Incidence Rate [IR] and Incidence Rate Ratio [IRR]: Analysis of Rates in Clinical Epidemiology Using Stata

1. Introduction

In clinical epidemiology, outcomes often occur over time, and individuals may contribute different lengths of follow-up. In such settings, simple risks or proportions are inadequate. Instead, incidence rates (IRs) and incidence rate ratios (IRRs) are appropriate measures of disease incidence and exposure effects.

Stata’s ir command is designed for this purpose and is widely used in cohort studies, occupational epidemiology, pharmacoepidemiology, and registry-based research.

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2. Incidence Rate (IR)

Definition

The incidence rate quantifies how rapidly new events occur in a population:

• Numerator: number of incident events (e.g., deaths, infections)
• Denominator: accumulated person-time (person-years, person-months)

Key property (dominant concept)

IR measures speed of occurrence, not probability

Two groups may have the same number of events, but if their follow-up times differ, their incidence rates differ.

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3. Person-Time: the Role of Time

Each individual contributes time until:

• the event occurs,
• loss to follow-up,
• or end of study.

Example

• Total events = 2
• Total person-time = 4.0 person-years

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4. Incidence Rate Ratio (IRR)

Definition

The incidence rate ratio compares incidence rates between two groups:

Interpretation

• IRR = 1 → no association
• IRR > 1 → higher event rate in exposed
• IRR < 1 → protective association

Example:

IRR = 2.0 → exposed group experiences events twice as fast

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5. Why IR / IRR are Dominant Measures

IR and IRR are dominant when:

1. Follow-up time varies between individuals
2. Entry into cohort is staggered
3. Loss to follow-up occurs
4. Population is dynamic
5. Outcome is recurrent or count-based
6. Interest is in rate, not cumulative risk

They outperform:

• Risk ratios when follow-up is unequal
• Odds ratios for longitudinal data

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6. Data Structure Required for ir

Each observation must contain:

Data may be individual-level or aggregated.

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7. Stata ir Command

Basic syntax

ir cases exposed time

What Stata does internally:

1. Splits data by exposed

2. Computes:

3. Calculates IRR and confidence intervals

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8. Worked Example (Individual-Level Data)

Example dataset

clear
input id exposed events time
1 1 1 1.2
2 1 0 2.0
3 1 1 0.8
4 0 0 3.0
5 0 1 1.5
6 0 0 2.5
end

Run incidence rate analysis

ir events exposed time

Interpretation

• Stata reports:
  • IR_exposed
  • IR_unexposed
  • IRR with 95% CI
• Time is automatically used as the denominator

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9. Manual Verification (Recommended for Understanding)

collapse (sum) events time, by(exposed)
gen ir = events / time
list

This produces two rows:

• exposed = 1 → IR_exposed
• exposed = 0 → IR_unexposed

IRR = ratio of these two IRs.

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10. Immediate Form: iri (Aggregated Data)

When you already have totals:

iri events_exposed events_unexposed time_exposed time_unexposed

Example

iri 12 8 240 400

Equivalent to the full ir command.

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11. Stratified Analysis

ir events exposed time, by(agegroup)

• Computes stratum-specific IRRs
• Combines them using Mantel–Haenszel weights
• Optional standardization (istandard, estandard)

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12. Relationship to Poisson Regression

ir is mathematically equivalent to:

poisson events exposed, exposure(time) irr

Differences:

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13. Common Misinterpretations (Important)

❌ IR is not a proportion ❌ IRR is not an odds ratio ❌ IRR is not a hazard ratio

✔ IRR compares rates per time ✔ HR compares instantaneous hazards

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14. How to Report in a Paper

“Incidence rates were calculated as events per person-year. Incidence rate ratios (IRRs) with 95% confidence intervals were estimated using person-time denominators.”

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15. Take-Home Message

• Incidence Rate (IR) = events ÷ person-time
• Incidence Rate Ratio (IRR) = ratio of two IRs
• Time is central, not optional
• ir provides clean, transparent rate comparisons
• Best used when follow-up is unequal or dynamic