Why Parameterize Selection With an Odds Ratio?
A logistic selection model describes how dropout odds vary with the current outcome, conditional on the history and covariates in the model. Its slope is useful, but its numerical value depends on the outcome’s units: a coefficient per point is not the same number as a coefficient per ten points. An odds-ratio parameterization makes the comparison explicit by naming the two outcome values being compared.
Let \(D_t=1\) indicate dropout at visit \(t\) among participants still at risk of dropout, and let \(Y_t\) be the outcome at that visit. Let \(H_t\) denote the covariates and observed history on which the dropout model conditions. Write \(p_t(y,h)=P(D_t=1\mid Y_t=y,H_t=h)\). The current outcome may be unobserved for people who drop out; the conditional probability is part of the selection model, not necessarily a probability directly estimable from complete observed records.
From the Odds Ratio to the Logistic Coefficient
Consider a logistic selection model with a common current-outcome slope across histories:
Here \(y_0\) is a chosen reference outcome value. The term \(\alpha_t(h)\) is the log dropout odds at \(y_0\) for history \(h\), and \(\beta_t\) is the change in log dropout odds per one outcome unit. Subtracting the logit at \(y_0\) from the logit at \(y_1\) gives the comparison directly:
Thus the odds ratio for a one-unit increase is \(\exp(\beta_t)\), while the odds ratio for a \(k\)-unit increase is \(\exp(k\beta_t)\). If the model includes an interaction between the current outcome and a covariate, the slope—and therefore the odds ratio—can vary with that covariate. The comparison must then state the covariate value as well as the two outcome values.
Deriving Probabilities From a Reference Value
An odds ratio can be paired with a reference dropout probability to obtain a probability at another outcome value. This conversion is useful because an odds ratio by itself does not reveal the absolute dropout probability. Let \(p_0=p_t(y_0,h)\), and let \(\psi=\operatorname{OR}_t(y_1,y_0\mid h)\). The odds at \(y_0\) are \(p_0/(1-p_0)\); multiplying by \(\psi\) gives the odds at \(y_1\). Converting those odds back to a probability yields:
The denominator is essential: multiplying a probability by an odds ratio does not generally produce the new probability. The formula also shows that the same odds ratio can imply different probability changes at different reference risks.
Worked Example: Comparing Dropout Odds Across Ten Outcome Points
Suppose a hypothetical model has a logistic coefficient of \(-0.18\) per outcome point. At a fixed history, the modeled dropout probability at \(Y_t=60\) is \(0.20\). Find the odds ratio and dropout probability at \(Y_t=70\).
The outcome contrast is \(70-60=10\) points. The conditional odds ratio is \( \exp\{-0.18(10)\}=\exp(-1.8)=0.1653 \). Thus, at the same history, the dropout odds at 70 points are about \(0.1653\) times those at 60 points.
At 60 points, the reference odds are \(0.20/(1-0.20)=0.25\). The odds at 70 points are \(0.25(0.1653)=0.04133\). Converting to a probability gives \(0.04133/(1+0.04133)=0.0397\).
The model therefore gives a dropout probability of about \(3.97\%\) at 70 points, compared with \(20\%\) at 60 points, conditional on the stated history. This is a substantial probability difference, but the odds ratio itself is not a risk ratio.
Worked Example: Starting With a Reference Probability and Odds Ratio
In a hypothetical analysis, the dropout probability at \(Y_t=50\) and a specified history is \(0.12\). Suppose the conditional dropout odds ratio for a five-point increase is \(1.50\). Calculate the probability at \(Y_t=55\), and express the corresponding logistic slope per point.
The reference odds are \(0.12/0.88=0.1363636\ldots\). Multiplying by the five-point odds ratio gives odds of \(0.1363636\ldots(1.50)=0.204545\ldots\approx0.2045\) at 55 points. The probability is \(0.2045/(1+0.2045)=0.1698\), or about \(17.0\%\).
For the logistic slope \(\beta_t\), the five-point contrast satisfies \(\exp(5\beta_t)=1.50\). Therefore, \(\beta_t=\log(1.50)/5=0.08109\) per point. The positive sign matches the odds ratio above one: higher values correspond to higher conditional dropout odds in this model.
Outcome Units and Reference Values
Changing the outcome’s units changes the coefficient’s numerical value but not the modeled dropout probabilities or a correctly specified contrast. Suppose \(Y_t\) is measured in points and the model is \(\operatorname{logit}\{p_t(y,h)\}=\alpha_t(h)+\beta_t(y-y_0)\). If the rescaled variable is \(Z_t=Y_t/10\), then \(Y_t=10Z_t\). Substitution shows that the coefficient per ten-point unit is \(10\beta_t\):
For example, a slope of \(-0.12\) per point becomes \(-1.2\) per ten-point unit. It is ten times the per-point coefficient, not ten times the per-ten-point coefficient. A 15-point contrast is 1.5 ten-point units, so its log odds ratio remains \((-1.2)(1.5)=-1.8\) and its odds ratio remains \(\exp(-1.8)=0.1653\). Rescaling does not alter the prediction for a specified outcome contrast.
A reference-value change also affects the intercept but not the slope or any contrast-specific odds ratio. For instance, if the linear predictor is written at reference \(y_0\) and then re-expressed at \(y_1\), the new intercept is the old logit evaluated at \(y_1\). The model’s fitted probabilities stay the same; only the point at which the intercept is interpreted changes.
Worked Example: Checking a Rescaled Coefficient
Suppose a hypothetical selection model has slope \(-0.12\) per point. Express the slope per ten-point unit and find the odds ratio for a 15-point increase.
Let \(Z_t=Y_t/10\). A ten-point increase is a one-unit increase in \(Z_t\), so the slope per ten-point unit is \(10(-0.12)=-1.2\). The same 15-point increase is \(15/10=1.5\) units of \(Z_t\). Its log odds ratio is \((-1.2)(1.5)=-1.8\), giving an odds ratio of \(\exp(-1.8)=0.1653\).
Using the original scale gives the same result: \((-0.12)(15)=-1.8\), so the odds ratio is again \(0.1653\). If the two calculations disagree, the contrast or the units have likely been mishandled.
Assumptions, Diagnostics, and Limitations
The odds-ratio interpretation is conditional on the stated selection model and its conditioning variables. The logistic form assumes that the log dropout odds are linear in the current outcome over the modeled range, unless nonlinear terms are added. A single \(\beta_t\) further assumes that the outcome contrast has the same log-odds effect across histories; interactions relax that restriction. The comparison also requires a consistent definition and timing of dropout and current outcome.
Useful checks include confirming the outcome’s units and reference value, calculating odds ratios for contrasts that matter scientifically, and converting those contrasts to probabilities at plausible reference dropout risks. If predicted probabilities change implausibly across the outcome range, inspect the assumed linear log-odds relationship and the range over which the model is being used. Sparse data can also make fitted coefficients unstable. These checks assess the model’s implications; they do not establish that the dependence on an unobserved current outcome is correct.
In particular, when current outcomes are missing for participants who drop out, the outcome-dependent selection parameter is generally not learned from the observed dropout records alone without additional assumptions. It may therefore serve as a sensitivity parameter: analysts can examine how conclusions change across scientifically plausible values. This is an identification issue, not a reason to interpret a chosen odds ratio as an observed association.
| Parameterization | What it states | Common caution |
|---|---|---|
| Logistic coefficient \(\beta_t\) | Change in log dropout odds per outcome unit, conditional on the model’s history and covariates | Its magnitude depends on the outcome scale |
| Conditional odds ratio \(\exp\{\beta_t(y_1-y_0)\}\) | Ratio of dropout odds at two specified outcome values, holding history fixed | It is not a probability ratio or a probability difference |
| Reference probability plus odds ratio | Absolute dropout probability at one outcome value and a relative odds comparison to another | The probability change depends on the reference probability |
A risk-scale description may be easier to communicate when absolute probabilities are the main interest, but a constant odds ratio does not imply a constant risk difference or risk ratio. Alternative link functions, such as a probit selection model, encode a different relationship between the linear predictor and dropout probability; they do not make the logistic coefficient and odds ratio interchangeable with a risk-scale effect.
Common Mistakes
- Reversing the comparison. \(\operatorname{OR}(y_1,y_0)\) has odds at \(y_1\) in the numerator. Reversing the values gives its reciprocal.
- Forgetting the contrast. \(\exp(\beta_t)\) is the odds ratio for one outcome unit, not automatically for a ten-point or other larger change. Use \(\exp\{\beta_t(y_1-y_0)\}\).
- Multiplying probability by an odds ratio. Multiply the reference odds by the odds ratio, then convert the resulting odds to a probability.
- Ignoring scale changes. If the outcome is expressed in ten-point units instead of points, the coefficient is ten times the per-point coefficient. Keep the contrast in the same units as the coefficient.
- Reading conditional odds as marginal risks. The odds ratio holds the modeled history fixed. It is not automatically a comparison of overall dropout probabilities across groups with different histories.
Check Your Understanding
Answer each question, then compare your work with the solutions.
- A logistic selection coefficient is \(0.10\) per outcome point. What is the conditional dropout odds ratio for a six-point increase?
Solution: The log odds ratio is \(0.10(6)=0.60\), so the odds ratio is \(\exp(0.60)=1.822\). At the same modeled history, dropout odds are about 1.82 times as large at the higher outcome value. - At a reference outcome value, dropout probability is \(0.25\). The odds ratio for a specified contrast is \(0.50\). What is the probability at the comparison value?
Solution: Reference odds are \(0.25/0.75=0.3333\). Comparison odds are \(0.3333(0.50)=0.1667\), giving probability \(0.1667/(1+0.1667)=0.1429\), or about \(14.3\%\). - A slope is \(-0.04\) per point. What is the corresponding slope per ten-point unit, and what is the odds ratio for a 20-point increase?
Solution: The per-ten-point slope is \(10(-0.04)=-0.40\). The 20-point contrast is two such units, so the log odds ratio is \((-0.40)(2)=-0.80\), and the odds ratio is \(\exp(-0.80)=0.4493\). - Why does an odds ratio of 2 not mean that dropout probability doubles?
Solution: It means the dropout odds double. The probability corresponding to doubled odds depends on the reference probability. For reference probability \(0.10\), odds are \(0.10/0.90=0.1111\); doubled odds are \(0.2222\), giving probability \(0.2222/(1+0.2222)=0.1818\), not \(0.20\).