Inverse-Intensity Weighting for Informative Observation

Definition and scope

Weighting of irregular longitudinal measurements using a model for observation or visit intensity, under assumptions linking the visit process, outcome, and observed history. The target population, use, and analysis remain protocol decisions; this card explains the choices and their consequences rather than approving a study.

Operational specification

Define the target trajectory or regression parameter, time scale, at-risk observation period, visit events, scheduled versus triggered observations, outcome collection, and covariates available before visits. Model the visit intensity rather than treating each recorded value as an equally sampled measurement of the population trajectory. Distinguish irregular visits from terminal dropout and treatment/censoring mechanisms.

Analysis and diagnostics

Construct observation histories including eligible people without later visits, fit a justified intensity model, and use compatible weighted estimating equations with appropriate dependence/variance handling. Assess conditional independence assumptions, weight stability, exposure-related measurement, and informative dropout. Adding every visit predictor can increase variance and does not guarantee an adequate adjustment set.

When NOT to use and common errors

Do not infer unbiasedness from balanced visit counts, omit all people with no follow-up by convenience, use current unobserved outcomes as if available to the visit model, or assume visit weighting also handles terminal dropout or unmeasured observation drivers.

Worked example and implementation boundary

Two synthetic recorded outcomes are 10 and 20, with observation intensities 1 and 2 in a common arbitrary unit. Calculate their inverse-intensity weighted mean as a teaching step. The teaching weighted mean is approximately 13.333, compared with unweighted 15. The intensities are assumed inputs, not a fitted observation model. The code performs that explicitly stated calculation with synthetic inputs. It is not a complete fitted analysis and does not demonstrate the identification assumptions, estimate sampling uncertainty, or certify application in another population.

Data-source and reporting requirements

Preserve the provenance, dates, units, and observation rule for each input. Claims capture services and transactions under observable benefits; absence of a record is not proof that an event or resource use did not occur. EHR and registry measurements depend on the collection process and may miss care outside the system. Linked sources add reconciliation and selection questions. Primary collection can improve a missing domain but still requires valid measurement and follow-up. Report which of these limitations affects the stated use, which evidence supports the chosen inputs, and what remains uncertain.

Worked example

Scenario

Two synthetic recorded outcomes are 10 and 20, with observation intensities 1 and 2 in a common arbitrary unit. Calculate their inverse-intensity weighted mean as a teaching step.

Dataset

Synthetic inputs with units defined in the scenario; no patient data.

Steps

Result

The teaching weighted mean is approximately 13.333, compared with unweighted 15. The intensities are assumed inputs, not a fitted observation model.

Trade-offs

Code example

y1 = 10
y2 = 20
intensity1 = 1
intensity2 = 2
result = (y1/intensity1+y2/intensity2)/(1/intensity1+1/intensity2)
assert abs(result - 13.333333333333334) < 1e-9
print(result)

y1 <- 10
y2 <- 20
intensity1 <- 1
intensity2 <- 2
result <- (y1/intensity1+y2/intensity2)/(1/intensity1+1/intensity2)
stopifnot(abs(result - 13.333333333333334) < 1e-9)
print(result)

data worked_example;
  y1 = 10;
  y2 = 20;
  intensity1 = 1;
  intensity2 = 2;
  result = (y1/intensity1+y2/intensity2)/(1/intensity1+1/intensity2);
  put result=;
run;

Citations

1. [1]Pullenayegum EM, et al. Inverse-intensity weighted generalized estimating equations for longitudinal data subject to irregular observation: which variables should be included in the visit rate model?. Biometrics. 2025.