasi-continuous, as are some of the outcome measures analyzed in this study. Therefore, the results contained herein should be interpreted bearing such a limitation in mind. 4. Because databases such as the TBIMS NDB are always evolving, this and similar studies should be repeated to reestablish relationships between outcome measure and time. Conclusions Fig 6 Average trajectory for FIM Cognitive with individual trajectories. We identified models that best describe idealized trends for outcome measures contained in the TBIMS database. The negative exponential fits the FIM Cognitive, FIM Motor, and DRS best because measures were administered during inpatient rehabilitation and follow-up. Outcomes measured after inpatient rehabilitation exhibit less extreme change and are best modeled by linear, quadratic, or cubic trends. Specifically, a quadratic trend best describes satisfaction with life, while cubic trends best describe the GOS-E, SRS, and weeks of employment per year. This study offers researchers and clinicians a detailed account of how the selected outcomes progress over time. Additionally, the modeling techniques can be applied to different timelines within each outcome and to outcomes not investigated. More importantly, after establishing an outcome s trajectory, IGC analyses can be used to explore a robust set of hypotheses regarding the influence of

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