Cumulative incidence

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Here is a thorough explanation of cumulative incidence:

Cumulative Incidence

Definition

Cumulative incidence (also called incidence proportion or attack rate in outbreak settings) is the proportion of a disease-free population that develops a new outcome over a defined time period.

"The numerator would be the number of subjects who developed the disease during follow-up and the denominator would be the total number of subjects at the beginning of the study."

Scott-Brown's Otorhinolaryngology Head & Neck Surgery

Formula

$$\text{Cumulative Incidence} = \frac{\text{Number of new cases during time period}}{\text{Population at risk at the start of the period}}$$

Result: A dimensionless proportion (0 to 1, or expressed as %)
Always tied to a specific time window (e.g., "1-year cumulative incidence of 5%")

Example: 1,000 smokers followed for 5 years; 15 develop head and neck cancer.

Cumulative incidence = 15/1,000 = 1.5% over 5 years

Key Characteristics

Feature	Cumulative Incidence
Type	Proportion (dimensionless)
Time	Fixed observation period specified
Denominator	Persons at risk at start
Assumption	All subjects followed for full period
Range	0 to 1 (or 0% to 100%)

Relationship to Incidence Rate

The incidence rate (incidence density) uses person-time in the denominator instead of persons, accounting for variable follow-up durations (dropouts, late entries).

Incidence rate = new cases / total person-time at risk (units: cases per person-year)
Same example: 15 cases / (5 years × 1,000 people) ≈ 3 cases per 1,000 person-years

When incidence is low, a useful approximation holds:

Cumulative incidence ≈ Incidence rate × Average follow-up time

This approximation breaks down when cumulative incidence is high (>10%) or follow-up is long.

Cumulative Incidence vs. Prevalence

	Cumulative Incidence	Prevalence
Measures	New cases only	All existing cases
Denominator	At-risk persons at start	Total population at a point in time
Time	Period (longitudinal)	Point or period
Use	Measuring disease risk	Measuring disease burden

Competing Risks and the Cumulative Incidence Function (CIF)

In survival analysis, when subjects can experience multiple types of events (e.g., death from cancer vs. death from other causes), standard Kaplan-Meier estimation overestimates cumulative incidence because it treats competing events as censored observations - violating the independent censoring assumption.

The solution is the Cumulative Incidence Function (CIF), which accounts for competing risks:

$$\text{CIF}(t) = \int_0^t S(u^-) \cdot h_k(u) , du$$

Where:

S(u⁻) = overall survival probability just before time u
h_k(u) = cause-specific hazard for the event of interest k

Key points about competing risks:

The sum of all cause-specific CIFs cannot exceed 1
Fine-Gray subdistribution hazard model is commonly used for regression with competing risks
CIF should always be used instead of Kaplan-Meier when competing events are present

Clinical Applications

Cohort studies: Primary measure of disease risk over follow-up
Outbreak investigations: Called the "attack rate" (e.g., 40% of attendees developed gastroenteritis)
Clinical trials: Reporting event rates over trial duration
Cancer epidemiology: Lifetime cumulative incidence (e.g., 1-in-8 lifetime risk of breast cancer)
Survival analysis: Plotted as 1 - Kaplan-Meier curve when no competing risks exist

Common Pitfalls

Omitting the time period - cumulative incidence is meaningless without specifying the time window
Using K-M when competing risks exist - leads to overestimation; use CIF instead
Confusing with incidence rate - incidence rate has units (per person-time); cumulative incidence is unitless
Ignoring losses to follow-up - if substantial dropout occurs, person-time methods (incidence rate) are more appropriate

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