Why the Probability Range Matters
A transition probability \(P_{ij}(s,t)\) is the probability of being in state \(j\) at time \(t\), conditional on being in state \(i\) at time \(s\). Its estimate \(\hat P_{ij}(s,t)\), such as an Aalen–Johansen estimate, must lie between zero and one. A confidence interval, however, is built to describe sampling uncertainty, and a procedure based on a normal approximation can produce endpoints outside that range.
This is especially likely when the estimated probability is close to zero or one, or when its standard error is large relative to its distance from a boundary. The issue is not merely cosmetic: an interval such as \([-0.03,0.11]\) cannot be interpreted literally as a range of possible probabilities. Clipping it to \([0,0.11]\) restores the range but does not, by itself, establish that the resulting interval has the intended coverage.
The procedures below presume that the target and time points were specified, that subjects are sampled independently under the study design, and that the transition-probability estimator and its variance estimate are valid under the applicable censoring and observation conditions. Approximate normal intervals additionally require a sufficiently accurate normal approximation to the estimator. A bootstrap interval requires a resampling scheme that reflects the sampling design and a bootstrap distribution that adequately approximates sampling variation. These conditions matter: a bounded interval is not automatically a well-calibrated one.
Direct Normal Intervals and Their Boundary Problem
If \(\hat P_{ij}(s,t)\) is approximately normal with estimated standard error \(SE\), the direct, or Wald, interval is
where \(z_{1-\alpha/2}\) is the corresponding standard normal quantile. For a 95% interval, \(z_{0.975}\approx1.96\). The formula is simple and can be reasonable when the estimate is well inside \([0,1]\), its sampling distribution is approximately symmetric, and the standard error is estimated reliably. Near a boundary, symmetry is often a poor approximation: the estimate cannot vary below zero or above one, while a normal distribution has no such restriction.
Worked Example: A Small Probability and an Invalid Wald Endpoint
Suppose a hypothetical study estimates the probability of transition from state A to state B by a fixed horizon as \(\hat P_{AB}=0.040\), with \(SE=0.035\). Assume independent subjects and, for this calculation, that the variance estimate is appropriate for the study design. The 95% direct normal interval is
\[ 0.040\pm1.96(0.035)=0.040\pm0.0686. \]
Thus the endpoints are \(0.040-0.0686=-0.0286\) and \(0.040+0.0686=0.1086\), giving \([-0.0286,0.1086]\). The lower endpoint is not a possible probability. This calculation signals that the direct normal approximation is questionable for this estimate; it does not justify reporting a negative probability or silently replacing it with zero.
Transforming to Respect the Range
A common alternative is to transform the estimate to a scale on which the parameter is unrestricted, construct an approximate interval there, and transform its endpoints back. For a probability strictly between zero and one, the logit transformation is
If the estimated standard error of \(\hat p\) is \(SE\), the delta method gives an approximate standard error for \(g(\hat p)\) of \(SE/\{\hat p(1-\hat p)\}\). A logit-based 95% interval is therefore formed by adding and subtracting \(1.96\) times this transformed-scale standard error from \(g(\hat p)\), then applying \(g^{-1}\) to both endpoints. Because \(g^{-1}\) always lies strictly between zero and one for finite inputs, the returned endpoints are within the probability range.
This is an approximation, not a guarantee of nominal coverage. The derivative grows near zero and one, so uncertainty on the logit scale can be large; the transformed sampling distribution may still be poorly approximated by a normal distribution. Also, if \(\hat p=0\) or \(\hat p=1\), its logit is undefined. Adding an arbitrary small constant to make the calculation work changes the procedure and needs separate justification.
Worked Example: A Logit Interval Near One
In a hypothetical multi-state analysis, suppose \(\hat P_{ij}(s,t)=0.960\) and \(SE=0.0250\). Assume the estimate is strictly between zero and one, the standard error is appropriate, and the transformed-scale normal approximation is being used as an approximation. First calculate
\[ g(0.960)=\log(0.960/0.040)=\log(24)=3.178054, \qquad SE_g=\frac{0.0250}{0.960(0.040)}=0.651042. \]
The 95% logit-scale endpoints are \[ 3.178054\pm1.96(0.651042) =3.178054\pm1.276042, \] so they are approximately \(1.902012\) and \(4.454095\). Transforming back gives \[ \left[ \frac{1}{1+e^{-1.902012}}, \frac{1}{1+e^{-4.454095}} \right] \approx[0.8701,\,0.9885]. \]
For comparison, the direct interval is \(0.960\pm1.96(0.0250)=[0.9110,1.0090]\), whose upper endpoint exceeds one. The logit interval stays in the probability range, but its lower endpoint is substantially below the estimate because the transformed uncertainty is not symmetric when returned to the original scale. Neither the bounded endpoints nor their asymmetry alone establish that this interval has accurate 95% coverage.
Bootstrap Intervals Near a Boundary
A bootstrap interval uses repeated resamples to approximate the sampling distribution of the estimator. For transition probabilities, resampling is typically performed at the independent-subject level, with the transition-probability estimator recalculated in each resample. A percentile interval uses the empirical \(\alpha/2\) and \(1-\alpha/2\) quantiles of those bootstrap estimates. Its endpoints are within \([0,1]\) when every resampled estimate is, but boundary values and skewness can make the bootstrap distribution discrete or concentrated at zero or one.
Other bootstrap constructions, including basic or bias-corrected and accelerated intervals, can yield different endpoints. None removes the need to select a resampling method appropriate to the design, use enough resamples for stable tail quantiles, and assess whether the bootstrap distribution represents the estimator’s sampling behavior. A percentile interval that looks plausible can still have poor coverage, particularly with sparse transitions or an estimate at a boundary.
Worked Example: A Bootstrap Interval Near Zero
Suppose a hypothetical analysis estimates a fixed-horizon transition probability as \(\hat P_{AB}=0.015\). Investigators resample independent subjects 2,000 times, preserving each resampled subject’s observed record and recalculating the probability estimate. Assume the study design supports subject-level resampling and the estimator is valid under the applicable observation conditions. The 2.5th and 97.5th empirical percentiles of the 2,000 estimates are 0.000 and 0.078.
The resulting 95% percentile interval is \([0.000,0.078]\). It stays in the probability range and reflects the nonnegative, potentially skewed bootstrap distribution better than a symmetric interval might. The zero lower endpoint does not mean the probability is known to be zero; it indicates that the lower empirical tail of the resampled estimates reaches the boundary. With many resamples equal to zero, the endpoint may be sensitive to a small number of observed transitions and to the resampling design. Reporting the number of zero bootstrap estimates and checking endpoint stability across resampling runs can help diagnose this problem.
Choosing and Checking an Interval
No interval method dominates in every setting. A direct normal interval is transparent and often adequate for an interior estimate with a stable standard error and a roughly symmetric sampling distribution. A logit interval respects the range and can accommodate asymmetry on the probability scale, but requires an interior estimate and relies on a transformed-scale approximation. A bootstrap interval can represent skewness without imposing symmetry, but its quality depends on the resampling scheme and on having enough information in the observed data to reproduce sampling variation.
| Approach | Range of endpoints | Key condition or concern |
|---|---|---|
| Direct normal (Wald) | May extend below zero or above one | Needs an adequate symmetric normal approximation |
| Logit transformation | Strictly between zero and one for finite endpoints | Undefined at estimates of zero or one; relies on transformed-scale approximation |
| Bootstrap percentile | Within zero and one if all bootstrap estimates are | Depends on design-appropriate resampling and informative empirical tail quantiles |
Useful diagnostics include checking whether the direct interval crosses a boundary, examining the estimate’s distance from the boundaries relative to its standard error, and inspecting the bootstrap distribution for skewness, ties, or a large point mass at a boundary. Also verify that the same probability is being estimated throughout: the origin state, destination state, and time interval must match across the point estimate, standard error, and resampling calculation. If a probability is one entry of a transition matrix, an interval for that entry does not by itself guarantee that a set of separately constructed intervals obeys all matrix constraints, such as row sums.
The main limitation is that range-respecting endpoints address only one feature of an interval. They do not correct bias, a misspecified variance, sparse-data instability, or an invalid sampling model. When the estimate is exactly zero or one, standard logit intervals are unavailable, and a bootstrap may repeatedly reproduce the same boundary estimate. Such cases call for careful reporting of the data support and method, not an automatic continuity correction or a claim of certainty.
Common Mistakes
- Keeping an impossible Wald endpoint. A negative lower limit is not a possible transition probability. Treat it as a diagnostic of the approximation, not as a literal bound.
- Clipping and calling the result a valid interval. Replacing a negative endpoint by zero enforces the range but does not establish the interval’s coverage.
- Assuming a transformed interval must be better calibrated. The logit interval is bounded, but its accuracy still depends on the approximation and the standard error.
- Applying the logit at zero or one. The transformation is undefined there. An added constant changes the analysis and is not a universal solution.
- Reading a bootstrap boundary as certainty. A bootstrap percentile of zero can reflect sparse data or resampling ties; it is not proof that the true probability equals zero.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A direct 95% interval for a transition probability is \([-0.02,0.14]\). What does the negative endpoint tell you, and what does it not justify?
- For \(\hat p=0.80\) and \(SE=0.04\), calculate the 95% direct normal interval. Does it cross a boundary?
- Why can a logit interval not be calculated in the usual way when \(\hat p=0\)?
- A bootstrap percentile interval is \([0,0.06]\), and many resampled estimates equal zero. Give one interpretation and one diagnostic to report.
Solutions. (1) It signals that the direct normal approximation may be unsuitable near the boundary. It does not justify interpreting negative probabilities or assuming that clipping the endpoint gives a calibrated interval. (2) \(0.80\pm1.96(0.04)=0.80\pm0.0784\), so the interval is \([0.7216,0.8784]\); it does not cross zero or one. (3) The logit requires \(\log\{\hat p/(1-\hat p)\}\), which is undefined at zero. (4) The lower endpoint shows that the empirical bootstrap lower tail reaches the boundary, not that the true probability is certainly zero. Report the number or proportion of zero bootstrap estimates and examine whether the endpoint is stable across resampling runs.