Rescale a Probability-Scale Effect Size

Transforms a probability-scale effect size (and, optionally, its standard error, confidence interval, and null value) between four scales: "probability", "difference", "logodds", and "odds".

This function serves as both a standalone utility and the internal engine for brunner_munzel(scale = ...).

Usage

trans_rank_prob(
  estimate,
  se = NULL,
  ci = NULL,
  null = NULL,
  from = c("probability", "difference", "logodds", "odds"),
  to = c("probability", "difference", "logodds", "odds")
)

Arguments

estimate

numeric; the point estimate to transform.

se

numeric or NULL; standard error of estimate. Transformed via the delta method.

ci

numeric vector of length 2 or NULL; confidence interval endpoints. Because every scale conversion is monotonic, CI endpoints are transformed directly (coverage is preserved without the delta method).

null

numeric (scalar or vector) or NULL; the null-hypothesis value(s) to transform (e.g., a single null, or two equivalence bounds).

from

character; the scale estimate is currently on. One of "probability", "difference", "logodds", "odds".

to

character; the target scale. One of "probability", "difference", "logodds", "odds".

Value

A list with components:

estimate

transformed point estimate

se

transformed standard error (or NULL)

ci

transformed CI endpoints (or NULL)

null

transformed null value(s) (or NULL)

from

the from scale (echoed back)

to

the to scale (echoed back)

Details

The four scales and their relationship to a probability \(p\) are:

Scale	Domain	Formula	Null at stochastic equality
probability	\((0, 1)\)	\(p\)	0.5
difference	\((-1, 1)\)	\(2p - 1\)	0
logodds	\((-\infty, \infty)\)	\(\log[p / (1-p)]\)	0
odds	\((0, \infty)\)	\(p / (1-p)\)	1

All conversions are routed through the probability scale internally.

Standard errors are transformed via the delta method:

$$\mathrm{SE}_{\mathrm{target}} = \mathrm{SE}_{\mathrm{original}} \times \left|\frac{dp}{dx}\right| \times \left|\frac{dy}{dp}\right|$$

where \(x\) is the original scale and \(y\) is the target scale.

Confidence intervals are transformed by applying the monotonic mapping directly to each endpoint, which preserves coverage exactly.

At the boundaries (\(p = 0\) or \(p = 1\)), transformations to the logodds scale return \(\pm\infty\), and the delta-method SE is infinite. This is mathematically correct behaviour; no clamping or warning is applied.

See also

brunner_munzel(), ses_calc()

Other effect sizes: boot_ses_calc(), boot_smd_calc(), rank_diff(), ses_calc(), smd_calc()

Examples

# Probability to difference (rank-biserial)
trans_rank_prob(0.7, se = 0.05, ci = c(0.6, 0.8),
               null = 0.5, from = "probability", to = "difference")
#> $estimate
#> [1] 0.4
#> 
#> $se
#> [1] 0.1
#> 
#> $ci
#> [1] 0.2 0.6
#> 
#> $null
#> [1] 0
#> 
#> $from
#> [1] "probability"
#> 
#> $to
#> [1] "difference"
#> 

# Probability to odds
trans_rank_prob(0.7, from = "probability", to = "odds")
#> $estimate
#> [1] 2.333333
#> 
#> $se
#> NULL
#> 
#> $ci
#> NULL
#> 
#> $null
#> NULL
#> 
#> $from
#> [1] "probability"
#> 
#> $to
#> [1] "odds"
#> 

# Round-trip: logodds -> probability -> logodds
lo <- trans_rank_prob(0.8473, from = "logodds", to = "probability")
trans_rank_prob(lo$estimate, from = "probability", to = "logodds")
#> $estimate
#> [1] 0.8473
#> 
#> $se
#> NULL
#> 
#> $ci
#> NULL
#> 
#> $null
#> NULL
#> 
#> $from
#> [1] "probability"
#> 
#> $to
#> [1] "logodds"
#> 

# Apply to brunner_munzel output
res <- brunner_munzel(mpg ~ am, data = mtcars)
#> Sample size in at least one group is small. Permutation test (test_method = 'perm') is highly recommended.
trans_rank_prob(as.numeric(res$estimate),
               se = res$stderr,
               ci = as.numeric(res$conf.int),
               null = as.numeric(res$null.value),
               from = "probability", to = "logodds")
#> $estimate
#> [1] -1.58534
#> 
#> $se
#> [1] 0.5481483
#> 
#> $ci
#> [1] -4.6886990 -0.7038185
#> 
#> $null
#> [1] 0
#> 
#> $from
#> [1] "probability"
#> 
#> $to
#> [1] "logodds"
#>