A Paradox in Hospitalized Patient Outcomes: Why Do High-Risk Patients Do Better than Low-Risk?

Ben Lengerich
4 min readJun 14, 2023

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Healthcare intertwines biological causal effects and systematized interventions. As a result, a statistical paradox commonly emerges when analyzing real-world evidence: harmful risk factors may be associated with better outcomes. This paradox is especially prevalent with systematized procedures which follow standardized protocols. For example, admission policies play a crucial role in determining which patients are selected for hospitalization; as a result, statistical analyses of hospitalized patients will produce very different estimates of risks than statistical analyses of the general population. In this brief post, we’ll explore the limitations of traditional risk assessment models, present concrete examples that illustrate the paradox, and discuss alternative approaches.

I. Admission Policies as Additive Models: Understanding the Relationship between Risk Factors and Outcomes

Medical decision policies can often be interpreted as additive models. For example, nomograms summing evidence from a list of known risk factors to calculate an overall risk assessment are additive models. Factors such as age, co-morbidities, and specific medical conditions are assigned weights or scores that are combined to determine the likelihood of adverse outcomes, and this cumulative risk score guides the decision-making process.

Intuitively, one would expect patients labeled as “high risk” based on these known risk factors to have a higher probability of experiencing poor outcomes once hospitalized. Conversely, those with a low risk score would be expected to have better outcomes or potentially not require hospitalization at all.

However, the statistical paradox emerges when analyzing the actual outcomes of hospitalized patients. Empirical evidence consistently reveals that patients labeled as “high risk” according to measured risk factors often exhibit unexpectedly favorable outcomes, while individuals without known “high risk” factors can face more adverse outcomes.

One example that highlights this paradox is the association between high body mass index (BMI) and better outcomes in hospitalized COVID-19 patients. Patients hospitalized with higher BMI, considered a risk factor for severe illness, actually exhibited better outcomes than those hospitalized with lower BMIs. Why does this happen?

II. The Paradox Unveiled: Reconciling Admission Policies and Outcomes

The statistical paradox can be attributed to two factors: systemic and human biases.

Firstly, systemic biases lead to statistical paradoxes. Model-based decision functions capture the impacts of known risk factors but overlook the impacts of unmeasured risk factors. Since many of these decision functions are implicitly additive models, the threshold for decision-making could be reached either due to measured risk factors or unmeasured risk factors. This leads to a strong selection bias: patients who reached the decision threshold without many measured risk factors must have a large impact from unmeasured risk factors.

Measured and unmeasured risk factors combine to influence decisions such as hospital admission. Conditioned on hospital admission, patients with few measured risk factors tend to have more unmeasured risk factors.

For example, in the COVID-19 pandemic, patients were hospitalized with attention paid to co-morbidities and BMI; if a patient without co-morbidities or high BMI were hospitalized they had a more severe infection. As these unaccounted-for risks can be the most pernicious, patients with low expected risk but severe disease often have surprisingly poor outcomes.

Risk factors associated with in-hospital mortality of hospitalized COVID-19 patients. Of note, Charlson score (a measure of co-morbidities) and BMI exhibit either no association or anti-correlation with mortality. Reproduced from [Lengerich et al 2022].

Secondly, human biases, such as threshold effects, influence the classification of patients into risk categories. Patients who meet the threshold for admission based on specific risk factors may receive better care and have better outcomes, while those just below the threshold may miss out on care and experience worse outcomes. These biases contribute to threshold-based paradoxes and complicate the relationship between risk factors and outcomes.

III. Unveiling the Complexity of Patient Outcomes: Limitations and Optimism in Model-Based Decision-Making

The paradox of anti-correlation between risk factors and real-world outcomes arises from the limitations of model-based decision-decision-making. No model-based decision function will ever be perfect, inevitably leading to paradoxes that arise when inspecting patients that passed a decision bottleneck.

However, instead of being pessimistic, this perspective provides optimism for enhancing healthcare outcomes. Any paradox representing an unmeasured risk presents an opportunity for improvement. Identifying and understanding these paradoxes allows us to delve deeper into the complexities of patient outcomes, propelling us towards more comprehensive and individualized risk assessment approaches.

While no model can encompass all the intricacies of human health, ongoing advancements in personalized medicine show promise. By integrating genetic information, biomarkers, patient-specific characteristics, and social determinants of health, we can enhance the accuracy and effectiveness of risk assessments. This shift towards individualized assessments reflects a more nuanced and inclusive understanding of patient outcomes.

In essence, the statistical paradox reminds us that healthcare is a complex and multifaceted domain. Embracing the imperfections of model-based decision-making empowers us to leverage these paradoxes for continuous care improvement. Through ongoing research, technological advancements, and a holistic approach to risk assessment, we can strive for more accurate predictions, improved patient outcomes, and a healthcare system that constantly evolves to meet individuals’ needs.

Conclusion: Embracing a Nuanced Approach to Risk Assessment

The statistical paradox surrounding admission policies in healthcare highlights the limitations of traditional risk assessment models and calls for a paradigm shift in evaluating patient outcomes. Healthcare systems must acknowledge the complexities of patient responses to treatment and care to embrace a more nuanced approach to risk assessment.

In particular, iterative improvement of health systems is crucial. Risk assessment is an ongoing process that necessitates continuous evaluation and refinement, and responds to systems of current practice. Healthcare systems should foster a culture of iterative improvement, analyzing data and outcomes to enhance risk assessment models in alignment with evolving understanding. By embracing a nuanced approach to risk assessment, healthcare systems can enhance patient outcomes, optimize resource allocation, and ensure that admission policies align with the complexities of patient care.

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Ben Lengerich

Postdoc @MIT | Writing about ML, AI, precision medicine, and quant econ