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Many statistical problems fall between the cracks of traditional methods. An important example of this is the analysis of clinical data gathered from patients' records over time. We might be interested, for example, in studying how a characteristic changes over time but the timing and number of measurements vary considerably from patient to patient.
The classical methods (e.g., multiple regression, analysis of variance, multivariate repeated measures) do not quite work without ignoring portions of the data. Mixed effects models, however, that combine aspects of regression, analysis of variance and repeated measures can be used to handle these problems. We describe two research projects in Psychology at York in which clinical data are analyzed with mixed models to recover information that would be difficult to obtain otherwise.
One of the authors, Pauline Wong, spent many years gathering data on the recovery of mental function among patients who had survived comas of various durations. The data include 345 WAIS-R (IQ) scores obtained from 207 patients, 97 of whom were tested on two or more occasions.
The purpose of the analysis is to study the shape of the recovery curve over time and the relationship between coma duration and both short-term and long-term recovery levels. The 97 patients with multiple test results provide information on the shape of the recovery curve. A mixed model analysis allows this information to be combined with the data from patients with single test results so that all data contribute to the estimation of long-term recovery levels.
We will use one of the subscales of the WAIS-R, the Object Assembly subscale, as an example. The first step is to identify a model that incorporates the known features of IQ recovery and conforms with the data. After considering a number of models, we settled on an 'exponential asymptotic regression' model which is used to represent some forms of growth including that of pine trees. The model for one patient has the form:
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where t is time post-coma in days, A is the patient's long-term recovery level, B is the 'initial deficit,' and C is a parameter that captures the speed of recovery, namely the proportion of the remaining deficit recovered per day.
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These parameters vary from patient to patient and we are primarily interested in the 'effect' of coma duration on their values. Figure 1 shows that with longer comas the recovery curve starts at a lower level and approaches a lower long-term recovery level. The analysis suggests that the parameter C is relatively constant across all patients. Its value of 0.53% corresponds to a half-recovery time of about 130 days for this particular subscale.
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| Figure 1: Post-coma recovery of WAIS-R Object Assembly subscale using an exponential asymptotic regression model. |
The average long-term level for patients with one day in coma is 10.1, which is comparable with the normed population mean of 10. For patients with 50 days in coma, it is 8.1, and, with 100 days, 7.2. Thus, the model captures the extent of the long-term deficit that can be expected for various lengths of coma.
Figure 2 shows the expected score as a function of both days post-coma and the duration of coma. Note that duration of coma is a between-subject variable and days post-coma is a within-subject variable. Patients with longer comas have a larger deficit after awakening and their long-term deficit is more profound. An additional day in coma is associated with a larger effect when the duration is short than when the duration is long.
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| Figure 2: Mean WAIS-R Object Assembly score as a function of days post-coma and duration of coma. |
The curves in Figures 1 and 2 are population averages. Mixed models allow individual predictions that take into account partial information known about a subject. If one test result is available for a patient, the mixed model combines this information with the fitted model to produce an individual's predicted long-term recovery level.
Tammy Kostecki-Dillon is studying the clinical records of 133 migraine sufferers who enrolled in a non-drug headache program involving four weekly sessions. Given the nature of the program, patients were not expected to show an immediate improvement.
The data include daily records of headache severity which patients kept for varying periods. Forty-four patients also kept records before the start of the program. The goal of the analysis is to estimate the pattern of change in headache severity during the program and, for patients with pre-program data, to compare pre- and post-program severity levels.
An ironic feature of the program is that patients appear to worsen at the very beginning. Improvement only becomes evident after a week. The model must therefore incorporate this iatrogenic jump. A modification of the exponential asymptotic regression model seems very promising. The mathematical form of the model is:
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where t is time in days with day 1 the first day of the program, A is the pre-program level, B is the long-term change compared with the pre-program level, and D is the change at the start of the program. Finally, C reflects the rate of change as a proportion per day. Figure 3 is a graphical rendition of the model.
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| Figure 3: Mean migraine severity before, during and after a treatment program. |
These two applications of mixed models exemplify their versatility in analyzing longitudinal observational data that are too 'messy' or unbalanced for convenient analysis with traditional methods. These models have become popular during the last decade under a variety of names: multi-level linear models, random-coefficient regression models, covariance component models, or hierarchical linear models. They help solve many types of problems characterized by hierarchical random structures: children within families, students within classes within schools, measurements within patients, sibling dyads within families, etc. Since these structures are typical of many current research projects in the social sciences at York, we expect some expansion in their use.
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