# Reduction of sampling bias of odds ratios for vertebral fractures using propensity scores

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## Abstract

### Introduction

Assessment of the predictive power of a newly introduced diagnostic technique with regard to fracture risk is frequently limited by the enormous costs and long time periods required for prospective studies. A preliminary estimate of predictive power usually relies on cross-sectional case-control studies in which bone measurements of normal and fractured subjects are compared. The measured discriminatory power is taken as an estimate of predictive power. Because of possible sample selection bias, study participants may have different bone mineral density (BMD) values, and fractured patients may have fractures of different severity levels. The same diagnostic techniques for the measured discriminatory power, expressed as odds ratios, will differ among studies with different patient and control populations.

### Methods

In this paper, we propose a weighted logistic regression approach to adjust the odds ratio in order to reduce the effect of sampling bias. The weight is derived from age, deformity severity, BMD, and the interactions of these, using the propensity score theory and reference population data.

### Results

Simulation examples using data from the Osteoporosis and Ultrasound Study (OPUS) demonstrate that such a procedure can effectively reduce the estimation bias of odds ratios introduced by sampling differences, such as for dual x-ray absorptiometry (DXA) scans of the spine and hip as well as various quantitative ultrasound techniques. The derived estimated odds ratios are substantially less biased, and the corresponding 95% confidence intervals contain the true odds ratios from the population data.

### Conclusions

We conclude that a statistical correction procedure based on propensity scores and weighted logistic regression can effectively reduce the effect of sampling bias on the odds ratios calculated from cross-sectional case-control studies. For a new diagnostic technique, hip BMD and deformity severity information are necessary and likely sufficient to derive the propensity scores required to adjust the measured standardized odds ratios.

## Keywords

Bias correction Osteoporosis Propensity score Spine fracture Weighted logistic regression## Notes

### Acknowledgements

This research was supported by a grant from the International Osteoporosis Foundation. We would also like to thank the OPUS investigator group for providing us with the data used to develop our methodology.

## References

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