Offset Components in the Linear Predictor

Why Include an Offset?

A generalized linear model may need to incorporate a known contribution to the predictor in addition to the effects represented by estimated coefficients. For example, a model may have a scientifically specified baseline adjustment that must be included but is not to be estimated from the current responses. An offset provides a direct way to include that contribution. It changes the linear predictor, and therefore can change the conditional mean, while leaving the offset value fixed.

The key distinction is what is allowed to vary during model fitting: coefficients attached to covariates are estimated, whereas the offset is supplied as known. The offset is not a second response, a weight, or a coefficient that the model learns from the data.

Definition: For observations $i=1,\ldots,n$, let $Y_i$ have conditional mean $\mu_i=E[Y_i\mid x_i,o_i]$, where $x_i$ is a $p$-dimensional covariate vector and $o_i$ is a known real-valued offset. A GLM with coefficient vector $\beta$ and link $g$ has the offset form $$ g(\mu_i)=\eta_i=x_i^{\mathsf T}\beta+o_i. $$ The offset is the known additive term $o_i$; it is not multiplied by an unknown coefficient. The predictor must lie in the range of $g$, so that the corresponding mean is in the permitted mean space.

Here and below, the covariates and offsets are treated as observed and fixed when describing the conditional distribution of the responses. As in any GLM, $g$ maps an allowable mean to the linear-predictor scale. When $g$ is one-to-one, write $g^{-1}$ for its inverse on the predictor range. Then the conditional mean follows by applying that inverse to the whole predictor, including the offset.

From the Offset to the Conditional Mean

Result: Suppose $g$ is one-to-one on the allowable mean space and the predictor $x_i^{\mathsf T}\beta+o_i$ belongs to the range of $g$. Under the offset model, $$ E[Y_i\mid x_i,o_i]=\mu_i=g^{-1}\!\left(x_i^{\mathsf T}\beta+o_i\right). $$

Proof. The model specifies $g(\mu_i)=x_i^{\mathsf T}\beta+o_i$. Since the predictor lies in the range of $g$, apply $g^{-1}$ to both sides. This gives $\mu_i=g^{-1}(x_i^{\mathsf T}\beta+o_i)$. By definition, $\mu_i=E[Y_i\mid x_i,o_i]$. $\square$

This result is simple, but it prevents a common misinterpretation: an additive offset on the link scale need not be an additive change in the mean. Its effect on the mean depends on the inverse link. With the identity link, adding to the predictor adds the same amount to the mean. With a nonlinear link, the corresponding mean-scale change generally depends on the existing predictor.

Worked Example: A Fixed Log-Odds Adjustment

Suppose a binary response has a logit link and predictor $\eta_i=x_i^{\mathsf T}\beta+o_i$. For a particular observation, the estimated covariate contribution is $x_i^{\mathsf T}\beta=-0.2$, and the known offset is $o_i=\log 2$. Find the conditional success probability.

For the logit link, $g(\mu)=\log\{\mu/(1-\mu)\}$, so its inverse is $g^{-1}(\eta)=1/\{1+\exp(-\eta)\}$. First add the two predictor components: $$ \eta_i=-0.2+\log 2\approx -0.2+0.693147=0.493147. $$

Apply the inverse link: $$ \mu_i=\frac{1}{1+\exp(-0.493147)} \approx \frac{1}{1+0.61070} \approx 0.6208. $$

Thus the model's conditional success probability is approximately $0.6208$. The offset contributes $\log 2$ to the log odds; it does not contribute $\log 2$ directly to the probability. In particular, adding the offset to $-0.2$ and then treating the result as a probability would be incorrect.

What the Offset Is—and Is Not

A covariate effect and an offset can both appear as additive terms in the linear predictor, but they have different statistical roles. If $z_i$ is a scalar covariate with an estimated coefficient $\gamma$, its contribution is $z_i\gamma$, and $\gamma$ is part of the unknown parameter vector. An offset $o_i$, by contrast, is already specified and enters as $o_i$, with no unknown coefficient to estimate.

Predictor component	Form	What is unknown?
Estimated covariate effect	$z_i\gamma$	The coefficient $\gamma$ is estimated from the data
Offset	$o_i$	The value $o_i$ is supplied as known

The distinction is not merely a difference in notation. If $o_i$ is replaced by $z_i\gamma$, the model allows the size of the contribution to be determined from the response data. That is a different specification. Conversely, describing a covariate effect as an offset is appropriate only when its value is genuinely fixed by information external to the model being fitted, rather than an unknown effect the analysis aims to estimate.

Worked Example: A Known Term Under a Log Link

Consider a positive conditional mean with log link and predictor $$ \log(\mu_i)=\beta_0+\beta_1x_i+o_i. $$ For one observation, suppose $\beta_0+\beta_1x_i=0.4$ and the known offset is $o_i=\log 3$. Calculate the mean.

The full predictor is $$ \eta_i=0.4+\log 3\approx 0.4+1.098612=1.498612. $$

The inverse of the log link is the exponential function, so $$ \mu_i=\exp(1.498612)=\exp(0.4)\exp(\log 3) =3\exp(0.4) \approx 3(1.491824)=4.4755. $$

The known offset has been included before applying the inverse link. If it were omitted, this calculation would instead give $\exp(0.4)\approx1.4918$, which is not the mean specified by the stated model. Here the offset changes the mean multiplicatively because the inverse link is exponential. That conclusion follows from this link choice; it is not a general rule for all offsets and links.

Offset Changes and Link-Scale Comparisons

The additive role of an offset is always exact on the link scale. To see what this does and does not imply, compare two settings with the same covariate vector and coefficient vector but possibly different known offsets.

Result: Suppose two settings $a$ and $b$ have the same $x$ and $\beta$, with offsets $o_a$ and $o_b$. If their means are $\mu_a$ and $\mu_b$, then $$ g(\mu_a)-g(\mu_b)=o_a-o_b. $$ If $g^{-1}$ exists on the relevant predictor values, then $\mu_a=g^{-1}(x^{\mathsf T}\beta+o_a)$ and $\mu_b=g^{-1}(x^{\mathsf T}\beta+o_b)$. The difference in means is therefore determined by the inverse link, not by assuming it equals the difference in offsets.

Proof. By the model specification, $g(\mu_a)=x^{\mathsf T}\beta+o_a$ and $g(\mu_b)=x^{\mathsf T}\beta+o_b$. Subtract the second equality from the first. The common term $x^{\mathsf T}\beta$ cancels, leaving $g(\mu_a)-g(\mu_b)=o_a-o_b$. Applying the inverse link separately to each predictor gives the stated expressions for the means. $\square$

Worked Example: Same Covariates, Different Offsets

Suppose a logit model has covariate contribution $x^{\mathsf T}\beta=0$. Compare two settings with offsets $o_a=0$ and $o_b=\log 2$. Find each conditional probability and compare the changes on the two scales.

For setting $a$, the predictor is $\eta_a=0+0=0$, so $$ \mu_a=\frac{1}{1+\exp(0)}=0.5. $$ For setting $b$, the predictor is $\eta_b=0+\log 2\approx0.693147$, so $$ \mu_b=\frac{1}{1+\exp(-\log 2)} =\frac{1}{1+1/2} =\frac{2}{3}\approx0.6667. $$

The link-scale difference is exactly $$ g(\mu_b)-g(\mu_a)=\log 2-0=\log 2. $$ The probability difference, however, is $2/3-1/2=1/6\approx0.1667$, not $\log 2$. This verifies why an additive offset should first be interpreted on the link scale, then translated through the inverse link.

Another Link, Another Mean-Scale Effect

An offset is not restricted to a log or logit link. It can be included with any link for which the resulting predictor corresponds to an allowable mean. The model's family and link determine how the same kind of additive predictor term is translated into a conditional mean.

Worked Example: An Offset With the Identity Link

Suppose a model uses the identity link and has predictor $$ \mu_i=\beta_0+\beta_1x_i+o_i. $$ For an observation with $\beta_0=1$, $\beta_1=0.5$, $x_i=4$, and known offset $o_i=2$, calculate the conditional mean and check that the mean is admissible for a normal response.

The estimated-coefficient contribution is $\beta_0+\beta_1x_i=1+(0.5)(4)=3$. Adding the offset gives $$ \eta_i=3+2=5. $$

Because the inverse identity link is itself, $\mu_i=5$. A normal response permits any real mean, so this value is admissible. For a response family whose mean must be positive, the same identity-link calculation would require a separate check that the resulting predictor is positive.

Common Mistake: Treating the Offset as an Estimated Effect

Incorrect reasoning: “The offset is a predictor, so I should attach an unknown coefficient to it and estimate how much it matters.” This changes the model. With an unknown coefficient $\gamma$, the term is $\gamma o_i$, which can be adjusted using the response data. It is not the fixed-offset model $o_i$.

Correction: Include the offset as its supplied value, added to the other predictor terms. Estimate the coefficients that the model specifies as unknown, and use the inverse link on the full predictor to obtain the conditional mean. Also check that the resulting predictor lies in the range of the link, so the implied mean is valid.

Key takeaway: An offset is a known additive component of the linear predictor. Its addition is exact on the link scale; the conditional mean is obtained by applying the inverse link to the entire predictor, and its mean-scale effect depends on that inverse link.

Check Your Understanding

Answer each question by identifying the predictor components and, where appropriate, applying the inverse link.

In $g(\mu_i)=x_i^{\mathsf T}\beta+o_i$, which quantity is supplied as known, and which quantities are estimated?
A logit model has $x_i^{\mathsf T}\beta=0.3$ and offset $o_i=-0.3$. What is its linear predictor and conditional mean?
Under a log link, explain why an additive offset on the predictor scale need not produce an additive change in the mean.
Two records have the same covariates and coefficients, but their offsets differ by $c$. What is the difference between their link-transformed means?
Why does replacing an offset $o_i$ by an estimated term $\gamma o_i$ specify a different model?

Predictor component	Form	What is unknown?
Estimated covariate effect	\(z_i\gamma\)	The coefficient \(\gamma\) is estimated from the data
Offset	\(o_i\)	The value \(o_i\) is supplied as known