Selection Model

Concept

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Theoretically, Selection Model is a statistical framework for analysing incomplete data by jointly modelling the outcome process and the mechanism determining whether observations are observed or missing. It factorises the joint distribution of outcomes and missingness into the marginal distribution of the outcomes and the conditional distribution of missingness given the outcomes. Selection models are particularly important for sensitivity analysis under Missing Not at Random assumptions, where the probability of missingness depends on unobserved outcomes after conditioning on observed data.

Mathematically, a Selection Model represents the joint distribution of outcome data and missingness indicators as the product of an outcome model and a missingness model. The outcome component may be specified using regression, longitudinal, survival, or generalised linear models, while the missingness mechanism is commonly represented using logistic or probit regression. Because Missing Not at Random models are generally not identified from observed data alone, estimation requires identifying restrictions, exclusion restrictions, distributional assumptions, or sensitivity parameters.

In practice, Selection Models are estimated using maximum likelihood, Bayesian methods, numerical integration, or simulation-based procedures. In health economics they are used to examine the sensitivity of treatment-effect, cost, utility, and quality-adjusted life-year estimates to informative dropout, non-response, and other forms of non-random missingness in clinical trials and observational studies.

Purpose

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Used to model outcomes jointly with the probability of observation, assess departures from Missing At Random assumptions, and quantify the effect of informative missingness on clinical and economic estimates.

Mathematical Formulae

Primary Formula

f(Y, R | X) = f(Y | X) ? P(R | Y, X)

where:

• Y = outcome data
• R = missingness indicator or pattern
• X = observed covariates
• f(Y | X) = outcome model
• P(R | Y, X) = missingness model

Supporting Formulae

A common logistic missingness model is:

logit[P(R? = 1 | Y?, X?)] = �? + �?Y? + �?X?

where:

logit(p) = ln[p � (1 ? p)]

Observed-data likelihood:

L(?, �) = ? ? f(Y? | X?; ?) ? P(R? | Y?, X?; �) dY????

For the Heckman sample-selection model:

Y? = X?? + �?

S?* = Z?? + u?

S? = 1 if S?* > 0

Corr(�?, u?) = ?

Related Mathematical Methods

• Missing Not at Random
• Pattern Mixture Model
• Shared Parameter Model
• Heckman Selection Model
• Maximum Likelihood Estimation
• Bayesian Estimation
• Logistic Regression
• Sensitivity Analysis

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Example

A trial-based economic evaluation records 12-month QALYs, but participants with poor health outcomes are more likely to withdraw before completing follow-up.

The outcome model is:

QALY? = ?? + ??Treatment? + ??BaselineUtility? + �?

The missingness model is:

logit[P(R? = 1)] = �? + �?QALY? + �?Treatment?

A negative estimate of �? indicates that participants with lower unobserved QALYs are less likely to provide complete follow-up data. The jointly estimated treatment effect can then be compared with the estimate obtained under Missing At Random assumptions to assess the sensitivity of the economic conclusions to informative dropout.

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Excel Implementation

Function	Example Formula	Health Economics Application
EXP	=1/(1+EXP(-(A2+B2*C2+D2*E2)))	Calculate the modelled probability that an outcome is observed.
LN	=LN(F2/(1-F2))	Convert an observation probability to the logit scale.
IF	=IF(B2="",0,1)	Create a missingness indicator for cost, utility, or QALY outcomes.
SUMPRODUCT	=SUMPRODUCT(LogLikelihoodRange)	Aggregate observation-level log-likelihood contributions.
SOLVER	Maximise TotalLogLikelihood by changing ParameterCells	Estimate simplified selection-model parameters by numerical optimisation.

VBA (Optional)

Automate construction of missingness indicators, likelihood evaluation, and sensitivity analyses across alternative selection-model parameters.

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Sources

• Heckman JJ. Sample Selection Bias as a Specification Error. Econometrica. 1979.
• Diggle P, Kenward MG. Informative Drop-Out in Longitudinal Data Analysis. Journal of the Royal Statistical Society: Series C. 1994.
• Little RJA, Rubin DB. Statistical Analysis with Missing Data. 3rd ed.
• Molenberghs G, Kenward MG. Missing Data in Clinical Studies.
• Carpenter JR, Kenward MG. Multiple Imputation and its Application.
• Faria R, Gomes M, Epstein D, White IR. A Guide to Handling Missing Data in Cost-Effectiveness Analysis Conducted within Randomised Controlled Trials. Pharmacoeconomics. 2014.
• NICE. Health Technology Evaluations: The Manual.