Semi-tail Units in Statistical Testing

• Semi-tail units are statistical measures defined as s = -log2(p), offering a universal scale for quantifying extremeness in hypothesis tests.
• They enable direct additivity by summing s-values, which facilitates the combination of evidence from independent tests.
• They standardize critical thresholds and measure efficiency improvements, providing clearer comparisons across various test statistics.

A semi-tail unit is a quantile-based measure for expressing extremeness, efficiency, and evidence in statistical hypothesis testing. Each semi-tail unit corresponds to a halving of the tail area under a null distribution, and this logarithmic base-2 scale permits direct additivity, uniform critical value progression, and natural efficiency quantification. Semi-tail units have been proposed as a universal, interpretable alternative to p-values, z-scores, and test-specific quantiles, applicable to any ordered statistical distribution (Vos, 28 Jun 2025).

1. Definition and Mathematical Formulation

Let $X$ be a real-valued test statistic, with $P = P(X \ge x_{\text{obs}})$ the right-tail probability under the null hypothesis. The semi-tail unit, denoted $s$, is defined by

$s = -\log_2 P$

which quantifies how many doublings of rarity have occurred in the observed tail probability. For example, $s=3.32$ corresponds to $P=0.10$, $s=4.32$ to $P=0.05$, and $s=5.32$ to $P=0.025$. In other words, each increment of 1 in $P = P(X \ge x_{\text{obs}})$0 represents a halving of $P = P(X \ge x_{\text{obs}})$1. For two-tailed tests, the $P = P(X \ge x_{\text{obs}})$2-value is employed: $P = P(X \ge x_{\text{obs}})$3 where $P = P(X \ge x_{\text{obs}})$4 is the median under $P = P(X \ge x_{\text{obs}})$5.

This formulation enables direct mapping between percentile, tail probability, and semi-tail unit via

$P = P(X \ge x_{\text{obs}})$6

for upper-tail percentile $P = P(X \ge x_{\text{obs}})$7. For instance, a result at the 99th percentile yields $P = P(X \ge x_{\text{obs}})$8.

2. Interpretive Properties and Practical Benchmarks

The primary interpretive property is additivity and halving: every $P = P(X \ge x_{\text{obs}})$9 increment in $s$0 divides the tail probability by two. Table 1, mapping selected probabilities and percentiles to $s$1:

Tail Probability $s$2	Semi-tail $s$3	Percentile
$s$4	$s$5	$s$6
$s$7	$s$8	$s$9
$s = -\log_2 P$0	$s = -\log_2 P$1	$s = -\log_2 P$2
$s = -\log_2 P$3	$s = -\log_2 P$4	$s = -\log_2 P$5

For two-tailed tests, the two-tailed $s = -\log_2 P$6-value is $s = -\log_2 P$7. This allows writing all critical values as equally spaced points on the $s = -\log_2 P$8 (or $s = -\log_2 P$9) scale, eliminating the distribution-specific tables common to classical hypothesis testing.

3. Conversion from and to Classical Test Statistics

To convert any test statistic:

1. Compute the classical $s=3.32$0-value under the null.
2. Set $s=3.32$1 (one-tailed) or $s=3.32$2 (two-tailed).

Examples:

• Normal $s=3.32$3-test, $s=3.32$4: two-tailed $s=3.32$5.
• $s=3.32$6-test with $s=3.32$7, $s=3.32$8: two-tailed $s=3.32$9.
• $P=0.10$0 test, $P=0.10$1: one-tailed $P=0.10$2.
• Poker hand "royal flush": $P=0.10$3 (tail prop $P=0.10$4).

All critical $P=0.10$5-levels become $P=0.10$6, so achieving a more stringent level is an arithmetic progression in $P=0.10$7.

4. Evidence Combination and Additivity

If $P=0.10$8 are independent one-tailed $P=0.10$9-values, their joint tail probability is $s=4.32$0, so the aggregate $s=4.32$1-value is

$s=4.32$2

This allows for direct addition of semi-tail units when accumulating evidence across studies or tests. For example, three $s=4.32$3-values of $s=4.32$4, $s=4.32$5, and $s=4.32$6 yield $s=4.32$7, $s=4.32$8, and $s=4.32$9, so $P=0.05$0.

5. Semi-tail Units and Bahadur Efficiency

Semi-tail units naturally encode efficiency in terms of Bahadur slopes. The Bahadur exact slope for a sequence $P=0.05$1 under an alternative $P=0.05$2 can be written as

$P=0.05$3

and the asymptotic Bahadur slope becomes $P=0.05$4 with $P=0.05$5. Thus, the $P=0.05$6-value per sample is an interpretable "bits" or "semi-tail units per observation" rate.

Semi-tail efficiency difference between two tests $P=0.05$7 is $P=0.05$8. A positive $P=0.05$9 means test $s=5.32$0 drives the sample $s=5.32$1 units deeper into the tail (i.e., $s=5.32$2 smaller $s=5.32$3-value) compared to test $s=5.32$4, an interpretable "distance" in evidence (Vos, 28 Jun 2025).

6. Unification and Advantages over Classical Scales

Semi-tail units provide a universal, interpretable, and additive scale for expressing the extremeness of results across all testing contexts. Key unifying features:

• All decision thresholds are linear and equally spaced ($s=5.32$5 increases by $s=5.32$6 when $s=5.32$7 halves).
• No distribution-specific lookup tables are required.
• Additivity for combining independent evidence (direct sum of $s=5.32$8).
• Simple efficiency comparison: differences in $s=5.32$9 per sample correspond to multiplicative changes in tail probability/explanatory power.
• Applicable to any ordered distribution, not just those with tabulated quantiles (e.g., permutation tests, empirical distributions, and discrete combinatorial ranks).

These properties, as demonstrated in applications from normal and $P=0.025$0-tests to poker hand rankings, distinguish the semi-tail unit from p-values, z-scores, and conventional quantile measures (Vos, 28 Jun 2025).

7. Worked Examples and Applications

Percentile to $P=0.025$1-value mapping:

• $P=0.025$2
• $P=0.025$3
• $P=0.025$4
• $P=0.025$5

Discrete ranking (combinatorial, e.g., poker hands):

Let $P=0.025$6 be total outcomes, $P=0.025$7 the rank or better, $P=0.025$8. Royal flush: $P=0.025$9.

Combining $P = P(X \ge x_{\text{obs}})$00-values:

Three experiments give $P = P(X \ge x_{\text{obs}})$01-values $P = P(X \ge x_{\text{obs}})$02, $P = P(X \ge x_{\text{obs}})$03, $P = P(X \ge x_{\text{obs}})$04; sum is $P = P(X \ge x_{\text{obs}})$05, corresponding to $P = P(X \ge x_{\text{obs}})$06.

Efficiency:

The difference in semi-tail slope per sample ($P = P(X \ge x_{\text{obs}})$07) quantifies how much more rapidly one test accumulates evidence, providing a bits-scale improvement directly interpretable (e.g., $P = P(X \ge x_{\text{obs}})$08 is a $P = P(X \ge x_{\text{obs}})$09 reduction in $P = P(X \ge x_{\text{obs}})$10 per observation in the asymptote).

Semi-tail units constitute a universal and principled system for standardization and comparison of test statistics, evidence accumulation, and efficiency reporting in modern statistical research (Vos, 28 Jun 2025).