Perturbing Subject-Specific Selection Probabilities

Why Perturb Selection Log-Odds?

In a selection model, the probability that an outcome is observed can depend on the outcome itself. A subject whose outcome is unusual may therefore have a different chance of being observed than a subject with an otherwise similar history. A small, subject-specific change to the selection log-odds asks a focused diagnostic question: how would the fitted outcome and response parameters move if the model assigned this subject slightly different response odds?

This is not the same question as what happens when a subject receives less weight in the likelihood. An offset changes a specified part of the response mechanism. Case weighting changes the subject’s whole likelihood contribution. The distinction matters especially in a selection model, where the response mechanism and outcome distribution are linked through the likelihood.

Define the Perturbation in the Selection Model

Let \(Y_i\) denote subject \(i\)'s outcome history and \(R_i\) its response or observation pattern. Write \(\beta\) for outcome-model parameters and \(\gamma\) for response-model parameters; let \(\theta=(\beta^\mathsf{T},\gamma^\mathsf{T})^\mathsf{T}\) collect the fitted parameters. For a response indicator \(R_{ij}\) at occasion \(j\), a selection-model logit may be written

Here \(R_{ij}=1\) means that the outcome is observed. A subject-level offset \(\omega_i\) adds a constant to the log-odds for the response events specified in advance. For example, if the offset applies to all relevant occasions for subject \(i\),

Thus, a positive \(\omega_i\) raises the conditional log-odds of response by \(\omega_i\), holding the other model inputs fixed. It multiplies the odds by \(e^{\omega_i}\); it does not add \(\omega_i\) to the response probability. If the model instead defines the event as dropout, the sign interpretation reverses. The event and the occasions affected by an offset must be stated explicitly.

For incomplete outcomes, the likelihood contribution is an observed-data contribution, not a likelihood that treats missing outcomes as observed. In a general selection model it has the form

The integral is over the missing components of \(Y_i\), with the observed components fixed at their recorded values. The model must define the outcome distribution and response mechanism, and the integral or its numerical approximation must be well defined. A local derivative further requires a differentiable likelihood and a locally identifiable, interior fitted solution with nonsingular observed information. If the response model has boundary estimates, the fit is not locally regular, or the Hessian is singular, the usual derivative may not exist or may be unstable.

Deriving the Local Parameter Change

Let \(\hat\theta(\omega)\) be the maximum-likelihood estimate under the offsets, and let \(U(\theta,\omega)=\partial\ell(\theta;\omega)/\partial\theta\) be the score vector. At an interior maximum, \(U(\hat\theta(\omega),\omega)=0\). Differentiate this equation with respect to one offset \(\omega_i\), then evaluate at the unperturbed fit \(\hat\theta=\hat\theta(0)\). Define \(H=\partial^2\ell/\partial\theta\partial\theta^\mathsf{T}\) and \(c_i=\partial^2\ell/\partial\theta\partial\omega_i\), both evaluated at \((\hat\theta,0)\). The chain rule gives

If the observed information \(J=-H\) is nonsingular, solving this equation yields the local influence direction:

This is a derivative, so for a sufficiently small change \(\delta\), the first-order approximation is \(\hat\theta(\delta)-\hat\theta(0)\approx\delta J^{-1}c_i\). The formula separates two ingredients: the offset’s effect on the score, \(c_i\), and the fitted likelihood’s curvature, represented by \(J^{-1}\). Since \(\theta\) includes both \(\beta\) and \(\gamma\), the resulting vector reports local changes in both outcome and response parameters.

A useful justification of the cross-derivative comes from the complete-outcome response contribution. For a single Bernoulli response event with logit \(\eta=x^\mathsf{T}\gamma+\omega_i\), its log-likelihood is \(r\eta-\log(1+e^\eta)\), and its score for \(\gamma\) is \(x(r-\pi)\), where \(\pi=\operatorname{expit}(\eta)\). Differentiating that score with respect to the offset gives

This direct response-score change can be nonzero even when the offset has no direct derivative in the outcome score. Nevertheless, the outcome-parameter estimate can move because \(J^{-1}\) reflects dependence between the fitted parameters. With missing outcomes, the observed-data cross-derivative must be computed from the integrated likelihood; it is not generally obtained by simply substituting unobserved outcomes into the complete-data expression.

Distinguishing Selection Offsets From Case Weights

A case-weight perturbation changes the multiplier on subject \(i\)'s entire log-likelihood contribution. A selection offset changes the response mechanism for the specified subject and events, while leaving the outcome distribution itself unperturbed as a model component. These perturbations therefore have different score cross-derivatives and answer different diagnostic questions.

Diagnostics, Interpretation, and Limitations

A large coordinate of \(J^{-1}c_i\) indicates that a small log-odds change for subject \(i\) can produce a large local movement in that fitted parameter, in the parameter’s own units. It does not establish that the subject is erroneous, that the assumed offset is plausible, or that the response model is correct. In particular, an outcome-parameter sensitivity can arise through parameter coupling even if the offset’s direct outcome-score cross-derivative is zero.

Before interpreting a calculation, check that the offset has the intended sign and applies to the intended response event and occasions. Verify that the fitted observed-data likelihood is differentiable at the estimate, the information matrix is nonsingular, and the derivatives use the same likelihood and parameterization as the fit. As a basic numerical check, refit at a small positive and negative offset and compare the finite-difference slope with \(J^{-1}c_i\). Large disagreement can indicate an offset that is not small enough, inaccurate derivatives, an ill-conditioned information matrix, or likelihood curvature that makes the linear approximation unreliable.

A single offset also imposes a specific perturbation pattern: it shifts each selected logit by the same amount wherever it is applied. Different patterns, such as occasion-specific changes, require a vector of offsets and a corresponding matrix of cross-derivatives. A scalar result should not be interpreted as covering patterns it did not perturb. Furthermore, a selection-offset diagnostic is conditional on the chosen selection and outcome model. It does not resolve uncertainty about missing outcomes that is not identified from observed data, and it does not validate the assumed dependence between outcomes and response.

A common mistake is to say, “A positive response-log-odds offset must increase every fitted response-model coefficient.” The correction is that the offset directly raises the specified subject’s response odds, but the refitted coefficients move according to \(J^{-1}c_i\). Curvature and parameter dependence can make a coefficient’s fitted change negative, and can transmit change to outcome parameters. Another mistake is to call this a case-weight diagnostic because both methods perturb one subject. The correction is to identify what was perturbed: response odds in one case, versus the entire likelihood contribution in the other.