Likelihood Ridges Between Outcome and Response Effects

Why Likelihood Ridges Matter

A selection model specifies both an outcome distribution and a model for whether the outcome is observed. These components can compensate: a parameter setting with a different outcome distribution may produce much the same observed cases if its response probabilities also change. A likelihood contour then stretches along a direction in which the fit changes little—or, in some models, not at all. The result is important because a good observed-data fit does not by itself guarantee that the outcome and response effects are separately determined.

The earlier material on weak identification in selection-model likelihoods motivates looking beyond the optimizer’s reported point estimate. Here the focus is on the geometry of the likelihood: how a ridge arises, how to recognize the trade-off, and what can and cannot be inferred from it. A binary example makes the compensation exact and transparent.

A Binary Selection Model and Its Observed Likelihood

This model assumes a common outcome prevalence and response rule across individuals, conditional independence across individuals, and the stated logistic form for response given the outcome. It does not assume that response is independent of the outcome: \(\gamma\) is the outcome-associated response effect on the log-odds scale. As usual for a selection coefficient, that association is not, by itself, a causal effect.

The three observable category probabilities are \(a=(1-p)q_0\), \(b=pq_1\), and \(c=(1-p)(1-q_0)+p(1-q_1)=1-a-b\). If the observed counts in these categories are \(n_{10},n_{11},n_0\), the likelihood, apart from the multinomial coefficient that does not depend on the parameters, is \(a^{n_{10}}b^{n_{11}}c^{n_0}\). Thus the observed data determine information about \(a\) and \(b\), but the three model parameters \(p,\alpha,\gamma\) need not be separately determined.

Deriving the Ridge

The calculation in the example extends to every interior parameter setting that reproduces fixed observed probabilities \(a\) and \(b\). Choose a value of \(p\), then solve the observed-probability equations for the response probabilities:

For these to be valid probabilities, both must lie between zero and one. For positive \(a\) and \(b\), this requires \(p>b\) and \(p<1-a\). For every \(p\) in that interval, the logistic parameters are then determined by

Substitution verifies the claim: \((1-p)q_0=a\), \(pq_1=b\), and therefore the missing category probability is \(1-a-b\) all along this curve. The likelihood is constant on the curve. In the three-dimensional parameter space \((p,\alpha,\gamma)\), it is an exact likelihood ridge, not merely a numerical optimizer’s difficulty.

The observed distribution has only two freely varying category probabilities, since the third must make the probabilities sum to one. The model has three parameters. In this example, the ridge makes that dimension count concrete: one parameter direction remains undetermined by the observed distribution. More generally, dimension counting can alert us to possible identification problems, but it does not prove a ridge in every model; that requires examining how parameters map to observed-data probabilities.

Likelihood Contours and Inferential Consequences

A contour plot is a visual representation of parameter values with the same likelihood, or the same change in log likelihood from a reference value. In this model, contours in the full \((p,\alpha,\gamma)\) space include the exact ridge derived above. If one looks at a two-parameter slice instead, contours may look curved or narrow simply because the slice cuts across the ridge; a plot of one slice does not establish that all parameter directions are well determined.

Along an exact ridge, the log likelihood is constant, so its derivative in the tangent direction is zero. At the maximum matching the empirical category proportions, the score is zero along the ridge, and differentiating the score along the ridge shows that the observed-information matrix has a null tangent direction. Away from a stationary point, constancy along a curved ridge alone does not imply that the observed-information matrix has a null direction; the second derivative along the curve also involves the score and the curve's acceleration. In a less exact but nearly flat ridge, the curvature is small rather than zero. Then modest changes in the data, numerical tolerance, or model specification can cause large changes in the estimated parameter combination along that direction.

Diagnostics, Alternatives, and Limitations

Useful diagnostics ask whether the fitted likelihood is informative in the parameter directions that matter, not merely whether an optimizer converged. Inspect likelihood contours or evaluate the log likelihood along deliberate parameter perturbations. Check whether very different outcome prevalences and response coefficients produce nearly unchanged likelihoods. Also verify that implied response probabilities stay in the feasible range across plausible parameter values, and distinguish a broad, shallow contour from a genuinely constant ridge. Numerical integration or optimization problems can also produce unstable fits, but a computational warning and structural non-identification are different problems; changing the algorithm cannot create information absent from the observed-data distribution.

The binary model gives an exact ridge because the observed categories reveal only \(a\) and \(b\), while \(p,\alpha,\gamma\) can vary together to preserve them. In richer settings, repeated measurements, covariates, restrictions shared across groups, or external information may add constraints and alter the geometry. Whether they actually identify the parameters depends on the model and the observed-data distribution; it should be checked rather than presumed. A pattern-mixture model expresses the outcome distribution by response pattern instead of specifying response conditional on outcome, so it offers a different parameterization and does not automatically supply information that the data lack. The earlier material on observed-data likelihoods is useful for seeing which parts of either formulation enter the observed likelihood.

The main limitation is interpretive as well as numerical: if several outcome and response parameter combinations fit equally well, reporting one optimizer output as though the data had uniquely established it is misleading. External assumptions can select among points on a ridge, but conclusions then depend on those assumptions. A likelihood ridge is evidence about what the observed data identify under the specified model, not evidence that one of the ridge points is true.

Common Mistake