Signature
u_prev = (I_1 * D_1 * u_1 + I_2 * D_2 * u_2) / (I_1 * D_1 + I_2 * D_2); u_inc = (I_1 * u_1 + I_2 * u_2) / (I_1 + I_2)
| Inputs | Definition | Unit |
|---|---|---|
I_1 | New cases of form 1 per person-year at risk | cases per person-year |
D_1 | Mean time from diagnosis to recovery or death for form 1 | years |
u_1 | Utility of people with form 1, constant over the disease course | utility |
I_2 | New cases of form 2 per person-year at risk | cases per person-year |
D_2 | Mean time from diagnosis to recovery or death for form 2 | years |
u_2 | Utility of people with form 2, constant over the disease course | utility |
u_prev | Mean utility of everyone living with the condition on the survey date | utility, 1 equals full health |
|---|---|---|
u_inc | Mean utility of newly diagnosed cases | utility, 1 equals full health |
Function
Prevalence estimation from a cross-sectional survey and its relation to incidence and duration
Estimates the prevalence of a condition from a single survey round, weighting respondents for how the sample was drawn, and links point prevalence to incidence and mean duration in a steady state. The same link shows why averages taken over prevalent cases weight each form of a condition by incidence times duration, not by incidence alone. Notation follows the Cross-Sectional Design article and its prevalent-case utility example.
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Implementations
Excel
Prevalence-weighted and incidence-weighted mean utility from named ranges
With the incidence, mean duration and utility of each form in ranges named IncRates, Durations and Utils, in the same order, the formulas return the prevalent-case and incident-case means, held in UtilPrev and UtilInc.
=SUMPRODUCT(IncRates,Durations,Utils)/SUMPRODUCT(IncRates,Durations); =SUMPRODUCT(IncRates,Utils)/SUM(IncRates)
Assumptions
Utility constant over the disease course within each form
Each form has one utility whatever the time since diagnosis, as in the article; if utility declines with time since diagnosis, long-duration cases are sampled late in their course and the gap can reverse.
Steady state for each form of the condition
Incidence and mean duration of each form are stable, so the prevalent pool holds the forms in proportion to I_k times D_k (HE-FM-XSD-002).
Worked examples
Mild and severe forms of a chronic condition
With incidence 0.001 per person-year for each form, durations of 10 and 2 years and utilities of 0.80 and 0.50, the prevalent mean is 0.009 / 0.012, or 0.75, and the incident mean 0.0013 / 0.002, or 0.65, so the survey overstates the utility at diagnosis by 0.10, as in the article.
I_1 = 0.001; I_2 = 0.001; D_1 = 10; D_2 = 2; u_1 = 0.8; u_2 = 0.5; u_prev = 0.75; u_inc = 0.65
Equal durations of two forms remove the prevalent-case utility gap
If both forms lasted five years, the prevalent and incident means would both be 0.65 (computed here for illustration).
I_1 = 0.001; I_2 = 0.001; D_1 = 5; D_2 = 5; u_1 = 0.8; u_2 = 0.5; u_prev = 0.65; u_inc = 0.65
Equal durations remove the gap
If both forms lasted five years, the prevalent and incident means would both be 0.65 (computed here for illustration).
I_1 = 0.001; I_2 = 0.001; D_1 = 5; D_2 = 5; u_1 = 0.8; u_2 = 0.5; u_prev = 0.65; u_inc = 0.65
Severe form three times as common at onset
With incidences of 0.0005 and 0.0015 and the article's durations and utilities, the incident mean is 0.575 and the prevalent mean 0.6875, a gap of 0.1125 (computed here for illustration).
I_1 = 0.0005; I_2 = 0.0015; D_1 = 10; D_2 = 2; u_1 = 0.8; u_2 = 0.5; u_prev = 0.6875; u_inc = 0.575
Common errors
Taking a prevalent-case survey mean as the baseline of an incident cohort
The survey over-represents long-lasting forms, which Simon described as length-biased sampling of living patients; in the article's example it overstates the utility of newly diagnosed patients by 0.10 and shows one severe case in six instead of one in two.
Expecting survey weights to remove the prevalent-case gap
Survey weights correct how the sample was drawn, not who is alive and ill on the survey date, so the weighted prevalent mean is still 0.75; the gap is removed by reweighting each form by the inverse of its mean duration.
Sources
Length-biased sampling of living patients
Simon R. American Journal of Epidemiology. 1980;111(4):444-452. doi:10.1093/oxfordjournals.aje.a112920 (abstract read). Abstract: describes the problem of estimating and comparing the frequency of a characteristic in newly diagnosed patients from a length biased sample of living patients; the estimator depends on the proportion of living patients with the characteristic and on the survival of patients with and without it.
Steady-state relation used to weight forms by incidence times duration
Reichenheim ME, Coutinho ESF. Measures and models for causal inference in cross-sectional studies: arguments for the appropriateness of the prevalence odds ratio and related logistic regression. BMC Medical Research Methodology. 2010;10:66. doi:10.1186/1471-2288-10-66 (full text read). Equation (3): P_i = ID_i T_i / (ID_i T_i + 1), with stationarity and no selective survival among the structuring conditions. Applying the relation to each form separately, with a common population at risk, makes the prevalent counts proportional to incidence times duration (step derived in this record).
Canonical Identity
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