Initial Values with Offsets, Trials, and Exposure

Why margins must match the observation structure

An initial intercept is useful when it gives the optimizer fitted means on a sensible scale before slopes are updated. For ordinary binary responses without offsets, the response proportion often provides a direct starting point. With grouped binomial data, however, observations can represent different numbers of trials; with Poisson data, counts can correspond to different exposure durations or populations at risk. In both cases the model’s total expected response depends on those trial counts or exposures. An unweighted average of the recorded responses generally does not match that total.

This tutorial derives intercept starts by setting the slopes to zero and matching the model’s fitted total to the observed total. The resulting start is a computational choice, not a claim that the intercept-only model describes all the structure in the data. Offsets remain part of the linear predictor: they are known contributions and are not estimated as coefficients.

Binomial counts: match successes across trials

If all offsets are zero, setting the intercept so that each fitted probability is the same makes the fitted expected number of successes equal the observed total when \( \sum_i m_i p_i=\sum_i Y_i. \) Thus the probability to use is the pooled proportion of successes, not the mean of the observation-specific proportions. When offsets differ, a single probability cannot be assigned to every group: the intercept must instead be chosen so that the offset-adjusted probabilities have the right trial-weighted total.

Binomial offsets: solve a weighted intercept equation

With offsets, the intercept-only fitted probabilities are \(p_i(\alpha)=\operatorname{logit}^{-1}(\alpha+o_i)\), where \(\operatorname{logit}^{-1}(u)=1/(1+\exp(-u))\). The margin-matching equation is \( \sum_i m_i\operatorname{logit}^{-1}(\alpha+o_i)=\sum_i Y_i. \) Except in special cases, it does not reduce to taking the logit of one pooled proportion and subtracting a common offset. The probabilities differ across observations, and each is weighted by its number of trials.

To prove this, define \(F(\alpha)=\sum_i m_i\operatorname{logit}^{-1}(\alpha+o_i)\). Each inverse-logit term is continuous and strictly increasing in \(\alpha\), so \(F\) is continuous and strictly increasing. Because each offset is finite, \( \lim_{\alpha\to-\infty}F(\alpha)=0,\qquad \lim_{\alpha\to+\infty}F(\alpha)=M. \) The condition \(0<Y<M\) therefore places \(Y\) strictly between these limits. The intermediate value theorem gives a solution to \(F(\alpha)=Y\), and strict increase gives uniqueness. A scalar root finder, such as bisection on a bracket with \(F(\alpha_{\rm low})<Y<F(\alpha_{\rm high})\), can compute it. \(\square\)

The interior-total condition matters. If \(Y=0\), fitted probabilities approach zero only as \(\alpha\to-\infty\); if \(Y=M\), they approach one only as \(\alpha\to+\infty\). In either case there is no finite intercept that exactly matches the total.

Poisson counts: account for exposure

The fitted total under this intercept-only start is \(\sum_i t_i\exp(\alpha)\). Setting it equal to the observed count total gives a closed-form intercept whenever the total count is positive: \( \hat\alpha_{\rm start}=\log\left(\frac{\sum_iY_i}{\sum_i t_i}\right). \) This is the log of the pooled count rate. It is not, in general, the log of the unweighted average of the individual rates \(Y_i/t_i\).

For fixed slopes equal to zero, the log-likelihood, omitting terms that do not depend on \(\alpha\), is \( \ell(\alpha)=\sum_i\{Y_i(\alpha+\log t_i)-t_i\exp(\alpha)\}+C, \) where \(C\) is constant in \(\alpha\). Its derivative is \( \ell'(\alpha)=\sum_iY_i-\exp(\alpha)\sum_i t_i =Y_\cdot-\exp(\alpha)T_\cdot. \) Setting this derivative to zero gives \(\exp(\alpha)=Y_\cdot/T_\cdot\). The second derivative is \( \ell''(\alpha)=-\exp(\alpha)T_\cdot<0 \), so the stationary point is the unique maximum. If \(Y_\cdot=0\), the derivative is negative for every finite \(\alpha\); the likelihood increases toward its supremum as \(\alpha\to-\infty\), and no finite intercept matches the total. \(\square\)

Offsets and current slope values

The same margin principle applies if slopes have already been assigned starting values rather than set to zero. For a binomial logit model, replace \(o_i\) in the intercept equation by \(o_i+x_i^\mathsf{T}\beta^{(0)}\), where \(\beta^{(0)}\) is the chosen starting slope vector. For a Poisson log-link model, the fitted means become \(t_i\exp(\alpha+x_i^\mathsf{T}\beta^{(0)})\), and matching the total gives \( \hat\alpha_{\rm start} =\log\left\{\frac{\sum_iY_i}{\sum_i t_i\exp(x_i^\mathsf{T}\beta^{(0)})}\right\}, \) provided \(\sum_iY_i>0\). These formulas make explicit that the intercept compensates for the offsets and any nonzero starting linear predictors; it should not simply be copied from an offset-free calculation.

A practical sequence is to choose the slope starts, retain all known offsets, and then solve for the intercept that makes the total fitted mean agree with the observed total. For Poisson log-link models the equation is explicit. For binomial logit models with varying offsets, it is a one-dimensional monotone equation. In both cases the target is a total count: successes across trials for binomial data, and events across exposure for Poisson data.

Common mistake: averaging the recorded responses

A tempting shortcut is to average the observed proportions \(Y_i/m_i\) equally for binomial data, or the rates \(Y_i/t_i\) equally for Poisson data, and then transform that average into an intercept. That reasoning treats each row as if it represented the same number of trials or the same exposure. Unless the trial counts or exposures are equal, it can produce fitted means whose sum misses the observed total. The correction is to match the total directly: weight binomial probabilities by \(m_i\), and weight Poisson rates by \(t_i\). When offsets vary in a binomial model, solve the weighted equation rather than applying one pooled proportion to every group.