Pseudo-Random Number

Pseudo-Random Number: Concept Architecture

Orientation and learning roadmap

The page follows one generated value from an initialized algorithm into a health simulation. It then separates the role of the seed from the event probability, illustrates repeatable draws, and explains how stream management and validation affect a model comparison.

From a seed to a sequence

A computer algorithm begins from an internal state, often initialized through a seed, and applies a specified update rule to generate successive values. Given the same implementation, initial state, and sequence of calls, it produces the same outputs again. Those outputs can mimic selected statistical properties of random draws even though the underlying process is deterministic.

An abstract generator can be written as:

$$S_{n+1}=f(S_n),\qquad U_n=g(S_n),$$

where $S_n$ is the generator state, $f$ advances it, and $g$ maps state to an output $U_n$ on a defined scale. The algorithm, seed or full initial state, stream identifier, and number and order of draws all matter for reproducing a sequence. The output number is not itself a probability estimate; the model uses it with a probability rule to select an outcome.

Many simulation routines provide values treated as approximately uniform on the interval $[0,1)$. A uniform draw is then transformed or compared with a threshold to simulate a Bernoulli event, a categorical transition, a waiting time, or another random variable. The suitability of that use depends on the generator's statistical properties and implementation, not on whether a short excerpt of the sequence looks irregular to a reader.

A worked event draw

Suppose a model assigns a 0.20 probability of an adverse event for a particular patient in one cycle. If $U$ is a suitably generated uniform draw, the model can record an event when $U<0.20$. Under the modeling assumptions, the event probability is 0.20 across repeated hypothetical draws; one patient's outcome remains either event or no event.

$$E=\begin{cases}1,&U<0.20,\\0,&U\geq0.20.\end{cases}$$

For illustration, draws of 0.08, 0.73, 0.19, and 0.44 produce outcomes 1, 0, 1, and 0. Two events in four draws do not change the specified 0.20 probability to 0.50; a tiny realized sample can differ substantially from its generating probability. These four numbers are teaching inputs, not evidence of the output of a named generator.

A model may also draw a value from a distribution for an uncertain parameter, then use separate draws to simulate individual events conditional on that parameter. The first represents parameter uncertainty; the second represents individual stochastic variation. Keeping the two levels distinct prevents a simulation result from being misinterpreted as a parameter estimate.

Reproducibility is more than recording a seed

A fixed seed can help reproduce a run, but it does not make its results statistically reliable. Changing the algorithm, software version, parallel scheduling, or order of random-number calls can change the output even when a seed number is reused. A reproducible analysis should record the generator and software environment, initialization method, stream design, model version, and analysis code alongside the seed or state.

A single run may also contain simulation noise. For a decision comparison, check whether incremental outcomes remain stable across sufficiently many individuals and independent replications or seeds. State a prespecified precision or convergence target appropriate to the decision rather than assuming that a repeatable result is an accurate expectation.

Compare strategies with well-managed streams

An individual-level economic model may simulate the same target population under two strategies. If unrelated random sequences generate patient characteristics or background events in each strategy, some variation in their difference comes from the draw assignment rather than the strategies. Common random numbers can align corresponding sources of variation and reduce the Monte Carlo variance of an incremental estimate when implemented correctly.

This requires careful stream design: draw for the same underlying event from the corresponding stream in each strategy, while preserving separate streams for distinct event types. Simply resetting one global seed before each arm can lose alignment when the arms consume different numbers of draws. Conversely, streams intended to be independent must be created by a method supported by the generator rather than by assuming adjacent seed integers always guarantee independence.

The variance identity explains the potential benefit of positively correlated outcomes $Y_A$ and $Y_B$ generated with common draws:

$$\operatorname{Var}(Y_A-Y_B)=\operatorname{Var}(Y_A)+\operatorname{Var}(Y_B)-2\operatorname{Cov}(Y_A,Y_B).$$

If the common draws produce a positive covariance while preserving the correct marginal simulations, the variance of the difference can fall. This technique improves simulation precision for a comparison; it does not eliminate uncertainty in the clinical evidence or automatically favor either strategy.

What to validate before relying on draws

A generator's output should be assessed for the intended use, including its period, distributional behavior, stream separation, and any software-specific guarantees. Passing a few statistical tests is not proof of perfect randomness or model validity. A simple demonstration generator can be useful for teaching but should not be assumed suitable for a large or parallel decision simulation.

1. Match the generator to the task. Document the algorithm and use a tested implementation appropriate to the simulation, rather than an ad hoc recurrence.
2. Check transformations. Confirm that draws map to the intended event probabilities or distributions with correct interval boundaries and units.
3. Design streams deliberately. Separate mechanisms and parallel runs, and document when corresponding strategies share random inputs.
4. Assess simulation error. Repeat runs or increase simulation size to establish that the decision-relevant results have adequate numerical precision.
5. Preserve provenance. Record software, version, seed or state, stream mapping, and model code needed to reproduce the outputs.

Simulation-grade pseudorandom numbers and cryptographically secure generators serve different purposes. A health economic model needs sound statistical and reproducibility properties for its simulation; a security-sensitive application may additionally require resistance to prediction. The fact that an algorithm is deterministic does not by itself tell the analyst whether it is fit for either task.

Evidence used for this explanation

NIST describes the deterministic generation and seed relationship, while model-specific guidance explains why streams and repeated runs matter in health economic simulation. The sources support these methodological distinctions; they do not certify an unnamed generator used in a particular model.

• NIST Computer Security Resource Center. Pseudorandom number generator glossary entry.
• NIST Dictionary of Algorithms and Data Structures. Pseudo-random number generator.
• Karnon J et al. Modeling using discrete event simulation: ISPOR-SMDM Good Research Practices Task Force 4. Value in Health (2012).
• Using Common Random Numbers in Health Care Cost-Effectiveness Simulation Modeling. Medical Decision Making (2013).