Case Fatality Rate

Case Fatality Rate as a Measure of Severity in Decision Models

The case fatality rate (CFR) tells a modeller how likely a case of a disease is to end in death, and in economic evaluations of treatment, vaccination or outbreak control it often becomes the probability on the death branch that follows each case. Despite its name the measure is a proportion rather than a rate, and it differs from a population mortality rate and from the infection fatality ratio. Crude estimates made while an outbreak is still running can be too high or too low.

A proportion of cases over a stated period

The CDC's teaching text on epidemiology defines the case fatality rate as the share of people with a particular condition who die from that condition, and describes it as a measure of severity. Its numerator is restricted to deaths among the cases in its denominator. That restriction makes it a proportion, as follows.

$$CFR = \frac{D}{N_c}$$

where $CFR$ is the case fatality rate, $D$ is the number of deaths from the disease among the cases counted, and $N_c$ is the number of cases that meet the case definition. The CDC text works through an outbreak of hepatitis A with 555 cases and 3 deaths, a case fatality rate of about 0.5%. Read this way, the case fatality rate is an absolute risk of death for one group, the cases, with no comparison group.

The CDC text calls the measure a proportion, not a true rate, and notes that some epidemiologists prefer the term case fatality ratio. The WHO scientific brief on estimating mortality from COVID-19 agrees, noting that a rate implies a time component, which the CFR lacks. Time still enters through the period over which deaths among the cases are counted. Lipsitch and colleagues observe that this period is rarely stated and that a precise definition should include it, for example a one-month case fatality risk.

The CDC text also separates the case fatality rate from the death-to-case ratio, which divides deaths in a period by new cases in the same period, so the deaths need not come from those cases. A routine statistic of that kind cannot be read as the probability that a newly diagnosed case dies.

Case fatality, mortality rate and infection fatality ratio

These measures share a numerator of deaths but answer different questions because their denominators differ. A mortality rate divides deaths by the population in which they occurred, and for one disease the cause-specific mortality rate does the same with deaths from that cause, so it reflects how common the disease is as well as how often it kills. The case fatality rate divides by cases, so a rare disease with a high case fatality rate can still produce a low population mortality rate.

For an outbreak in a closed population, the two are linked through the attack rate, as follows.

$$M = AR \times CFR$$

where $M$ is the proportion of the population that dies of the disease over the outbreak, $AR$ is the attack rate (cases divided by the population at risk) and $CFR$ is the case fatality rate among those cases. The identity holds only when both measures use the same case definition and period, the attack rate's denominator is the population at risk to which $M$ refers, and every death from the disease counted in $M$ occurs among the counted cases. Deaths among infections never ascertained as cases, or deaths that fall after the counting window, break it. With illustrative figures, $M = 0.20 \times 0.03 = 0.006$, or 6 deaths per 1,000 people over the outbreak, a cumulative incidence of death for the period rather than a rate per person-year.

The WHO brief separates two fatality measures by what counts as a case. The infection fatality ratio (IFR) divides deaths by all infected individuals, while the CFR divides deaths by identified, confirmed cases. Estimating the IFR needs a count of all infections, and the brief names serological testing of a representative random sample of the population as an important way to obtain one. Because many infections are mild or asymptomatic and never detected, the CFR among confirmed cases is normally higher than the IFR for the same infection.

Why crude estimates mislead during an outbreak

A crude case fatality rate taken from surveillance counts part-way through an outbreak is exposed to two biases that push in opposite directions. The WHO brief states that reliable estimates are generally obtained at the end of an outbreak, once every case has died or recovered, and that the crude formula assumes both that detection of cases and deaths stays consistent over the outbreak and that every detected case has resolved. The brief was written for COVID-19 in August 2020 and set to expire after two years, but Lipsitch and colleagues report both biases in earlier outbreaks, including pandemic influenza in 2009 and Ebola.

The first bias is preferential ascertainment of severe cases. Early in an outbreak, surveillance captures people ill enough to seek care, so milder cases are missed and the case fatality rate is overestimated; Lipsitch and colleagues note that every proposed correction needs an auxiliary data source, such as serological surveys or contacts followed from before they became ill. The ISPOR and SMDM task force on dynamic transmission modelling makes the same point for economic models. Natural history estimated from surveillance data is likely to understate incidence and overstate severity, hospitalisation and case fatality.

The second bias comes from the delay between becoming a case and dying. Some detected cases will die later, so cumulative deaths divided by cumulative cases understates the final value, and the WHO brief notes that the effect is accentuated in fast-growing epidemics. Lipsitch and colleagues give a heuristic for exponential growth, given below.

$$B \approx 2^{d / T_d}$$

where $B$ is the factor by which the crude estimate understates the case fatality rate, $d$ is the delay between a case being reported and its death being reported, and $T_d$ is the epidemic doubling time in the same units. In their example of a three-week delay and a two-week doubling time, $B = 2^{1.5} \approx 2.8$. The heuristic assumes one fixed delay, whereas real delays vary.

One simple response, set out in the WHO brief, is to restrict the calculation to resolved cases, as follows.

$$CFR_r = \frac{D}{D + R}$$

where $CFR_r$ is the case fatality rate among resolved cases, $D$ is the number of deaths and $R$ is the number of recoveries among detected cases. The brief warns that this is still biased. If people who die typically do so sooner than others recover, the estimate is too high, and if the reverse holds it is too low. Survival methods that treat death and recovery jointly, such as the Kaplan-Meier based method that Ghani and colleagues tested on data from the 2003 SARS epidemic in Hong Kong, also use unresolved cases, but need individual-level data. The brief also asks for estimates within risk groups, because severity varies with age, sex and comorbidities and a single overall figure hides differences in case mix.

Comparing case fatality between treatments or settings

Economic evaluations sometimes need a treatment effect on mortality before trial evidence exists, and a difference in case fatality between treated and untreated patients in outbreak data can look like one. Lipsitch and colleagues caution against reading such a relative case fatality as causal. Patients who die before they can be admitted cannot appear in the admitted group (survivorship bias), the sickest patients may be the ones treated (confounding), and admission and death can both make a case more likely to enter a surveillance database (selection bias). The last two can act in either direction, so a model that takes its treatment effect from such data should test it through scenario or bias sensitivity analysis, as Lipsitch and colleagues suggest for selection bias. Systematic bias of this kind is not sampling error, so neither a one-way sensitivity analysis over a confidence interval nor a standard treatment of parameter uncertainty captures it.

Using case fatality as a model input

In a decision tree, the case fatality rate usually enters directly as the probability on the death branch that follows each case, and expected deaths are cases multiplied by that probability. Each death then carries the costs and health losses of dying at that age, namely expected deaths multiplied by remaining life expectancy give years of life lost, or the same deaths are valued in discounted QALYs. Because the figure covers the whole illness episode, the tree does not need to know when in the episode a death occurs.

A Markov model needs a transition probability for each cycle, so the period over which the case fatality rate was measured has to be matched to the cycle length. Where the only transitions are death or staying in the state, the standard approach described in Gidwani and Russell's tutorial assumes a constant underlying rate over the period (they describe the deaths as occurring evenly through it), as follows.

$$p_c = 1 - (1 - p)^{1/n}$$

where $p_c$ is the probability for one model cycle, $p$ is the probability over the measured period and $n$ is the number of cycles in that period. The same result comes from probability-to-rate conversion with $r = -\ln(1 - p)/t$ followed by rate-to-probability conversion with $p_c = 1 - e^{-r t_c}$, where $r$ is the rate, $t$ is the measured period and $t_c$ is the cycle length.

Two cautions apply. Cases usually leave the illness state by recovering as well as by dying, and Gidwani and Russell show that the two-state conversion gives incorrect probabilities when three or more transitions can occur in a cycle; applied alongside recovery, a weekly death probability derived this way produces fewer deaths than the case fatality rate implies. A tunnel state sequence that follows each case through its illness, or a branch applied when a case occurs, keeps the total right. Deaths after onset are also rarely spread evenly in time, so the constant-rate assumption places them only approximately.

The case definition behind the fatality figure should match the case count it multiplies. A CFR among confirmed cases goes with confirmed cases, and an IFR with all infections, which a dynamic transmission model usually tracks. Fatality should be age-specific where possible, because the ISPOR and SMDM task force notes that lowering infection risk raises the average age at infection, which for many diseases, such as hepatitis A and varicella, increases complication risk and mortality. In probabilistic analysis the case fatality rate is a probability, so a beta distribution fitted to deaths and survivors among cases is the usual choice. Fitted to mid-outbreak counts, that distribution reflects sampling error only, not the ascertainment and delay biases.

Worked example of one outbreak with three estimates and three cost-effectiveness results

The figures below are illustrative. Eight weeks into an outbreak, surveillance has recorded 2,000 confirmed cases, of which 40 have died, 1,000 have recovered and 960 are still ill. By the end, 20 of the 960 have also died, giving 60 deaths among the 2,000 cases. In this outbreak, deaths occur sooner after onset than recoveries.

1. Crude estimate at week 8. Deaths are divided by all confirmed cases.

$$CFR_{crude} = \frac{40}{2000} = 0.020$$

where $CFR_{crude}$ is the crude case fatality rate at week 8. It is too low, because 20 of the unresolved cases will still die.

2. Resolved-case estimate at week 8. Deaths are divided by deaths plus recoveries.

$$CFR_r = \frac{40}{40 + 1000} = \frac{40}{1040} \approx 0.0385$$

where $CFR_r$ is the case fatality rate among resolved cases at week 8. It is too high, because the unresolved cases are mostly people who will recover.

3. Final estimate. Once every case has resolved, the estimate is as follows.

$$CFR = \frac{60}{2000} = 0.030$$

where $CFR$ is the final case fatality rate among confirmed cases. If a serological survey suggested five infections for every confirmed case, there would have been 10,000 infections, and if the 60 deaths are all the deaths the infection caused, the infection fatality ratio would be 60 / 10,000 = 0.006, or 0.6%.

4. Effect on a cost-effectiveness result. A control programme is expected to prevent 1,000 confirmed cases at a net cost of £4,000,000, and each death averted is assumed to save 12 discounted QALYs; health losses from non-fatal illness are left out for simplicity. With the final estimate the results are as follows.

$$\frac{4000000}{1000 \times 0.03 \times 12} = \frac{4000000}{360} \approx 11111$$

where the denominator is the cases prevented multiplied by the case fatality rate, giving deaths averted, and by the QALYs saved per death averted. The crude estimate gives 20 deaths averted, 240 QALYs and about £16,667 per QALY, and the resolved-case estimate gives about 38.5 deaths averted, 461.5 QALYs and about £8,667 per QALY.

5. Interpret the result. The cost per QALY nearly doubles between the resolved-case and crude estimates from the same outbreak. At an illustrative threshold of £10,000 per QALY the programme looks cost-effective only with the resolved-case estimate, which is biased upwards here. A mid-outbreak case fatality rate should therefore enter a model with the likely direction of its bias stated and a range wide enough to cover both mid-outbreak estimates.

6. Convert to a weekly probability. Suppose the final 3% refers to deaths within four weeks of becoming a case, and a Markov model with weekly cycles keeps each case ill for four weeks, with death as the only exit during that time and all survivors moving to recovery at the end of week four. The weekly probability is as follows.

$$p_w = 1 - (1 - 0.03)^{1/4} \approx 0.0076$$

where $p_w$ is the weekly probability of death while ill. Four cycles return the 3%, as the four-week risk is $R_4 = 1 - (1 - 0.0076)^4 \approx 0.030$. Had the 3% been treated as an annual probability spread over 52 weekly cycles, the weekly figure would be about 0.00059 and a four-week illness would carry a death risk of about 0.23%, roughly one thirteenth of the intended value.

Common misreadings of case fatality figures

The most frequent error is to read the case fatality rate as a rate per unit of time, or as the chance that a member of the general population dies of the disease, when it applies only to cases over the period in which their deaths were counted. Figures from different countries or outbreaks are also compared without checking the case definition, the testing strategy and how far the outbreak had progressed, each of which changes the numerator or the denominator.

Carrying a figure from surveillance of severe cases into a model of a population that includes mild illness overstates deaths averted and flatters an intervention, while applying an infection fatality ratio to a count of confirmed cases understates deaths. A difference in case fatality between treated and untreated patients in outbreak data is not by itself evidence of a treatment effect, and a model that uses it as one should say so and test it.

Sources

• Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice, 3rd ed. Lesson 3: Measures of Risk, Section 3 (Mortality frequency measures). Atlanta: CDC; 2006, page last reviewed 2012. https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/section3.html
• World Health Organization. Estimating mortality from COVID-19: scientific brief, 4 August 2020. WHO reference number WHO/2019-nCoV/Sci_Brief/Mortality/2020.1. Geneva: WHO; 2020. https://www.who.int/publications/i/item/WHO-2019-nCoV-Sci-Brief-Mortality-2020.1
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