New Single-Subject and Small- n Design in Occupational Therapy: Application to Weight Loss in Obesity

Abstract

Ottenbacher (1986) showed the usefulness of single-subject design (SSD) in occupational therapy. However, SSD methodology is not regarded by the wider research community as providing statistically reliable and valid evidence of effectiveness of treatment partly because of its observational nature. Although statistical estimations can also be made from least squares regression or by a trend line, a new methodology has great potential to influence research in occupational therapy. The new model enables the use of initial client data from the beginning of treatment (for single subjects or small groups) to determine a point in the linear regression at which predictions can be made for the number of treatments needed for stability or improvement. This model is invaluable for third-party payment as well as for client motivation. The purpose of this article is to present this new methodology, the semiparametric ratio estimator (SPRE), illustrated by case application to treatment of obesity.

Keywords

biostatistics models,statistical obesity outcome assessment (health care)research design

Our objective in this article is to present a new statistical method that addresses the needs of and problems in quantitative research identified in occupational therapy over the past 30 yr (Banigan et al., 2008; Davies & Case-Smith, 1998; Ellenberg, 1996; Ottenbacher & York, 1984; Rogers, 2010). Ottenbacher’s (1986) work exposed occupational therapy to single-subject design (SSD). Although this methodology helped the profession with research involving small numbers of participants, SSD is observational in nature and unable to predict future outcomes, nor is it able to indicate the number of treatment sessions in which the client demonstrates a change and at what point in treatment the client will stabilize his or her performance.

In this article, we present a new methodology, a semiparametric ratio estimator (SPRE), with a case application to the obesity population. This new model, used in primary or secondary analysis of the outcomes of the participant during treatment or for the predicted response to treatment, addresses the stated goal of the Centennial Vision for the profession of occupational therapy to be science driven and evidence based (American Occupational Therapy Association [AOTA], 2005).

The challenges to attaining this goal are many. Not only are “the inherent difficulties associated with describing and investigating true outcomes of intervention complicated by the nature of occupational therapy—the use of occupation as therapy” (Robertson & Colborn, 2000, p. 541), but also, “when using the classical sums-of-squares linear approach to statistics, the contribution of one variable is examined at one time, independent of the contribution of other variables” (Davies & Case-Smith, 1998, p. 524) in which the ordinary least squares approach constitutes the total analysis and in which this analysis does not coincide with occupational therapy’s holistic approach. The challenge in occupational therapy, then, is to translate improvement in a client-centered discipline to evidence, in this case using an SSD with internal and external validity that will be understandable to third-party payers.

Up to this point, the statistics used in occupational therapy have generally been borrowed from the social sciences and public health, in which these methodologies are dependent on large samples. Moreover, preintervention–postintervention designs do not identify the changes that take place during intervention over time. Difficulties in establishing external validity for the data in SSDs mean that many datasets are not able to be statistically generalized to a population for the classic SSD (Bordens & Abbott, 2008). Classic SSDs, as extensions of pre–post designs, are not considered statistically rigorous and lack external validity (Ottenbacher, 1986). In a classic SSD, when experimental conditions are the same from phase to phase, the classic design can establish internal reliability to the point of determining the effectiveness of the treatment. However, it remains observational and cannot statistically determine correlation, and it will be difficult to generalize to the larger population from which the sample was drawn. Another factor in an SSD is a fundamental assumption that in the ABA or other design, the treatments are the same for a single participant. Although few randomized controlled trials have been conducted in occupational therapy (Clark et al., 1997, 2011), most occupational therapists do not work with large groups, and randomized controlled trials rely on the mean of outcomes from a large number of people. As Johnston and Smith (2010, p. 5) stated, “Smaller, more practical research designs are needed to advance knowledge in many—perhaps most—areas of OT practice.”

At least two major choices can be made in deriving a new method to deal with a statistical representation of data from occupational therapy: (1) Try to collect a participant’s different responses (e.g., weight loss in terms of body mass index, waist circumference, waist-to-hip ratio, physiological responses) or (2) derive a method in which the occupational therapy measure used for that treatment modality includes subsets that measure all the relevant responses for the intended treatment and yields a number that represents all the potential variable responses relating to that person at that time. “Occupations are complex, individualized and essential for health and well-being. Evaluation of occupational performance, the outcomes of doing occupation is enhanced by the development of a broad measurement perspective” (Law, Baum, & Dunn, 2005, p. 2). Law et al. (2005) went on to state that “measurement will focus on both the subjective experience and the desirable qualities of occupational performance” (p. 8). Fawcett (2009) provided extensive analyses on outcome measurement and models of function. What the occupational therapist is intending to improve is the person’s ability to reach his or her goals (i.e., the person’s response at that time).

We intend for the SPRE model, which requires an outcome measure over time (or treatment number), to be used with an occupational therapy measure to obtain relevant occupational therapy outcomes. These broad concepts are also relevant to the behavioral therapy for weight loss we analyze later in this article using the SPRE model.

An important objective of this article is to provide the statistical analysis methods and an example of results used in the SPRE model, in which the total person response is given as a single outcome that can then be predicted for future outcomes over time. Through the use of mathematics and statistics, Deborah Weissman-Miller has developed a new, innovative, and powerful tool through which occupational therapists can now accurately predict outcomes of therapy for a single participant or a small group of participants. This predictive ability was originally developed in continuum mechanics for materials in which the mechanics model is comparable to blood, bone, and structures in the human body and serves as a basic model for human response to treatment of disease (Miller, Weissman-Berman, & Martin, 2008; Weissman-Berman & Classen, 2005; Weissman-Miller, 2010). In this model, a change point is derived to statistically predict where the participant turns the corner in treatment, from which a ratio in the SPRE model predicts further outcomes.

In many cases, statistical estimations can be made from initial data from a least squares regression, by predicting a trend line or, in time series, forecasting when the use of a model to forecast future events is based on known past events from a model used in the past. These models cannot estimate a change point in the participant’s response to occupational therapy treatment from the beginning of current treatment, nor can they predict outcomes in the form of an exponential curve function (Cleves, Gould, & Gutierrez, 2004) and for survival (time-to-event data; Hosmer, Lemeshow, & May, 2008) or a parametric cumulative distribution (a Weibull function similar to an extended exponential model as given in Ross, 2007). Therefore, this new method has been developed to predict long-term outcomes using initial data from the client or small group of clients from the beginning of current treatment and has great potential to influence research in occupational therapy by enabling the use of initial client data to determine outcomes and prediction of intervention effects.

Method

In quantitative measures, patients’ health response to treatment of disease often has an initial, effectively linear response as the body responds to treatment. The purpose, procedure, and outcomes for this method are outlined in Table 1, with the data collection given as an example for weight loss.

Table 1.

Summary of Semiparametric Ratio Estimator (SPRE) Method

Purpose	Procedure	Outcome
Data collection	Weight-loss data	Data observed.
Data graphing	Graph raw data.	Graph progress through therapy.
Find the change point.	Use backward stepwise regression method.	Determine an F statistic and R ².
Highest and lowest F statistic for R ²	Tabulate F statistic for R ² for each dataset.	Identify highest or lowest F statistic (and R ²).
Predict outcomes after change point.	Analyze Weibull distribution ratio.	Calculation yields the SPRE ratio.
Calculate predictive outcomes.	Multiply new ratio by prior outcome.	Calculation yields predicted outcome.

Procedural Outline for Estimation of Occupational Therapy Data

Before using SPRE with occupational therapy clinical data, three conditions have to be met:

From the clinical data, make sure the data are approximately linear and the raw data are statistically ordered—that is, the numbers follow in sequence necessary for the occupational therapy outcomes, and that the time or session numbers are ordered together with the outcomes. Data should be at least an ordinal level of measurement. In occupational therapy, when the initial response is not linear, statistical transform procedures may be used to linearize the data as given in Berk (2004), provided more extensively by Weisberg (2005), and applied to occupational therapy by Stewart, Chisvo, Hutcherson, and Smith (2011).

Use least squares linear regression. Calculate the highest or lowest F statistic, which is computed by the analysis of variance (ANOVA; Kraus & Olson, 2005) to predict the change point in the SPRE model. This F statistic defines a statistical change point which is the data point at which the character of the regression changes from linear to a curve, resembling an extended exponential curve which can estimate the “belly” of the curve from the slope of the initial data to accurately predict future outcome points. The outcome has internal validity if the residuals are normal (i.e., the errors are random) at the change point, which indicates that the outcome at that point is a population parameter (Miller et al., 2008; Weissman-Miller, 2010). In this article, the population parameter derived from the least squares estimates at the change point is nearly (asymptotically) normal and centered around the true parametric values if the error terms are normally distributed. The normality of the residuals can be checked by graphing a residuals plot in R Commander (Fox, 2005; R Development Core Team, 2010).

Use SPRE to predict future outcomes. Outcomes are predicted from the change point in the linear data, which then have both internal validity and external validity, where internal validity is given by the least squares method, resulting in an unbiased estimate, and external validity is given when the statistical population parameter θ at the change point indicates outcome parameters over time for that single participant or a population mean for a small group. For a small group, if p ≤ .05 for the value of F at the change point, then inferences may be made to similar groups in the population. The predictions have statistical validity for that subject (Weissman-Miller, 2010).

First Step: Determining the Most or Least Significant F Statistic

The F statistic is computed from a backward stepwise elimination from results given by least squares regression analysis from a linear regression equation (Berk, 2004):

where β ₁ is the slope and e_i is the error term. This method is used to determine the F statistics for each dataset, starting, for the weight loss example, at an assumed full model at 10 mo and working backward, usually two time or session numbers (given here as months), as summarized in the third step in Table 1.

Second Step: Evaluating the Change Point

The F statistics can be evaluated using R Commander for the statistical language R displayed in Excel and are given in R through Excel under linear models (Fox, 2005; Heiberger & Neuwirth, 2009; R Development Core Team, 2010). An evaluation of the data for the weight loss application of SPRE for the highest or lowest F statistic shows that two measured high values of F statistics exist for the application of SPRE to weight loss. In occupational therapy, the treatment should be carried out past a first mode (or highest F statistic) to be sure that all relevant modes such as a second mode of response (or second highest F statistic) have been recorded. For this reason, data collection is usually carried out for 14–20 time values or session numbers, which, in practice, captures the highest and, if relevant, a second highest F statistic. The terms first mode and second mode refer to the relative high values of the participant’s response measured by the derived F statistic (Table 2). In general, most SSDs analyzed using SPRE have only one highest F statistic.

Table 2.

Determination of Highest or Lowest F Statistic for Behavior Therapy

No. of Initial Data	Time, Mo	Weight Loss, Kg	R ²	F Statistic
1	0	0	—	—
2	0.25	0.5	—	—
3	0.5	1.0	—	—
4	1.0	2.5	—	—
5	1.5	3.5	—	—
6	1.75	4.5	—	—
7	2.0	6.0	.9754	198.10
8	2.5	6.5	.9818	323.49
9	3.0 a τ	7.5	.9841	432.76^b
10	3.5	8.0	.9774	346.11
11	4.0^c	9.0	.9769	380.33^d
12	4.5	9.5	—	—
13	5.0	10.0	.9668	320.20
14	5.5	10.5	—	—
15	6.0	11.0	.957	289.28
16	6.5	10.8	—	—
17	7.0	10.75	.924	182.73
18	7.5	10.6	—	—
19	8.0	10.5	.875	118.97
20	9.0	10.4	—	—
21	10.0	10.25	.7844	69.11

Note. Relative high values of the participant’s response measured by the derived F statistic are shown in bold. — = R ² and F statistic not calculated.

Value of τ for highest F statistic. ^bResidual, not normal. ^cStart prediction. ^dNormal.

Third Step: Estimation Function

The third step is to estimate health outcomes forward in time from the change point. This estimation is done by inputting values of time into the SPRE ratio of Weibull distributions. The Weibull distribution, really an extended exponential function, represents a mechanics model that includes a spring representing the structure of the human body and a damper that represents bodily fluids as given originally in continuum mechanics to predict outcomes over time (Flugge, 1967; Weissman-Berman, 1992). The Weibull distribution can now predict the belly of the prediction curve (Weissman-Miller, 2010), which tells the client, occupational therapy administrator, and third-party payers how many treatments are needed over how much time:

In the SPRE model to determine health outcomes, the Weibull distribution relates to both variables: outcomes and time, given as θ and t. Those two values are derived from the initial data at the change point. For example, the change point may be indicated by a point in time or a session number, given on the time axis corresponding to weight loss. One of the two values derived from the change point is the shape parameter labeled k in Equation 2. This shape function determines the belly of the predicted outcomes curve. The shape parameter is determined by the slope of the least squares regression and is solved by k = ln |(1 − · w _τ)|, given is the estimate of the slope in a formal regression model also given as the absolute value, and w _τ is the displacement of the change point on the time axis (or axis of treatment numbers). The resulting value of k is given as the absolute value and represents the belly of the curve. The slope can be determined from the statistics program R or by a command in Excel. An example is the value of k for a change point at 4 mo as given later in this article.

Final Frontier: Estimating Long-Term Outcomes

To predict long-term outcomes of an intervention, the Weibull distribution in Equation 2 is implemented as the outcome SPRE ratio, given in Equation 3, which enables the clinician–researcher to predict outcomes over time until a predetermined cutoff point (e.g., a score on a specific measure) or the Weibull ratio equals 1, indicating that the client’s performance is stable and may be given with statistical validity.

The point estimation of weight loss may be given as

(3)

Here, is the estimated outcome, R is a ratio of Weibull distributions, and θ ₁ is the prior outcome. The ratio estimator, R, is expanded using Equation 2 as a ratio times the prior outcome value θ ₁ as shown in Equation 3 as given in Miller et al. (2008), Weissman-Berman (2009), Weissman-Miller (2010), and Weissman-Miller and Miller (2011).

At a time greater than τ, here the time or session number at the change point, t_i _{+ 1} is the increased value of time for increasing outcomes and t_i is the prior time:

An example follows, illustrated by weight loss data tabulated and presented at the end of this article. The change point for weight loss is given as 4 mo (see Table 1); then, to predict for the next time step (e.g., 4.5 mo), compute a ratio as in Equation 4. The numerator value t_i _{+ 1}= 4.5, τ = 3, and k = 2.113. The denominator value t_i = 4, τ = 3, the value for the highest F statistic, and k = 2.113.

The prior outcome—in this case, at = 9 kg; the new outcome at t_i ₊₁ = 4.5 mo is 9.69 kg lost. This equation steps the outcome forward in time, enabling the clinician–researcher to predict when there would be no further change in the outcome measure.

The point estimates (predictions) indicate sequential estimates of the parameter for that participant. This estimator can be calculated in Maple 13 (Maplesoft, Waterloo, Ontario), a symbolic mathematics program, the statistical language R (R Development Core Team, 2010), or by any scientific hand calculator. The estimations for weight loss are in Table 3 and Figure 1.

Table 3.

Point Estimations for Behavior Therapy

No. of Estimation	Time, Mo	τ at Change Point	R Ratio (Weibull Ratio)	Outcomes Over Time
1	3 a (τ)	3	—	7.5^b
2	4^c	3	1	9^d
3	4.5	3	1.0768	9.69
4	5	3	1.0462	10.14
5	5.5	3	1.0263	10.41
6	6	3	1.0141	10.56
7	7	3	1.0103	10.67
8	8	3	1.0020	10.69
9	9	3	1.0003	10.69
10	10	3	1	10.69

Value of τ for highest F statistic. ^bMode 1. ^cStart prediction. ^dMode 2.

Figure 1.

Point estimates correlated to patients’ outcomes in months.

Unbiased Semiparametric Ratio Estimator Properties

In this analysis, the value of R at the change point and when R = 1.00 is statistically unbiased. Outcomes estimated by this ratio estimator, when the ratio = 1.00, are also shown to be asymptotically consistent and robust as given by Weissman-Miller (2010). The statistical bias ratio of this single ratio estimator will typically be small in practice, as given in Meng (1993), whether the estimator is for a clinical trial or the long-term estimated outcomes for a single participant. Additionally, the values of the predicted outcome statistic, on the real axis, estimate the population parameter θ (Freund & Walpole, 1987).

This use of internally and externally validated statistics indicates that predictions made from a single participant are germane to that specific individual. A further condition in this analysis is that at the change point has random residuals. This is a sufficient condition when the change point occurs at low numbers of time or session number. This personalized form of analysis is more valid, accurate, and substantive than using large group means that assume homogeneity, normal distribution, and large sample sizes and that contradict the client-centered nature of occupational therapy.

Internal and External Validity of Regression Data

At the change point τ, the least squares regression mean μ is the normal distribution parameter for the response, provided that the errors are randomly distributed for the outcomes, given as the population parameter θ. What this means is that the regression data for a group have statistical inference and that the estimations derived from the change point will also have statistical generalizability to a wider population, similar to regression analysis. The only requirement for this analysis is that the participant’s initial data be nearly linear so that a highest or lowest F statistic can be derived and that 14–20 data points be taken for the analysis. Therefore, the number of participants in a study is irrelevant.

Application of SPRE

A secondary analysis of data from a weight loss study using behavior therapy demonstrates one application of SPRE as it could be used in clinical occupational therapy practice and in research. In this application, the efficacy of weight loss using controlled diet and behavior therapy can be generalized depending on random error functions at the change point, where residuals fit a normal distribution, rather than the construction of a statistical mean.

In quantitative measures, patients’ health response to treatment of disease often has an initial effectively linear response as the body responds to treatment, as shown in treatment of obesity (Aronne, 2004; Brownell & Wadden, 1998; Wadden, Stunkard, Brownell, & Day, 1984), where weight loss from behavior therapy is modeled in this article from the primary dataset and expanded in Smoller, Wadden, and Brownell (1988) to observed outcomes using this model based initially on data gathered from a single patient or a small group. The behavior therapy “included monitoring eating behavior (amount, time, etc.) and increasing physical activity by walking or stationary bicycling” (Wadden et al., 1984, p. 692), which are aspects often included in occupational therapy treatment for obesity (AOTA, 2007; Mosley, Jedlicka, LeQuieu, & Taylor, 2008; Clark, Reingold, & Salles-Jordan, 2007).

There were 17 female participants in the original dataset, and 16 completed the treatment. All were overweight or obese. Participants received individual behavior therapy and were treated in small groups of 4–6. In the primary dataset, weight was averaged at specific 2-mo time intervals from 0 to 18 mo.

We should note that the SPRE model can be used for a wide variety of occupational therapy data, for example, changes in treatment as measured on the Comprehensive Occupational Therapy Evaluation scale (Brayman, Kirby, Misenheimer, & Short, 1976), even when the treatment is varied from session to session (Stewart et al., 2011) to improve the participant’s response. Variations in treatment within a class of treatments are often used in occupational therapy to further enable the client to achieve his or her goals.

Sessions and treatments in obesity are similar to frequency and/or duration in many occupational therapy settings. The results of least squares regression to determine a change point are given in Table 2. The change point is determined by the transition from the lower to the highest value of the F statistic ~~of F~~ and again to a lower value of the F statistic. Predictions can be made from the identified change point if the residuals at that value are normal. For patients’ outcomes having two modes of response, one should predict from the mode in which the data have normally distributed residuals (where errors are random), as shown in Table 2. Randomness is indicated by values that are scattered and have no defined shape and can be determined by graphical functions in R Commander (Fox, 2005). At 3 mo, the F statistic is the highest value so that the value for τ is given at τ = 3. However, further analysis shows that the residuals are not normally distributed. Therefore, the predictions are not made from this value. The F statistic at 4 mo is the second highest value in Table 2, and the residuals are normally distributed, indicating that this value is the change point from which predictions are made.

The estimations for behavior therapy are in Table 3 and Figure 1. In this obesity example, k = 1.939 at 3 mo (first mode) and 2.133 at 4 mo (second-mode weight loss resulting from behavior therapy). Here, k = ln | (1 −2.360 · 4) | = 2.133. The data region for the predicted outcomes is given in Figure 1 as solid triangles, where predictions are made from 4 to 18 mo, based on the initial data from the first 4 mo of treatment.

Conclusion

The estimations in this article are based on the calculation of a change point from a least squares regression of initial data. The point estimates predict events over time and are based on the values of previous events. For this reason, this particular statistical technique is especially useful in predicting weight loss in obesity and weight control programs or any other behaviors that are initially linear or can be transformed to be linear (e.g., numerical scores on a particular measure such as the Comprehensive Occupational Therapy Evaluation Scale or other criterion-reference measures used in occupational therapy). By using initial data and evaluating from a least squares regression, correlation of the outcome to time can easily be determined for a single subject or a small group. For a single subject, the estimated outcomes are statistically valid for that participant. For a small group, and particularly one with treatment and control participants, the resulting point estimates from each group are generalizable to a population. Finally, the SPRE model is a useful and accurate method for an occupational therapy measure that uses a numerical scale (including a Likert scale).

Implications for Occupational Therapy Practice

This new statistical methodology provides several new benefits to occupational therapy practice that continue the ethos of being client centered while becoming more science driven, evidence based, and reimbursed for services.

Occupational therapists can now provide additional motivation for clients by showing the amount of improvement they can make in the long term.

Occupational therapists can now reliably justify continued funding for therapy from third-party payers.

Clients and family members can be motivated by demonstrating the number of visits until stability is achieved.

Occupational therapists can also now provide statistically valid evidence for the efficacy of occupational therapy intervention and create evidence needed to validate occupational therapy interventions while accurately showing and predicting change over time.

Future Work

The accuracy of the predictions from the SPRE model has recently been enhanced by a novel maximum spacing for point estimates, which derives accurate spacing for the intervals on the time axis, as given by Weissman-Miller (2011). Further studies using this maximum spacing function with SPRE should be conducted using occupational therapy measures and outcomes.

We also suggest that computer simulation studies be performed to further validate this new statistics, that is, to see how precisely this statistical procedure will estimate the population parameter.

Footnotes

Acknowledgments

Our thanks to our colleagues and students at Brenau University and to all our friends in occupational therapy who have supported this work since 2005. Thanks to the American Occupational Therapy Association for the presentation of this model as a poster by Weissman-Berman and Classen in 2005; as a research paper presentation by Miller, Weissman-Berman, and Martin in 2008; and as a research paper presentation by . Detailed information on the SPRE is available from Mary P. Shotwell at Brenau University, mshotwell@brenau.edu.

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