Baseline Covariates in the Selection Equation

Why Add Baseline Covariates?

A dropout equation that depends on the current outcome and previous outcome can still omit important baseline differences between participants. For example, treatment assignment, age, or a baseline severity score may be associated with the chance of leaving follow-up. Including such variables describes dropout differences between participants with the same modeled outcome history. This is especially important when those variables also help describe the outcome distribution.

The focus here is their role in the selection equation. A baseline coefficient describes a conditional association in the specified dropout model; it does not by itself establish that changing the covariate would change dropout. In particular, a treatment coefficient is not automatically a causal effect of treatment on dropout.

Definition: Let \(D_{it}=1\) indicate that participant \(i\), while still under follow-up, drops out before the scheduled measurement at time \(t\); let \(D_{it}=0\) indicate continued follow-up at that occasion. Let \(Y_{it}\) be the current outcome, which may be unobserved when \(D_{it}=1\), and let \(Y_{i,t-1}\) be the previous outcome. Write \(A_i\) for baseline treatment assignment and \(Z_i\) for a vector of other baseline covariates. A current-outcome selection model specifies the dropout hazard conditional on \(Y_{it}\), the previous outcome, and baseline covariates.

The hazard is conditional on being at risk: participant \(i\) has not dropped out before time \(t\). The timing convention matters. If the decision to drop out occurs before the current measurement, then \(Y_{it}\) is latent for a dropout at \(t\), even though it appears in the selection equation. It is not a value that can simply be inserted from the observed data for those participants.

A Logistic Selection Equation With Baseline Predictors

One common specification uses a logistic link and a time-specific intercept:

\operatorname{logit}\{P(D_{it}=1\mid D_{i,t-1}=0,Y_{it},Y_{i,t-1},A_i,Z_i)\} = \alpha_t+\gamma_C Y_{it}+\gamma_L Y_{i,t-1} +\beta_A A_i+\boldsymbol{\beta}_Z^{\mathsf T}Z_i.

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The parameters \(\gamma_C\) and \(\gamma_L\) describe current- and lagged-outcome associations, respectively. The parameter \(\beta_A\) is the treatment coefficient, and \(\boldsymbol{\beta}_Z\) contains coefficients for the baseline covariates. A time-specific intercept \(\alpha_t\) allows the baseline dropout odds to differ by occasion. Continuous predictors should be expressed in units that make their coefficients interpretable; centering can make the intercept correspond to a meaningful reference profile.

This equation assumes a monotone dropout process, a correctly specified hazard among those still at risk, and the stated functional form for each predictor. The baseline covariates must be measured and available for participants who later drop out. The equation also assumes that the displayed effects are additive on the log-odds scale: unless interactions are included, the treatment odds ratio is the same at each fixed value of the other predictors and outcomes. These are modeling assumptions, not consequences of including covariates.

Interpretation: Holding the occasion, current outcome, previous outcome, and all other covariates fixed, increasing a continuous baseline covariate \(Z_{ij}\) by one unit multiplies the conditional dropout odds by \(\exp(\beta_{Z_j})\). For a binary treatment indicator coded \(A_i=1\) versus \(A_i=0\), the corresponding conditional odds ratio is \(\exp(\beta_A)\).

The conditioning is essential. These odds ratios compare participants at the same values of the outcomes in the equation and the other included variables. They are not, in general, ratios of marginal dropout probabilities. Nor does adjustment make the coefficient causal: causal interpretation requires a design and assumptions appropriate to the causal question, in addition to a correctly specified selection model.

Worked Example: Specifying Baseline Effects

Suppose a hypothetical study models dropout at each occasion using treatment assignment, age, baseline severity, the current outcome, and the previous outcome. Treatment is coded 1 for the active arm and 0 for control. Age is recorded in decades above age 50, and baseline severity \(P_i\) is standardized so that one unit is one standard deviation. A proposed equation is

\operatorname{logit}(h_{it}) = \alpha_t+0.30\frac{Y_{it}-10}{5} +0.20\frac{Y_{i,t-1}-10}{5} +0.50A_i+0.08\frac{\text{age}_i-50}{10}+0.40P_i,

where \(h_{it}\) denotes the conditional dropout hazard. The treatment coefficient implies a conditional odds ratio of \(\exp(0.50)=1.649\): at the same occasion, outcomes, age, and severity, the active-arm dropout odds are estimated to be 1.649 times the control-arm odds. The age coefficient is per decade, not per year: a ten-year increase multiplies conditional dropout odds by \(\exp(0.08)=1.083\). A one-standard-deviation increase in baseline severity multiplies them by \(\exp(0.40)=1.492\), holding the other terms fixed.

These are interpretations of the specified model. They do not say that assignment, age, or severity causes dropout, and the treatment interpretation is conditional on the current outcome even when that outcome is latent for dropouts.

Deriving the Conditional Odds-Ratio Interpretation

The coefficient interpretation follows directly from the logistic model. Let \(\eta_0\) be the linear predictor for a participant with \(A_i=0\), and hold all other predictors fixed. For treatment \(A_i=1\), the linear predictor is \(\eta_1=\eta_0+\beta_A\). Since the logistic model gives odds \(h/(1-h)=\exp(\eta)\), the ratio of the treatment and control conditional odds is:

\frac{h_1/(1-h_1)}{h_0/(1-h_0)} = \frac{\exp(\eta_0+\beta_A)}{\exp(\eta_0)} = \exp(\beta_A).

The cancellation works because the comparison holds all other terms constant and because the model has no treatment interaction. If an interaction were included, the treatment odds ratio would depend on the interacting variable. That is a different specification and should not be silently inferred from this additive equation.

Odds ratios also do not directly give risk differences. Given a particular linear predictor \(\eta\), the corresponding probability is \(h=1/\{1+\exp(-\eta)\}\). Changing a baseline covariate shifts \(\eta\), and the resulting change in probability depends on the starting value of \(\eta\). Thus the same conditional odds ratio can correspond to different absolute risk changes for different profiles.

Worked Example: Conditional Treatment Odds and Dropout Probabilities

Use the hypothetical equation above at a particular occasion where \(\alpha_t=-1.20\). Consider participants with current and previous outcomes both equal to 10, age 50, and baseline severity \(P_i=0\). The outcome and covariate contributions beyond the intercept and treatment are all zero. For control, the linear predictor is \(-1.20\), so:

h_0=\frac{1}{1+\exp(1.20)} =\frac{1}{1+3.3201} =0.2315,

The fitted control-arm conditional dropout probability is therefore 23.15% for this profile and occasion. For active treatment, the linear predictor is \(-1.20+0.50=-0.70\), giving:

h_1=\frac{1}{1+\exp(0.70)} =\frac{1}{1+2.0138} =0.3318.

The fitted active-arm probability is 33.18%. The probabilities differ by about 10.03 percentage points in this example. The conditional odds ratio is \(\exp(0.50)=1.649\), not \(0.3318/0.2315\); that probability ratio is a different quantity. All comparisons hold current and previous outcome, age, severity, and occasion fixed. Because the current outcome is latent for someone who drops out before measurement, these are model-based conditional predictions, not a direct comparison of observed outcomes among dropouts.

Calculating a Covariate Profile

To obtain a fitted hazard for a participant profile, substitute the values into the entire linear predictor and transform it with the inverse-logit function. This calculation illustrates how the baseline covariates contribute while the current- and lagged-outcome terms remain in the model.

Worked Example: Combining Age and Baseline Severity

Use the same hypothetical equation and \(\alpha_t=-1.20\). Consider a control participant aged 60, with baseline severity one standard deviation above the mean, current outcome \(Y_{it}=15\), and previous outcome \(Y_{i,t-1}=10\). Age contributes \(0.08(60-50)/10=0.08\), severity contributes \(0.40(1)=0.40\), the current outcome contributes \(0.30(15-10)/5=0.30\), and the previous outcome contributes zero. Therefore:

\eta_{it}=-1.20+0.08+0.40+0.30=-0.42, \qquad h_{it}=\frac{1}{1+\exp(0.42)} =\frac{1}{1+1.522}=0.3965.

The fitted conditional dropout probability is 39.65% for this profile at this occasion. The calculation is conditional on the specified current outcome and previous outcome. The current outcome would not be observed for a participant who actually dropped out before the measurement; in that case the hazard contributes through the selection model and must be handled with the joint model for outcomes and dropout rather than treated as an observed predictor.

How Baseline Covariates Enter the Observed-Data Model

Adding baseline predictors changes the selection part of the model, but it does not remove the need to model the outcomes. In a joint selection-model formulation, the outcome process is modeled conditional on baseline variables and treatment, while the dropout process is modeled conditional on outcomes and those same baseline variables. The observed-data likelihood uses the model for observed outcomes and integrates over any outcome values that are unobserved because of dropout. For example, a dropout contribution involving the latent \(Y_{it}\) requires averaging over its distribution conditional on the modeled history and baseline covariates.

This has a practical consequence: a fitted hazard for a specified value of \(Y_{it}\) is not automatically a marginal dropout probability for a treatment group or covariate subgroup. A marginal probability averages over the relevant outcome distribution and, where applicable, histories and baseline covariate distributions. Because the logistic function is nonlinear, applying it to an average linear predictor generally does not equal averaging the probabilities:

E\!\left[\frac{1}{1+\exp(-\eta)}\right] \ne \frac{1}{1+\exp\{-E(\eta)\}} \quad\text{in general}.

This distinction is important when presenting results. State whether a reported quantity is a conditional odds ratio, a profile-specific hazard, or a marginal risk obtained by averaging over a stated distribution.

Diagnostics, Comparisons, and Limitations

A baseline covariate should be included for a scientific or modeling reason, not merely because it is available. Before fitting, check coding, units, and reference categories. A coefficient for age in years is numerically different from one for age in decades, even though correctly converted interpretations agree. Continuous predictors may need a more flexible functional form if a straight-line effect on the log-odds scale is implausible; inspect residual and calibration patterns where possible and compare plausible specifications.

Useful checks include the following:

Data support: tabulate treatment and categorical covariate levels by dropout occasion. Sparse or empty combinations can produce unstable estimates or separation.
Functional form: examine whether observed dropout patterns across a continuous baseline predictor are compatible with its modeled log-odds effect. A single linear coefficient can be inadequate.
Calibration: compare predicted and observed dropout frequencies among participants at risk, using appropriate groups or smooth summaries. Calibration among observed risk sets does not verify the model's dependence on a latent current outcome.
Stability: check convergence, standard errors, sensitivity to starting values, and whether estimates change substantially under reasonable alternative specifications.
Model comparison: compare a covariate-adjusted model with a simpler model using a justified criterion and inspect whether the added terms improve fit without relying on unsupported extrapolation.

The main alternative is to omit a baseline predictor from the selection equation, or to use a different link or covariate functional form. Omitting a predictor may be defensible if it has no relevant conditional association under the modeling assumptions, but an omitted variable associated with dropout can distort other fitted coefficients when the model is misspecified. Conversely, adding many weakly supported terms can make estimates imprecise. Baseline adjustment in a selection equation also differs from simply comparing observed dropout proportions: observed proportions do not condition on latent current outcomes.

Most importantly, because \(Y_{it}\) is unobserved for participants who drop out before time \(t\), the dependence of dropout on that current outcome is not directly checked using their observed data. Good fit to observed dropout frequencies does not establish that the current-outcome selection assumption is correct. Estimates can depend on the assumed outcome distribution and selection specification; report this uncertainty and examine sensitivity to plausible assumptions. Baseline covariates can improve the model and describe conditional dropout patterns, but they do not by themselves resolve the fundamental information missing about outcomes after dropout.

Common pitfall: Saying “the treatment group is 1.649 times as likely to drop out” because \(\exp(\beta_A)=1.649\) is incorrect. The model gives a conditional odds ratio, holding the specified outcomes and covariates fixed. The corresponding probability ratio depends on the starting hazard, and the current outcome may be latent for dropouts. State the estimand and conditioning variables explicitly.

Check Your Understanding

Answer each question, then compare with the solutions below.

If a baseline severity coefficient is 0.30 per one-unit increase, what is the conditional odds ratio for a two-unit increase, assuming no interaction?
Why is a treatment odds ratio from the selection equation not automatically a marginal risk ratio?
In a current-outcome dropout model, why can the current outcome not be inserted as an observed covariate for someone who drops out before the measurement?
Name two checks that could reveal problems with baseline covariate specification or estimation.

Solutions. (1) The two-unit conditional odds ratio is \(\exp(2\times0.30)=\exp(0.60)=1.822\), holding the other model terms fixed. (2) The odds ratio compares conditional odds, whereas probabilities depend on the starting linear predictor; marginal risks additionally average over outcome and covariate distributions. (3) Under the stated timing, dropout occurs before the scheduled measurement, so that participant’s current outcome is latent, not observed. The joint model must account for it rather than treating it as recorded. (4) Suitable checks include tabulating treatment and categorical covariate levels by dropout occasion to identify sparse cells, checking the functional form of continuous predictors, assessing calibration among risk sets, and checking convergence and estimate stability.