Checking Implied Dropout Probabilities Across Outcome Ranges

Why Inspect Implied Probabilities?

A selection model can have coefficients that look modest on the log-odds scale yet imply striking dropout probabilities for some patients. Conversely, a coefficient that sounds large may produce only a small probability change over the outcome range that matters clinically. The useful diagnostic is therefore not just the sign or size of a selection coefficient: it is the probability pattern the fitted equation implies across realistic outcomes and covariate profiles.

This inspection is a model-checking step. For outcomes missing because of dropout, the relevant outcome value is not observed at the dropout visit. The model nevertheless specifies what dropout probability would correspond to each possible value of that outcome, conditional on the recorded history. Calculating this curve does not make the missing value known or establish that the model is true. It shows the consequences of the specification.

Definition: Let \(D_t=1\) indicate dropout at visit \(t\), before the outcome at that visit is observed, among individuals still at risk of dropout. Let \(Y_t\) be the current outcome and let \(H_t\) denote the recorded history and covariates used in the dropout equation. The model-implied conditional dropout probability is \(q_t(y,h)=P(D_t=1\mid Y_t=y,H_t=h,D_{t-1}=0)\). For a logistic selection model, \(q_t(y,h)=\operatorname{expit}\{\eta_t(y,h)\}=1/[1+\exp\{-\eta_t(y,h)\}]\), where \(\eta_t\) is the specified linear predictor.

The probability is conditional on still being at risk at visit \(t\); it is not the probability of having dropped out at any time by visit \(t\). The selection equation and outcome model together provide the model-based setting in which \(Y_t\) can be considered at values even when it is unobserved for dropouts. For this diagnostic, the selection equation must first be stated, including its link, outcome transformation, covariates, interactions, and coding conventions. The calculations below assume the displayed logistic equations are the specified models, the stated covariate profiles are meaningful, and the outcome values lie in ranges that are clinically plausible for those profiles.

Translate the Equation Into a Probability Curve

A useful general form for a continuous current outcome is

\eta_t(y,h,a)=\alpha_0+\alpha_Y g(y)+\alpha_A a+\alpha_{YA}a\,g(y)+\alpha_H^{\mathsf T}h, \qquad q_t(y,h,a)=\frac{1}{1+\exp\{-\eta_t(y,h,a)\}}.

Here \(a\) is a treatment indicator, \(h\) collects other covariates, and \(g(y)\) is the outcome scale actually used in the fitted model. If \(g(y)=(y-c)/s\), the equation uses an outcome centered at \(c\) and scaled by \(s\); substitute that transformed value rather than the raw measurement. For a model without treatment interaction, \(\alpha_{YA}=0\). The probability calculation is otherwise direct: form the linear predictor at the chosen outcome and profile, then apply the inverse-logit function.

The logit link implies that a one-unit increase in \(g(y)\), at a fixed covariate profile, multiplies dropout odds by \(\exp(\alpha_Y)\) when \(a=0\), and by \(\exp(\alpha_Y+\alpha_{YA})\) when \(a=1\). These are conditional model associations, not causal effects. Probability changes are not constant across the curve: the same change in linear predictor can produce different changes in probability at different starting probabilities.

Worked Example: A Continuous Biomarker Across Its Clinical Range

Suppose a hypothetical fitted model uses \(g(y)=(y-50)/10\), where biomarker values are measured in units, and has \(\eta=-1.4+0.8g(y)\). Consider plausible values from 30 to 70 units for a fixed covariate profile. The model assumes a logistic link and this linear outcome effect throughout that range.

At \(y=30\), \(g(y)=-2\), so \(\eta=-1.4+0.8(-2)=-3.0\). Thus \(q=1/[1+\exp(3.0)]=0.0474\), or a 4.74% conditional dropout probability.

At \(y=50\), \(g(y)=0\), so \(\eta=-1.4\) and \(q=1/[1+\exp(1.4)]=0.1978\), or 19.8%.

At \(y=70\), \(g(y)=2\), so \(\eta=-1.4+1.6=0.2\) and \(q=1/[1+\exp(-0.2)]=0.5498\), or 55.0%.

The model therefore implies a steep increase in conditional dropout probability across these 40 biomarker units. Whether this is plausible requires clinical and design knowledge; the numbers alone do not establish that the specification is wrong. They do show what should be scrutinized: in particular, whether a probability above one-half at 70 units is credible for this visit and profile.

Compare Covariate Profiles and Interactions

A single probability curve can conceal important differences between profiles. When the equation includes treatment, baseline severity, or an interaction, calculate predictions for profiles that differ in those covariates while holding the others fixed. A treatment-by-outcome interaction changes the slope of the linear predictor across treatment arms; consequently, curves may have different directions or steepness. Inspecting several points on each curve makes that implication visible.

Worked Example: Checking a Treatment-by-Outcome Interaction

Consider the hypothetical model \(\eta=-2+0.5a+0.7z-0.6az+0.9h\), where \(a=1\) denotes the active-treatment profile, \(a=0\) the comparison profile, \(z=(y-50)/10\), and \(h=1\) indicates a specified baseline-severity category. First compare treatment profiles at \(h=0\), over \(z=-1,0,1\), corresponding to outcomes 40, 50, and 60 units. The logistic link is assumed, and each value is within the clinically plausible range for both profiles.

For \(a=0\), the linear predictors at \(z=-1,0,1\) are \(-2.7,-2.0,-1.3\). Applying the inverse logit gives dropout probabilities \(0.0630,0.1192,0.2142\), or 6.30%, 11.9%, and 21.4%.

For \(a=1\), the outcome slope is \(0.7-0.6=0.1\), and the predictors are \(-1.6,-1.5,-1.4\). The corresponding probabilities are \(0.1680,0.1824,0.1978\), or 16.8%, 18.2%, and 19.8%.

The fitted equation implies a much flatter outcome pattern in the active-treatment profile over these values. It also implies higher probabilities there at all three selected outcomes. These are model-based comparisons conditional on the specified profiles; they do not show that treatment causes dropout or that the interaction has a causal interpretation. Before judging the pattern, verify that the coding of \(a\), \(z\), and \(h\) matches the intended profile.

The \(h=1\) profile adds \(0.9\) to each linear predictor. For example, at \(a=0,z=0\), the predictor changes from \(-2.0\) to \(-1.1\), giving \(q=1/[1+\exp(1.1)]=0.2497\), rather than \(0.1192\). This illustrates why profiles should be stated explicitly: a probability is conditional on all covariates in the model, not on outcome alone.

Why the Probability Curve Behaves This Way

For a logistic model with a linear outcome term, the probability curve has a precise direction. If \(\eta(y)=c+\gamma y\), differentiation gives

\frac{dq(y)}{dy} = \frac{\exp\{\eta(y)\}}{[1+\exp\{\eta(y)\}]^2}\gamma = q(y)\{1-q(y)\}\gamma.

Since \(q(y)\{1-q(y)\}\) is positive for probabilities strictly between zero and one, the sign of the probability slope is the sign of \(\gamma\). Thus a positive outcome coefficient forces dropout probability to increase with the modeled outcome, and a negative coefficient forces it to decrease, for a fixed profile. The derivative also explains why probability changes flatten near zero and one: \(q(1-q)\) becomes small near either boundary. With a treatment interaction, use the profile-specific slope, such as \(\alpha_Y+\alpha_{YA}\) for \(a=1\), rather than the main-effect coefficient alone.

For discrete or nonlinear outcomes, evaluate the model at clinically relevant values using its actual specified form. Do not infer a linear trend where the model includes a nonlinear transformation, category indicators, or a different link. The same principle applies to ordinal or count outcomes: calculate probabilities at valid outcome categories or counts, and examine the pattern implied by the particular equation rather than treating the outcome as continuous without justification.

Worked Example: Inspecting a Count-Outcome Range

Suppose a hypothetical selection equation for an overdispersed count outcome uses \(\eta=-3+0.45y\), where \(y\) is the count, with a logistic link. For the profile under review, assume counts from 0 through 12 are clinically plausible. The equation specifies a linear effect in the count itself; no additional count-distribution assumption is needed to calculate these conditional probabilities.

At \(y=0\), \(\eta=-3\), so \(q=1/[1+\exp(3)]=0.0474\). At \(y=4\), \(\eta=-3+0.45(4)=-1.2\), giving \(q=1/[1+\exp(1.2)]=0.2315\). At \(y=8\), \(\eta=0.6\), giving \(q=1/[1+\exp(-0.6)]=0.6457\). At \(y=12\), \(\eta=2.4\), giving \(q=1/[1+\exp(-2.4)]=0.9168\).

The model thus predicts that a count of 12 is associated with a 91.7% conditional dropout probability for this visit and profile. This is a prompt to check the clinical interpretation and the equation’s fit over the range, not an automatic reason to reject it. If values near 12 are rare or outside the relevant clinical range, predictions there may chiefly reflect extrapolation. If they are common and the implied probability is not credible, the assumed linear predictor or other model components may need reconsideration.

Range, Visit, and Model Checks

A disciplined check uses a grid rather than a single “typical” patient. Choose outcome values spanning the plausible range at the visit, then repeat the calculation for clinically meaningful covariate profiles. State the visit, units, profile, and outcome transformation so that every probability has a clear interpretation. A compact table or curve can reveal reversals, sharp changes, and probabilities close to zero or one that a coefficient table does not make obvious.

The per-visit probability \(q_t\) must not be confused with cumulative dropout. Along a specified sequence of covariate and outcome values, if \(q_j\) is the conditional dropout probability at visit \(j\) among those still at risk, the model-implied probability of remaining observed through visit \(T\) is \(\prod_{j=1}^{T}(1-q_j)\). The corresponding probability of dropout by then is \(1-\prod_{j=1}^{T}(1-q_j)\). This calculation is conditional on the specified path; it is not obtained by simply adding the visit-specific probabilities.

Useful warning signs include probabilities that are clinically implausible over common outcome values, unexpectedly steep curves, treatment profiles that behave differently from the intended specification, and predictions that change dramatically under small coding or scaling changes. Check whether the apparent problem is caused by a mistaken unit conversion, reversed indicator coding, omitted centering, or applying a coefficient to the wrong scale. Then distinguish predictions within the supported clinical range from extrapolations beyond it.

A pattern-mixture specification describes outcome distributions by missingness pattern, whereas a selection specification models response or dropout conditional on outcomes and covariates. These approaches parameterize the missing-data relationship differently. Probability-curve inspection is especially direct for a selection equation, but it does not resolve whether its unobserved-outcome dependence is identified by observed data. Nor does a plausible curve prove the assumed model; conclusions still depend on the stated specification and assumptions.

Common Mistake and Limitations

Common pitfall: A reader sees a positive outcome coefficient and concludes that dropout probability rises by a fixed number of percentage points per outcome unit. That is incorrect: the coefficient is a change in log odds per unit of the modeled outcome, and the corresponding probability change depends on the starting probability. Calculate \(q\) at relevant outcomes and profiles instead. A second error is to interpret a model-implied probability at a missing outcome as an observed fact about that person; it is a consequence of the assumed model, not directly verified information.

These checks are descriptive diagnostics of model implications, not formal tests of model adequacy. The observed data generally cannot directly validate the selection relationship at values of the current outcome missing because of dropout. Clinical plausibility also depends on visit timing, the definition of dropout, and the populations represented by each profile. A probability near zero or one can be appropriate in some settings; the issue is whether it is credible for the stated context and whether it arises over values that matter.

State the conditional event.
Confirm that the probability concerns dropout at the visit among those still at risk, rather than cumulative dropout or nonresponse at another time.

Specify the model and profiles.
Record the link, outcome scale and transformation, covariate coding, and interactions before substituting values.

Calculate across a plausible grid.
Compute the linear predictor and inverse link at several outcomes and relevant covariate profiles, retaining units and context.

Judge the implications, not just the coefficients.
Look for unexpected directions, steep changes, extreme probabilities, and extrapolation; check coding and clinical plausibility before drawing conclusions.

Check Your Understanding

Answer each question using the stated model assumptions. The solutions follow the questions.

A logistic selection model has \(\eta=-1+0.5z\). Calculate the dropout probability at \(z=0\) and \(z=2\), and describe the implied direction.
A model has outcome slope \(-0.4\) for one profile. Must its implied dropout probability decrease as the outcome increases? State the condition under which your answer holds.
At two successive visits, a specified path has conditional dropout probabilities \(0.20\) and \(0.30\). Calculate the probability of dropout by the second visit along that path.
Why is a predicted probability of 0.95 at an outcome value rarely seen in the relevant clinical population not, by itself, decisive evidence that a model is unacceptable?

Solutions. (1) At \(z=0\), \(\eta=-1\), so \(q=1/[1+\exp(1)]=0.2689\). At \(z=2\), \(\eta=0\), so \(q=0.5000\). The positive slope implies an increasing probability as \(z\) increases. (2) Yes, provided the model has a logistic link and the linear predictor has that negative slope throughout the outcome range considered, with other covariates held fixed. Since \(dq/dy=q(1-q)(-0.4)<0\), probability decreases. (3) The probability of remaining observed through both visits is \((1-0.20)(1-0.30)=0.8(0.7)=0.56\); thus dropout by the second visit is \(1-0.56=0.44\), or 44%, conditional on the specified path. (4) That value may be an extrapolation into a sparse or clinically irrelevant part of the outcome range. Inspect predictions over common, plausible values and profiles, and assess whether the extreme prediction reflects a credible specification, coding, or extrapolation issue.