Point estimation is particularly important in predicting weight loss in individuals or small groups. In this analysis, a new health response function is based on a model of human response over time to estimate long-term health outcomes from a change point in short-term linear regression. This important estimation capability is addressed for small groups and single-subject designs in pilot studies for clinical trials, medical and therapeutic clinical practice. These estimations are based on a change point given by parameters derived from short-term participant data in ordinary least squares (OLS) regression. The development of the change point in initial OLS data and the point estimations are given in a new semiparametric ratio estimator (SPRE) model. The new response function is taken as a ratio of two-parameter Weibull distributions times a prior outcome value that steps estimated outcomes forward in time, where the shape and scale parameters are estimated at the change point. The Weibull distributions used in this ratio are derived from a Kelvin model in mechanics taken here to represent human beings. A distinct feature of the SPRE model in this article is that initial treatment response for a small group or a single subject is reflected in long-term response to treatment. This model is applied to weight loss in obesity in a secondary analysis of data from a classic weight loss study, which has been selected due to the dramatic increase in obesity in the United States over the past 20 years. A very small relative error of estimated to test data is shown for obesity treatment with the weight loss medication phentermine or placebo for the test dataset. An application of SPRE in clinical medicine or occupational therapy is to estimate long-term weight loss for a single subject or a small group near the beginning of treatment.

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. Weissman-Miller, D., Shotwell, M. P., & Miller, R. J. (2012). New single-subject and small-n design in occupational therapy: Application to weight loss in obesity. American Journal of Occupational Therapy, 66, 455-462. http://dx.

<p>Prostate cancer is a condition of public health significance in the United States. A new method for predicting survival is derived for the domain around the change point from a semiparametric ratio estimator (SPRE) to predict survival in response to treatment for prostate cancer. Using an extended maximum spacing estimator, the geometric mean of sample spacings from a uniform distribution <span style="font-family: Times New Roman; font-size: medium;"> is derived </span>with known endpoints given at 0 and at the value of the change point from an ordinary least squares (OLS) regression for SPRE. To determine the maximum interval on the ‘x’ axis between point estimates, the maximum spacing estimation method is derived from a continuous univariate distribution where spacing will be defined as gaps between ordered values of the distribution function. The maximum is defined as a single value in the neighborhood of the change point and spacing defined as a function of time. This maximum spacing defines the gaps between point estimates at each time-dependent predicted outcome from the change point and results in a semiparametric ratio estimator that is reliable and repeatable. Performance is discussed through a simulation of change point values for a real application in clinical medicine and, using SPRE, in personalized medicine for a single prostate cancer patient.</p>

Fear of falling (fof) and falls are increasingly severe worldwide public health problems. The Falls Weight Function (FWF) uses a new scale to incorporate fear of falling (fof) into analyses with participants who have already experienced a fall. FWF is a weight function in the semiparametric ratio estimator (SPRE) to predict a change point and point estimations. FWF data is a discrete set of numbers that is finite or countable. In a study using Stepping On R ⃝ fall prevention program, initial data from fof responses were counted, summed as increments of .10 values (ranging from .1 to 1.0), and then multiplied by 1 fall. The un-weighted value of 1 fall was multiplied by the weight function for fof. Then, the scale of falls and fof is the same, and represented on a continuum from fear of falling to having a fall, so that all participants can be treated and analyzed together.

Translational research is redefined in this paper using a combination of methods in statistics and data science to enhance the understanding of outcomes and practice in occupational therapy. These new methods are applied, using larger data and smaller single-subject data, to a study in hippotherapy for children with developmental disabilities (DD). The Centers for Disease Control and Prevention estimates DD affects nearly 10 million children, aged 2–19, where diagnoses may be comorbid. Hippotherapy is defined here as a treatment strategy in occupational therapy using equine movement to achieve functional outcomes. Semiparametric ratio estimator (SPRE), a single-subject statistical and small data science model, is used to derive a “change point” indicating where the participant adapts to treatment, from which predictions are made. Data analyzed here is from an institutional review board approved pilot study using the Hippotherapy Evaluation and Assessment Tool measure, where outcomes are given separately for each of four measured domains and the total scores of each participant. Analysis with SPRE, using statistical methods to predict a “change point” and data science graphical interpretations of data, shows the translational comparisons between results from larger mean values and the very different results from smaller values for each HEAT domain in terms of relationships and statistical probabilities.