Calculates the non-overlap of all pairs index (Parker & Vannest, 2009).

NAP(
A_data,
B_data,
condition,
outcome,
baseline_phase = NULL,
intervention_phase = NULL,
improvement = "increase",
SE = "unbiased",
confidence = 0.95,
trunc_const = FALSE
)

## Arguments

A_data

vector of numeric data for A phase. Missing values are dropped.

B_data

vector of numeric data for B phase. Missing values are dropped.

condition

vector identifying the treatment condition for each observation in the series.

outcome

vector of outcome data for the entire series.

baseline_phase

character string specifying which value of condition corresponds to the baseline phase. Defaults to first observed value of condition.

intervention_phase

character string specifying which value of condition corresponds to the intervention phase. Defaults to second unique value of condition.

improvement

character string indicating direction of improvement. Default is "increase".

SE

character value indicating which formula to use for calculating the standard error of NAP, with possible values "unbiased" for the exactly unbiased estimator, "Hanley" for the Hanley-McNeil estimator, "null" for the (known) variance under the null hypothesis of no effect, or "none" to not calculate a standard error. Defaults to "unbiased".

confidence

confidence level for the reported interval estimate. Set to NULL to omit confidence interval calculations.

trunc_const

logical value indicating whether to return the truncation constant used to calculate the standard error.

## Value

A data.frame containing the estimate, standard error, and/or confidence interval.

## Details

NAP is calculated as the proportion of all pairs of one observation from each phase in which the measurement from the B phase improves upon the measurement from the A phase, with pairs of data points that are exactly tied being given a weight of 0.5. The range of NAP is [0,1], with a null value of 0.5.

The unbiased variance estimator was described by Sen (1967) and Mee (1990). The Hanley estimator was proposed by Hanley and McNeil (1982). The null variance is a known function of sample size, equal to the exact sampling variance when the null hypothesis of no effect holds. When the null hypothesis does not hold, the null variance will tend to over-estimate the true sampling variance of NAP.

The confidence interval for NAP is calculated based on the symmetrized score-inversion method (Method 5) proposed by Newcombe (2006).

## References

Hanley, J. A., & McNeil, B. J. (1982). The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143, 29--36. doi:doi:10.1148/radiology.143.1.7063747

Mee, W. (1990). Confidence intervals for probabilities and tolerance regions based on a generalization of the Mann-Whitney statistic. Journal of the American Statistical Association, 85(411), 793–800. doi:doi:10.1080/01621459.1990.10474942

Newcombe, R. G. (2006). Confidence intervals for an effect size measure based on the Mann-Whitney statistic. Part 2: Asymptotic methods and evaluation. Statistics in Medicine, 25(4), 559--573. doi:doi:10.1002/sim.2324

Parker, R. I., & Vannest, K. J. (2009). An improved effect size for single-case research: Nonoverlap of all pairs. Behavior Therapy, 40(4), 357--67. doi:doi:10.1016/j.beth.2008.10.006

Sen, P. K. (1967). A note on asymptotically distribution-free confidence bounds for PX<Y, based on two independent samples. The Annals of Mathematical Statistics, 29(1), 95-102. https://www.jstor.org/stable/25049448

## Examples

A <- c(20, 20, 26, 25, 22, 23)
B <- c(28, 25, 24, 27, 30, 30, 29)
NAP(A_data = A, B_data = B)
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9166667 0.06900656 0.5973406 0.9860176

# Example from Parker & Vannest (2009)
yA <- c(4, 3, 4, 3, 4, 7, 5, 2, 3, 2)
yB <- c(5, 9, 7, 9, 7, 5, 9, 11, 11, 10, 9)
NAP(yA, yB)
#>    ES       Est         SE  CI_lower  CI_upper
#> 1 NAP 0.9636364 0.03192622 0.7499741 0.9950729