Confidence Intervals for Multiplicative Factors

Why Confidence Intervals for Relativities Start on the Log Scale

In a GLM with a log link, a coefficient commonly describes a multiplicative change in expected frequency, severity, or another modeled quantity. For a categorical rating factor, the exponential of a non-reference-level coefficient is the estimated relativity against the reference level, with other modeled variables held fixed. For a continuous predictor, it is the estimated relativity for a one-unit increase in that predictor.

An estimated coefficient is uncertain, so its exponentiated value is uncertain too. A standard approach is to form a confidence interval for the coefficient on the log scale and exponentiate each endpoint. The result is a confidence interval for the multiplicative factor. This interval is generally not symmetric around the estimated factor, because the exponential transformation is nonlinear.

Definition: Suppose a GLM has a log link and a coefficient \(\beta\) whose multiplicative interpretation is \(R=\exp(\beta)\). If \(\hat{\beta}\) is the estimated coefficient and \(\widehat{\operatorname{SE}}(\hat{\beta})\) is its estimated standard error, a two-sided Wald interval for \(\beta\) at confidence level \(1-\alpha\) is

\left[ \hat{\beta}-z_{1-\alpha/2}\widehat{\operatorname{SE}}(\hat{\beta}), \quad \hat{\beta}+z_{1-\alpha/2}\widehat{\operatorname{SE}}(\hat{\beta}) \right].

The corresponding interval for \(R\) is obtained by exponentiating the two coefficient endpoints.

Here \(z_{1-\alpha/2}\) is the standard normal critical value: approximately 1.96 for a 95% interval and 1.645 for a 90% interval. The estimated standard error is the one appropriate to the fitted model and its stated variance and dependence assumptions. The procedure is a large-sample, normal-approximation method; it is not an exact interval for every sample size or every data-generating process.

Constructing the Interval

The calculation has two scales. First construct the interval for \(\beta\), where the normal approximation is applied. Then transform the endpoints to the relativity scale. Since the exponential function is strictly increasing, the lower coefficient endpoint maps to the lower relativity endpoint and the upper coefficient endpoint maps to the upper relativity endpoint.

Identify the coefficient and its meaning.
Confirm that the coefficient is on the log scale and that its exponential is the particular multiplicative factor of interest.

Choose the critical value.
Use the value corresponding to the requested two-sided confidence level, such as 1.96 for 95% or 1.645 for 90%.

Calculate the coefficient endpoints.
Subtract and add the critical value times the coefficient standard error. Keep unrounded endpoints for the transformation.

Exponentiate both endpoints.
Report the transformed endpoints in increasing order, with rounding applied only at the final reporting step.

A confidence interval is not a range containing a fixed proportion of future individual losses. Under the model and approximation, it describes uncertainty in the estimated coefficient, and hence in the estimated relativity. It also does not automatically account for model misspecification, omitted variables, or uncertainty from a model-selection process.

Why the Endpoint Transformation Works

Assume the fitted coefficient has an approximately normal sampling distribution, with standard error estimated consistently. The Wald construction then gives an approximate interval with nominal coverage \(1-\alpha\) for the coefficient. Exponentiating both endpoints preserves coverage for the relativity because the transformation is one-to-one and increasing.

Justification: Let \(L=\hat{\beta}-z_{1-\alpha/2}\widehat{\operatorname{SE}}(\hat{\beta})\) and \(U=\hat{\beta}+z_{1-\alpha/2}\widehat{\operatorname{SE}}(\hat{\beta})\). The Wald event that the true coefficient lies in the interval is \(L\leq\beta\leq U\). Applying the increasing function \(\exp(\cdot)\) to all three terms gives \(\exp(L)\leq\exp(\beta)\leq\exp(U)\). Since \(R=\exp(\beta)\), the transformed interval is \([\exp(L),\exp(U)]\), with the same approximate coverage as the coefficient interval.

This reasoning also gives a useful check: a relativity of 1 corresponds exactly to a coefficient of 0, because \(\exp(0)=1\). Thus, a coefficient interval includes 0 if and only if the transformed relativity interval includes 1. This is a relationship between interval endpoints, not a claim that a factor near 1 is necessarily practically unimportant.

Worked Example: A 95% Interval for a Territory Relativity

A hypothetical log-link frequency model uses a reference territory and an indicator for a second territory. The estimated coefficient for the second territory is \(\hat{\beta}=0.1823\), with standard error \(0.0800\). Construct a 95% confidence interval for its relativity using \(z=1.96\), and interpret it.

Method. The territory relativity is \(R=\exp(\beta)\). Construct the Wald interval on the coefficient scale first, then exponentiate its endpoints. The estimate is approximately \(\exp(0.1823)=1.2000\).

Coefficient interval. The margin of error is \(1.96(0.0800)=0.1568\). Therefore the endpoints are \(0.1823-0.1568=0.0255\) and \(0.1823+0.1568=0.3391\). The coefficient interval is \([0.0255,0.3391]\).

Relativity interval. Exponentiating the unrounded coefficient endpoints gives \(\exp(0.0255)\approx1.0258\) and \(\exp(0.3391)\approx1.4037\). Rounded to four decimal places, the 95% relativity interval is \([1.0258,1.4037]\).

Interpretation and check. Under the model and Wald approximation, the second territory’s relativity against the reference territory is estimated at about 1.2000, with a 95% interval from 1.0258 to 1.4037. The interval excludes 1; equivalently, the coefficient interval excludes 0. The endpoints are in the same order as the coefficient endpoints because exponentiation is increasing.

Worked Example: A 90% Interval Below the Reference Level

A hypothetical coverage indicator has log-link coefficient \(\hat{\beta}=-0.2231\) and standard error \(0.1200\). Construct a two-sided 90% relativity interval using \(z=1.645\). State whether the interval includes the reference relativity.

Method. The coverage relativity is \(\exp(\beta)\), and the estimated factor is approximately \(\exp(-0.2231)=0.8000\). Use the requested 90% critical value to form the coefficient interval.

Coefficient interval. The margin is \(1.645(0.1200)=0.1974\). The lower endpoint is \(-0.2231-0.1974=-0.4205\), and the upper endpoint is \(-0.2231+0.1974=-0.0257\). Thus the coefficient interval is \([-0.4205,-0.0257]\).

Relativity interval. Exponentiating the endpoints gives \(\exp(-0.4205)\approx0.6567\) and \(\exp(-0.0257)\approx0.9746\). The 90% relativity interval, rounded to four decimal places, is \([0.6567,0.9746]\).

Interpretation and check. The interval lies below 1, so it does not include the reference relativity. This agrees with the coefficient interval lying below 0. It indicates an estimated relativity below the reference, though the interval alone does not establish that the model is correctly specified.

Worked Example: Avoiding Premature Rounding

A hypothetical rating factor has estimated coefficient \(\hat{\beta}=0.2624\), standard error \(0.0900\), and a requested 90% interval with \(z=1.645\). Construct the relativity interval, showing why the exponentiation should use unrounded coefficient endpoints.

Method. Form the coefficient Wald interval and exponentiate its endpoints. The estimated relativity is approximately \(\exp(0.2624)=1.3000\).

Coefficient interval. The margin is \(1.645(0.0900)=0.14805\). The unrounded endpoints are \(0.2624-0.14805=0.11435\) and \(0.2624+0.14805=0.41045\). To four decimal places these are 0.1144 and 0.4105, but retain 0.11435 and 0.41045 for the next calculation.

Relativity interval. Exponentiating the unrounded endpoints gives \(\exp(0.11435)\approx1.1211\) and \(\exp(0.41045)\approx1.5075\). The 90% relativity interval is \([1.1211,1.5075]\), rounded to four decimal places.

Interpretation and check. The interval lies above 1, consistent with both coefficient endpoints being positive. Exponentiating the displayed rounded coefficient endpoints instead can change a last reported digit; that is why the transformation uses the unrounded endpoints.

Interpreting Width and the Baseline

The coefficient interval has equal distances above and below \(\hat{\beta}\), but the relativity interval generally does not have equal distances above and below \(\exp(\hat{\beta})\). The lower and upper factor endpoints are obtained by multiplying the estimated relativity by different amounts:

\left[ \exp(\hat{\beta})\exp\!\left(-z_{1-\alpha/2}\widehat{\operatorname{SE}}(\hat{\beta})\right), \quad \exp(\hat{\beta})\exp\!\left(z_{1-\alpha/2}\widehat{\operatorname{SE}}(\hat{\beta})\right) \right].

This makes the interval asymmetric on the relativity scale. A higher confidence level or a larger standard error widens the coefficient interval and therefore widens the transformed interval. It does not follow that the interval is centered at the estimated relativity.

For a coefficient coded against a reference level, the reference relativity is 1 by definition. An interval entirely above 1 indicates that the coefficient interval is entirely above zero; an interval entirely below 1 indicates that it is entirely below zero. If the interval crosses 1, it also crosses the reference value on the coefficient scale. These statements concern the particular coefficient and coding being interpreted.

Common Mistake: Making the Factor Interval Symmetric

A tempting but incorrect calculation is to take the estimated factor and add or subtract the coefficient-scale margin of error. For the first worked example, the estimated factor is about 1.2000 and the margin is 0.1568. The incorrect calculation would be \([1.2000-0.1568,\ 1.2000+0.1568]=[1.0432,1.3568]\). This is wrong because 0.1568 is a distance on the log-coefficient scale, not on the relativity scale.

The correction is to apply the margin on the coefficient scale: obtain \([0.0255,0.3391]\), then exponentiate to get \([1.0258,1.4037]\). The mistaken symmetric interval is not the transformed Wald interval and may lead to incorrect conclusions about whether the factor interval includes 1.

Another avoidable error is rounding coefficient endpoints before exponentiating. Keep the full intermediate precision available from the estimate and standard error, exponentiate those endpoints, and round only the final relativity limits. Report the confidence level and the scale of each interval so that the result is unambiguous.

Key takeaway: For a log-link coefficient whose exponential is a multiplicative factor, construct the Wald interval on the coefficient scale and exponentiate both unrounded endpoints. The resulting relativity interval is generally asymmetric, and it includes 1 exactly when the coefficient interval includes 0.

Check Your Understanding

For each problem, construct the requested coefficient interval before transforming it. Round final relativity endpoints to four decimal places.

A hypothetical log-link model estimates a class coefficient of \(0.0953\) with standard error \(0.0400\). Use \(z=1.96\) to find the 95% relativity interval. (Answer: coefficient interval \([0.0169,0.1737]\); relativity interval \([1.0170,1.1897]\).)
A hypothetical predictor has coefficient \(-0.1054\) and standard error \(0.0600\). Using \(z=1.645\), find the 90% relativity interval and state whether it includes 1. (Answer: coefficient interval \([-0.2041,-0.0067]\); relativity interval \([0.8154,0.9933]\); it does not include 1.)
A hypothetical log-link coefficient is \(0.4055\), with standard error \(0.1100\). Use a 95% critical value of 1.96. Find the coefficient and relativity intervals. (Answer: coefficient interval \([0.1899,0.6211]\); relativity interval \([1.2091,1.8610]\).)
A hypothetical rating coefficient is \(-0.0200\), with standard error \(0.0150\). Use \(z=1.96\), and determine whether the transformed interval includes the reference factor. (Answer: coefficient interval \([-0.0494,0.0094]\); relativity interval \([0.9518,1.0094]\); it includes 1.)

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