Confidence Intervals for Proportions and Rates

Why the Scale of an Interval Matters

A confidence interval for a weighted proportion should stay within zero and one, while a confidence interval for a rate should not imply a negative value. A symmetric interval formed by adding and subtracting a margin on the original scale can violate these limits, especially when the estimate is close to a boundary. A transformation can provide an interval that respects the parameter’s scale while still using a design-consistent standard error.

The central ingredient is the standard error from the survey design, not a formula that treats observations as an independent simple random sample. The variance estimator must reflect the design features used for inference, such as stratification, clustering, unequal weights, and any appropriate degrees of freedom. The earlier material on design-consistent standard errors and confidence intervals for weighted totals and means provides the general interval framework; here the emphasis is on proportion and rate scales.

Estimators and the Design-Based Interval Framework

Definition: For a binary outcome $y_i$, where $y_i=1$ indicates the characteristic of interest and $y_i=0$ otherwise, the weighted proportion estimator is $$ \hat{p}=\frac{\sum_{i\in s}w_i y_i}{\sum_{i\in s}w_i}, $$ where $s$ is the sample and $w_i$ is the survey weight. For an exposure-based rate, the weighted rate estimator is $$ \hat{R}=\frac{\sum_{i\in s}w_i e_i}{\sum_{i\in s}w_i t_i}, $$ where $e_i$ is an event count and $t_i$ is the corresponding exposure. The rate’s units are events per unit of exposure.

In an actual survey, a proportion’s denominator may be restricted to eligible records or to a specified domain. The numerator and denominator must refer to the same target population and eligibility rules. For a rate, the event and exposure totals must likewise refer to the same population and time period. These checks are essential: an interval cannot repair a mismatched estimand.

Let $\widehat{SE}(\hat{p})$ or $\widehat{SE}(\hat{R})$ denote a standard error estimated with the survey design. For a two-sided confidence level $1-\alpha$, a common large-sample interval is the estimate plus or minus a reference critical value times its standard error. The reference value may be a standard normal quantile or a $t$ quantile with design-based degrees of freedom $df$, depending on the variance procedure and inferential convention. The examples use $t$ critical values and treat the supplied standard errors and degrees of freedom as design-based quantities.

Assumptions and conditions: The weights and variance estimator must be appropriate for the stated survey target and sampling design; the denominator must be positive; and the design-based standard error must estimate sampling variability adequately. The normal or $t$ approximation also requires enough information for the estimator’s sampling distribution to be reasonably approximated by that reference distribution. The intervals below address sampling uncertainty, not measurement error, coverage error, or unaddressed nonresponse bias.

When an Original-Scale Interval Is Reasonable

For a proportion that is not close to zero or one, the original-scale interval can be useful:

\hat{p}\ \pm\ t_{1-\alpha/2,df}\widehat{SE}(\hat{p}).

Its endpoints are plausible only if they lie between zero and one. Plausibility also depends on whether the symmetric shape is reasonable for the estimate and its uncertainty; merely landing inside the bounds is not a guarantee of good coverage. The same idea applies to a positive rate, but a symmetric interval can extend below zero, which is not a plausible rate.

Worked Example: A Proportion Away From the Boundaries

Suppose a hypothetical survey estimates that 42% of eligible adults in a defined population meet a health-screening guideline. The weighted numerator and denominator give $\hat{p}=0.420$. A design-based variance procedure gives $\widehat{SE}(\hat{p})=0.035$ and $df=24$. Assume the design and variance conditions stated above are adequate, and use the 95% critical value $t_{0.975,24}=2.064$.

The margin of error is $2.064(0.035)=0.07224$. Thus the original-scale interval is $\,0.420\pm0.07224=(0.34776,0.49224)$, or approximately $(0.3478,0.4922)$.

Both endpoints are between zero and one, and the estimate is not especially close to either boundary. On these facts, a symmetric interval is a reasonable summary. The interval estimates that between 34.78% and 49.22% of eligible adults in the target population meet the guideline, under the stated 95% design-based procedure.

Logit Intervals for Proportions

A proportion is bounded, so a symmetric interval on the proportion scale can be unsuitable near zero or one. A logit transformation maps values strictly between zero and one to the whole real line. Its inverse maps any real number back to a value strictly between zero and one:

g(p)=\log\left(\frac{p}{1-p}\right), \qquad g^{-1}(z)=\frac{1}{1+\exp(-z)}.

Using the delta method, the estimated standard error on the logit scale is approximately $\widehat{SE}\{g(\hat{p})\}=\widehat{SE}(\hat{p})/[\hat{p}(1-\hat{p})]$. The resulting interval is formed on that scale and then mapped back:

\left[ g^{-1}\left\{g(\hat{p})-t_{1-\alpha/2,df}\widehat{SE}\{g(\hat{p})\}\right\}, g^{-1}\left\{g(\hat{p})+t_{1-\alpha/2,df}\widehat{SE}\{g(\hat{p})\}\right\} \right].

The construction is approximate because the transformed estimator’s sampling distribution is approximated by a normal or $t$ reference distribution. The back-transformation does not make that approximation exact. It does ensure that both endpoints are between zero and one when the estimated proportion is strictly between zero and one.

Justification: If the transformed parameter $g(p)$ has an approximately valid interval $[L,U]$, then $P\{L\leq g(p)\leq U\}$ is approximately the stated confidence level. Since $g$ is strictly increasing on $(0,1)$, this event is equivalent to $g^{-1}(L)\leq p\leq g^{-1}(U)$. Therefore, back-transforming both endpoints preserves the interval’s coverage approximation and gives endpoints on the proportion scale.

Worked Example: A High Weighted Proportion

In a hypothetical survey, the weighted proportion of eligible respondents who report receiving a recommended service is $\hat{p}=0.960$. The design-based standard error is $0.030$, with $df=18$. Assume the weights, denominator, and variance procedure are appropriate, and use $t_{0.975,18}=2.101$ for a 95% interval.

A symmetric interval on the original scale would be $0.960\pm2.101(0.030)=0.960\pm0.06303$, or $(0.8970,1.0230)$. Its upper endpoint exceeds one, so this interval is not a plausible proportion interval.

The estimated logit is $\log(0.960/0.040)=\log(24)=3.1781$. The delta-method standard error on the logit scale is $0.030/[0.960(0.040)]=0.78125$, approximately $0.7813$. Using these displayed rounded values, the logit interval is $3.1781\pm2.101(0.7813)$. The margin is $1.6415113$, giving endpoints $1.5365887$ and $4.8196113$, or $(1.5366,4.8196)$ to four decimal places.

Back-transforming gives $g^{-1}(1.5366)\approx0.8230$ and $g^{-1}(4.8196)\approx0.9920$. The resulting 95% interval is approximately $(0.8230,0.9920)$, or 82.30% to 99.20%. It is asymmetric around $0.960$ on the proportion scale, as expected from the nonlinear transformation, and both endpoints are within the allowable range.

Log Intervals for Positive Rates

A rate is positive, so a log transformation is a natural way to avoid negative interval endpoints. For a positive estimate $\hat{R}$, the delta method gives $\widehat{SE}\{\log(\hat{R})\}\approx\widehat{SE}(\hat{R})/\hat{R}$. Form a symmetric interval for $\log(\hat{R})$, then exponentiate both endpoints. The result is positive and generally asymmetric on the rate scale.

\left[ \exp\left\{\log(\hat{R})-t_{1-\alpha/2,df}\frac{\widehat{SE}(\hat{R})}{\hat{R}}\right\}, \exp\left\{\log(\hat{R})+t_{1-\alpha/2,df}\frac{\widehat{SE}(\hat{R})}{\hat{R}}\right\} \right].

Changing the unit of the rate, such as from events per person-year to events per 1,000 person-years, multiplies both the estimate and its standard error by the same positive constant. Their ratio, and hence the log-scale standard error, stays the same. The interval endpoints must still be reported in the chosen rate unit.

Worked Example: A Weighted Rate Per 1,000 Person-Years

A hypothetical survey-based analysis estimates 84,000 weighted events over 1,200,000 weighted person-years. The estimated rate is $84{,}000/1{,}200{,}000=0.0700$ events per person-year, or $70.0$ events per 1,000 person-years. Suppose a design-based calculation gives a standard error of $12.0$ events per 1,000 person-years, with $df=15$. Assume the event and exposure totals match the target population and the variance estimator accounts for the sampling design. Use $t_{0.975,15}=2.131$.

On the per-1,000-person-years scale, the estimated log rate is $\log(70.0)=4.2485$, and its standard error is $12.0/70.0=0.17143$. The log-scale margin is $2.131(0.17143)\approx0.3653$. Thus the log interval is approximately $(3.8832,4.6138)$.

Exponentiating the endpoints gives $(\exp(3.8832),\exp(4.6138))\approx(48.6,100.9)$ events per 1,000 person-years. This 95% interval is positive and allows more room above the estimate than below it. In context, it indicates uncertainty about the population event rate, not about the number of events in a particular future period.

Choosing and Reporting the Interval

The choice of scale should follow the parameter, estimate, and uncertainty rather than a preference for one formula. A symmetric interval may be adequate when its endpoints are plausible and the estimate is not near a boundary. For proportions near zero or one, a logit interval often provides more credible endpoints. For positive rates, a log interval avoids negative values. In every case, report the point estimate, interval, confidence level, units or percentage scale, and the design basis for the standard error.

A 95% confidence procedure has approximately 95% coverage over repeated samples drawn under the specified survey design, when its assumptions and variance approximation hold. It does not mean there is a 95% probability that the fixed population proportion or rate lies in the particular interval after it has been calculated. The interval quantifies sampling uncertainty under the stated procedure.

Common pitfall: Clipping an implausible symmetric interval to the parameter’s limits is not the same as constructing an interval with justified coverage. For example, replacing an upper proportion endpoint of 1.0230 with 1.0000 changes the procedure without accounting for the skewness near the boundary. Instead, choose a suitable transformed-scale interval and report its method and assumptions.

Check Your Understanding

Use the definitions and interval procedures above to answer each question.

A weighted proportion is $0.18$ with design-based standard error $0.07$. What issue should be checked before reporting an original-scale symmetric interval?
Why does applying the inverse logit to both endpoints produce a proportion interval within zero and one?
A rate is estimated as $25$ events per 1,000 person-years with standard error $5$. What is the delta-method standard error on the log scale?
What design features must a standard error reflect for a design-consistent interval?
In repeated-sample terms, what does a 95% confidence level mean, and what does it not mean for a calculated interval?