* This article is based on an actual analysis project; however, in light of the confidentiality agreement (NDA) with the client, while maintaining the framework of the medical and healthcare field, specific details such as disease names and variables have been substantially altered from the actual case. We ask for your understanding in advance.

This case evaluated how the intensity of multifaceted behavioral interventions (exercise guidance, nutritional guidance, thorough self-management recording, etc.) for a group of patients with a lifestyle-related disease (such as type 2 diabetes) affects a clinical evaluation indicator (the degree of improvement in laboratory values such as HbA1c). In actual clinical practice, increasing the number of a particular treatment or guidance sessions does not raise the treatment effect indefinitely. There is a “dose-response relationship” in which the effect is weak until a certain amount of intervention is reached, the effect appears sharply once a certain threshold is exceeded, and ultimately a physiological limit (a plateau) is reached.

Through this analysis, rather than assuming a simple linear relationship, I aimed to apply a “sigmoid function” that can mathematically express the period of sharp increase in effect and the point of the plateau, and to extract the true contribution of each factor under conditions where multiple intervention elements are complexly intertwined. By building a multivariate sigmoid model using non-linear least squares, Dr.DataScience provided advanced objective evidence directly linked to the allocation of resources in the clinical setting—namely, which intervention should be increased and to what extent to obtain the treatment effect most efficiently.

Background and objective

In this case, what was required was to quantitatively clarify, among the various behavioral-intervention programs provided to patients, “which guidance item should be preferentially strengthened to most efficiently expect improvement in laboratory values.”

Conventional evaluation tended to assume a simple proportional (linear) relationship along the lines of “the more guidance sessions, the more improvement.” However, the actual response of patients follows an S-shaped trajectory: a “rising period” in which a few sessions do not lead to behavior change and produce no effect, a “period of sharp increase” in which guidance takes hold and the figures improve dramatically, and a “limit period (plateau)” in which the margin of improvement shrinks no matter how much more guidance is added.

The main objective of the analysis was to use a sigmoid function expressing this S-shaped trajectory to score the influence of each factor when considering not only a single intervention element but multiple intervention elements simultaneously, and thereby to scientifically derive the optimal combination of interventions and the target number of sessions.

Data and variables

This analysis used per-patient intervention records and periodic test data carried out at multiple medical facilities. The main variables used in the analysis were as follows.

1. • Response variable (clinical evaluation indicator): the improvement score of a particular blood test value (such as HbA1c) after a certain period (continuous variable).
   • Upper-bound parameter (K): the upper limit to which the sigmoid function ultimately converges. In this analysis, the mean value of the top 5% of patients who showed the best improvement for each evaluation indicator was set as the physiological limit of improvement (K).
   • Explanatory variables (multifaceted intervention indicators): more than ten types of intervention-intensity indicators within a certain period, such as “the number of face-to-face nutritional-guidance sessions,” “the number of exercise-guidance sessions by a physical therapist,” “the number of phone calls confirming living conditions,” “the number of submissions of self-management record sheets,” and “the number of times health information was viewed through digital devices.”

Analytical methods

In this case, to accurately capture the plateau phenomenon of the treatment effect and to determine the priority of multiple intervention elements, I selected and applied the following statistical methods.

1. Modeling the non-linear dose-response relationship
   • Adopted method: multivariate sigmoid analysis using non-linear least squares
     I adopted non-linear regression analysis that fits the influence of each intervention element on the response variable to an S-shaped curve (a sigmoid function) with an upper bound (K). Specifically, using an optimization algorithm called the Levenberg–Marquardt method, I estimated the coefficient of each variable that minimizes the error between predicted and observed values. This made it possible to quantify “which variable makes the rise of the S-curve how steep.”
     
     (Figure: Example of a sigmoid curve)
   • Method not adopted: ordinary multiple regression analysis (linear model)
     Ordinary multiple regression assumes as a fundamental premise that the variables have a linear relationship. However, fitting a straight line to the data in this case carries an extremely high risk of predicting an “impossible degree of improvement exceeding the physiological limit” in patients with an extremely large number of intervention sessions, or of overestimating the effect in regions with little intervention; I therefore clearly judged it inappropriate and did not adopt it.
2. Optimization of explanatory variables and model selection
   • Adopted method: elimination of multicollinearity using VIF (variance inflation factor) and variable selection based on AIC (Akaike Information Criterion)
     Among the many intervention indicators were some with extremely strong mutual correlation (e.g., “the number of phone confirmations” and “the number of face-to-face guidance sessions” always moving together). Entering these into the model simultaneously makes the coefficient estimates unstable, so I evaluated multicollinearity using VIF and excluded redundant variables in advance. I then created all model patterns for the combinations of the remaining variables (from 2 variables up to a maximum of 11 variables) and identified the optimal combination of variables that minimized AIC—an objective index that evaluates the balance between predictive accuracy and model complexity—that is, the best fit.

Overview of the main results and clinical considerations

As a result of the analysis using the optimal multivariate sigmoid model, it was confirmed that, among the multiple intervention elements, “the number of face-to-face nutritional-guidance sessions” and “the number of submissions of self-management record sheets” had an independent and strong influence on the improvement of laboratory values (significance level p < 0.050).

Examining in detail the shape of the constructed sigmoid curve (a distribution evaluation using the probability density function) revealed an extremely practical threshold, both visually and numerically: “the number of face-to-face nutritional-guidance sessions” begins to rapidly increase its improvement effect once it exceeds a particular number (for example, 3 times a month), and once it exceeds 5 times the effect reaches a plateau and levels off.

On the other hand, “the number of times health information was viewed through digital devices,” which appeared to have an extremely strong effect in univariate analysis (when evaluated with only one intervention element), had its coefficient substantially shrink when incorporated into the multivariate model. This shows that there was confounding (the influence of another underlying factor): patients who frequently view information also tend to have a high “number of submissions of self-management record sheets.” By performing multivariate sigmoid analysis, I was able to narrow down the intervention elements that should truly be strengthened, without being fooled by apparent effects.

Dr.DataScience’s contribution

In this case, Dr.DataScience fused medical background knowledge with advanced non-linear modeling techniques to create concrete guidance for making the most of limited medical resources.

1. Building a mathematical model that matches clinical reality
   • By incorporating the physiological principle that “treatment effect always has a limit” into the model as a formula (the upper-bound parameter K of the sigmoid function), I eliminated unrealistic predictions. By avoiding the careless application of a linear model and appropriately implementing the high-difficulty technique of non-linear regression analysis, I dramatically enhanced the reliability of the results.
2. Scoring complex influences and visualizing priorities
   • I standardized the coefficient of each intervention element calculated by the multivariate model into an “influence score” that healthcare workers in the field can intuitively understand. This made it easy to formulate practical action plans—namely, “which guidance item’s number of sessions should we focus on increasing first.”
3. Providing a foundation for predictive simulation
   • As the final deliverable, I provided the mathematical parameters of the optimal model based on AIC. This enabled the client to perform evidence-based simulations of treatment plans—such as “for this patient, if we add 2 more sessions of guidance A, by how much will the improvement score rise”—directly contributing to improving the quality of care.