The dataset was obtained from the National Health and Nutrition Examination Survey (NHANES), a cross-sectional research program conducted by the Centers for Disease Control and Prevention (CDC) since 1999. NHANES has systematically monitored the health and nutritional status of the U.S. population through standardized questionnaires, physical examinations, and laboratory tests since the 1960s. For model development, we analyzed data from two consecutive NHANES survey cycles (2011–2014) that implemented Physical Activity Monitor (PAM) examinations, comprising a total of 19,931 participants with complete activity monitoring records. We focused on adults aged 60 years or older and applied the following exclusion criteria: participants younger than 60 years (n = 16,299), those with insufficient accelerometer data (n = 863), and individuals missing key covariates including household income, marital status, BMI, smoking status, and alcohol use status (n = 372). The final derivation cohort comprised 2,397 eligible participants aged ≥60 years.

External validation was performed using NHANES 2005–2006 data (N = 10,348) through temporal validation (15, 16). This validation cohort utilized a different version of the Physical Activity Monitor (PAM) for assessing physical activity compared to the development cohort. Exclusion criteria mirrored the derivation phase: participants younger than 60 years (n = 8,778), those with missing accelerometer data (n = 328), and cases with incomplete covariate data (n = 169). The validation cohort ultimately included 1,073 participants. To ensure computational reproducibility, we fixed the random seed at 42 during sample extraction.

The NHANES 2011–2014 dataset employed wrist-worn ActiGraph GT3X+ accelerometers to continuously monitor participants’ 24-h physical activity patterns over seven consecutive days. The devices recorded triaxial acceleration data at 80 Hz while simultaneously collecting ambient light data at 1 Hz sampling frequency. Wear time metrics included: effective awake minutes (PAXWWMD), effective sleep minutes (PAXSWMD), and undifferentiated minutes where activity state (awake/asleep) could not be determined. Since undifferentiated minutes potentially introduce measurement error, we implemented stringent quality control criteria. A valid monitoring day required ≥1,296 minutes (17) of total wear time (PAXWWMD + PAXSWMD), excluding undifferentiated minutes. Participants were included only if they provided ≥4 valid days with at least one valid weekend day.

For NHANES 2005–2006, physical activity monitoring used the hip-mounted ActiGraph AM-7164 uniaxial accelerometer worn for seven consecutive days. This earlier protocol differed fundamentally by: (1) using single-axis rather than triaxial measurement, and (2) lacking sleep monitoring capability. We maintained comparable rigor by defining valid minutes as meeting both wear status (PAXSTAT = 1) and calibration criteria (PAXCAL = 1), with identical daily wear time requirements (≥1,296 valid minutes/day). The same inclusion threshold of ≥4 valid days was applied to ensure data quality parity across survey cycles.

The construction of the sleep duration classification variable in the derivation cohort was first based on the calculation of the effective average daily sleep duration ( Sleepdaily ) from wrist accelerometer data, where ∑ d = 1 D _ valid PAXSWMD represents the total sleep wear time across valid days and ∑ d = 1 D _ valid total _ wear _ time denotes the total wear time in minutes over valid days. The calculation was performed as follows:

In the external validation cohort, due to the lack of objective sleep monitoring device data, the self-reported sleep duration from questionnaires was used as the average daily sleep duration.

After calculating the effective average daily sleep duration, we referred to the sleep duration classification established by the National Sleep Foundation (18), which categorizes sleep duration for older adults aged 65 and above into a binary variable: 0 = bad (<5 or >9 h) and 1 = appropriate (5 to 9 h).

The average daily physical activity volume ( PAdaily ) was calculated as follows. In the derivation cohort, the data were expressed in MIMS (Monitor-Independent Movement Summary) (19) units—a novel aggregation method for raw sub-second accelerometer data, where minute-level triaxial acceleration values (x-, y-, and z-axes) are converted into MIMS units. Here, ∑ d = 1 D _ valid PAXMTSD represents the sum of daily triaxial MIMS values across valid days ( D valid ), while ∑ d = 1 D _ valid total _ wear _ time denotes the total wear time in minutes over valid days. The calculation formula was:

For the validation cohort, where physical activity volume lacked standardized units, daily activity was computed by first calculating the daily total activity ( DailyTotal d , d where subscript d represents day d). The total valid activity volume was obtained by summing daily activity across valid days: ∑ d = 1 D valid DailyTotal d . The average daily physical activity volume was then derived by dividing the total valid activity by the number of valid days:

This continuous variable represented participants’ average daily physical activity volume. To mitigate the influence of extreme values, we applied 1% winsorization to both tails of the distribution.

The study incorporated the following demographic covariates: age, sex (0 = male, 1 = female), and race/ethnicity categorized as 1 = Hispanic (including Mexican American and other Hispanic), 2 = non-Hispanic White, 3 = non-Hispanic Black, and 4 = other races. BMI was classified into 1 = BMI < 25, 2 = 25 ≤ BMI < 30, and 3 = BMI ≥ 30, with measurements obtained from physical examination in the derivation cohort and calculated from self-reported height and weight in the validation cohort. Education level was categorized as 1 = less than high school, 2 = high school graduate or equivalent, and 3 = some college or college graduate. Marital status was grouped as 1 = married/living with partner and 2 = widowed/divorced/separated/never married. Household income was stratified by poverty-to-income ratio (1 = PIR ≤ 1, 2 = 1 < PIR < 4, 3 = PIR ≥ 4) (20). Smoking status was defined as 1 = never smoker (<100 cigarettes lifetime), 2 = former smoker (≥100 cigarettes but quit), and 3 = current smoker. Alcohol drinkers (0 = non-drinker, 1 = drinker) were determined based on affirmative responses to the question: “During any 1 year, have you had at least 12 drinks of any type of alcoholic beverage?" (21).

All data analyses were performed using SPSS and Python software. Following NHANES recommendations, we incorporated the Mobile Examination Center (MEC) examination weights (WTMEC2YR) in all analyses, as some study variables were collected during MEC examinations. Participant characteristics were described using weighted means (standard deviation, SD) and frequencies (weighted percentages). Group differences were assessed using weighted t-tests for continuous variables and weighted chi-square tests for categorical variables.

In the derivation cohort, weighted multivariable logistic regression was initially employed to explore the association between physical activity volume and sleep duration, with adjustments made for sociodemographic, behavioral, and other confounding factors. Two adjusted models were developed to verify result robustness. Variables showing p < 0.05 in univariate analyses were subsequently incorporated into four machine learning algorithms: Random Forest (RF), Gradient Boosting Decision Tree (GBDT), eXtreme Gradient Boosting (XGBoost), and Light Gradient Boosting Machine (LightGBM). Optimal hyperparameters were determined through GridSearchCV with 5-fold cross-validation, and models were internally validated using test datasets.

Model selection was based on comparison of the area under the receiver operating characteristic curve (AUC) in test datasets, with additional evaluation of predictive performance using accuracy, precision, recall, and F1-score metrics (22). The final model underwent external validation using the independent validation cohort. All tests were two-sided with significance level α = 0.05.

For model interpretation, we applied SHAP (SHapley Additive exPlanations) values, an algorithm developed by Lundberg and Lee based on Shapley values from cooperative game theory (23). This approach quantifies each variable’s contribution to model predictions, providing additive feature importance measures for interpreting the “black-box” machine learning model.

In summary, the study flowchart is presented in Figure 1.

The distribution of the primary study factor, sleep duration, among the two groups of older adults is presented in Table 1. Statistically significant differences (p < 0.05) were observed between the groups for the following variables: age (p < 0.001), sex (p < 0.001), race/ethnicity (p < 0.001), BMI (p < 0.001), education level (p < 0.001), marital status (p < 0.001), household income (p < 0.001), smoking status (p < 0.001), alcohol consumption (p < 0.001), and total daily physical activity volume (p < 0.001). The comparison of baseline characteristics between groups in the external validation cohort is provided in Supplementary Table 1.

Table 2 presents the odds ratios (ORs) and 95% confidence intervals (CIs) for the association between total physical activity volume and sleep duration. The crude analysis showed an OR of 1.000525 (95% CI: 1.000524–1.000525). After adjustment for age and race/ethnicity, the OR for appropriate sleep duration was 1.000539 (95% CI: 1.000539–1.000539). Further adjustment for additional covariates including BMI, household income, smoking status, and alcohol consumption yielded an OR of 1.000563 (95% CI: 1.000563–1.000564), with all models showing statistical significance (p < 0.001). Furthermore, interpreting the odds ratio in the context of physical activity measurement scale is essential. To evaluate the clinical relevance of our findings, we employed the minimal clinically important difference (MCID) approach. While MCID can be determined through various methods including criterion, distribution-based, literature analysis, and expert consensus approaches, the limited existing research on MIMS units constrained the applicability of literature analysis and expert consensus methods. The criterion approach would also require additional clinical validation. Therefore, we adopted the distribution-based method for MCID estimation. Refer to existing research (24), we defined MCID as 0.5 times the baseline standard deviation (SD). The analysis revealed an SD of 3437.27 for physical activity levels, yielding an MCID of 1718.64 MIMS (0.5 × SD). This indicates that for every MCID-unit (1718.64 MIMS) increase in physical activity, the odds of achieving appropriate sleep duration improve by 2.6309-fold compared to the reference group with poorer sleep duration. The calculation formula is as follows.

Finally, to ensure the robustness of our findings, we conducted sensitivity analyses by redefining appropriate sleep duration as 7–8 h (representing 20.8% of the total sample) as the positive outcome. After performing weighted multivariate logistic regression using the same analytical pipeline, the sensitivity results remained consistent with our primary findings, demonstrating a positive association between physical activity volume and the probability of appropriate sleep duration (OR = 1.000207, p < 0.001).

These results demonstrate that higher physical activity volume remained significantly associated with increased probability of appropriate sleep duration after comprehensive adjustment for potential confounding factors.

The predictive model for sleep duration classification was constructed using 10 variables identified as significant (p < 0.05) in univariate analyses as input features, with continued application of WTMEC2YR weighting. To enhance model interpretability and avoid imposing spurious ordinal relationships, categorical variables were processed as follows: nominal features underwent one-hot encoding while ordinal features received label encoding. The dataset was partitioned into training (80%) and independent test (20%) sets while preserving the weighted survey design.

For hyperparameter optimization, we implemented an exhaustive grid search strategy (GridSearchCV) with 5-fold cross-validation across four machine learning algorithms: Random Forest (RF), Gradient Boosting Decision Tree (GBDT), eXtreme Gradient Boosting (XGBoost), and Light Gradient Boosting Machine (LightGBM). This systematic approach evaluated all possible parameter combinations within predefined search spaces to identify globally optimal configurations, thereby maximizing model generalizability and computational efficiency. Key parameters were optimized as shown in Table 3, while non-critical parameters retained their default values.

As shown in Table 4, bootstrap resampling (n = 100 iterations) was employed for model performance evaluation. The results demonstrated that all four models achieved AUC values exceeding 0.80 on the test set: RF (AUC = 0.851, 95% CI: 0.800–0.896, p < 0.001), GBDT (AUC = 0.859, 95% CI: 0.821–0.897, p < 0.001), XGBoost (AUC = 0.852, 95% CI: 0.813–0.893, p < 0.001), and LightGBM (AUC = 0.847, 95% CI: 0.807–0.883, p < 0.001). These results indicate excellent discriminative ability in predicting sleep duration among older adults, with the GBDT model showing superior overall predictive performance compared to other individual models.

To further enhance generalization capability, we developed a two-level stacked ensemble architecture using GBDT as the meta-learner and RF, XGBoost, and LightGBM as base-learners. The ensemble model achieved an AUC of 0.837 (95% CI: 0.790–0.871, p < 0.001), which was slightly lower than the standalone GBDT model (ΔAUC = 0.022). This may be attributed to the similarity among the tree models used for integration, but it further confirms the optimal generalization capability of traditional GBDT models in our specific research context.

Figure 2 displays the receiver operating characteristic (ROC) curves for the four prediction models. The x-axis represents the false positive rate (FPR, 1-specificity), while the y-axis indicates the true positive rate (TPR, sensitivity). In ROC analysis, points closer to the upper-left corner correspond to higher model accuracy. The area under the ROC curve (AUC) serves as a comprehensive performance metric, where larger AUC values indicate superior predictive accuracy.

To further assess model stability and technical robustness while minimizing generalization risks, we performed systematic sensitivity analysis on six critical hyperparameters of the optimal GBDT model. Centered on the grid-search-optimized parameters, we evaluated one incremental step above and below each parameter value, generating 729 distinct parameter combinations as specified in Table 5. The analysis revealed remarkable parameter stability, with the model achieving a mean AUC of 0.8641 and a standard deviation of merely 0.0023, corresponding to a negligible coefficient of variation of 0.26%. The AUC values remained consistently high across all tested configurations, ranging from 0.8585 to 0.8678 with a minimal total variation of 0.93%. These findings demonstrate that the model’s predictive performance is highly resilient to hyperparameter adjustments, thereby confirming the reliability of our research conclusions against potential parameter fluctuations.

The GBDT model developed in the derivation cohort was applied to predict outcomes in a weighted external validation set. The validation set revealed significant class imbalance between adequate sleep duration (92%) and poor sleep duration (8%). To address this imbalance, we implemented the improved SMOTE algorithm Borderline-SMOTE (25). For sensitivity analysis, we employed the hybrid SMOTE-ENN (26) technique (combining oversampling and undersampling) rather than single undersampling methods to preserve critical information while enhancing sample purity. All sampling procedures used a fixed random seed of 42 to ensure reproducibility. The final ROC curves are presented in Figure 3. Although differences in physical activity measurement units and sleep duration assessment methods between the derivation and validation cohorts might theoretically compromise predictive performance, the validation results demonstrated robust performance: AUC = 0.707 (95% CI: 0.690–0.730) with Borderline-SMOTE balancing and AUC = 0.754 (95% CI: 0.716–0.792) with SMOTE-ENN balancing, indicating stable predictive outcomes. However, the observed reduction in generalizability compared to internal validation may be attributable to: (1) different accelerometer models with incompatible measurement units between cohorts, and (2) potential reporting bias in the externally-collected self-reported sleep data, which might lead participants to report more socially desirable sleep durations.

Following model development, we conducted SHAP interpretability analysis on the final GBDT model. Figure 4A ranks predictive variables by mean absolute SHAP values, demonstrating that physical activity volume exerts the most significant influence on appropriate sleep duration. Figure 4B’s beeswarm plot displays individual samples (y-axis) and their impact on predictions (x-axis), where the color gradient (red = high values, blue = low values) reveals a positive association between higher physical activity levels and increased probability of appropriate sleep duration—consistent with logistic regression results. Figure 4C’s heatmap visualizes SHAP values for each participant’s physical activity volume: red/blue regions, respectively, indicate increased/decreased probability of appropriate sleep duration prediction, while the f(x) curve demonstrates a clear declining trend in overall probability as physical activity decreases (color gradient transitioning from red to blue).

The analysis of variable importance and directionality in Figures 4A–C demonstrates the association between physical activity volume and sleep duration in terms of magnitude and direction, but cannot explain potential threshold effects or nonlinear relationships across different physical activity volume ranges. Therefore, this study further generated SHAP feature dependency plots to visualize the dynamic response of model outputs (predicted values) to varying physical activity volume, as shown in Figure 4D. The horizontal axis represents physical activity volume values (MIMS), while the vertical axis indicates SHAP values. When SHAP values exceed zero, the model demonstrates a stronger propensity to predict positive class outcomes (appropriate sleep duration). The dependency plot reveals distinct threshold effects in the physical activity volume-sleep duration relationship. When daily physical activity volume was below 7,000 MIMS, the model tended to predict bad sleep duration with relatively small effects of physical activity volume changes on SHAP values, indicating limited marginal effects. In the 7,000 − 11461.51 MIMS range, while the model still predominantly predicted bad sleep duration, the slope of SHAP value changes increased significantly, demonstrating that physical activity volume increases could more effectively improve the probability of appropriate sleep duration during this phase. When daily physical activity volume exceeded 11461.51 MIMS, the model was more likely to predict appropriate sleep duration in older adults. As physical activity volume further increased beyond 15,000 MIMS, subsequent SHAP value changes stabilized, suggesting gradually diminishing marginal effects of additional physical activity volume.