Blocking does not always increase power: the within-pair correlation crossover

With n pairs, effect delta, unit standard deviation sigma and t critical values \(c_p\) (paired, \(n-1\) df) and \(c_u\) (two-sample, \(2n-2\) df), the paired design has more approximate power iff the within-pair correlation exceeds \(\rho^* = 1 - (1 + (c_p-c_u)/k)^{-2}\) with \(k = \delta\sqrt{n/2}/\sigma\) (Research.P3.Blocking.paired_better_iff); \(\rho^* > 0\) (crossover_pos).

Usage

morie_blocking_crossover(delta, sigma, n, alpha = 0.05, rho = NULL)

Arguments

delta, sigma

Effect size and unit standard deviation.

n

Number of pairs (at least 3).

alpha

One-sided level for the critical values.

rho

Optional within-pair correlations to classify.

Value

A list: k, c_paired, c_unpaired, crossover and, with rho, paired_better and the two power deviates.

References

Weisburd D, Britt C (2014). Statistics in Criminal Justice, 4th ed., ch. 9. Springer.

Examples

morie_blocking_crossover(delta = 0.5, sigma = 1, n = 10, rho = c(0.05, 0.3))
#> $k
#> [1] 1.118034
#> 
#> $c_paired
#> [1] 1.833113
#> 
#> $c_unpaired
#> [1] 1.734064
#> 
#> $crossover
#> [1] 0.156142
#> 
#> $z_paired
#> [1] -0.6860343 -0.4968067
#> 
#> $z_unpaired
#> [1] -0.6160296
#> 
#> $paired_better
#> [1] FALSE  TRUE
#> 
#> attr(,"theorem")
#> [1] "Research.P3.Blocking.paired_better_iff"