Two Ways to Represent Exposure
In a count study, exposure measures the opportunity for events to occur: person-time at risk, population size, or another relevant denominator. A larger exposure will usually produce a larger expected count even if the underlying rate is unchanged. The modeling choice is whether to encode that proportional relationship as a fixed assumption or estimate how counts scale with exposure.
Let \(Y_i\) be the event count for observation \(i\), \(E_i>0\) its exposure, and \(x_i\) a vector of predictors that does not include an intercept. Assume the counts are conditionally independent with Poisson distributions. Both specifications below use a log link; they differ in the role assigned to \(\log E_i\).
An estimated-exposure model specifies \(\log \mu_i=\beta_0+x_i^\mathsf{T}\beta+\gamma\log E_i\), estimating the coefficient \(\gamma\) from the data.
The offset model is often described as a rate model because dividing its mean by exposure gives \(\mu_i/E_i=\exp(\beta_0+x_i^\mathsf{T}\beta)\). Thus, at the same predictor values, its expected rate does not depend on exposure. The estimated-exposure model instead allows that rate to depend on exposure. This distinction is about the model’s assumptions and estimands, not simply about two computational ways to fit the same model.
Worked Example: A Fixed Proportional Rate
Suppose a Poisson model for event counts among two groups uses person-years as exposure. Group 0 has \(x=0\), group 1 has \(x=1\), and the fitted offset model is \(\log\mu_i=\log E_i+\beta_0+\beta_1x_i\), with \(\exp(\beta_0)=0.02\) events per person-year and \(\exp(\beta_1)=1.5\).
For group 0, the fitted rate is \(0.02\). For group 1, the fitted rate is \(0.02\exp(\beta_1)=0.02(1.5)=0.03\) events per person-year. The rate ratio is therefore \(0.03/0.02=1.5\).
If a group 0 observation contributes \(1{,}000\) person-years, its expected count is \(1{,}000(0.02)=20\). If a group 1 observation contributes \(1{,}500\) person-years, its expected count is \(1{,}500(0.03)=45\). The different expected counts reflect both the different exposures and the different rates. Within either group, doubling exposure doubles the expected count while leaving the expected rate unchanged.
The Rate Relationship Implied by an Estimated Coefficient
When the exposure coefficient is estimated, it is useful to separate the count mean into its exposure and rate components. Dividing \(\mu_i\) by \(E_i\) gives the expected count per unit of exposure, conditional on the predictors.
Proof: Exponentiating the linear predictor yields \(\mu_i=\exp(\beta_0+x_i^\mathsf{T}\beta+\gamma\log E_i)\). Since \(E_i>0\), \(\exp(\gamma\log E_i)=E_i^\gamma\), giving the stated mean. Dividing this expression by \(E_i\) gives the rate formula. At fixed predictor values, the factor \(\exp(\beta_0+x_i^\mathsf{T}\beta)\) is the same for observations \(a\) and \(b\), so it cancels in the ratio of means. It also cancels in the ratio of rates, leaving the exposure ratio raised to \(\gamma-1\). \(\square\)
The coefficient \(\gamma\) is the elasticity of the expected count with respect to exposure: a 1% increase in exposure corresponds locally to a \(\gamma\)% increase in the mean count, with other predictors fixed. The corresponding rate elasticity is \(\gamma-1\). In particular, \(\gamma=1\) gives a count proportional to exposure and a rate that is constant across exposure levels at fixed predictors. If \(\gamma<1\), the expected rate declines as exposure increases; if \(\gamma>1\), it increases.
Worked Example: Interpreting an Estimated Exposure Effect
Suppose the estimated model has \(\gamma=0.75\). Compare two observations with the same predictors and exposures \(E_a=4\) and \(E_b=16\), so the second exposure is four times the first.
The ratio of expected counts is \((16/4)^{0.75}=4^{0.75}=\sqrt{8}\approx2.828\). The second expected count is about 2.83 times the first, not four times as large.
The ratio of expected rates is \((16/4)^{0.75-1}=4^{-0.25}=1/\sqrt{2}\approx0.707\). Thus the rate at exposure 16 is about 70.7% of the rate at exposure 4. For an explicit scale, suppose \(\exp(\beta_0+x^\mathsf{T}\beta)=0.01\). Then at exposure 4, the mean is \(0.01(4^{0.75})\approx0.0283\), and the rate is \(0.0283/4\approx0.00707\). At exposure 16, the mean is \(0.01(16^{0.75})=0.08\), and the rate is \(0.08/16=0.005\). The rate ratio \(0.005/0.00707\) is approximately \(0.707\), as derived.
What Fixing the Coefficient at One Means
The offset model is exactly the estimated-exposure model with the restriction \(\gamma=1\). The restriction is scientifically meaningful: it says that, conditional on the other predictors, expected counts scale proportionally with exposure. It is not the claim that exposure has no association with counts. Under this restriction, multiplying exposure by a factor multiplies the expected count by that factor.
Proof: Substituting \(\gamma=1\) into \(\log\mu_i=\beta_0+x_i^\mathsf{T}\beta+\gamma\log E_i\) gives \(\log\mu_i=\log E_i+\beta_0+x_i^\mathsf{T}\beta\), exactly the offset specification. To obtain the score, differentiate the Poisson log-likelihood. Since \(\partial\log\mu_i/\partial\gamma=\log E_i\) and \(\partial\mu_i/\partial\gamma=\mu_i\log E_i\), the derivative of the \(i\)th contribution is \(y_i\log E_i-\mu_i\log E_i=(y_i-\mu_i)\log E_i\). Summing over observations gives the displayed score. At an interior maximum of the unrestricted likelihood, this score is zero; the offset model instead fixes \(\gamma=1\) rather than estimating it. \(\square\)
This score equation also indicates what information estimates \(\gamma\): the model relates exposure values to the observed-minus-fitted counts, weighted by \(\log E_i\). If exposure barely varies, the data may provide little information about its coefficient. Estimation also does not make the coefficient a causal effect automatically; an association between exposure and event risk, or differences in how exposure is measured, can affect the fitted relationship.
Worked Example: Comparing the Specifications on Two Records
Consider two independent Poisson counts with no additional predictors: \(E_1=1,\ y_1=2\) and \(E_2=4,\ y_2=4\). This tiny example illustrates the different fitted relationships; it is not a basis for reliable scientific inference.
In the estimated-exposure model, \(\mu_i=\exp(\beta_0)E_i^\gamma\). There are two parameters and two observations, and the fitted means can equal the observed counts. Setting \(\exp(\beta_0)=2\) matches \(\mu_1=2\). Matching \(\mu_2=4\) then requires \(2(4^\gamma)=4\), so \(4^\gamma=2\) and \(\gamma=\log(2)/\log(4)=0.5\). The fitted rates are \(2/1=2\) and \(4/4=1\); this model allows the rate to decline with exposure.
For the offset model, write \(\mu_i=\lambda E_i\), where \(\lambda=\exp(\beta_0)\) is the common rate. The log-likelihood derivative with respect to \(\lambda\) is \((y_1+y_2)/\lambda-(E_1+E_2)\). Setting it to zero gives \(\hat\lambda=(2+4)/(1+4)=6/5=1.2\). The fitted means are \(1.2\) and \(4.8\), and both fitted rates are \(1.2\). The two specifications differ because one permits exposure-dependent rates and the other requires a common rate at the same predictor values.
Choosing the Specification and Avoiding a Common Mistake
An offset is appropriate when exposure is the amount of opportunity for events and proportional scaling is part of the scientific model: for example, when a rate is intended to be comparable across populations of different sizes. An estimated exposure coefficient answers a different question: how does the expected count scale with exposure in the chosen model, and does that scaling depart from proportionality? It may be useful when there is a substantive reason to allow nonproportional scaling and enough informative variation in exposure.
The choice should follow the target question and study design, not be made solely because software makes both options available. If exposure is a denominator that defines the rate of interest, estimating its coefficient changes that rate relationship. If exposure is itself a predictor whose association with counts is of interest, fixing its coefficient at one may impose a restriction that does not answer the intended question. Neither choice, by itself, resolves confounding or makes an estimated association causal.
- Common mistake: “Exposure appears in both models, so both must estimate the same rate relationship.”
- Correction: An offset fixes the exposure coefficient at one and implies a rate independent of exposure at fixed predictors. Estimating \(\gamma\) allows the rate to vary with exposure as \(E^{\gamma-1}\).
- Common mistake: “An estimated \(\gamma\) different from one proves that exposure causes the rate to change.”
- Correction: The fitted coefficient describes a conditional model relationship. A causal interpretation requires additional design and assumptions beyond the Poisson specification.
Check Your Understanding
Use the formulas and distinctions developed here to answer the following questions.
- In an estimated-exposure model with \(\gamma=1.2\), what are the elasticities of the expected count and expected rate with respect to exposure?
- At fixed predictors, exposure increases from 5 to 20. Under an offset model, what is the ratio of expected counts, and what is the ratio of expected rates?
- Why is the offset model a restricted version of the estimated-exposure model?
- What does the score for \(\gamma\) imply about the observed and fitted counts at an interior maximum?
- Give one scientific reason to use an offset and one reason an estimated exposure coefficient might be relevant.