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.

*Centennial Vision*for the profession of occupational therapy to be science driven and evidence based (American Occupational Therapy Association [AOTA], 2005).

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. |

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. |

- 1.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) .
- 2.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). - 3.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).

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

*β*

_{1}is the slope and

*e*is the error term. This method is used to determine the

_{i}*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.

*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.

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.

*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.×

*τ*for highest

*F*statistic.

^{b}Residual, not normal.

^{c}Start prediction.

^{d}Normal.

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.

*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.×

*τ*for highest

*F*statistic.

^{b}Residual, not normal.

^{c}Start prediction.

^{d}Normal.

*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.

*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) .

*t*

_{i}_{+ 1}is the increased value of time for increasing outcomes and

*t*is the prior time:

_{i}*t*

_{i}_{+ 1}= 4.5,

*τ*= 3, and

*k*= 2.113. The denominator value

*t*= 4,

_{i}*τ*= 3, the value for the highest

*F*statistic, and

*k*= 2.113.

*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.

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 |

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 |

*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).

*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.

*F*statistic

*F*

*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.

*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.

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.

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