Confidence Interval from Hypothesis Test (Duality)

Extracts a confidence interval from a hypothesis test object, exploiting the fundamental duality between hypothesis tests and confidence intervals.

# S3 method for class 'hypothesis_test'
confint(object, parm = NULL, level = 0.95, ...)

# S3 method for class 'wald_test'
confint(object, parm = NULL, level = 0.95, ...)

# S3 method for class 'z_test'
confint(object, parm = NULL, level = 0.95, ...)

Arguments

object

A hypothesis_test object.

parm

Ignored (for compatibility with generic).

level

Numeric. Confidence level (default 0.95).

...

Additional arguments (ignored).

Value

A named numeric vector with elements lower and upper.

Details

Hypothesis tests and confidence intervals are two views of the same underlying inference. For a test of \(H_0: \theta = \theta_0\) at level \(\alpha\), the \((1-\alpha)\) confidence interval contains exactly those values of \(\theta_0\) that would not be rejected.

This duality means:

• A 95% CI contains all values where the two-sided test has p > 0.05

• The CI boundary is where p = 0.05 exactly

• Inverting a test "inverts" it into a confidence set

Available Methods

Confidence intervals are currently implemented for:

• wald_test: Uses \(\hat{\theta} \pm z_{\alpha/2} \cdot SE\)

• z_test: Uses \(\bar{x} \pm z_{\alpha/2} \cdot \sigma/\sqrt{n}\)

Tests without stored estimates (like lrt or fisher_combined_test) cannot produce confidence intervals directly.

Examples

# Wald test stores estimate and SE, so CI is available
w <- wald_test(estimate = 2.5, se = 0.8)
confint(w)              # 95% CI
#>     lower     upper 
#> 0.9320288 4.0679712 
confint(w, level = 0.99) # 99% CI
#>     lower     upper 
#> 0.4393366 4.5606634 

# The duality: 2.5 is in the CI, and testing H0: theta = 2.5
# would give p = 1 (not rejected)
wald_test(estimate = 2.5, se = 0.8, null_value = 2.5)
#> Hypothesis test ( wald_test )
#> -----------------------------
#> Test statistic:  0 
#> P-value:  1 
#> Degrees of freedom:  1 
#> Significant at 5% level:  FALSE 

# z-test also supports confint
z <- z_test(rnorm(50, mean = 10, sd = 2), mu0 = 9, sigma = 2)
confint(z)
#>     lower     upper 
#>  9.326016 10.434739