Prevalence Ratio

Prevalence Ratio in Cost-of-Illness and Budget Impact Work

The prevalence ratio (PR) compares how common a condition is in two groups measured at the same time, dividing the prevalence in one group by the prevalence in the other. It is the usual measure of association when the evidence is a snapshot, such as a health survey, a registry extract or a primary care database, and for health economists such sources can be the only local evidence linking a risk factor or population characteristic to a condition. The PR therefore ends up inside attributable cost estimates and the population sizing of budget impact models. The prevalence odds ratio exaggerates the PR when the condition is common, and in an illustrative example using the wrong ratio raises an attributable cost by 27%.

From a two-by-two table to a ratio of prevalences

A cross-sectional study classifies each person by exposure and by whether the condition is present at the time of measurement. No follow-up is needed, only four counts. Arranged as a two-by-two table, the counts give one prevalence for each exposure group, as shown below.

$$PR = \frac{a/(a+b)}{c/(c+d)} = \frac{P_1}{P_0}$$

where $a$ and $b$ are the exposed people with and without the condition, $c$ and $d$ are the unexposed people with and without it, $P_1$ is the prevalence among the exposed and $P_0$ is the prevalence among the unexposed. This is the layout Tamhane and colleagues use for a cross-sectional study.

A PR of 1 means the condition is equally common in both groups. Above 1 it is more common among the exposed, and below 1 it is less common. Because $P_1$ cannot exceed 1, the PR cannot exceed the reciprocal of $P_0$. Where 15% of the unexposed already have the condition, no PR above about 6.67 is possible. In health economic work the exposure is often a population characteristic, such as a deprivation group or another long-term condition, and both groups must share a case definition and timing, whether point prevalence or period prevalence.

Why a prevalence ratio is not an incidence measure

Prevalence counts everyone who has the condition when it is measured, so it carries both how many people developed the condition and how long they kept it. The CDC's Principles of Epidemiology states that prevalence is based on both incidence and duration of illness, so a high prevalence may reflect high incidence, prolonged survival without cure or both. A PR above 1 can therefore arise because exposed people develop the condition more often, because they live with it for longer, or because unexposed people with the condition die or recover sooner. An exposure that improves survival with a disease raises prevalence without raising incidence, so reading a PR as a ratio of risks of becoming ill needs further assumptions about duration and survival.

Prevalence ratio and the prevalence odds ratio

Logistic regression is the default model for binary outcomes and reports an odds ratio, which in cross-sectional data is called the prevalence odds ratio (POR). Both are computed from the same four cells. They differ by a factor that depends on the two prevalences, as follows.

$$POR = \frac{a d}{b c} = PR \times \frac{1 - P_0}{1 - P_1}$$

where $POR$ is the prevalence odds ratio, $a$, $b$, $c$ and $d$ are the cells of the two-by-two table, and $P_1$ and $P_0$ are the prevalences among the exposed and unexposed. Tamhane and colleagues give the same identity.

Zocchetti, Consonni and Bertazzi showed that, whenever the two prevalences differ, the POR lies further from 1 than the PR, and that the gap depends much more on the prevalence of the condition than on the prevalence of the exposure. When both prevalences are low the two nearly agree, so that with 2% and 1% the PR is 2.00 and the POR is 2.02. Barros and Hirakata observed that with frequent outcomes the odds ratio can strongly overestimate the PR, and that reading it as a relative risk can mislead, particularly when it is used alongside true relative risks to set policy priorities.

The two measures can also lead to different conclusions about confounding and effect modification, as Thompson, Myers and Kriebel found. The PR lacks the symmetry of the odds ratio, since the PR for not having the condition is not the reciprocal of the PR for having it. Tamhane and colleagues note that such reversals are nonetheless made in cost-effectiveness analyses that convert results to a common direction, so a model needing the complement should compute it from the two prevalences.

Estimating adjusted prevalence ratios by regression

A crude PR from a table ignores differences between the groups in age, sex and other factors, so most analyses estimate an adjusted PR by regression. Two families of model estimate it directly.

The log-binomial model is a generalised linear model with a log link and a binomial error, in which the exponentiated coefficient of each covariate is its adjusted PR. Because the log of a probability cannot exceed 0, estimation has to be constrained and may fail to converge, which Barros and Hirakata report is most likely with a continuous covariate, several categorical covariates or a high prevalence of the outcome.

Poisson regression with a log link estimates the same ratio, but its variance assumption does not fit binary data, so its confidence intervals are too wide. A robust standard error from the sandwich estimator corrects this. Zou proposed this modified Poisson approach for relative risks in prospective studies and, in a limited simulation study, found it reliable with total samples as small as 100. Barros and Hirakata found Poisson regression with robust variance and log-binomial regression performed equally well when correctly specified, though the robust estimate tends to understate variability in moderately sized samples, and recommended both over logistic regression for cross-sectional studies because the PR is easier to interpret and communicate.

Prevalence ratios in cost-of-illness and budget impact work

In health economic analyses PRs usually serve as inputs, converting a survey association into numbers of cases and then into costs or eligible populations. Two uses recur.

Attributable cost from cross-sectional evidence

Jo describes the top-down, or attributable risk, approach to cost-of-illness studies. It applies a population attributable fraction (PAF) to aggregate costs, computed from the prevalence of the exposure (in Jo's notation, condition A) and the unadjusted relative risk of the outcome condition B for people with A compared with people without A. Jo describes prevalence-based studies as estimating the economic burden of a condition over a specific period, usually a year, and cautions that the formula holds only where other factors do not affect the association, since uncontrolled confounding can bias the relative risk, and so the PAF, upwards.

When the evidence is cross-sectional, the crude PR is the natural input, because the formula then reproduces the share of prevalent cases in excess of the unexposed prevalence, as shown below.

$$\frac{e(PR - 1)}{1 + e(PR - 1)} = \frac{P - P_0}{P}$$

where $e$ is the share of the population exposed, $P$ is the overall prevalence and $P_0$ is the prevalence among the unexposed. The result refers to prevalent cases and their annual costs, not to new cases and lifetime costs.

Sizing eligible populations for budget impact

The ISPOR good practice report on budget impact analysis lists estimating the size of the eligible population among the key elements and recommends data that reflect the decision maker's own population. Local records often give an overall prevalence and the population mix by some characteristic, while a published PR gives the relative prevalence between groups; combining them recovers subgroup prevalences, as step 6 below shows. The same calculation can set subgroup baseline prevalences in decision models and support subgroup analysis in a cost-effectiveness analysis. Where a PR describes health inequality, it gives the relative gap, and analyses informing resource allocation benefit from reporting the prevalence difference as well, since excess cases drive costs.

Worked example of one survey, an attributable cost and a budget impact population

The figures are illustrative. A health survey of 2,000 adults finds that 600 have an exposure and 1,400 do not. A chronic condition is present in 180 of the exposed and 210 of the unexposed adults.

1. Prevalences and the prevalence ratio.

$$PR = \frac{180/600}{210/1{,}400} = \frac{0.30}{0.15} = 2.00$$

where 0.30 and 0.15 are the prevalences among exposed and unexposed adults. The prevalence difference is 15 percentage points.

2. The prevalence odds ratio.

$$POR = \frac{180 \times 1{,}190}{420 \times 210} = \frac{214{,}200}{88{,}200} = 2.43$$

where 420 and 1,190 are the exposed and unexposed adults without the condition. The identity gives the same result, since 2.00 × 0.85 / 0.70 = 2.43, about 21% above the PR.

3. The ratio for the complement. The PR for being free of the condition is 0.70 / 0.85 = 0.82, which is not the reciprocal of the PR (1 / 2.00 = 0.50).

4. Attributable cost in a local population. A local health economy has 500,000 adults, 40% of them exposed, with the survey prevalences. Prevalent cases are 200,000 × 0.30 = 60,000 among the exposed and 300,000 × 0.15 = 45,000 among the unexposed, 105,000 in total. With $e = 0.40$ and the PR of 2.00, the population attributable fraction is as follows.

$$PAF = \frac{0.40 \times (2.00 - 1)}{1 + 0.40 \times (2.00 - 1)} = \frac{0.40}{1.40} = 0.2857$$

where 0.40 is the share exposed and 2.00 is the PR. Attributable cases are (0.40/1.40) × 105,000 = 30,000, the same as the excess among the exposed, 200,000 × 0.15 = 30,000. Assuming each prevalent case costs the health service an illustrative GBP 1,200 a year in both groups, the attributable annual cost is 30,000 × 1,200 = 36,000,000, or GBP 36.0 million.

5. Using the odds ratio by mistake.

$$PAF_{OR} = \frac{0.40 \times (2.4286 - 1)}{1 + 0.40 \times (2.4286 - 1)} = \frac{0.5714}{1.5714} = 0.3636$$

where 2.4286 is the unrounded POR entered in place of the PR. Attributable cases become (0.5714/1.5714) × 105,000 = 38,182 and the annual cost GBP 45.8 million, 27% more than the figure based on the PR.

6. Subgroup prevalences for a budget impact model. A neighbouring area with 200,000 adults knows that 21.75% have the condition and 45% are exposed. Writing $P = e P_1 + (1 - e) P_0$ with $P_1 = PR \times P_0$ gives the following.

$$P_0 = \frac{P}{1 + e(PR - 1)} = \frac{0.2175}{1 + 0.45 \times (2.00 - 1)} = \frac{0.2175}{1.45} = 0.15$$

where $P$ is the overall prevalence, $e$ is the share exposed and $P_0$ is the prevalence among the unexposed, so $P_1$ is 0.30. The area has 90,000 × 0.30 = 27,000 exposed cases and 110,000 × 0.15 = 16,500 unexposed cases, 43,500 in total. A treatment indicated only for exposed people with the condition would have an eligible population of 27,000, or 62% of prevalent cases.

7. What the numbers show. Because the condition is common, using the POR adds GBP 9.8 million a year to the attributable cost. Both figures read the PR as causal and unconfounded, and they may include excess cases that reflect longer duration among exposed people as well as more frequent onset. Step 6 assumes the survey PR holds in the neighbouring area.

Assumptions and misreadings when a prevalence ratio enters a model

A PR taken from a survey and placed in a cost or budget model carries assumptions the model inherits. Most cannot be tested within the survey, so they belong in the model's stated assumptions and sensitivity analysis.

• Confounding. An unadjusted PR between groups that differ in age or deprivation mixes the exposure with those differences.
• Temporal order. Exposure and condition are recorded at the same time, so a condition that changes behaviour after diagnosis can produce an association in the wrong direction.
• Reading a PR as a risk ratio. A PR reflects onset and duration together, so using it as the effect on new cases can misstate the benefit of prevention.
• Treating an attributable cost as avoidable. Removing the exposure would reduce prevalence only as existing cases resolve or die, and only to the extent that the association is causal.
• Transportability. Applying a PR from one population to another assumes the relative prevalence carries over.
• Switching models silently. When a log-binomial model fails to converge, falling back to logistic regression changes the measure from a PR to a POR.

Prevalence ratio among neighbouring measures

The table sets the PR beside the measures most often confused with it. Each answers a different question.

Sources

• Barros AJD, Hirakata VN. Alternatives for logistic regression in cross-sectional studies: an empirical comparison of models that directly estimate the prevalence ratio. BMC Medical Research Methodology. 2003;3:21. doi:10.1186/1471-2288-3-21
• Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice: An Introduction to Applied Epidemiology and Biostatistics. 3rd ed. Lesson 3: Measures of Risk, section 2. Atlanta: CDC; page last reviewed 18 May 2012.
• Jo C. Cost-of-illness studies: concepts, scopes, and methods. Clinical and Molecular Hepatology. 2014;20(4):327-337. doi:10.3350/cmh.2014.20.4.327
• Sullivan SD, Mauskopf JA, Augustovski F, Jaime Caro J, Lee KM, Minchin M, et al. Budget impact analysis-principles of good practice: report of the ISPOR 2012 Budget Impact Analysis Good Practice II Task Force. Value in Health. 2014;17(1):5-14. doi:10.1016/j.jval.2013.08.2291
• Tamhane AR, Westfall AO, Burkholder GA, Cutter GR. Prevalence odds ratio versus prevalence ratio: choice comes with consequences. Statistics in Medicine. 2016;35(30):5730-5735. doi:10.1002/sim.7059. Correction: Stat Med. 2017;36(23):3760. doi:10.1002/sim.7375
• Thompson ML, Myers JE, Kriebel D. Prevalence odds ratio or prevalence ratio in the analysis of cross sectional data: what is to be done? Occupational and Environmental Medicine. 1998;55(4):272-277. doi:10.1136/oem.55.4.272
• Zocchetti C, Consonni D, Bertazzi PA. Relationship between prevalence rate ratios and odds ratios in cross-sectional studies. International Journal of Epidemiology. 1997;26(1):220-223. doi:10.1093/ije/26.1.220
• Zou G. A modified Poisson regression approach to prospective studies with binary data. American Journal of Epidemiology. 2004;159(7):702-706. doi:10.1093/aje/kwh090