Discrete Data

Discrete Data and Event Counts in Health Economics

Much of the evidence behind an economic evaluation arrives as counts. Trials and routine datasets record how often each patient was admitted, saw a GP or had an exacerbation, and those counts are costed or turned into rates and probabilities for a decision model. Counts are whole numbers with a floor at zero, are often dominated by patients with no events and vary more between patients than simple models expect, so the choice of summary and model changes both the estimated treatment effect and the cost difference.

Counts among the outcome data types

Ali and Bhaskar divide numerical data into discrete and continuous measurements. Discrete numerical data are recorded as whole numbers such as 0, 1, 2 or 3, and in their wording observations that can be counted constitute discrete data, while observations that can be measured constitute continuous data. The Cochrane Handbook lists counts and rates, built from the number of events each individual experiences, as one of five common outcome types. Rose and colleagues add that counts cannot be analysed without the exposure over which they were counted, usually follow-up time.

In health economics the counted events fall into two families. Clinical events such as relapses or exacerbations are outcomes in their own right. Resource-use events, such as admissions, outpatient attendances, GP consultations or prescriptions, are the quantities that costing multiplies by unit costs. Outcomes that place each patient in one of a set of groups are covered under categorical data, and measured quantities such as cost or utility under continuous data.

Rates per person-time and the rate ratio

A count means little without the time over which it accrued. The Cochrane Handbook defines rates as counts related to the time during which events could have happened, and gives the example of 18 myocardial infarctions over 314 person-years of follow-up, a rate of 0.057 per person-year. Two arms are compared by the rate ratio, which is analysed on the log scale.

$$RR = \frac{E_E / T_E}{E_C / T_C}, \qquad SE(\ln RR) = \sqrt{\frac{1}{E_E} + \frac{1}{E_C}}$$

where $E_E$ and $E_C$ are the total events in the experimental and comparator arms and $T_E$ and $T_C$ the total person-time at risk in each arm. Rose and colleagues note that this standard error depends on the numbers of events, not on sample sizes. A rate ratio computed this way between two groups is the incidence rate ratio.

A rate differs from a probability in its denominator. Gidwani and Russell describe a probability as events in a period divided by the number of people followed, ranging from 0 to 1, and a rate as events divided by total time at risk, ranging from 0 to infinity. The Cochrane Handbook adds that in a randomised trial a rate ratio is often close to the risk ratio obtained by dichotomising patients, but the two differ when an intervention changes how likely some patients are to have several events.

Poisson models, overdispersion and the negative binomial

The Cochrane Handbook notes that counts of rare events are often called Poisson data, after the Poisson distribution. Rose and colleagues state that Poisson regression can be used if events occur randomly at a constant rate and independently of each other, that the mean and variance of a Poisson distribution are equal, and that this is often implausible because one event can change the rate of later events in the same person. Counts that vary more than the Poisson allows are overdispersed.

The negative binomial distribution relaxes the equal mean and variance. Lee and colleagues describe it as a gamma mixture of Poisson distributions, in which the Poisson mean varies between patients.

$$\operatorname{Var}(X) = m + \frac{m^2}{r}$$

where $m$ is the expected count per patient and $r$ is the dispersion or shape parameter. As $r$ grows large the variance approaches $m$ and the model reduces to the Poisson. In negative binomial regression, as in the Poisson case, unequal follow-up can enter as an offset, and Lee and colleagues included follow-up days this way. Mihaylova and colleagues note that generalized linear models are used for resource use with Poisson and negative binomial specifications.

Keene and colleagues reanalysed exacerbation data from two large trials in chronic obstructive pulmonary disease, TRISTAN and ISOLDE, and reported that the Poisson model does not satisfactorily account for variability between patients, while a negative binomial model, which assumes a separate Poisson parameter for each patient, gave higher estimated exacerbation rates and wider confidence intervals without changing the trials' conclusions. They described the negative binomial approach as the method of choice at the time.

Excess zeros in resource-use counts

Resource-use counts pile up at zero, because many patients use none of a given service during follow-up. Lee and colleagues list clinic visits, admissions and days in hospital as typical zero-inflated counts, with a large share of zeros, right skew and overdispersion.

Two families of models target the zeros. Zero-inflated models mix a point mass at zero with a count distribution, and two-part or hurdle models first estimate the probability of any use and then model the count among users. Mihaylova and colleagues observe that zeros are expected when resource use is modelled with discrete distributions, but that two-part modelling may still be indicated when zeros are numerous. In simulations based on a telemedicine trial, Lee and colleagues found little difference in bias for the treatment effect between the negative binomial and its zero-inflated version, and advised against choosing a zero-inflated model on the share of zeros alone.

From counts of resource use to costs

Špacírová and colleagues describe the valuation step of bottom-up methods as multiplying the number of units of each resource used by a unit cost, which is the core of bottom-up costing and micro-costing. The NICE manual (PMG36, section 4.4.3) states that estimates of resource use may include healthcare service use outcomes such as length of stay, number of hospitalisations, and outpatient or primary care consultations.

Because each patient's cost is a sum of counts times prices, the cost distribution inherits the features of the counts, including a mass at zero and a long right tail. Thompson and Barber explain that a budget estimate comes from multiplying the arithmetic mean cost by the number of patients to be treated, so the mean count, not the median, is the quantity a cost-effectiveness analysis needs.

Event counts as decision model inputs

Decision models usually take event rates and convert them to per-cycle probabilities. Gidwani and Russell give the standard identities for a node with two possible transitions.

$$r = \frac{-\ln(1 - p)}{t}, \qquad p = 1 - \exp(-rt)$$

where $p$ is the probability of at least one event in a period of length $t$ and $r$ is the constant rate. The same conversion appears under rate to probability conversion, and the resulting transition probability feeds a Markov model. Gidwani and Russell warn that this conversion fails when three or more transitions can occur in a cycle. Repeated events add a second trap, because $p$ is the chance of at least one event, so a model that costs at most one event per patient per cycle undercounts patients with several.

Worked example with exacerbation counts in a two-arm trial

All numbers in this example are illustrative. A trial randomises 300 patients to a new inhaler and 300 to usual care, follows every patient for one year, and records 240 exacerbations needing hospital care in the new arm and 360 under usual care.

1. Rates and rate ratio. Each arm has 300 person-years, so the rates are 240/300 = 0.80 and 360/300 = 1.20 exacerbations per person-year. The rate ratio is 0.80/1.20 = 0.667, and $\ln 0.667 = -0.405$.

2. Poisson interval. The standard error is $\sqrt{1/240 + 1/360} = \sqrt{0.004167 + 0.002778} = 0.0833$. The 95% interval is $\exp(-0.405 \pm 1.96 \times 0.0833)$, which is 0.566 to 0.785.

3. Check the dispersion. Usual-care counts have mean 1.2 and variance 3.0, well above the Poisson value. Setting $m + m^2/r$ equal to 3.0 with $m = 1.2$ gives $r = 1.44/1.8 = 0.8$. A Poisson model predicts $\exp(-1.2) = 30.1\%$ of patients with no exacerbation, while the negative binomial predicts $(0.8/2.0)^{0.8} = 48.0\%$.

4. Negative binomial interval. Treating $r = 0.8$ as known and common to both arms, and applying the delta method to the mean count of $N$ patients, the variance of the log mean in each arm is $\frac{1}{Nm} + \frac{1}{Nr}$. This gives 1/360 + 1/240 = 0.006944 for usual care and 1/240 + 1/240 = 0.008333 for the new inhaler, a standard error of $\sqrt{0.015278} = 0.1236$ and an interval of 0.523 to 0.849. The point estimate is unchanged, but the interval is about half as wide again as the Poisson one.

5. Costs. At an illustrative unit cost of GBP 2,000 per exacerbation, expected costs are 1.20 × 2,000 = GBP 2,400 and 0.80 × 2,000 = GBP 1,600 per patient-year, a saving of GBP 800. Per 1,000 patients that is 400 exacerbations and GBP 800,000 a year.

6. A model that allows one event per cycle. With an annual cycle, $p = 1 - \exp(-1.2) = 0.699$ under usual care and $p = 1 - \exp(-0.8) = 0.551$ with the inhaler, so costing one event per cycle gives GBP 1,398 and GBP 1,101 and a saving of GBP 296, about 63% below the saving from the counts. A monthly cycle gives 12 × 0.0952 = 1.142 and 12 × 0.0645 = 0.774 events a year and a saving of GBP 736, closer to GBP 800 but still short.

The rate fixes the expected cost, the dispersion fixes the uncertainty around the treatment effect, and the model structure decides whether repeated events are counted.

Discrete data compared with nearby data types

Counts sit between binary and continuous outcomes and are often forced into one or the other. The Cochrane Handbook calls treating count data as dichotomous a common error, since the total number of events can exceed the number of patients, and Rose and colleagues note that an arm with more events than participants cannot hold a dichotomous outcome. The Handbook allows counts of common events to be treated as continuous data with a mean difference, and Rose and colleagues add that the normal approximation behind this is reasonable for large counts but poor for small ones.

Time-to-first-event analysis, covered under survival analysis, discards events after the first. The Cochrane Handbook also cautions that analysing counts as rates assumes a constant risk across participants and over time.

Limitations and common misreadings

Most errors with discrete data come from ignoring time, ignoring variation between patients or capping counts at one. Each changes either the effect estimate or the cost.

1. Dividing events by patients, not person-time. Unequal follow-up then distorts the comparison between arms.
2. Reading a rate as a probability. A rate of 1.2 per person-year cannot be a probability and must be converted.
3. Fitting a Poisson model to overdispersed counts. Confidence intervals come out too narrow.
4. Choosing a zero-inflated model on the share of zeros alone. Simulation evidence suggests the plain negative binomial often performs as well.
5. Using the median count for costing. Budgets and cost-effectiveness depend on the mean.
6. Allowing one event per model cycle. Repeated events are lost and the savings from reducing them are understated.

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