Selection Models for Nonignorable Panel Attrition

Why Model the Outcome and Attrition Together?

When panel response depends on an outcome that is itself missing, the observed records do not directly reveal how outcomes among respondents compare with outcomes among people who dropped out. A model that uses only observed histories cannot, by itself, learn this relationship. A selection model makes the relationship explicit by specifying both an outcome distribution and a response probability conditional on the outcome.

This joint specification is useful because it makes the assumptions needed for inference visible. It does not make nonignorable attrition harmless: conclusions can depend strongly on how the outcome and response processes are modeled, and some features of those processes are not determined by observed data alone.

Selection-Model Definition

Definition: Let \(Y_{it}\) be the outcome for panel unit \(i\) at time \(t\), and let \(R_{it}=1\) if that outcome is observed and \(R_{it}=0\) otherwise. A selection model specifies an outcome distribution \(f(\mathbf y_i\mid\mathbf x_i;\theta)\) and an observation-process distribution \(p(\mathbf r_i\mid\mathbf y_i,\mathbf x_i,\mathbf z_i;\gamma)\). It is nonignorable when response probabilities depend on outcomes that are missing, after conditioning on the modeled observed information.

Here \(\mathbf y_i=(Y_{i1},\ldots,Y_{iT})\) denotes the outcome path, \(\mathbf r_i=(R_{i1},\ldots,R_{iT})\) the observation pattern, and \(\mathbf x_i\) observed covariates and histories. The vector \(\mathbf z_i\) may include variables used in the response model but excluded from the outcome model. The parameters \(\theta\) govern the outcome distribution; \(\gamma\) governs response. Both distributions, including their conditioning information, must be specified before forming a likelihood.

For example, with monotone dropout, a unit observed at time \(t-1\) may either respond at \(t\) or drop out. One possible transition model is

P(R_{it}=1\mid R_{i,t-1}=1,\mathbf h_{i,t-1},Y_{it},X_{it},Z_{it}) =\operatorname{logit}^{-1} \left(\gamma_0+\gamma_yY_{it}+\gamma_h^\top\mathbf h_{i,t-1} +\gamma_x^\top X_{it}+\gamma_z^\top Z_{it}\right),

where \(\mathbf h_{i,t-1}\) is the observed history available before the response decision. If \(\gamma_y\ne0\), the chance of responding depends on the current outcome even when that outcome is unobserved for dropouts. This is an MNAR specification. In contrast, a response probability that depends only on observed history is a different assumption; likelihood methods for ignorable missingness do not justify using that assumption when current unobserved outcomes drive response.

From the Full-Data Model to the Observed-Data Likelihood

The joint model gives a full-data density for outcomes and response indicators. Since outcomes are unavailable when \(R_{it}=0\), their values must be integrated out of the likelihood. In a single response occasion, with covariates held fixed, the full-data density is

f(y\mid x;\theta)\, \pi(y,x,z;\gamma)^r \{1-\pi(y,x,z;\gamma)\}^{1-r},

where \(\pi(y,x,z;\gamma)=P(R=1\mid Y=y,x,z)\). If \(R=1\), the outcome is observed and the likelihood contribution is the outcome density multiplied by its response probability. If \(R=0\), only the event of nonresponse is observed, so the likelihood contribution sums over or integrates across possible outcome values.

L_{\mathrm{obs}}= \begin{cases} f(y\mid x;\theta)\,\pi(y,x,z;\gamma), & r=1,\ y\ \text{observed},\\[4pt] \displaystyle\int f(y\mid x;\theta)\{1-\pi(y,x,z;\gamma)\}\,dy, & r=0,\ y\ \text{unobserved}. \end{cases}

The second line is the key point: it averages the nonresponse probability over the outcome distribution, rather than evaluating it at an invented or observed value. In a panel, the corresponding likelihood integrates over every missing outcome path, with the response probabilities conditioned on the modeled histories. If dropout is monotone, the response process is typically represented as a sequence of conditional transitions. The integration must respect those dependencies; multiplying single-wave contributions that assume independent response decisions is not generally valid.

Worked Example: Response Rates Under an Outcome-Dependent Selection Model

Consider a hypothetical population with a binary outcome \(Y\), where \(P(Y=0)=0.6\) and \(P(Y=1)=0.4\). Let \(Z\) be a binary variable excluded from the outcome distribution, and suppose the response model is

\operatorname{logit}P(R=1\mid Y,Z)=-1+0.5Y+0.8Z.

Assume this model is correctly specified for the hypothetical example, \(Z\) is observed for everyone, and the outcome distribution is the same at both values of \(Z\). Because the response probability changes with \(Y\), response is nonignorable when \(Y\) is missing.

For \(Z=0\), the response probabilities are \(\operatorname{logit}^{-1}(-1)=0.2689414\) for \(Y=0\), and \(\operatorname{logit}^{-1}(-0.5)=0.3775407\) for \(Y=1\). Averaging over the outcome distribution gives

P(R=1\mid Z=0) =0.6(0.2689414)+0.4(0.3775407) \approx 0.1614+0.1510 =0.3124.

For \(Z=1\), the corresponding logits are \(-0.2\) and \(0.3\). Thus

P(R=1\mid Z=1) =0.6(0.4501660)+0.4(0.5744425) \approx 0.2701+0.2298 =0.4999.

Using the unrounded probabilities, the difference is \(0.4998766-0.3123811=0.1874955\), or about \(0.1875\). Under this model, the response rate is about 0.1875 higher at \(Z=1\). This is a model-based comparison, not evidence that \(Z\) is a valid exclusion restriction in an actual application.

Exclusion Restrictions and Functional Form

An exclusion restriction is a substantive assumption that a variable affects the response process but has no direct effect on the outcome distribution, conditional on the included covariates. In the example, treating \(Z\) as excluded means that the outcome model does not contain \(Z\), while the response model does. If \(Z\) varies across units and predicts response, it can provide information about the response mechanism without directly changing the modeled outcome distribution.

The restriction is not guaranteed by giving a variable a different name in the two equations. It requires a defensible argument about the data-generating process, and it must be paired with adequate variation and a response effect strong enough to be informative. If \(Z\) does not predict response, it supplies little leverage. If it affects the outcome through an omitted pathway, treating it as excluded can distort the fitted joint model. An exclusion restriction is therefore an identifying assumption, not a fact established merely by estimating the model.

A model can also rely on functional-form assumptions. For instance, a normal outcome distribution combined with a logistic response probability may imply relationships between observed outcome values and response patterns that a more flexible model would not impose. Such curvature or tail restrictions may distinguish parameter values within the chosen model, even without an excluded variable. That is model-based identification: it depends on the specified distributions being credible, not solely on a design feature that independently shifts response.

Worked Example: Different Latent Prevalences, Identical Observed Data

Suppose a hypothetical binary outcome is recorded only for respondents. The observed data show \(P(R=1)=0.6\) and \(P(Y=1\mid R=1)=0.5\). Hence the observed joint probabilities are \(P(R=1,Y=1)=0.3\) and \(P(R=1,Y=0)=0.3\); the remaining probability \(0.4\) is nonresponse.

Let \(p=P(Y=1)\), \(r_1=P(R=1\mid Y=1)\), and \(r_0=P(R=1\mid Y=0)\). The observed respondent probabilities impose

p r_1=0.3,\qquad (1-p)r_0=0.3.

These are two constraints on the three unknown quantities \(p,r_1,r_0\). For \(p=0.4\), they imply \(r_1=0.3/0.4=0.75\) and \(r_0=0.3/0.6=0.50\). The response rate is \(0.4(0.75)+0.6(0.50)=0.60\), and the observed respondent positive fraction is \(0.4(0.75)/0.60=0.50\).

A different latent prevalence, \(p=0.6\), gives \(r_1=0.3/0.6=0.50\) and \(r_0=0.3/0.4=0.75\). It produces the same response rate, \(0.6(0.50)+0.4(0.75)=0.60\), and the same respondent positive fraction, \(0.6(0.50)/0.60=0.50\). Both parameter sets generate the same observed probabilities, although the population outcome prevalence differs. Without additional restrictions, observed data alone do not determine that prevalence.

A parametric selection model can restrict the possibilities, and valid exclusions or sufficiently strong distributional assumptions may identify parameters within that model. The example shows why such identification cannot simply be attributed to the observed respondent outcomes.

What a Model Can Identify

Key distinction: A parameter is identified by a model if distinct parameter values imply distinct distributions of the observed data under that model. Model-based identification does not mean that the data have verified the model’s assumptions, nor that the parameter is identified without those assumptions.

In an MNAR selection model, observed outcomes among respondents inform the outcome distribution and response probabilities jointly, but they do not directly reveal the outcome distribution among nonrespondents. Depending on the design and model, parameters that may remain weakly determined or undetermined by observed data alone include:

the mean or prevalence of outcomes among units who did not respond;
the strength of dependence between the missing outcome and response, such as \(\gamma_y\);
the outcome-distribution parameters that depend on the unobserved part of the population; and
the separate outcome and selection parameters when multiple combinations yield the same observed-data distribution.

These are not claims that every selection model has the same identification failure. Rather, identification depends on the model, covariate support, response patterns, and restrictions supplied by the design or functional form. Even when parameters are formally identified, limited response variation or weak exclusion variables can leave estimates imprecise. A numerical optimizer returning an estimate and standard error does not establish that the assumptions are credible or that the data contain substantial information about every parameter.

Worked Example: A Respondent’s Likelihood Contribution

For illustration, suppose a single-wave outcome follows \(Y\mid X\sim N(8,2^2)\), and the response probability is \(\operatorname{logit}^{-1}(-1+0.3Y+0.8Z)\). Treat the normal model and logistic response model as correctly specified for this hypothetical calculation. For a respondent with observed \(Y=10\) and \(Z=1\), the outcome density is

f(10\mid X)=\frac{1}{2\sqrt{2\pi}} \exp\left[-\frac{(10-8)^2}{2(2^2)}\right] =\frac{1}{2\sqrt{2\pi}}e^{-0.5} \approx 0.1210.

The linear predictor for response is \(-1+0.3(10)+0.8(1)=2.8\), so the response probability is \(\operatorname{logit}^{-1}(2.8)\approx0.9427\). The respondent’s observed-data density contribution is therefore

f(10\mid X)P(R=1\mid Y=10,Z=1) \approx 0.11405.

For a nonrespondent with \(Z=1\), \(Y\) is unknown. The likelihood contribution is not obtained by substituting an arbitrary value for \(Y\); it is the integral \(\int f(y\mid X)\{1-\operatorname{logit}^{-1}(-1+0.3y+0.8)\}\,dy\). This integral uses the assumed outcome and response distributions to account for all possible missing outcomes.

Common Pitfalls and Practical Implications

Treating a fitted selection model as if it revealed missing outcomes. It supplies predictions under its joint assumptions; the unobserved outcomes are not directly recovered from the observed records.
Calling a response predictor an exclusion restriction without justification. It must be plausibly excluded from the outcome distribution conditional on modeled covariates, as well as informative about response.
Confusing functional-form identification with design-based information. A parametric model may distinguish parameters because of distributional shape. That distinction can disappear when the assumed form is wrong or relaxed.
Ignoring the panel response history. Dropout probabilities generally condition on prior observed history and risk sets. A one-wave response formula cannot automatically be applied as independent across time.
Reporting only a point estimate. For MNAR models, the assumptions governing selection are part of the interpretation. State the modeled outcome and response processes and make the identifying restrictions explicit.

Selection models matter because they turn otherwise implicit beliefs about attrition into explicit, inspectable assumptions. They can support coherent likelihood-based estimation, but they do not eliminate the need to ask what observed data determine, what the model supplies, and what relies on substantive restrictions.

Check Your Understanding

Use the model definitions and likelihood contributions above to answer these questions.

What feature of a response model makes attrition nonignorable when an outcome is missing?
Why does the likelihood contribution for a nonrespondent integrate over the missing outcome?
In the binary-outcome example, verify that both parameter sets yield the same three observed probabilities for respondent-positive, respondent-negative, and nonresponse outcomes.
What two conditions make an excluded response predictor potentially useful for identification, and why does its exclusion still require substantive justification?
Explain the difference between identification supplied by functional form and information supplied by an exclusion restriction.