Why Identifiability Matters
A shared-parameter model connects repeated outcomes and dropout through a latent subject-specific effect. That connection can represent dependence between the outcome process and dropout, but it does not guarantee that the observed data determine the latent distribution or the strength of the connection. Two different specifications can imply exactly the same distribution for everything observed, while implying different outcomes for participants who dropped out.
The key question is not merely whether a model can be fitted. It is whether the features being interpreted are identified by the observed-data distribution, or whether their apparent precision depends mainly on restrictions imposed by the model. This distinction matters when explaining an association parameter, reporting sensitivity results, or deciding which model features the study design can support.
In this tutorial, \(Y_{ij}\) denotes outcome \(j\) for subject \(i\), \(R_{ij}=1\) indicates that the outcome is observed, and \(b_i\) is a shared latent effect. A typical specification gives the outcomes and dropout process separate conditional models given \(b_i\), then integrates over its distribution. Identification concerns the resulting distribution of observed outcomes and observation indicators—not the complete-data distribution, which includes outcomes that were never observed.
An Exact Example of Nonidentification
Consider a deliberately simple shared-effect model. Each subject has an unobserved class \(U\), with \(U\in\{0,1\}\) and \(P(U=0)=\pi\). The outcome is \(Y=U\), but it is observed only when \(R=1\). Let \(q_0=P(R=1\mid U=0)\) and \(q_1=P(R=1\mid U=1)\). Assume subjects are independent and that these probabilities fully specify the dropout process. The observed data consist of \(R\), and \(Y\) when \(R=1\).
Worked Example: The Same Observed Data, Different Associations
Suppose hypothetical observed-data probabilities are \(P(R=1,Y=0)=0.25\), \(P(R=1,Y=1)=0.25\), and \(P(R=0)=0.50\). The model implies
One explanation is \(\pi=0.50\), \(q_0=0.50\), and \(q_1=0.50\). It gives \(0.50(0.50)=0.25\) in each observed outcome class, and dropout probability \(0.50\). Under this explanation, observation is unrelated to the outcome class.
A different explanation is \(\pi=0.30\), \(q_0=0.25/0.30=0.8333\), and \(q_1=0.25/0.70=0.3571\). It gives the same two observed-cell probabilities. Its dropout probability is \(0.30(1-0.8333)+0.70(1-0.3571)=0.0500+0.4500=0.5000\), also exactly the same. Yet the population proportion with \(Y=1\) is \(1-\pi=0.70\), rather than \(0.50\).
In the second explanation, the observation odds are \(0.8333/0.1667=5.00\) for \(Y=0\) and \(0.3571/0.6429=0.5556\) for \(Y=1\). Their ratio is \(5.00/0.5556=9.00\). Thus the observed data are compatible both with no association and with a strong association, as well as with different population outcome distributions. The likelihood for these observed data cannot distinguish those explanations.
This is structural nonidentifiability, not a small-sample problem: even infinitely many observations from the stated observed-data distribution would not distinguish the parameter settings. The example also shows why observed outcome distributions among those retained do not, without further restrictions, determine the distribution in the full population.
What Repeated Measurements Can Add
Repeated outcomes can provide information about shared-effect variation because measurements on the same subject may be dependent through \(b_i\). The following moment calculation makes the role of that dependence explicit. Its assumptions are important: outcomes are collected before dropout can affect whether either measurement is observed; the shared effect and measurement errors are independent; and the errors are independent of one another and have mean zero. These conditions let the within-subject covariance reflect the shared effect rather than selection into a complete-pair sample.
Worked Example: Recovering Shared-Effect Variance From Two Outcomes
Suppose \(Y_{ij}=\mu_j+b_i+\varepsilon_{ij}\), with \(E(b_i)=0\), \(\operatorname{Var}(b_i)=\tau^2\), and independent errors having variances \(\sigma_1^2\) and \(\sigma_2^2\). Under the assumptions above,
For hypothetical population moments, suppose \(\operatorname{Var}(Y_{i1})=5.0\), \(\operatorname{Var}(Y_{i2})=5.2\), and \(\operatorname{Cov}(Y_{i1},Y_{i2})=3.0\), in squared outcome units. Then \(\tau^2=3.0\). Since \(\operatorname{Var}(Y_{ij})=\tau^2+\sigma_j^2\), the error variances are \(\sigma_1^2=5.0-3.0=2.0\) and \(\sigma_2^2=5.2-3.0=2.2\).
The repeated outcomes have separated shared variation from measurement-specific variation in this specified model. This calculation does not, by itself, identify how strongly dropout depends on \(b_i\). That requires information in the dropout histories and a specified dropout model; moreover, if dropout selects which pairs are observed, the covariance among observed pairs need not equal the population covariance used here.
More repeated measurements can reveal more of the dependence structure: for example, covariance across different measurement times can help distinguish a shared intercept from time-varying or subject-specific components. But this extra information is conditional on the model form and observation pattern. Measurements made before dropout can help characterize the latent effect among those still observed; they do not directly reveal later outcomes for those who have already dropped out.
Restrictions That Set the Scale
Shared effects are latent, so their scale and origin must be tied to the outcome and dropout equations. Consider a simplified model in which the linear predictors contain \(\lambda b_i\) for an outcome and \(\gamma b_i\) for dropout. If both loadings are free, replacing \(b_i\) by \(b_i^*=c b_i\), for nonzero \(c\), and replacing the loadings by \(\lambda^*=\lambda/c\) and \(\gamma^*=\gamma/c\) leaves both linear predictors unchanged. The variance of the latent effect becomes \(c^2\operatorname{Var}(b_i)\). Thus the latent scale, loadings, and variance cannot all be given separate interpretations without an anchoring restriction.
A common restriction fixes a loading or fixes the variance of \(b_i\); other choices can also anchor the scale, but they express different parameterizations. A sign convention may be needed as well: changing \(b_i\) to \(-b_i\) and changing both loadings’ signs leaves the predictors unchanged. Likewise, a latent-effect mean can be confounded with model intercepts unless its location is fixed or otherwise constrained. These are parameterization issues, not evidence that the data have discovered a unique latent scale.
Worked Example: A Scale Change That Leaves Predictions Unchanged
Suppose a hypothetical model has outcome predictor \(10+2b_i\) and dropout predictor \(-1+0.5b_i\). If \(b_i^*=2b_i\), then \(b_i=b_i^*/2\). The same predictors become \(10+1b_i^*\) and \(-1+0.25b_i^*\). The latent-effect variance is multiplied by \(2^2=4\), but all conditional outcome and dropout predictions are unchanged.
Consequently, the numerical values of the two loadings and the latent variance cannot be compared across these parameterizations as if they were invariant features. The model must fix a scale before those quantities are separately interpreted. A parameter such as the dropout loading is meaningful only relative to that chosen scale.
Restrictions can also concern the distributional shape of \(b_i\), the outcome link, or the way dropout depends on the shared effect. Such restrictions can make a parameterization estimable or give a unique model-based completion of the missing outcomes. They do not turn the assumed shape or link into an empirically verified fact. The distinction is between information in the observed distribution and information supplied by the specification.
Worked Example: An Anchor Can Resolve a Toy Model
Return to the two-class example, but suppose an external design feature or a justified outcome model fixes \(\pi=0.50\). If the observed probabilities are instead \(P(R=1,Y=0)=0.30\), \(P(R=1,Y=1)=0.20\), and \(P(R=0)=0.50\), the two equations give \(q_0=0.30/0.50=0.60\) and \(q_1=0.20/0.50=0.40\). The dropout probabilities are \(0.40\) in class 0 and \(0.60\) in class 1, and their odds ratio for observation is
This calculation identifies the class-specific observation probabilities within the restricted model because \(\pi\) was fixed. If \(\pi\) were free, the same observed cells would instead give \(q_0=0.30/\pi\) and \(q_1=0.20/(1-\pi)\), allowing a range of compatible values subject to probabilities lying between 0 and 1. The anchor supplies information; it is not a consequence of those observed cells.
Assessing Information and Comparing Approaches
Structural identifiability asks whether two parameter settings can yield exactly the same observed-data distribution. Practical identifiability asks whether the study has enough information to distinguish settings that yield slightly different distributions. A model can pass the first test and still produce broad, unstable estimates in a modest sample, especially when few subjects have long outcome histories or when dropout events are sparse.
Useful warning signs include very large standard errors, strong parameter correlations, estimates that change substantially with starting values or small specification changes, and a likelihood that is nearly flat along combinations of latent variance and dropout loading. These are reasons to inspect the information, not automatic proof of structural nonidentifiability. Numerical convergence alone does not establish that a parameter is well determined by observed data.
Repeated measurements, variation in measurement times, and observed outcomes close to dropout can help distinguish shared-effect structures. In contrast, early dropout, few repeated outcomes, and little variation in dropout histories can leave association weakly informed. For a continuous latent effect, the observed-data implications also depend on its assumed distribution and on the chosen links. A simple parametric distribution may yield a unique fitted parameter vector while other plausible distributions imply different unobserved outcomes.
The main alternatives organize assumptions differently. A selection model specifies the outcome distribution and the probability of observation conditional on outcomes or covariates. A pattern-mixture model specifies outcome distributions within missingness patterns. A shared-parameter model instead links outcome and dropout processes through a latent effect. Each approach can require restrictions to describe unobserved outcomes; the parameterization alone does not make those outcomes observable. These alternatives are useful contrasts, not automatic solutions to weak identification.
The practical aim is therefore to separate features supported by repeated observed data from features that depend on the shared-effect distribution, scale conventions, or link functions. When multiple plausible specifications remain compatible with the observations, conclusions should make that dependence clear rather than presenting a single association estimate as directly observed fact.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In the two-class example, what does structural nonidentifiability mean, and why would a larger sample from the same observed distribution not resolve it?
- Under what assumptions does the covariance of two repeated outcomes identify the shared-effect variance in the Gaussian random-intercept example?
- What changes when both the outcome and dropout loadings on a shared effect are free and the latent scale is not anchored?
- Name two features of a study that can make outcome-dropout association weakly identified even when a model is structurally identifiable.
Solutions. (1) Distinct parameter values imply the same distribution of observed outcomes and observation indicators; a larger sample estimates that same distribution more precisely but cannot distinguish its equivalent explanations. (2) The shared effect must enter both outcomes as specified, have finite variance, and be independent of mean-zero measurement errors that are independent across measurements; the paired outcomes must not be selected by dropout in a way that alters their covariance. (3) A rescaling of the latent effect can be offset by inverse rescaling of both loadings, leaving predictors unchanged; a scale restriction is needed for separate interpretation. (4) Examples include few repeated measurements before dropout, sparse dropout histories, and early dropout that leaves little observed outcome information per subject.