Rates, Ratios and Risks in Medical Research

Overview

Words such as rate, ratio, risk, prevalence and odds are sometimes used loosely in clinical reports. In epidemiology and biostatistics, they have distinct meanings.

The denominator is central:

• a proportion has a numerator contained within its denominator;
• a risk is a proportion of initially event-free people who develop an event during a defined period;
• an incidence rate divides events by person-time at risk;
• odds divide events by non-events; and
• a ratio compares two quantities and need not contain the numerator within the denominator.

These measures answer different questions and should not be substituted casually.

Why this matters

Suppose 12% of exposed patients and 6% of unexposed patients experience readmission within one year.

The findings can be described as:

• risk difference: 6 percentage points;
• risk ratio: 2.0;
• odds ratio: approximately 2.14.

These numbers are not interchangeable.

The risk difference communicates absolute excess events. The risk ratio communicates relative risk. The odds ratio compares odds and is farther from 1 than the risk ratio when the event is not rare.

Person-time rates introduce another denominator and may differ when follow-up varies.

Ratio, proportion and percentage

A ratio divides one quantity by another. The numerator does not need to be part of the denominator.

Examples:

• nurse-to-patient ratio;
• male-to-female ratio in a sample;
• risk ratio;
• rate ratio.

A proportion is a ratio in which the numerator is included within the denominator.

A percentage is a proportion multiplied by 100.

The phrase “mortality rate of 10%” is common but mathematically imprecise when the measure is a risk or proportion rather than a person-time rate.

Prevalence

Prevalence is the proportion of a population with a condition at a specified time or during a specified period.

Point prevalence

Point prevalence refers to one time point:

$$\text{Point prevalence} = \frac{\text{existing cases at the specified time}} {\text{population assessed at that time}}$$

Period prevalence

Period prevalence includes anyone who had the condition at any time during a specified interval.

Prevalence reflects both:

• occurrence of new cases; and
• duration of the condition.

A chronic condition can have high prevalence even if incidence is modest because people live with it for a long time.

Prevalence is not the same as risk of developing disease.

Risk or cumulative incidence

Risk, also called cumulative incidence, is the proportion of initially at-risk individuals who develop an event during a specified period.

Risk must include a time horizon:

• 30-day mortality risk;
• one-year readmission risk;
• five-year cancer recurrence risk.

A risk ranges from 0 to 1.

In the cohort:

• high-risk group: $60/500=0.12$, or 12.0%;
• lower-risk group: $30/500=0.06$, or 6.0%.

This simple calculation assumes outcome status is known over the stated period for the denominator or that the analysis otherwise addresses incomplete follow-up appropriately.

Odds

Odds are the probability of an event divided by the probability of no event.

For the high-risk group:

$$\text{odds} = \frac{60}{440} = 0.136$$

This means 60 events for every 440 non-events, not a 13.6% risk.

Odds and risk are similar only when the event is uncommon. They diverge as the event becomes more frequent [1].

Incidence rate

An incidence rate, also called a person-time rate, divides the number of incident events by total time at risk.

The unit may be:

• events per 100 person-years;
• infections per 1,000 catheter-days;
• falls per 10,000 bed-days.

A rate is not bounded by 1. If recurrent events are counted, one person may contribute several events, but the analysis must then account for the recurrent-event structure.

In the cohort:

• high-risk: $60/450=0.1333$ per person-year, or 133.3 per 1,000 person-years;
• lower-risk: $30/480=0.0625$ per person-year, or 62.5 per 1,000 person-years.

Risk and rate differ because the rate incorporates observed time at risk.

Absolute comparisons

Risk difference

The risk difference compares two risks by subtraction.

For the cohort:

$$RD=0.12-0.06=0.06$$

The 12-month risk is 6.0 percentage points higher in the high-risk group.

Using the Newcombe score method, the 95% confidence interval is 2.5 to 9.6 percentage points.

Incidence-rate difference

The incidence-rate difference subtracts person-time rates:

$$133.3-62.5=70.8$$

The rate difference is 70.8 additional readmissions per 1,000 person-years.

Absolute measures are strongly influenced by baseline risk and often communicate public-health or clinical burden directly.

Relative comparisons

Risk ratio

The risk ratio divides two risks.

For the cohort:

$$RR=\frac{0.12}{0.06}=2.00$$

The 12-month readmission risk is twice as high in the high-risk group. The large-sample 95% confidence interval is 1.31 to 3.05.

Rate ratio

The incidence-rate ratio divides two person-time rates:

$$IRR= \frac{60/450}{30/480} = 2.13$$

The simulated 95% confidence interval is 1.38 to 3.31 using a large-sample Poisson approximation.

A rate ratio compares event occurrence per unit person-time. It is not automatically a risk ratio.

Odds ratio

The odds ratio divides two odds:

$$OR = \frac{60/440}{30/470} = 2.14$$

The 95% confidence interval is 1.35 to 3.37 using the standard large-sample log-odds method.

Calculations for the confidence intervals in this section are not shown here.

The odds ratio should be interpreted as an odds ratio. It may approximate a risk ratio when outcomes are rare under particular designs, but it should not be described routinely as “twice the risk” [1], [2].

Data structure

Simulated heart-failure cohort

simulated data

Table 1. Twelve-month readmission by adherence-risk group

Adherence-risk group	Readmitted	Not readmitted	Total	Risk
High risk	60	440	500	12.0%
Lower risk	30	470	500	6.0%

Note: Participants are classified observationally. The table does not establish causation.

Table 2. Person-time and incidence rates

Adherence-risk group	Readmissions	Person-years	Incidence rate per 1,000 person-years
High risk	60	450	133.3
Lower risk	30	480	62.5

Note: The incidence-rate calculation incorporates unequal observed time at risk.

Worked example

Common terminology traps

Attack rate

An attack rate used in outbreak investigation is usually a risk or cumulative incidence, despite the word rate.

Case-fatality proportion

The proportion of diagnosed cases who die from the condition over a stated period is a case-fatality proportion, often called a case-fatality rate in practice.

Mortality rate

A mortality measure may be:

• a risk over a fixed period; or
• a person-time mortality rate.

The denominator and unit must make the meaning clear.

Hazard

A hazard is an instantaneous event rate among those still event-free. A hazard ratio from a survival model is not a risk ratio and does not directly state the ratio of cumulative risks.

Why a rate cannot be converted to a risk without assumptions MMed extension

A rate uses person-time, while a risk uses people initially at risk.

Under a constant hazard model, a rate $\lambda$ can be related to risk over time $t$ by:

$$R(t)=1-e^{-\lambda t}$$

This relationship requires assumptions about event occurrence. Simply multiplying a rate by time is only an approximation when risk is small and the rate is stable.

Suggested reporting

The CDC’s epidemiology guidance distinguishes ratios, proportions, risks and person-time rates and emphasises explicit numerators and denominators [3]. Observational reports should also follow design-specific guidance such as STROBE [4].

Common mistakes

Knowledge check

What is the difference between a risk and an incidence rate?

Answer and reasoning

Risk is the proportion of initially at-risk people who experience an event over a defined period. Incidence rate divides events by person-time at risk and has a time unit.

What does a risk ratio of 2.0 mean in the simulated cohort?

Answer and reasoning

The observed 12-month readmission risk in the high-risk group is twice the risk in the lower-risk group. It does not prove that the classification caused the difference.

Why is the odds ratio of 2.14 not the same as the risk ratio of 2.00?

Answer and reasoning

Odds divide events by non-events, whereas risk divides events by all initially at-risk people. The measures diverge as event frequency increases.

Why should an incidence rate be written as per 1,000 person-years?

Answer and reasoning

The denominator is accumulated time at risk, not a number of people. The unit tells readers how the rate was scaled and permits correct interpretation.

Why report both risk difference and risk ratio?

Answer and reasoning

The risk difference communicates absolute excess burden, while the risk ratio communicates relative strength. Together they provide more clinically useful context than either alone.

Key take-home messages

References

1. Grimes DA, Schulz KF. Making sense of odds and odds ratios. Obstet Gynecol. 2008;111(2 Pt 1):423–426. https://doi.org/10.1097/01.AOG.0000297304.32187.5d
2. Davies HTO, Crombie IK, Tavakoli M. When can odds ratios mislead? BMJ. 1998;316(7136):989–991. https://doi.org/10.1136/bmj.316.7136.989
3. Centers for Disease Control and Prevention. Principles of epidemiology in public health practice. 3rd ed. Lesson 3: measures of risk. Atlanta (GA): CDC; 2012. https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/index.html
4. von Elm E, Altman DG, Egger M, Pocock SJ, Gøtzsche PC, Vandenbroucke JP, et al. The Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) statement: guidelines for reporting observational studies. PLoS Med. 2007;4(10):e296. https://doi.org/10.1371/journal.pmed.0040296