SIMULATION IN PRACTICE: THE BALANCING INTERCEPT

Simulation is an important tool within epidemiology for both learning and developing new methodology (1–5). Unfortunately, few epidemiology training programs teach basic simulation methods. Briefly, when conducting a simulation experiment, we generally follow the same basic steps. We first decide which variables to include, as well as their distributions and associations—often aided by a causal diagram. We then generate those variables by sampling from their specified distributions and estimate whatever target parameter is of interest (e.g., sample average or causal effect). We finally repeat the process multiple times, building a distribution for the target parameter from the estimates obtained in each replicate.

Table 1.

Estimated Marginal Probabilities Under Different Simulation Specifications, in a Single Simulation of 10,000 Units

Simulationa	Y Model Intercept	X Model Intercept	L Model Intercept	(L=1)	(X=1)	(Y=1)
Generate Y	0.3	N/A	N/A	N/A	N/A	0.295
Generate X and Y	0.3	0.5	N/A	N/A	0.490	0.400
		0.5	N/A	N/A	0.490	0.288
Generate L, X, and Y	0.3	0.5	0.8	0.799	0.577	0.497
			0.8	0.799	0.501	0.301

Abbreviations: , estimated probability of ; NA, not applicable.

^a Variables generated: Y, outcome; X, exposure; L, confounder.

Here, we briefly demonstrate one key simulation practice: the balancing intercept, which allows us to specify in the first step above the marginal mean of a variable and see that mean preserved (in expectation) in the simulated sample. This is attractive because the marginal probability is an easily interpretable quantity. Additionally, if the simulation is designed to mimic real data, the marginal probability of a variable is often a known, reported characteristic of the data (although, conversely, there could be areas in which researchers know more about the conditional probability of that variable). Use of a balancing intercept also makes validation of simulation code simple; regardless of how other simulation parameters change, as long as the balancing intercept is specified correctly, the marginal probability of a variable in the simulated data set should match the desired marginal probability.

Suppose we were only interested in generating a binary outcome for simulated individuals, from a Bernoulli distribution with probability . We can express this probability in terms of an intercept-only model: , where . It is then relatively easy to use any software program of one’s choosing to generate (see Web Table 1 and the Web Appendix, available at https://www.doi.org/10.1093/aje/kwab039, for example code). Although here we focus on binary , much of the discussion below also applies to continuous .

The simulation becomes slightly more complicated when we expand our simulation to include a binary exposure (with ), which affects . Now, when we generate , we need to account for the association has with . This is commonly done by specifying a model for the conditional probability of given , , rather than . One could also model the joint distribution directly. For example, we can use the linear model (specifically, fit with an identity link function):

where is the probability of when and is the risk difference for , which we set to be . The question here is: How should we determine the appropriate value of in this model? One approach would be to add to the intercept-only model above:

This is intuitive because when in the above formula, . We will refer to this intercept as the “standard intercept.”

The main drawback of the standard intercept is that the marginal probability of in our simulated sample, , will not be the same as the desired as long as . We see this in our simulation (Table 1). In the scenario where , was 0.400, rather than the desired 0.3. We can show why this occurs using the law of total probability:

If we do wish to preserve the specified marginal probability, we can replace the standard intercept with what we refer to here as the “balancing intercept.” To do this, we could specify in the model for

where is the desired marginal probability of . This intercept balances out the influence of the effect of on on the expectation of by including an offset that multiplies the effect size for by the expected value of . For binary variables, the expected value of is the marginal probability, ; if were continuous, the expected value would be the mean. Note that the sign is opposite of that in the formula for the conditional probability; that is, for (positive) , we include the term . In the simulation scenario in which , we estimated that , which is closer to the we desired. The difference here is largely due to random error, which decreases if we increase .

The approach presented here can be extended to account for more variables. Let us add to our simulation a binary confounder , with , which affects the exposure and the outcome . Accordingly, in our simulation, we model the conditional probabilities of and given , as follows:

Note that we first simulate based on , then based on and together. Furthermore, each equation includes a balancing intercept:

In our simulation, we specified and (i.e., no interaction between and on the additive scale) and estimated that and .

We can model the conditional probabilities using functions other than the linear model. In fact, when simulating a binary variable, we might prefer to use the inverse of the logit function (expit function), which will guarantee that will fall between (0, 1). A linear model, on the other hand, could lead to probabilities falling outside this range (e.g., a probability of 1.04). The expit model (see Web Table 2 and the Web Appendix for code) takes the form:

where is the intercept, is the conditional log odds ratio (log(OR)) for , and is the conditional log(OR) for . The standard intercept for this model would be the marginal log odds of :

We can instead specify a balancing intercept, as follows:

The balancing intercept allows a researcher to set a desired marginal probability for and then see that same probability manifested in the resulting simulation (in expectation). We should note, though, that the balancing intercept approach is an approximation and could fail as the model for the conditional probability of becomes extreme (e.g., when variables in the model have complex distributions). Even so, we believe that using a balancing intercept is a best practice for simulation design that deserves wider attention and adoption.