Calculator
Use this when you already have coefficient estimates (β) and standard errors from a fitted model.
Example data table
Illustrative output from a binomial model with a logit link. Coefficients below are example-only.
| Term | β | SE | exp(β) | exp(CI lower) | exp(CI upper) |
|---|---|---|---|---|---|
| Intercept | -1.2500 | 0.2200 | 0.2865 | 0.1862 | 0.4410 |
| TenureMonths | -0.0450 | 0.0100 | 0.9560 | 0.9374 | 0.9749 |
| SupportCalls | 0.1800 | 0.0550 | 1.1972 | 1.0749 | 1.3335 |
Formula used
For each coefficient estimate β with standard error SE, the Wald confidence interval is:
SEadj = SE × √φ(optional dispersion scaling)z = Φ⁻¹(1 − α/2)(standard normal critical value)CI = β ± z × SEadj
If you enable exponentiation, the calculator reports exp(β) and exp(CI).
This is common for odds ratios (logit) and rate ratios (log).
How to use this calculator
- Fit your GLM in your statistics tool and open the coefficient table.
- Copy each term’s estimate (β) and its standard error (SE).
- Select the family and link for documentation purposes.
- Set
α(0.05 equals 95% confidence). - Use dispersion
φonly if your model output suggests it. - Press Submit. Results will appear above the form.
- Use CSV or PDF to archive the interval report.
Why confidence intervals matter in GLMs
In generalized linear models, point estimates summarize direction and size, but intervals show precision. Narrow bounds usually indicate strong information in the sample, while wide bounds flag uncertainty that can affect decisions, ranking, and policy thresholds. Interval width is about 2 × z × SEadj.
Interpreting intervals across common links
With an identity link, the interval is on the response scale and reads as additive change. With log and logit links, exponentiating transforms effects to multiplicative ratios. For example, exp(β) near 1.12 suggests an estimated 12% increase per unit, with the interval describing plausible ratio values. If exp(CI) spans 0.98 to 1.28, the data allow anything from a small decrease to a meaningful increase.
Using α to control coverage
The calculator uses α to set the confidence level. α = 0.10 gives a 90% interval and a smaller critical value than α = 0.05. Typical z values are about 1.645 (90%), 1.960 (95%), and 2.576 (99%). As α decreases, intervals widen because the method must cover more probability mass in the tails of the reference distribution.
Dispersion scaling for realistic uncertainty
Some families allow dispersion beyond one. If your model reports an estimated dispersion or a scale parameter, multiplying SE by √φ inflates uncertainty consistently. This matters when comparing terms: two predictors with similar β can differ in practical certainty once φ is applied. For instance, moving from φ = 1 to φ = 1.44 multiplies SE by 1.2 and increases interval width by 20%.
Reading the plot and spotting instability
The interval plot highlights which terms cross a neutral value. On the β scale, crossing 0 indicates the data support both positive and negative effects. On an exponentiated scale, crossing 1 indicates ratios consistent with no change. Extremely long bars often signal collinearity, sparse outcomes, or separation-like behavior. When a single term dominates width, check coding, rare categories, and whether the predictor needs centering or rescaling.
Reporting for audits and stakeholders
Good reporting includes the model family, link, α, and whether exponentiation was applied. Exported tables simplify documentation and review. When presenting results, pair intervals with domain context, baseline rates, and sensitivity checks so readers understand both statistical and operational impact.
FAQs
1) Does this compute intervals for predictions or coefficients?
It computes Wald confidence intervals for coefficients using the estimates and standard errors you enter. Prediction intervals require fitted means, variance functions, and new-data design rows.
2) When should I exponentiate the results?
Exponentiate when your link implies multiplicative interpretation, such as logit or log. Then exp(β) and exp(CI) give ratios like odds ratios or rate ratios.
3) What does it mean if an interval crosses 0 or 1?
On the β scale, crossing 0 means the sign is uncertain at the chosen confidence level. On the exponentiated scale, crossing 1 means the ratio could be no effect.
4) How do I pick the dispersion value φ?
Use the scale or dispersion reported by your fitted model when available. If your output does not provide it, keep φ = 1 to avoid inflating or shrinking uncertainty arbitrarily.
5) Are these intervals exact for small samples?
No. They rely on large-sample normal approximations. For small samples or near-boundary estimates, consider profile likelihood or bootstrap intervals from your modeling tool.
6) Why do some intervals look extremely wide?
Wide intervals typically follow large standard errors caused by limited information, multicollinearity, sparse events, or separation. Review data balance, scaling, and model specification.