Complementary Log-Log Current-Outcome Hazards

Why Use a Complementary Log-Log Hazard?

A dropout model describes the chance that a participant stops contributing observed outcomes at a particular visit, among those still observed immediately before that visit. The link function determines how the model’s linear predictor maps to that conditional probability. The complementary log-log link is useful when a proportional-hazards interpretation on a cumulative-hazard scale is appropriate. It is asymmetric: unlike the logistic link, it does not treat probabilities near zero and near one symmetrically.

Here, $R_t=1$ means that the outcome is observed at visit $t$, and $R_t=0$ means that the participant has dropped out by that visit. Dropout is monotone: once $R_t=0$, all later indicators are zero. The dropout hazard at visit $t$ concerns participants with $R_{t-1}=1$. When dropout at visit $t$ depends on the current outcome $Y_t$, that outcome may be unobserved for a person who drops out at $t$; the model treats $Y_t$ as a well-defined, possibly latent, outcome for that person.

Definition: The visit-$t$ dropout hazard conditional on the current outcome is $$ h_t(y)=P(R_t=0\mid R_{t-1}=1,Y_t=y). $$ A complementary log-log model with a time-specific intercept is $$ \log\{-\log[1-h_t(y)]\}=\alpha_t+\gamma y, $$ where $\alpha_t$ is the visit-specific intercept and $\gamma$ is the current-outcome coefficient. Thus $$ h_t(y)=1-\exp\{-\exp(\alpha_t+\gamma y)\}. $$

This specifies a conditional probability at each visit, not a model for the outcome distribution $Y_t$. To construct a joint selection model, an outcome model and a dropout mechanism must be combined. In particular, the conditional hazard definition alone does not say that survival probabilities multiply after conditioning on an entire future outcome path. That stronger condition is stated explicitly below when needed.

From the Link to the Discrete-Time Probability

The complementary log-log link transforms a probability $h$ by taking the log of the negative log of its complement. Inverting that transformation gives the dropout probability directly:

\begin{aligned} \log\{-\log(1-h_t(y))\}&=\eta_t(y),\\ -\log(1-h_t(y))&=\exp\{\eta_t(y)\},\\ 1-h_t(y)&=\exp\{-\exp(\eta_t(y))\},\\ h_t(y)&=1-\exp\{-\exp(\eta_t(y))\}, \end{aligned}

where $\eta_t(y)=\alpha_t+\gamma y$. The quantity $-\log(1-h_t(y))$ is the discrete-interval cumulative hazard corresponding to the probability $h_t(y)$. A one-unit increase in $y$ multiplies this quantity by $\exp(\gamma)$:

\frac{-\log\{1-h_t(y+1)\}}{-\log\{1-h_t(y)\}} =\frac{\exp\{\alpha_t+\gamma(y+1)\}}{\exp\{\alpha_t+\gamma y\}} =\exp(\gamma).

This is a cumulative-hazard ratio, not a dropout risk ratio, odds ratio, or difference in probabilities. The probability difference depends on the starting value of $y$ and on $\alpha_t$. If $\gamma>0$, the hazard increases with the current outcome; if $\gamma<0$, it decreases. If $\gamma=0$, the specified hazard does not depend on $Y_t$.

Worked Example: Converting a Predictor to a Dropout Probability

Suppose, hypothetically, that at one visit the dropout model is $\log\{-\log[1-h_t(y)]\}=-2+0.7y$, where $y$ is measured in units of 10 outcome points. Calculate the hazard at $y=0$ and $y=1$.

At $y=0$, the predictor is $\eta=-2$. Therefore, $$ h_t(0)=1-\exp\{-\exp(-2)\} =1-\exp(-0.1353) \approx 1-0.8734 =0.1266. $$

At $y=1$, the predictor is $\eta=-2+0.7=-1.3$. Thus, $$ h_t(1)=1-\exp\{-\exp(-1.3)\} =1-\exp(-0.2725) \approx 1-0.7615 =0.2385. $$

In this hypothetical model, among participants still observed before this visit, the conditional dropout probability is about 12.7% at $y=0$ and 23.9% at $y=1$. The cumulative-hazard ratio for a 10-point increase is $\exp(0.7)\approx2.014$, or about 2.01. The probability itself does not double: it increases by about 11.2 percentage points for these two particular outcome values.

Interpreting the Outcome Effect and Visit Structure

With a common $\gamma$ across visits, each visit has its own baseline hazard through $\alpha_t$, while the outcome effect is constant on the cumulative-hazard scale. More precisely, for any two current-outcome values $y_a$ and $y_b$,

\frac{-\log\{1-h_t(y_b)\}}{-\log\{1-h_t(y_a)\}} =\exp\{\gamma(y_b-y_a)\}.

This comparison is conditional on still being observed at visit $t$. It is not a comparison of marginal dropout probabilities across all participants, since reaching visit $t$ depends on earlier dropout. If $Y_t$ is measured in points, $\gamma$ is the change in log cumulative hazard per point; rescaling the outcome changes the numerical coefficient but not the model’s fitted probabilities when the coefficient is transformed consistently.

There is a standard continuous-time motivation for this link. Suppose a latent dropout event has a constant event rate $\lambda_t(y)$ within a visit interval of length $\Delta_t$, conditional on the participant’s outcome and being at risk. The probability of no event during that interval is $\exp\{-\lambda_t(y)\Delta_t\}$. Hence the interval dropout probability is $1-\exp\{-\lambda_t(y)\Delta_t\}$, and

\log\{-\log[1-h_t(y)]\}=\log\{\lambda_t(y)\}+\log(\Delta_t).

If the rate follows $\lambda_t(y)=\lambda_{0t}\exp(\gamma y)$, this gives a complementary log-log model with intercept $\log(\lambda_{0t}\Delta_t)$. This derivation motivates the link; it does not establish that dropout truly follows a constant-rate continuous-time process. The discrete-time model can instead be used directly as a model for visit-specific probabilities.

Worked Example: Outcome Units and the Hazard Ratio

Suppose, hypothetically, that $\eta_t(y)=-1.5+0.4(y-5)$, where $y$ is an outcome score in points. Compare the hazards at scores 4 and 6.

At score 4, $\eta=-1.5+0.4(-1)=-1.9$, so $$ h_t(4)=1-\exp\{-\exp(-1.9)\} =1-\exp(-0.1496) \approx 0.1389. $$ At score 6, $\eta=-1.5+0.4(1)=-1.1$, giving $$ h_t(6)=1-\exp\{-\exp(-1.1)\} =1-\exp(-0.3329) \approx 0.2831. $$

The two-point increase multiplies the cumulative hazard by $\exp\{0.4(6-4)\}=\exp(0.8)\approx2.226$. The modeled conditional dropout probability rises from about 13.9% to 28.3%, a difference of 14.4 percentage points. These are visit-specific probabilities among those at risk, not overall probabilities of ever dropping out.

When Visit-Specific Survival Probabilities Multiply

A sequence of hazards can be used to obtain survival across visits, but the conditioning must be specified correctly. From the chain rule, conditional on a full outcome path $Y_{1:T}=(Y_1,\ldots,Y_T)$, the probability of remaining observed through $T$ is a product of conditional survival probabilities given that full path and survival so far. Those conditional probabilities need not equal $1-h_t(Y_t)$ merely because the local hazard $P(R_t=0\mid R_{t-1}=1,Y_t)$ has that value: conditioning on future outcomes can change them.

A common selection-model specification adds the assumption that, conditional on the full outcome path, the visit-$t$ dropout probability among those still at risk is exactly $h_t(Y_t)$. Under this explicit assumption, the chain rule gives:

Assumption and result: If, for every $t$, $$ P(R_t=0\mid R_{t-1}=1,Y_{1:T})=h_t(Y_t), $$ then $$ P(R_T=1\mid Y_{1:T})=\prod_{t=1}^{T}\{1-h_t(Y_t)\} =\exp\left[-\sum_{t=1}^{T}\exp\{\alpha_t+\gamma Y_t\}\right]. $$ The product is justified by the full-path conditional hazard assumption, not by the local hazard definition alone.

Worked Example: Survival Along a Specified Outcome Path

Consider a hypothetical two-visit path with $Y_1=0$ and $Y_2=1$. Suppose the fitted model gives $h_1(0)=0.10$ and $h_2(1)=0.20$. For this calculation, assume explicitly that, conditional on the full two-visit outcome path, the visit-specific dropout probabilities among those still at risk equal these hazards.

The probability of remaining observed at visit 1, conditional on the path, is $1-0.10=0.90$. Given that survival and the full path, the probability of remaining observed at visit 2 is $1-0.20=0.80$. Therefore, $$ P(R_2=1\mid Y_1=0,Y_2=1)=0.90(0.80)=0.72. $$

The probability of first dropping out at visit 2, conditional on this path, is $0.90(0.20)=0.18$; the probability of dropping out at visit 1 is $0.10$. These probabilities sum to $0.10+0.18+0.72=1.00$. Without the stated full-path assumption, the product $0.90(0.80)$ is not implied by the two local conditional hazards alone.

Assumptions, Diagnostics, and Alternatives

The complementary log-log specification is most useful when the outcome-dependent hazard structure is scientifically plausible and the outcome scale is meaningful. Its principal assumptions should be distinguished:

Correct risk set and timing. The hazard is conditional on $R_{t-1}=1$, and the model treats the current $Y_t$ as the outcome on which dropout at visit $t$ may depend. The timing must match the scientific process.
Correct link and linear predictor. The model assumes a linear relationship between the predictor and the log cumulative hazard. A common $\gamma$ further assumes the same outcome effect at each visit.
Outcome-path selection factorization, when used. Products for survival conditional on $Y_{1:T}$ require the full-path conditional assumption stated above. A marginal local hazard does not supply it automatically.
Outcome scale and model support. A one-unit effect depends on the units of $Y_t$. Implausible extrapolation at extreme outcomes can produce near-zero or near-one hazards.

Useful diagnostics include comparing observed and model-predicted dropout proportions within visits and across ranges of the current outcome among participants still at risk. Examine whether the fitted hazards change monotonically with the outcome as the model requires, whether residual discrepancies show curvature, and whether a small number of observations determine the slope. If few participants drop out at a visit, the visit-specific intercept and outcome effect can be imprecise; complete or near separation can also lead to unstable estimates. A good fit to observed risk sets does not verify the unobserved outcomes of people who dropped out or prove that the full-path assumption is correct.

The logistic model uses log odds as its linear predictor, while the probit model uses a standard-normal quantile of the probability. The complementary log-log model instead makes the log cumulative hazard linear. These links generally yield different probability curves and coefficient interpretations, even when all fit a similar range of observed probabilities. The choice should follow the intended hazard interpretation and be checked for sensitivity, rather than being made solely because one link gives a preferred result.

Link	Linear predictor scale	Interpretation of a one-unit outcome increase
Complementary log-log	Log cumulative hazard	Cumulative hazard is multiplied by $\exp(\gamma)$
Logistic	Log odds	Dropout odds are multiplied by $\exp(\gamma)$
Probit	Standard-normal quantile	Coefficient shifts the latent normal index; it is not a constant odds or risk ratio

Worked Example: Avoiding an Odds-Ratio Interpretation

Suppose, hypothetically, the current-outcome coefficient is $\gamma=0.4$. The cumulative-hazard ratio for a one-unit increase is $\exp(0.4)\approx1.492$. At a reference value where the hazard is $0.10$, the cumulative hazard is $-\log(0.90)=0.1054$. Multiplying by 1.492 gives $0.1573$, which corresponds to a new hazard $$ 1-\exp(-0.1573)\approx0.1455. $$

Thus, at this reference point, the hazard rises from 10.0% to about 14.6%. The odds ratio for these probabilities would be $\{0.1455/(1-0.1455)\}/\{0.10/(1-0.10)\}\approx1.532$, not 1.492. The coefficient’s direct interpretation is the cumulative-hazard ratio; the probability and odds comparisons depend on the reference hazard.

Common Pitfall and Practical Limits

Wrong reasoning: “The coefficient is 0.4, so each outcome unit multiplies the dropout odds by $\exp(0.4)$.” That is the usual interpretation of a logistic coefficient, not a complementary log-log coefficient. Correction: $\exp(0.4)$ multiplies the cumulative hazard $-\log(1-h_t)$; calculate the probability at the relevant outcome values to describe absolute changes.

Another important limit is that a current-outcome-dependent dropout model concerns selection into observation. It does not, by itself, identify the distribution of outcomes after dropout from observed data alone. Inferences about those outcomes require the outcome model and assumptions connecting missing outcomes to observed information. The model’s conditional interpretation and any full-path factorization must be reported clearly; fitting the link cannot validate those assumptions.

Key takeaway: A complementary log-log current-outcome hazard maps $\alpha_t+\gamma Y_t$ to $1-\exp\{-\exp(\alpha_t+\gamma Y_t)\}$. Its outcome coefficient describes a cumulative-hazard ratio. Multiplying visit-specific survival probabilities conditional on the full outcome path requires an explicit full-path hazard assumption.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If $\eta_t(y)=\log\{-\log[1-h_t(y)]\}$, solve for $h_t(y)$.
For $\gamma=-0.3$, what is the cumulative-hazard ratio for a one-unit increase in $y$, and what direction of association does the model imply?
If $h_1=0.20$ and $h_2=0.25$ along a fixed outcome path, what is the conditional probability of surviving both visits under the full-path hazard assumption?
Why does a local hazard conditional on $R_{t-1}=1,Y_t$ not, by itself, justify multiplying $1-h_t(Y_t)$ after conditioning on $Y_{1:T}$?

Solutions. (1) Inverting the link gives $h_t(y)=1-\exp\{-\exp(\eta_t(y))\}$. (2) The ratio is $\exp(-0.3)\approx0.741$; a one-unit increase is associated with about a 25.9% lower cumulative hazard, holding the visit fixed. (3) The probability is $(1-0.20)(1-0.25)=0.80(0.75)=0.60$, conditional on the full-path assumption. (4) Conditioning on future outcomes can change the conditional visit-specific survival probabilities; the local hazard does not assert that those probabilities, given the full path, equal $1-h_t(Y_t)$.

Link	Linear predictor scale	Interpretation of a one-unit outcome increase
Complementary log-log	Log cumulative hazard	Cumulative hazard is multiplied by \(\exp(\gamma)\)
Logistic	Log odds	Dropout odds are multiplied by \(\exp(\gamma)\)
Probit	Standard-normal quantile	Coefficient shifts the latent normal index; it is not a constant odds or risk ratio