Law of Large Numbers

Law of Large Numbers in Health Economic Estimation and Simulation

The law of large numbers is the probability result that allows an analyst to treat an average as an estimate of an expected value. In health economics it is the reason the mean cost of a large trial arm approaches the cost per patient a payer will face, and the reason the averages reported by a probabilistic sensitivity analysis stop moving once enough iterations have been run. The law says where an average ends up, but not how many observations are needed to get there, which Chebyshev's inequality and the central limit theorem address. It also rests on conditions that health data can break, chiefly independent observations and a finite expected value.

Weak and strong forms for a sample of patient costs

For a trial arm, $X_1, \dots, X_n$ denote the costs of $n$ patients drawn independently from the same population, each with expected value $\mu$. Their sample mean is the figure a trial reports.

$$\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i$$

where $\bar{X}_n$ is the sample mean cost, $X_i$ is the cost of patient $i$ and $n$ is the number of patients.

Grinstead and Snell record that James Bernoulli gave the first proof, in Ars Conjectandi, published in 1713, and that the result is often called the law of averages. The weak law states that, for any margin greater than zero, the chance that the sample mean misses $\mu$ by more than that margin shrinks to zero.

$$\lim_{n \to \infty} \Pr\left(\left|\bar{X}_n - \mu\right| > \varepsilon\right) = 0$$

where $\Pr$ denotes probability, $\mu$ is the population mean cost, $\varepsilon$ is any positive tolerance and $n \to \infty$ means the number of patients increases without limit.

The strong law makes a statement about the whole sequence of sample means.

$$\Pr\left(\lim_{n \to \infty} \bar{X}_n = \mu\right) = 1$$

where the probability applies to the entire sequence of sample means. Owen explains that under the strong law the absolute error eventually falls below any tolerance and then stays there.

For a binary outcome coded 1 when an event such as an unplanned admission occurs and 0 otherwise, $\mu$ is the event probability, and the observed proportion of patients with the event approaches it.

Grinstead and Snell prove the law for independent trials with a finite mean and variance. Owen requires only that the mean exists, and notes that with a finite mean but infinite variance the law still assures convergence, while the usual rate and the central limit theorem intervals are lost.

How fast the average settles, and why the law cannot say

Owen points out that neither form of the law says how large $n$ must be for the error to become small. When the variance of a single observation is finite, the variance of the mean is that variance divided by $n$, and Chebyshev's inequality turns this into a bound that holds for any distribution.

$$\Pr\left(\left|\bar{X}_n - \mu\right| \ge \varepsilon\right) \le \frac{\sigma^2}{n\varepsilon^2}$$

where $\sigma^2$ is the variance of one patient's cost, $\sigma^2 / n$ is the variance of the mean of $n$ independent costs and $\varepsilon$ is the tolerance. The right-hand side falls to zero as $n$ grows, which is how Grinstead and Snell prove the weak law, though they call the inequality very crude. Owen calculates that a 99% interval built on it uses about 15 times as much computation as the normal approximation needs. The square root of $\sigma^2 / n$ is the standard error, and the choice of simulation sample size is covered by Monte Carlo error.

Where economic evaluation relies on the law

Several routine steps in economic evaluation apply the law, each with its own unit of averaging. Each also has its own way of failing the conditions of independence and a finite mean.

Mean cost per patient in a trial

Thompson and Barber argue that the arithmetic mean is the informative measure for cost data in pragmatic trials, because multiplying the mean cost by the number of patients to be treated gives the total budget a health service needs. The law makes the sample mean a consistent guide to the population mean as recruitment grows. Skewness changes the speed at which the mean settles but leaves its limit unchanged.

Risk pools and insurance

Arrow observed in 1963 that, through the law of large numbers, the indivisibility of most objects of insurance creates a range of increasing returns, so insurance companies have to be above a certain size. The risk pooling page works through the variance arithmetic for independent claims. The law offers less when costs share a common cause.

$$\operatorname{Var}(\bar{X}_n) = \frac{\sigma^2}{n} + \frac{n-1}{n}\rho\sigma^2 = \sigma^2\left(\rho + \frac{1-\rho}{n}\right)$$

where $\operatorname{Var}(\bar{X}_n)$ is the variance of average annual cost across $n$ members, $\sigma^2$ is the variance of one member's cost and $\rho$ is the correlation between any two members' costs, assumed equal for every pair. As $n$ grows the variance falls towards $\rho\sigma^2$ and not to zero, so an epidemic or price shock that raises every member's costs together is not diversified away.

Probabilistic sensitivity analysis

A probabilistic analysis estimates expected costs and QALYs as averages over iterations of the model. Owen states that the primary justification for simple Monte Carlo is through the laws of large numbers, and Koehler, Brown and Haneuse write the Monte Carlo simulation estimate as the mean over replicates, which converges to its target by the strong law. The NICE manual (PMG36, section 4.7.11) asks that enough model simulations be used to minimise the effect of Monte Carlo error. Extra iterations shrink only this simulation error, and the spread of results across iterations, which reflects parameter uncertainty, does not narrow.

Patient-level simulation

An individual patient simulation adds a second layer of averaging, since each run averages outcomes over many simulated patients. O'Hagan, Stevenson and Madan note that the large number of patients required in each run means that only a relatively small number of probabilistic runs is possible, and they develop analysis of variance methods to reduce that computational burden. With too few patients per run, patient-level noise is mixed into the variation across parameter draws.

Markov chain Monte Carlo

Draws from Markov chain Monte Carlo are dependent, so the independent-trials form does not apply directly. Grinstead and Snell give a law of large numbers for ergodic Markov chains, in which the proportion of steps spent in a state converges to its long-run probability whatever the starting state. In a finite run, draws taken before the chain has settled can still distort the average, the concern of the convergence page.

The limit in a non-linear model is the expected output

The law guarantees convergence to the expectation of whatever is being averaged. In a probabilistic analysis that is the expected model output, which differs from the output calculated at mean inputs whenever the model is non-linear.

$$\frac{1}{N}\sum_{j=1}^{N} g(\boldsymbol{\theta}_j) \to \operatorname{E}\left[g(\boldsymbol{\theta})\right] \ne g\left(\operatorname{E}[\boldsymbol{\theta}]\right)$$

where $g$ is the model, $\boldsymbol{\theta}_j$ is the parameter set drawn in iteration $j$, $N$ is the number of iterations, $\operatorname{E}[g(\boldsymbol{\theta})]$ is the expected output and $g(\operatorname{E}[\boldsymbol{\theta}])$ is the deterministic output at mean parameter values. The two are equal for a linear model and generally differ otherwise. The NICE manual (PMG36, section 4.7.12) states that the committee's preferred cost-effectiveness estimate should be derived from a probabilistic analysis when possible unless the model is linear, and section 4.7.13 defines non-linear as having no straightforward linear relationship between changes in inputs and outputs.

Ratios need separate care. Owen notes that for the expected value of a ratio to be finite the denominator should ordinarily not have a positive density at zero, and Grinstead and Snell state that the ratio of two independent standard normal variables has a Cauchy density, for which the expected value is undefined and the law does not hold. An ICER computed in each iteration has this structure whenever the incremental QALY distribution has positive density at zero, so a running mean of iteration ICERs need not settle at all. Mean incremental cost and mean incremental QALYs each satisfy the law, the ICER is formed as the ratio of those two means, and net monetary benefit, being linear in costs and QALYs, can be averaged directly.

Worked example with a patient-level model and an uncertain hazard

All numbers here are illustrative. A patient-level model simulates lifetime costs, and the standard deviation of cost between simulated patients is GBP 18,000. The analyst wants the mean of one run to lie within GBP 500 of that run's expected value with probability at least 95%.

1. Patients per run guaranteed by Chebyshev's inequality. Setting the bound $\sigma^2 / (n\varepsilon^2)$ no higher than 0.05 gives $n \ge \sigma^2 / (0.05\varepsilon^2)$, so $n \ge 18{,}000^2 / (0.05 \times 500^2) = 324{,}000{,}000 / 12{,}500 = 25{,}920$ patients.

2. Patients per run under the normal approximation. Using the central limit theorem instead, $n = (1.96 \times 18{,}000 / 500)^2 = 70.56^2 = 4{,}978.7$, so 4,979 patients. The distribution-free guarantee asks for 25,920 / 4,979 = 5.2 times as many. The smaller figure comes from the central limit theorem and holds only once the run mean is close to normal.

3. What the probabilistic mean converges to. Patients leave a disease state at a constant annual hazard, so the mean time in the state is one divided by the hazard. The hazard is uncertain, either 0.10 or 0.30 a year with equal probability, so its mean is 0.20. At the mean hazard the deterministic time in state is 1 / 0.20 = 5.0 years. The expected time is half of 1 / 0.10 plus half of 1 / 0.30, that is 0.5 × 10 + 0.5 × 3.333 = 6.667 years. At GBP 2,000 a year, undiscounted, the deterministic cost is GBP 10,000 and the expected cost is 2,000 × 20 / 3 = GBP 13,333. A probabilistic running mean converges to 6.667 years, and no number of iterations moves it towards 5.0.

4. Averaging ratios and net benefit. Four iterations of another comparison give these results, with net monetary benefit at an illustrative GBP 25,000 per QALY.

Mean incremental cost is 4,200 / 4 = GBP 1,050 and mean incremental QALYs are 0.232 / 4 = 0.058, so the ICER of the means is 1,050 / 0.058 = GBP 18,103 per QALY. The mean of the four iteration ICERs is 497,750 / 4 = GBP 124,438, dominated by iteration 3, where the QALY gain is almost zero. Net monetary benefit averages to 1,600 / 4 = GBP 400, which matches the value from the means, since 25,000 × 0.058 = 1,450 and GBP 1,450 less GBP 1,050 is GBP 400.

5. Interpretation. The law fixes the destination of a run mean but leaves the sample size to other tools. In a non-linear model, more iterations converge on the expected output, the quantity PMG36 prefers, and ICERs are formed from means because iteration ICERs need not converge.

Related results and where each applies

The law of large numbers concerns where an average settles, while the central limit theorem describes the approximate normal spread around that limit. Owen calls simple Monte Carlo unbiased because the expected value of the estimate equals the target at every sample size, so unbiasedness holds even for one draw and is separate from convergence. Variance reduction methods such as Latin hypercube sampling aim at the same limit with less simulation error for a given number of model runs.

Common misreadings

Most misuses treat the law as a promise about a finite sample or about something other than the average. Six recur in cost analysis and simulation.

1. Expecting a run of results to even out. Later patients are no cheaper after a run of costly ones. The excess is diluted, and the gap between total cost and $n\mu$ can keep growing while the average converges.
2. Treating a large sample as free of bias. A large but unrepresentative cohort converges precisely to the wrong value, and bias is untouched by sample size.
3. Reading more iterations as less uncertainty. Extra iterations reduce Monte Carlo error only, not decision uncertainty.
4. Counting correlated units as independent. Clustered patients and successive MCMC draws add less information than their number suggests.
5. Averaging a quantity with no finite mean. Iteration ICERs can fail the law entirely.
6. Reading the limit as a sample size. The law names no $n$ at which an average is close enough.

Sources

• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. https://www.aeaweb.org/aer/top20/53.5.941-973.pdf
• Grinstead CM, Snell JL. Introduction to Probability. 2nd ed. American Mathematical Society; 2003. Chapter 8 and section 11.3. https://math.dartmouth.edu/~prob/prob/prob.pdf
• Koehler E, Brown E, Haneuse SJ. On the assessment of Monte Carlo error in simulation-based statistical analyses. The American Statistician. 2009;63(2):155-162. https://doi.org/10.1198/tast.2009.0030
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 4.7.11 to 4.7.13. https://www.nice.org.uk/process/pmg36
• O'Hagan A, Stevenson M, Madan J. Monte Carlo probabilistic sensitivity analysis for patient level simulation models: efficient estimation of mean and variance using ANOVA. Health Economics. 2007;16(10):1009-1023. https://doi.org/10.1002/hec.1199
• Owen AB. Monte Carlo Theory, Methods and Examples. 2013. Chapter 2. https://artowen.su.domains/mc/Ch-intro.pdf
• Thompson SG, Barber JA. How should cost data in pragmatic randomised trials be analysed? BMJ. 2000;320(7243):1197-1200. https://doi.org/10.1136/bmj.320.7243.1197