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Crude co-mention trends and standardized observed-to-expected ratios can move in opposite directions: a decomposition, with an application to US multiple-cause-of-death data

This paper demonstrates through a mathematical decomposition and US mortality data that crude co-mention trends can diverge from standardized over-representation ratios when the background frequency of a co-condition changes, arguing that interpreting one metric as a proxy for the other without accounting for this background shift can lead to misleading conclusions about comorbidity burden.

Original authors: Sekani Nicolas Boxill

Published 2026-09-11

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CC BY 4.0

Original authors: Sekani Nicolas Boxill

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When a person dies, the official record of their death is more than a simple cause; it is a list of every medical condition that contributed to the end of their life. Researchers use these lists, known as multiple-cause-of-death records, to understand how different diseases travel together. If a doctor lists both heart failure and kidney disease on the same certificate, it suggests a link between the two. By counting how often these pairs appear together over time, scientists try to see if the connection between specific diseases is growing stronger, weaker, or staying the same. This information is vital for public health, as it helps officials understand whether the burden of complex, overlapping illnesses is increasing among the population. However, there is a hidden trap in how these numbers are read. A simple count of how often two conditions appear together can rise, even if the actual medical link between them is fading. This happens because the background frequency of one of the conditions might be rising for reasons entirely unrelated to the other, creating a statistical illusion that the two are becoming more tightly bound when they are not.

A researcher named Sekani Nicolas Boxill set out to untangle this confusion by looking at real data from the United States spanning two decades. The study focused on four different serious infections and how often they appeared alongside chronic kidney disease on death certificates. The researcher examined the raw numbers, which simply count how many people died with both conditions, and compared them to a more refined calculation. This refined method asks a different question: given the age and sex of the people who died from the infection, how many would we expect to also have kidney disease based on the general population's trends? By separating the raw count into these two parts—the raw count and the expected background frequency—the researcher could see what was actually driving the trends.

The findings revealed a striking contradiction in the data for infective endocarditis, a serious infection of the heart. Between 2013 and 2019, the raw percentage of heart infection deaths that also mentioned kidney disease rose steadily, increasing by nearly one percentage point each year. A casual reading of this number would suggest that the two conditions were becoming more frequently linked. However, when the researcher applied the refined calculation to account for the rising background rate of kidney disease in the general population, the picture flipped completely. The measure of how much the two conditions were truly over-represented together actually fell by more than two and a half percentage points each year. The raw number went up because kidney disease was becoming more common on death certificates overall, not because the heart infection was suddenly attracting more kidney patients than before.

This same pattern of opposing trends appeared in other pairs as well. For sepsis, a severe blood infection, the raw co-mention rate showed a slight, statistically uncertain rise, while the refined measure showed a clear and consistent decline. In the case of pneumonia, both numbers rose, but the raw count climbed much faster than the refined measure, indicating that the increase was largely driven by the general rise in kidney disease mentions rather than a specific strengthening of the link between pneumonia and kidney failure. The study also highlighted a specific period in 2011 and 2012 where a change in how kidney disease was coded on death certificates caused a sudden jump in mentions across all categories. This coding change affected the raw numbers and the background expectations equally, proving that standard statistical adjustments cannot always fix sudden shifts in how data is recorded.

The researcher then looked at how other scientists have been interpreting similar data. A review of twenty-five recent studies found that most researchers reported the raw rise in co-occurring deaths as evidence of an increasing burden or a strengthening link between diseases. While many of these studies did compare their results to some form of background trend, none of them broke the raw number down into its two distinct components: the expected background frequency and the true relative over-representation. This means that the field has been missing a crucial step in its analysis. The study concludes that a rising raw number does not automatically mean a disease pair is becoming more dangerous or more linked. To understand the true story, researchers must look at the background frequency of the conditions and calculate the ratio of what was observed against what was expected. Without this separation, the data can easily tell a story of increasing connection where none actually exists.

Technical Summary: Crude Co-mention Trends and Standardized Observed-to-Expected Ratios

Problem Statement
Studies utilizing multiple-cause-of-death (MCOD) data frequently interpret rising age-adjusted mortality rates for condition pairs, or increasing proportions of index-condition deaths that also mention a co-condition, as evidence of "increasing enrichment" or a strengthening clinical link between the two conditions. However, the paper argues that these crude metrics can rise simply because the background frequency of the co-condition among all decedents is increasing, independent of any specific change in the relationship between the two diseases. The core problem is the conflation of a rising crude proportion with a rising relative over-representation, leading to potential misinterpretation of mortality trends.

Methodology
The study employs a mathematical decomposition applied to US MCOD data (CDC WONDER, 1999–2019) for adults aged ≥25. The author defines three key statistics for a pair of conditions (Index A and Co-condition B) across age-sex strata $s$ :

Crude Co-mention ( $C$ ): The proportion of index deaths ( $N_A$ ) that also mention the co-condition ( $O$ ). $C = O/N_A$ .
Background Frequency ( $Q$ ): The expected proportion of index deaths that would mention the co-condition based on the co-condition's age-sex-specific frequency among decedents without the index condition. $Q = E/N_A$ , where $E$ is the expected count.
Standardized Observed-to-Expected Ratio ( $ER$ ): The ratio of observed co-mentions to expected co-mentions ( $O/E$ ).

The paper establishes the exact arithmetic identity:
$C(t) = ER(t) \times Q(t)$
Taking logarithms, this becomes an additive relationship:
$\log C(t) = \log ER(t) + \log Q(t)$
Consequently, under Ordinary Least Squares (OLS), the slope of the log-crude trend is the sum of the slope of the log-standardized trend and the slope of the log-background trend:
$\text{slope}(\log C) = \text{slope}(\log ER) + \text{slope}(\log Q)$

The analysis focuses on four pairs involving Chronic Kidney Disease (CKD) as the co-condition: Infective Endocarditis (IE), Clostridioides difficile infection (CDI), Sepsis, and Pneumonia. The study accounts for data suppression in CDC WONDER (1–9 death counts) using a constrained allocation method to estimate expected values ( $E$ ) and treats the 2011–2012 period (a coding transition for CKD) as a discontinuity, excluding it from stable-period slope estimation.

Key Results

Mathematical Identity: The decomposition $C = ER \times Q$ held to machine precision across all data.
Divergent Trends (Infective Endocarditis): For IE and CKD, the crude co-mention proportion ( $C$ ) rose significantly (e.g., +0.93%/yr in 2013–2019), while the standardized ratio ( $ER$ ) fell significantly (−2.53%/yr). This divergence was driven by a sharp increase in the background frequency ( $Q$ ) of CKD mentions (+3.55%/yr).
Other Pairs:
- Sepsis × CKD: Crude proportion showed a non-significant increase, while the standardized ratio declined significantly.
- Pneumonia × CKD: Both crude and standardized ratios rose, but the crude rise was approximately double the standardized rise.
- CDI × CKD: Showed a structural break at the 2011–2012 transition, preventing a single interpretable slope.
Literature Audit: A review of 25 pair analyses from the literature found that 18 interpreted a rising pair statistic as evidence of increasing burden. While 13 papers provided some form of background context (e.g., citing a trend or reporting a component series), none explicitly decomposed the crude statistic into expected background and relative over-representation components.

Significance and Claims
The paper claims that the divergence between crude and standardized trends is not a statistical artifact but a mathematical certainty when the background frequency of a co-condition changes. The primary contribution is the explicit demonstration that:

A rising crude co-mention rate does not necessarily imply a strengthening clinical link or "enrichment"; it may merely reflect a rising background prevalence or documentation frequency of the co-condition.
The difference between the crude trend and the standardized trend is not an independent quantity to be tested; it is identical to the background trend ( $Q$ ).
Standardization against age-sex composition does not remove the effects of coding changes (e.g., the 2011 CKD code restructuring) or shifts in certification practices, which can affect both the numerator and denominator.

The author concludes that researchers investigating comorbidity trends must examine the contemporary background frequency ( $Q$ ) and report observed-to-expected ratios or equivalent decompositions to distinguish between changes in documented caseload and changes in relative over-representation. The study does not claim to reveal clinical relationships or causality, but rather provides a methodological correction for interpreting descriptive mortality trends.

Technical Summary: Crude Co-mention Trends and Standardized Observed-to-Expected Ratios

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