* 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 in detail how the “cumulative dose of a steroidal immunosuppressant over a long period” for a group of patients with a particular chronic autoimmune disease affects the “rate of decline in bone density (the degree of progression of osteoporosis)” as a side effect, and estimated the clinically acceptable safe limit of the dose.

In long-term observational studies in actual clinical practice, it is unavoidable that much missingness arises in the data due to patient transfers, tests not being performed, and so on. In addition, patients’ background factors (age, lifestyle, genetic factors, comorbidities, etc.) range widely from numerical data to categorical data, and these are complexly intertwined in the manifestation of side effects.

Through this analysis, I aimed to apply an advanced missing-value imputation technique (multiple imputation) to prevent bias from simply deleting data, and to perform a rigorous correlation analysis matched to the scale of the variables, thereby extracting the true factors that predict side effects. By combining multiple regression analysis based on the imputed data with calculation of the intersection of regression lines accounting for error (estimation of the threshold), Dr.DataScience provided objective evidence for a concrete safety reference value that field physicians need most—”at what cumulative dose does the fracture risk reach a clinically dangerous zone”—and contributed to formulating a safer long-term treatment plan.

Background and objective

In this case, in a patient group requiring long-term treatment, there was a strong demand to “establish a clear limit standard for the cumulative dose in order to prevent serious side effects (a marked decline in bone density) while maintaining the treatment effect.”

A major challenge in the clinical setting was that, although it was empirically known that bone density declines as the steroid dose increases, it had not been quantitatively clarified “at what specific number of milligrams the danger level requiring urgent fracture-prevention intervention (for example, a 5% rate of bone-density decline) is reached.” In addition, the collected long-term patient-record data contained many missing values in some blood-test results and interview items, raising concerns that simple analysis would produce serious distortions in the results.

The main objective of the analysis was to comprehensively grasp the relationships among patients’ wide variety of background factors after performing rigorous statistical handling of the missing values, and to build a high-precision predictive model that adjusts for their confounding. Furthermore, the ultimate goal was to use the regression line and confidence intervals obtained from that model to derive the concrete “cumulative-dose figure” at which the side effect exceeds the acceptable range, and its safety margin (lower limit).

Data and variables

This analysis used patient-record data of several hundred patients followed long-term at multiple specialized medical institutions. The collected variables were a complex mix of numerical and categorical data; the main variables analyzed were as follows.

1. • Response variable (side-effect indicator): the “rate of bone-density decline (%)” before and after the observation period (continuous variable).
   • Main explanatory variable (intervention indicator): the “cumulative dose (mg)” of the steroid within the observation period (continuous variable).
   • Patient background factors (continuous variables): age, blood vitamin D concentration, BMI, recent exercise-habit score, etc.
   • Patient background factors (nominal/ordinal variables): sex, presence of a particular underlying disease, climate zone of the area of residence, presence of a history of fracture, etc.
   • State of the data: about 15% of the continuous variables and about 10% of the categorical variables contained missing values (unmeasured/unanswered).

Analytical methods

In this case, to prevent serious information loss and bias in the estimates due to missing data, and to accurately capture the relationships among variables of differing scales, I selected and applied the following advanced statistical methods in stages.

1. Appropriate handling of missing values (application of multiple imputation)
   • Adopted method: multiple imputation and the CART algorithm
     To impute the many missing values present in the dataset, I adopted multiple imputation. This method uses the information of all other variables to calculate predicted values for the missing parts, and creates multiple different datasets (50 in this case) that account for uncertainty. In particular, for imputing categorical variables (such as the presence of a particular underlying disease), I adopted the CART (Classification and Regression Trees) algorithm, which can flexibly handle non-linear relationships and complex conditional branching, achieving imputation extremely close to the reality of actual clinical practice.
   • Methods not adopted: listwise deletion (complete-case analysis) and mean imputation
     “Listwise deletion,” which removes all data of any patient containing even a single missing value, not only substantially reduces the precious sample size, but also discards entirely the information of patients who, for example, could no longer come for tests because their symptoms worsened—creating serious selection bias in the results; I therefore strictly excluded it. Also, “mean imputation,” which simply fills missing parts with the overall mean, makes the data’s variability (variance) appear unduly small and carries a high risk of leading to erroneous significant differences for variables that are not truly significant, so I did not adopt it.
2. Comprehensive correlation analysis according to the scale of the variables
   • Adopted method: the use of Pearson, Spearman, Cramér’s V, and the correlation ratio
     To identify the confounding factors lurking in the patient backgrounds, I rigorously used different correlation-evaluation methods according to the variables’ data type (scale). I adopted “Pearson correlation analysis” or “Spearman’s rank correlation coefficient” between continuous variables, “Cramér’s V” for the association between nominal variables, and the “correlation ratio (η),” which shows the proportion of between-group variance, for the relationship between continuous and nominal variables. By calculating these comprehensively in a matrix, I identified and excluded in advance the variable pairs with strong correlation (such as R > 0.8) that cause multicollinearity.
   • Method not adopted: uniform application of the Pearson correlation coefficient to all variables
     Forcibly assigning numerical values to categorical data such as sex or the presence of a disease and calculating the Pearson correlation coefficient is statistically meaningless and a cause of erroneous interpretation, so I did not do it at all.
3. Multiple regression analysis based on the multiply-imputed data (pooling)
   • Adopted method: multiple regression analysis integrating the 50 datasets (integration based on Rubin’s rules)
     I built a multiple regression model with the rate of bone-density decline as the response variable and the cumulative dose and the carefully selected background factors as the explanatory variables. Here, rather than analyzing a single dataset, I ran the regression analysis on each of the 50 datasets created by multiple imputation, and statistically integrated (pooled) the results of those 50 runs (coefficients and standard errors) according to Rubin’s rules. This calculated extremely robust and reliable coefficients and p-values (such as p < 0.001) that correctly incorporate the uncertainty of the missing values.
4. Precise estimation of the clinical limit value (threshold)
   • Adopted method: inverse estimation using the regression line and confidence intervals (calculating x for a particular y)
     For the relationship between the cumulative dose (x-axis) and the rate of bone-density decline (y-axis), I created a scatter plot visualizing the per-patient variability (standard error) as error bars. On that, I mathematically calculated the x-coordinate of the intersection of the estimated regression line with a horizontal line indicating the target level of the clinical danger threshold, “a 5% rate of bone-density decline (y = 5.0).” Furthermore, by simultaneously calculating the intersections with the upper and lower lines of the regression line’s 95% confidence interval, I also derived a conservative limit value: “evaluated on the safest side, at how many mg should we consider the limit to be reached.”

Overview of the main results and clinical considerations

As a result of the robust multiple regression analysis that imputed the information loss using multiple imputation, it was proven that the “cumulative steroid dose” had an independent and extremely strong positive influence on the rate of bone-density decline, even after adjusting for other background factors such as the patient’s age and blood vitamin D concentration (integrated p = 0.002).

Furthermore, in the clinically most important “estimation of the threshold,” the intersection calculation from the simple regression showed, as a clear figure, that the average estimated cumulative dose at which the rate of bone-density decline reaches the danger level (5%) was “about 8,500 mg.”

Notably, the evaluation of the lower-limit intersection using the 95% confidence interval derived a stricter reference value: “evaluated on the safest side, accounting for patient variability, the danger level may be exceeded once the cumulative dose reaches about 6,800 mg.” This result provides a clear trigger (action criterion) for proactive preventive intervention in the clinical setting: “once the dose approaches 7,000 mg, the concomitant use of potent osteoporosis drugs should be strongly considered.”

In addition, the comprehensive prior correlation analysis confirmed that the higher a patient’s “recent exercise-habit score,” the more gradual the decline in bone density even with a large cumulative dose (a negative correlation), quantitatively suggesting that not only drug therapy but also parallel early exercise guidance is extremely effective in mitigating the side effect.

Dr.DataScience’s contribution

In this case, Dr.DataScience applied advanced statistical processing to make the most accurate use of incomplete real clinical data, elevating the medical field’s judgment criteria—which had relied on intuition—into safety standards based on clear formulas and visual evidence.

1. Providing cutting-edge missing-value handling that prevents information loss
   • Without carelessly discarding missing data, by applying multiple imputation using MICE and CART, I made 100% use of the information in the precious patient records. This built a robust analytical foundation that is unbiased, objective, and truly reflective of reality.
2. Eliminating multicollinearity through rigorous correlation evaluation according to scale
   • By eliminating haphazard correlation analysis that ignores data type and accurately using four different correlation indices (Pearson, Spearman, Cramér’s V, correlation ratio), I precisely mapped the complex relationships among variables. This preemptively prevented the collapse of the regression model due to entering the wrong variables.
3. Calculating a “numerical standard” directly linked to clinical decision-making
   • Rather than stopping at presenting a vague result that “more drug means more side effects,” by performing the mathematical intersection calculation of the regression line and confidence intervals, I calculated the clear figure the field wants most—”specifically, how many mg is the limit of safety.” By also presenting a “safe-side limit value (lower-limit intersection)” that accounts for variability, I directly contributed to formulating safer patient-management procedures (guidelines).