From a Full-Data Model to an Observed-Data Likelihood
A selection model specifies both how outcomes are distributed and how the chance of observing an outcome depends on the full outcome. The likelihood for analysis, however, must be built from what was actually recorded. When an outcome is missing, its value is not available to insert into either model component. Its possible values must instead be averaged over using the specified outcome distribution.
Begin with one outcome \(Y\), covariates \(X=x\) treated as observed, and a response indicator \(R\), where \(R=1\) means that \(Y\) is observed and \(R=0\) means that it is missing. Let \(f_\theta(y\mid x)\) be the specified outcome density or probability mass function, with parameter \(\theta\). Let \(\pi_\psi(y,x)=P(R=1\mid Y=y,X=x)\) be the response probability, with parameter \(\psi\). The response model is conditional on the full outcome, including when that outcome is missing in the recorded data.
These contributions follow from the joint model \(f_\theta(y\mid x)P(R=r\mid y,x)\). If \(R=1\), \(y\) is recorded, so the joint density is evaluated at that value. If \(R=0\), \(y\) is not recorded, so the joint density must be integrated over all possible values of \(y\). For independent individuals conditional on their observed covariates, the likelihood is the product of their individual contributions. Independence is an assumption about the sampling model, not a consequence of the likelihood formula.
Why Missing Outcomes Are Integrated Out
The marginalization can be seen directly from probability. For a missing record, the event observed in the data is \(R=0\), not a particular value of \(Y\). The probability of that event, given \(x\), is the sum or integral of its joint probability over every possible outcome:
For a discrete outcome, replace the integral by a sum over its support. For an outcome vector \(Y=(Y_{\mathrm{obs}},Y_{\mathrm{mis}})\) that is only partly observed, the same argument gives a contribution of the form \(\int f_\theta(y_{\mathrm{obs}},y_{\mathrm{mis}}\mid x)P_\psi(R=r\mid y_{\mathrm{obs}},y_{\mathrm{mis}},x)\,dy_{\mathrm{mis}}\). The integral is over the missing components; the observed values remain fixed at the values recorded. The response probability must describe the observed response pattern.
This is a likelihood for the joint observed data: it includes the probability of which outcomes were observed as well as the density of observed outcome values. It is not generally the same as the density of \(Y\) conditional on \(R=1\). That conditional density is \(f_\theta(y\mid x)\pi_\psi(y,x)/P(R=1\mid x)\); conditioning on response changes the normalization. Omitting response probabilities or using a conditional density without accounting for its denominator changes the likelihood and can change which parameters it informs.
Worked Example: A Binary Outcome With Outcome-Dependent Response
Suppose a hypothetical study has a binary outcome \(Y\in\{0,1\}\), with \(P(Y=1)=0.40\). Let the response probabilities be \(P(R=1\mid Y=1)=0.80\) and \(P(R=1\mid Y=0)=0.50\). In 100 independent individuals, the recorded data contain 24 observed ones, 30 observed zeros, and 46 missing outcomes. Treat the stated model probabilities as fixed for this likelihood calculation.
An observed one contributes \(P(Y=1)P(R=1\mid Y=1)=0.40(0.80)=0.32\). An observed zero contributes \(P(Y=0)P(R=1\mid Y=0)=0.60(0.50)=0.30\). A missing outcome contributes the sum over its two possible values:
Thus the likelihood for the recorded counts, up to a multinomial coefficient that does not depend on these fixed probabilities, is \(0.32^{24}0.30^{30}0.38^{46}\). The three possible recorded categories have probabilities \(0.32\), \(0.30\), and \(0.38\), which sum to 1. In particular, assigning each missing outcome the value 0 or 1 would not produce the likelihood of the recorded data.
Continuous Outcomes and Selection Probabilities
With a continuous outcome, an observed value contributes a density, not the probability of observing exactly that value. The missing-outcome contribution is an integral that averages the nonresponse probability over the outcome distribution. It need not have a closed form; numerical quadrature or Monte Carlo integration can be used to evaluate it. Such computation evaluates the specified integral, rather than replacing each missing outcome with a guessed or predicted value.
Worked Example: A Normal Outcome With Logistic Selection
Consider a hypothetical model with \(Y\sim N(0,1)\) and \(\pi(y)=\operatorname{expit}((\ln 3)y)=1/\{1+\exp(-(\ln 3)y)\}\). Suppose two independent individuals have been sampled: one has an observed outcome \(y=1\), and the other has a missing outcome. Evaluate their likelihood contributions under these specified parameters.
For the observed value, the normal density is \(f(1)=\frac{1}{\sqrt{2\pi}}\exp(-1/2)=0.24197\). Also, \(\pi(1)=1/(1+\exp(-\ln 3))=1/(1+1/3)=0.75\). Therefore the observed record contributes \(f(1)\pi(1)=0.24197(0.75)=0.1815\), to four significant figures.
For the missing record, the contribution is \(E\{1-\pi(Y)\}\) under \(Y\sim N(0,1)\). The normal distribution is symmetric about zero, and \(\operatorname{expit}(z)+\operatorname{expit}(-z)=1\). Pairing the contributions at \(y\) and \(-y\) therefore gives \(E\{\pi(Y)\}=0.5\), and hence \(E\{1-\pi(Y)\}=0.5\). The joint likelihood contribution for these two records is \(0.1815(0.5)=0.09074\), up to rounding.
The response probability depends on the outcome even though its average over this symmetric outcome distribution is one half. That average is appropriate for the missing record; inserting an outcome value into the response model for that person would instead treat an unobserved quantity as observed.
Likelihoods for Groups and Model Checking
When multiple records share a model, a convenient way to construct the likelihood is to identify the probability of each type of observed record and then raise each probability to its observed count. For discrete outcomes, the missing category probability is a sum over the outcome categories. It must include both the probability of each outcome and its probability of nonresponse.
Worked Example: Three Outcome Categories
In a hypothetical population, let a categorical outcome have probabilities \(0.50\), \(0.30\), and \(0.20\) for mild, moderate, and severe outcomes. Suppose their respective response probabilities are \(0.90\), \(0.60\), and \(0.25\). A sample of 20 independent individuals has 9 observed mild outcomes, 3 observed moderate outcomes, 1 observed severe outcome, and 7 missing outcomes. Assume these category probabilities and response probabilities are the specified model.
The observed-record probabilities are \(0.50(0.90)=0.45\) for mild, \(0.30(0.60)=0.18\) for moderate, and \(0.20(0.25)=0.05\) for severe. The missing-record probability is \(0.50(0.10)+0.30(0.40)+0.20(0.75)=0.05+0.12+0.15=0.32\). The likelihood, omitting the parameter-free multinomial coefficient, is therefore \(0.45^9(0.18)^3(0.05)^1(0.32)^7\).
As a basic check, \(0.45+0.18+0.05+0.32=1.00\). The likelihood accounts for all 20 records: the exponents sum to \(9+3+1+7=20\). A missing record has one contribution, \(0.32\), obtained by summing across all three possible unobserved outcome categories.
A useful diagnostic is to check that the probabilities of all possible recorded categories sum to one. For a discrete outcome, these categories include every observed outcome-response combination and the missing category. For a continuous outcome, integrate the observed-record density over possible recorded values and add the probability of a missing record; the result should be one. Failure can indicate an omitted response factor, an incorrect integration range, or a coding error.
Other checks concern whether the specification represents the data and the intended sampling process. Confirm that each observed record uses its recorded outcome and the response probability for \(R=1\), while each missing record uses the \(R=0\) probability integrated over the full outcome support. Confirm that covariates used in the model are observed as assumed, that probabilities lie between zero and one, and that the outcome density or mass function is normalized. For repeated or dependent observations, a product of individual contributions is not justified unless the model accounts for their dependence.
Assumptions, Alternatives, and Limits
The construction relies on a specified, normalized outcome distribution and a specified response model for the full outcome and observed covariates. It also relies on the assumed sampling structure, such as conditional independence across individuals when the likelihood is written as a product. If outcome or response parameters are shared, their dependence is retained by evaluating all contributions using the same parameter values. The model need not assume missing at random: dependence of \(\pi_\psi(y,x)\) on an unobserved \(y\) is permitted. But that dependence must be specified rather than learned directly from missing outcomes alone.
The likelihood is only as credible as these assumptions. Different combinations of outcome and response parameters can sometimes imply very similar distributions for the observed data. Thus, writing down and maximizing a likelihood does not itself establish that the parameters are well identified, that the model is correct, or that a selection effect has a causal interpretation. The fit should be checked against observed-data patterns, and substantive assumptions about unobserved outcomes should be made explicit.
The construction also clarifies how this approach differs from common alternatives. Inverse-probability weighting uses modeled response probabilities to weight observed records; it does not replace the likelihood contribution for each missing outcome with the integral above. Multiple imputation generates plausible missing values under an imputation model and analyzes completed data sets; it is a different inferential procedure. A likelihood conditional on the response pattern may be used for a specifically conditional analysis, but it is not automatically the joint observed-data likelihood. Each method has assumptions; none makes the unobserved outcomes directly available.
Common Pitfall
The distinction matters even when a predicted value seems reasonable. In general, \(E\{g(Y)\}\neq g\{E(Y)\}\), so replacing the integral of a response probability by its value at a predicted outcome changes the model. The correct contribution is determined by the specified joint model and the pattern actually observed.
Specify the outcome density or mass function and the response probability conditional on the full outcome and observed covariates.
For each record, distinguish observed outcome values from missing components and record the response pattern.
Use the joint density or probability for observed outcomes; integrate or sum the joint density over missing outcomes.
Multiply contributions under the assumed sampling structure and verify normalization, support, and parameter sharing.
Check Your Understanding
Answer each question using the distinction between observed values and missing values. Solutions follow the questions.
- For a discrete outcome, what terms are summed to obtain the contribution from a missing record?
- A binary outcome has \(P(Y=1)=0.25\), \(P(R=1\mid Y=1)=0.80\), and \(P(R=1\mid Y=0)=0.40\). What is the probability of a missing record?
- Why is the contribution for an observed continuous value a density multiplied by a response probability?
- What is the missing-record contribution when \(Y\sim N(0,1)\) and \(P(R=1\mid Y=y)=0.60\) for all \(y\)?
Solutions. (1) Sum \(P(Y=y\mid x)P(R=0\mid Y=y,x)\) over every possible outcome value \(y\). (2) The probability is \(0.25(0.20)+0.75(0.60)=0.05+0.45=0.50\). (3) The observed record includes both the value \(y\), whose density is \(f_\theta(y\mid x)\), and the event \(R=1\), whose conditional probability is \(\pi_\psi(y,x)\); their joint density is their product. (4) The contribution is \(\int f(y)(1-0.60)\,dy=0.40\int f(y)\,dy=0.40\), since the normal density integrates to one.