Treatment-Arm-Specific Dropout Equations

Why Specify Dropout Equations by Treatment Arm?

A treatment may change not only the outcome trajectory but also how dropout depends on the current outcome. For example, a high biomarker value might be associated with greater dropout in one arm and only weakly associated with dropout in another. A single common outcome slope would rule out that difference. Treatment-arm-specific dropout equations allow the association between the current outcome and dropout risk to vary by arm, while also allowing the arms to have different baseline dropout risks.

The key modeling choice is to give each arm its own intercept and current-outcome coefficient. This is a way to describe the conditional dropout mechanism, not a claim that treatment causally changes an individual’s probability of dropping out through a particular mechanism. The coefficients depend on the chosen link, outcome scale, covariates, and conditioning history.

Definition and Model

Definition: Let \(A_i\in\{0,1\}\) denote the treatment arm for person \(i\). At visit \(t\), let \(R_{it}=1\) indicate that the outcome is observed and \(D_{it}=1-R_{it}\) indicate dropout at that visit. The discrete-time dropout hazard is the probability of dropout at \(t\), conditional on being at risk at \(t\): \(P(D_{it}=1\mid R_{i,t-1}=1,\ldots)\). An arm-specific current-outcome selection equation gives that hazard a separate intercept and current-outcome slope in each arm.

For a logistic selection equation, let \(Y_{it}\) be the current outcome and \(Z_{it}\) a vector of covariates or observed history available for the conditioning set. Choose a centering value \(c\) and write \(X_{it}=Y_{it}-c\). One version with common coefficients for \(Z_{it}\) is

\operatorname{logit} P(D_{it}=1\mid R_{i,t-1}=1,A_i=a,Y_{it},Z_{it}) =\alpha_a+\beta_a X_{it}+\gamma^\mathsf{T}Z_{it},\qquad a\in\{0,1\}.

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\), \(\alpha_a\) is the arm-specific intercept, and \(\beta_a\) is the arm-specific coefficient for the current outcome. If desired, additional components of the equation, such as coefficients for history, can also vary by arm; that is a further modeling choice, not implied merely by allowing \(\alpha_a\) and \(\beta_a\) to differ. When the outcome is centered at \(c\), \(\alpha_a\) describes the log odds of dropout at \(Y_{it}=c\) and \(Z_{it}=0\), for a person at risk in arm \(a\).

The equation conditions on the current outcome even when that outcome is missing for a person who drops out at that visit. This is the defining selection-model feature: the dropout probability is modeled as depending on the outcome value, including values that are unobserved after dropout. It therefore needs to be considered together with an outcome model and assumptions about the unobserved outcomes for estimation. The equation alone does not supply those missing values.

Relationship to a Treatment-by-Outcome Interaction

A single equation with a treatment indicator and an interaction can express exactly the same two arm-specific intercepts and slopes. With the control arm coded \(A_i=0\), write

\operatorname{logit}P(D_{it}=1\mid\cdots) =\alpha_0+\delta_A A_i+\beta_0 X_{it} +\delta_{AX}A_iX_{it}+\gamma^\mathsf{T}Z_{it}.

Setting \(A_i=0\) gives intercept \(\alpha_0\) and outcome slope \(\beta_0\). Setting \(A_i=1\) gives intercept \(\alpha_0+\delta_A\) and slope \(\beta_0+\delta_{AX}\). Thus the conversion is

\alpha_1=\alpha_0+\delta_A,\qquad \beta_1=\beta_0+\delta_{AX}; \qquad \delta_A=\alpha_1-\alpha_0,\qquad \delta_{AX}=\beta_1-\beta_0.

The interaction coefficient is the difference between the two arms’ current-outcome slopes on the log-odds scale. It is not the treatment-arm slope itself. Setting \(\delta_{AX}=0\) imposes a common outcome slope; setting \(\delta_A=0\) imposes a common intercept at the chosen outcome reference and covariate values. These are distinct restrictions.

Worked Example: Translating an Interaction Equation

Suppose the hypothetical dropout equation is

\operatorname{logit}P(D_{it}=1\mid\cdots) =-2.0+0.5A_i+0.30X_{it}-0.20A_iX_{it},

where \(X_{it}=Y_{it}-c\). For arm 0, set \(A_i=0\): the equation is \(-2.0+0.30X_{it}\), so \(\alpha_0=-2.0\) and \(\beta_0=0.30\). For arm 1, set \(A_i=1\): the equation becomes \((-2.0+0.5)+(0.30-0.20)X_{it}=-1.5+0.10X_{it}\). Thus \(\alpha_1=-1.5\) and \(\beta_1=0.10\). The treatment-by-outcome interaction is \(-0.20=\beta_1-\beta_0\).

At \(X_{it}=2\), the arm 0 linear predictor is \(-2.0+0.30(2)=-1.40\), giving dropout probability \(1/(1+e^{1.40})\approx0.1978\). In arm 1 it is \(-1.5+0.10(2)=-1.30\), giving \(1/(1+e^{1.30})\approx0.2142\). These probabilities are conditional on the modeled history and on being at risk at that visit.

Interpreting and Calculating Arm-Specific Risks

For a logistic equation, increasing \(X_{it}\) by one unit in arm \(a\), with the other conditioning variables fixed, multiplies the conditional odds of dropout by \(e^{\beta_a}\). The corresponding probability change is not constant: it depends on the starting probability. A positive \(\beta_a\) means higher current outcomes are associated with higher conditional dropout odds in that arm; a negative value means lower odds. Neither statement is causal.

Worked Example: Comparing Current-Outcome Slopes

Consider hypothetical arm-specific equations with no additional \(Z_{it}\) terms:

\operatorname{logit}P(D_{it}=1\mid\cdots,A_i=0)=-2.0+0.40X_{it}, \qquad \operatorname{logit}P(D_{it}=1\mid\cdots,A_i=1)=-1.2+0.10X_{it}.

In arm 0, a one-unit increase in \(X_{it}\) multiplies conditional dropout odds by \(e^{0.40}\approx1.492\), an increase of about 49.2%. In arm 1 the multiplier is \(e^{0.10}\approx1.105\), about a 10.5% increase. These are odds comparisons per one outcome unit, not percentage-point changes in dropout probability.

At \(X_{it}=0\), the arm 0 log odds are \(-2.0\), so the probability is \(1/(1+e^2)\approx0.1192\). The arm 1 log odds are \(-1.2\), so its probability is \(1/(1+e^{1.2})\approx0.2315\). At \(X_{it}=5\), the linear predictors are \(0\) and \(-0.7\), giving probabilities \(0.5000\) and \(1/(1+e^{0.7})\approx0.3318\), respectively. The probabilities cross over these values: a comparison at one outcome level need not describe the comparison at another level when slopes differ.

The centering value matters for the intercept but not for the arm-specific slope. If \(X=Y-c\), the intercept is the log odds at \(Y=c\), under the stated covariate reference values. Choosing a meaningful \(c\), such as a clinically relevant value or a value near the middle of the observed range, makes the intercept easier to interpret and can reduce needless numerical separation between intercept and slope. Changing the centering value changes the intercepts; it does not change the fitted probabilities when the equation is transformed consistently.

Worked Example: Why Centering Clarifies the Intercepts

Suppose a hypothetical biomarker is measured in units for which \(50\) is a useful reference. The arm equations are

\operatorname{logit}P(D=1\mid A=0,Y)=-1.5+0.08(Y-50), \qquad \operatorname{logit}P(D=1\mid A=1,Y)=-1.0+0.03(Y-50).

At \(Y=50\), the predictors are \(-1.5\) and \(-1.0\), so the dropout probabilities are \(1/(1+e^{1.5})\approx0.1824\) and \(1/(1+e^{1.0})\approx0.2689\). These intercepts therefore compare the arms at a biomarker value that is meaningful in this hypothetical setting.

At \(Y=60\), arm 0 has predictor \(-1.5+0.08(10)=-0.70\); arm 1 has predictor \(-1.0+0.03(10)=-0.70\). Both probabilities are \(1/(1+e^{0.70})\approx0.3318\). The equality at this particular value does not imply equal equations: the arm slopes are \(0.08\) and \(0.03\), and the intercepts at \(Y=50\) differ.

Assumptions, Diagnostics, and Limitations

An arm-specific equation is meaningful only relative to its conditioning set and link. Its interpretation requires a correctly defined at-risk set, an appropriate time ordering for the covariates and outcomes, and a correctly specified conditional functional form over the outcome range where predictions are used. The logistic version assumes a linear relationship between the centered current outcome and the log odds of dropout within each arm, unless nonlinear terms are added. It does not assume that the outcome itself is logistic or binary.

The current-outcome dependence is especially consequential because the outcome is unavailable for those who drop out at that visit. Data observed among people who remain do not directly reveal the distribution of current outcomes among dropouts. A joint analysis therefore needs an outcome model and identifying assumptions; different assumptions about unobserved outcomes can support different selection coefficients. This equation is a model specification, not a guarantee that the coefficients are identified precisely or verified from observed data alone.

Useful checks include examining the outcome range and number of dropout events separately by arm, checking whether the assumed linear log-odds pattern is plausible, and calculating implied probabilities at low, central, and high outcome values. Sparse dropout events in one arm or little overlap in outcome ranges can make arm-specific slopes unstable or largely extrapolative. Implausible probabilities, very large coefficient estimates, or sharply changing results under modest specification changes are reasons to revisit the model and its assumptions, not reasons to interpret the coefficients more confidently.

The main alternatives impose different structure. A common-slope model allows different intercepts but assumes the same current-outcome log-odds association in both arms. A treatment-by-outcome interaction is algebraically equivalent to separate arm intercepts and slopes when both terms are included; it is often convenient for describing restrictions or contrasts. Separate arm equations make arm-specific predictions and interpretations explicit, but use more parameters. Neither form automatically solves weak information about dropout among people with missing current outcomes.

Specification	What it allows	Restriction or caution
Arm-specific intercepts and slopes	Both baseline log odds and outcome association vary by arm	Requires enough information in each arm; outcome effects remain conditional associations
Interaction form with treatment-by-outcome term	Same two intercepts and slopes, expressed through contrasts	The interaction is the difference in slopes, not an arm-specific slope
Different intercepts, common slope	Arm-specific baseline log odds at the reference outcome	Assumes the same outcome association in both arms

Common Mistake

A common error is to read the treatment-by-outcome interaction as the outcome slope in the treatment arm. In the interaction equation, the treatment-arm slope is \(\beta_0+\delta_{AX}\), while \(\delta_{AX}\) alone is the difference between treatment and control slopes. Another error is to call an arm-specific dropout coefficient a treatment effect on the outcome or a causal effect on dropout. It describes a conditional association in a specified selection model; causal interpretation requires additional design and assumptions and does not follow from this equation.

Key takeaway: Write an arm-specific current-outcome selection equation as \(\alpha_a+\beta_a(Y_{it}-c)\) on the chosen link scale. The arm intercepts describe risks at the reference outcome, while the arm slopes describe how conditional dropout odds change with the current outcome. In interaction form, the treatment-by-outcome coefficient is the difference between slopes.

Check Your Understanding

Answer each question, then compare with the brief solutions.

If \(\beta_0=0.20\) and \(\beta_1=-0.10\) in logistic arm-specific equations, what is the treatment-by-outcome interaction coefficient?
In arm 1, an outcome increase of two units occurs with \(\beta_1=0.30\). By what factor do the conditional dropout odds change?
Why does changing the outcome-centering value affect an intercept but not the fitted probabilities when the model is re-expressed correctly?
Does a positive arm-specific outcome coefficient establish that higher outcomes cause dropout? Explain.

Solutions. (1) The interaction is \(\beta_1-\beta_0=-0.10-0.20=-0.30\). (2) The odds multiplier is \(e^{2(0.30)}=e^{0.60}\approx1.822\), so the conditional odds are about 1.822 times as large. (3) Centering changes the zero point of the predictor, so the intercept must shift to preserve the same linear predictor at every \(Y\); consequently the probabilities remain unchanged. (4) No. The coefficient describes a conditional association under the specified selection model, not a causal effect.