Continuous Data

Continuous Data in Health Economics and Evidence Synthesis

Many of the numbers that enter a cost-effectiveness analysis are continuous measurements, including costs per patient, health state utility values, QALYs, length of stay and clinical measures such as blood pressure. The data type decides which summary statistics, effect measures, regression models and probability distributions are appropriate, so an early error in describing data carries through to the final ratio. The difficulty in health economics is that the continuous data analysts care most about are rarely well behaved, because costs pile up at zero with a long right tail and utilities are capped at full health.

Continuous outcomes among the five data types

The Cochrane Handbook names five common types of outcome data, dichotomous, continuous, ordinal, counts and rates, and time-to-event. For continuous data each individual's outcome is a measurement of a numerical quantity, and the Handbook adds that the term conventionally refers to a variable that can take any value in a specified range, with weight, area and volume as truly continuous examples.

The same methods are used for data that are not strictly continuous. The Handbook notes that measurement scales and counts of large numbers of events are often analysed this way, and that longer ordinal scales acquire properties similar to continuous outcomes. Costs, with their very many possible values, belong here. Continuous measurements sit on an interval or ratio scale, unlike an ordinal scale that only ranks.

Time to an event is continuous in nature but forms its own type. The Handbook warns that analysing it with methods for continuous outcomes, such as mean times to event, is not appropriate, because times are known only for participants who had the event and excluding censored participants almost certainly introduces bias. A continuous model is a different idea again, a decision model run in continuous time.

Summaries and effect measures for continuous outcomes

A continuous outcome is usually summarised by its mean and standard deviation. For meta-analysis the Cochrane Handbook asks for the mean, the standard deviation and the number measured in each group, and it calls misreading a standard error as a standard deviation a particularly misleading error.

Medians are close to means when the distribution is symmetrical but can differ greatly under skew, and the Handbook observes that medians are often reported precisely because the data are skewed. Its rough check for skewness applies when an outcome has a lowest possible value, such as zero for a cost. The mean minus that value is divided by the standard deviation, and a ratio below 2 suggests skew while a ratio below 1 is strong evidence of it.

Two groups are compared with an effect size built from the group summaries. The Handbook names the mean difference and standardised mean difference as the common measures, with the ratio of means as an option. Mean Difference and Standardised Mean Difference are covered in separate entries, and Hedges' g and Cohen's d are versions of the standardised form.

Costs as skewed continuous data

Mihaylova and colleagues describe resource use and cost data as non-negative, often with substantial positive skewness, sometimes heavy tails, and often multimodal, with a mass at zero for people who use no services. The mean is still the statistic a health economist needs.

Mihaylova and colleagues state that it is widely accepted in health economics that the estimated population mean cost is the statistic of interest to policy makers. Thompson and Barber give the budget reason, since the total cost of treating a population is the arithmetic mean cost multiplied by the number of patients, and other measures cannot provide that total. The incremental cost in an ICER is therefore a difference in arithmetic means.

Uncertainty in a mean cost can rest on the central limit theorem or on bootstrapping; Mihaylova and colleagues note that both are asymptotically valid but have uncertain properties in small, non-normal samples. Generalised linear models with a gamma specification model costs on the raw scale, so no back-transformation is needed, whereas a log transformation needs one and is not appropriate for zeros. Two-part models handle zeros by estimating the probability of incurring any cost in one part and the mean cost among those with some cost in the other.

Utilities and QALYs as bounded continuous data

NICE DSU Technical Support Document 12 describes health-related quality of life data, particularly the EQ-5D preferred by NICE, as not normally distributed, typically skewed, bimodal or trimodal, bounded by the limits of the index and often include negative values, with a lower limit of -0.594 for the UK EQ-5D index. These features affect any analysis of a health state utility value or a QALY.

Basu and Manca describe utility data as typically negatively skewed, with a spike at 1 that mirrors the spike at zero in costs, the statistical trace of a ceiling effect. Their simulations found that ordinary least squares can give substantially biased covariate effects when those effects are large or when there is a large spike at 1, and they propose beta-based regression with a two-part version. Modelling one minus the utility moves the spike to zero, where a two-part model applies directly.

Choosing distributions for continuous parameters in PSA

A probabilistic sensitivity analysis represents uncertainty in mean inputs, not variation between patients. The NICE manual (PMG36, section 4.7.10) asks for distributions that characterise the uncertainty in the precision of mean parameter values, chosen to represent the evidence and not chosen arbitrarily. The ISPOR-SMDM task force separates stochastic uncertainty, the variability between identical patients, from parameter uncertainty, which it likens to the standard error of an estimate in regression.

The task force states that with plenty of data the sampling distribution of the arithmetic mean is normal whatever the shape of the underlying data, and that otherwise standard choices apply, including gamma or log-normal distributions for right-skewed parameters. A gamma or log-normal distribution cannot produce a negative cost, which a wide normal distribution can. For utilities, Technical Support Document 12 judges that a normal distribution usually describes uncertainty in the mean adequately, and suggests sampling the decrement from full health from a log-normal or gamma distribution in exceptions such as patient-level simulation from a cohort with widely varying scores and a relatively low or high mean. The beta distribution is also common for utility weights, as in the tutorial by Alarid-Escudero and colleagues, which gives costs gamma distributions and utility weights beta distributions. The method of moments fits either family from a mean and standard error.

Worked example from patient costs to a gamma distribution

All numbers in this example are illustrative. Ten patients in a trial arm incur annual costs of GBP 0, 0, 150, 220, 310, 400, 520, 750, 1,900 and 6,250. The total is GBP 10,500, so the mean is GBP 1,050, the median is GBP 355 and the sample standard deviation is GBP 1,909.26.

1. Check the shape. The skew check gives $\frac{1050 - 0}{1909.26} = 0.55$, below 1, so the costs are strongly skewed.

2. Choose the summary for the decision. Treating 2,000 such patients costs GBP 2,100,000 on the mean but would appear to cost GBP 710,000 on the median. Only the mean gives the budget.

3. Move from patient spread to uncertainty in the mean. Suppose the full arm has 200 patients with the same mean and standard deviation.

$$SE = \frac{s}{\sqrt{n}} = \frac{1909.26}{\sqrt{200}} = 135.0$$

where $s$ is the sample standard deviation in GBP and $n$ is the number of patients.

4. Fit a gamma distribution by the method of moments. Matching the gamma mean to the sample mean and its variance to the squared standard error gives the parameters.

$$k = \frac{\bar{x}^2}{SE^2} = \frac{1050^2}{135.0^2} = 60.49, \qquad \theta = \frac{SE^2}{\bar{x}} = \frac{135.0^2}{1050} = 17.36$$

where $k$ is the shape, $\theta$ is the scale in GBP and $\bar{x}$ is the mean cost. Gamma(60.49, 17.36) has mean GBP 1,050 and a central 95% range of about GBP 802 to GBP 1,331, close to the normal interval of GBP 785 to GBP 1,315 because 200 patients make the mean nearly normal.

5. Avoid fitting to the standard deviation. Using GBP 1,909.26 in place of the standard error gives a shape of 0.30 and a 95% range from about GBP 0 to GBP 6,627. That describes how individual patients differ, and placing it on the mean input would greatly overstate parameter uncertainty. A normal distribution built on the standard deviation would put about 29% of draws below zero.

Dichotomising continuous measurements

Continuous outcomes are sometimes split into two groups, such as responders defined by a minimal clinically important difference. The Cochrane Handbook accepts that reviewers may dichotomise to express results as an odds ratio or risk ratio, to aid interpretation or because most studies reported results that way.

The price is information. Altman and Royston explain that splitting at the median reduces statistical power as much as discarding a third of the data, that patients just either side of the cut-point are treated as very different, and that choosing the cut-point giving the most convincing result overestimates the difference and should never be done. In an economic model a responder split is reasonable when costs or utilities genuinely change at the threshold and the cut-point is fixed in advance.

Continuous data compared with nearby data types

Categorical Data covers outcomes recorded as one of a set of categories, with dichotomous data as the two-category case. Short ordinal scales are often collapsed into two groups. Counts of common events can be treated as continuous, with the Handbook's warning that they are likely to be highly skewed, and otherwise as rates with Poisson regression or negative binomial regression. Time-to-event data need survival methods, such as parametric survival models, because of censoring.

Limitations and common misreadings

Most errors come from treating cost and utility data as normal and unbounded. Others come from confusing variation between patients with uncertainty in a mean.

1. Reporting medians for costs in place of means. A median describes the typical patient but cannot be multiplied up to a budget or used in an ICER.
2. Confusing the standard deviation with the standard error. A PSA distribution fitted to a standard deviation represents patient variation, not parameter uncertainty.
3. Assuming normality for small skewed samples. In small skewed samples both the central limit theorem and the bootstrap have uncertain properties, so the analysis model needs care while the arithmetic mean stays the target.
4. Ignoring the bounds of utility data. A spike at full health can bias estimated effects in linear models.
5. Reporting a back-transformed log mean as the mean cost. The exponentiated mean of logged costs is a geometric mean, which falls below the arithmetic mean for right-skewed costs.
6. Dichotomising without need. A cut-point discards information, and a data-driven cut-point exaggerates differences.

Sources

• Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, et al (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Sections 6.1.1, 6.5 to 6.8. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06
• Deeks JJ, Higgins JPT, Altman DG, McKenzie JE, Veroniki AA (editors). Chapter 10: Analysing data and undertaking meta-analyses. In: Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 10.5.3. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-10
• Mihaylova B, Briggs A, O'Hagan A, Thompson SG. Review of statistical methods for analysing healthcare resources and costs. Health Economics. 2011;20(8):897-916. https://doi.org/10.1002/hec.1653
• Thompson SG, Barber JA. How should cost data in pragmatic randomised trials be analysed? BMJ. 2000;320:1197-1200. https://doi.org/10.1136/bmj.320.7243.1197
• Ara R, Wailoo A. NICE DSU Technical Support Document 12: The use of health state utility values in decision models. Decision Support Unit, ScHARR, University of Sheffield; July 2011. Section 4.1. https://doi.org/10.15131/shef.data.33744907
• Basu A, Manca A. Regression estimators for generic health-related quality of life and quality-adjusted life years. Medical Decision Making. 2012;32(1):56-69. https://doi.org/10.1177/0272989X11416988
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://doi.org/10.1177/0272989X12458348
• Alarid-Escudero F, Krijkamp E, Enns EA, Yang A, Hunink MGM, Pechlivanoglou P, Jalal H. An introductory tutorial on cohort state-transition models in R using a cost-effectiveness analysis example. Medical Decision Making. 2023;43(1):3-20. https://doi.org/10.1177/0272989X221103163
• Altman DG, Royston P. The cost of dichotomising continuous variables. BMJ. 2006;332:1080. https://doi.org/10.1136/bmj.332.7549.1080
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 2022, last updated 13 October 2026. Section 4.7.10. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2