Percentage of all non-overlapping data

The pand() function calculates the percentage of all non-overlapping data (PAND; Parker, Hagan-Burke, & Vannest, 2007), an index to quantify a level increase (or decrease) in performance after the onset of an intervention.

Usage

pand(
  data,
  dvar,
  pvar,
  decreasing = FALSE,
  phases = c(1, 2),
  method = c("sort", "minimum")
)

# S3 method for sc_pand
print(x, ...)

# S3 method for sc_pand
export(
  object,
  caption = NA,
  footnote = NA,
  filename = NA,
  kable_styling_options = list(),
  kable_options = list(),
  round = 1,
  ...
)

Arguments

data: A single-case data frame. See scdf() to learn about this format.
dvar: Character string with the name of the dependent variable. Defaults to the attributes in the scdf file.
pvar: Character string with the name of the phase variable. Defaults to the attributes in the scdf file.
decreasing: If you expect data to be lower in the B phase, set decreasing = TRUE. Default is decreasing = FALSE.
phases: A vector of two characters or numbers indicating the two phases that should be compared. E.g., phases = c("A","C") or phases = c(2,4) for comparing the second to the fourth phase. Phases could be combined by providing a list with two elements. E.g., phases = list(A = c(1,3), B = c(2,4)) will compare phases 1 and 3 (as A) against 2 and 4 (as B). Default is phases = c(1,2).
method: Either "sort"" or "minimum". See details.
x: An object returned by pand()
...: Further arguments passed to the function.
object: An scdf or an object exported from a scan function.
caption: Character string with table caption. If left NA (default) a caption will be created based on the exported object.
footnote: Character string with table footnote. If left NA (default) a footnote will be created based on the exported object.
filename: String containing the file name. If a filename is given the output will be written to that file.
kable_styling_options: list with arguments passed to the kable_styling function.
kable_options: list with arguments passed to the kable function.
round: Integer passed to the digits argument internally used to round values.

Value


`pand`	Percentage of all non-overlapping data.
`method`	Calculation method.
`phi`	Effect size Phi based on expected and observed values.
`perc_overlap`	Percentage of overlapping data points.
`overlaps`	Number of overlapping data points.
`n`	Number of data points.
`N`	Number of cases.
`n_a`	Number of data points in phase A.
`n_b`	Number of data points in phase B.
`matrix`	2x2 frequency matrix of phase A and B comparisons.
`matrix_counts`	2x2 counts matrix of phase A and B comparisons.
`chi_test`	A Chi-squared analysis of expected and observed data (chisq.test()).
`fisher_test`	A Fisher exact test analysis of expected and observed data (fisher.test()).

Details

PAND was proposed by Parker, Hagan-Burke, and Vannest in 2007. The authors emphasize that PAND is designed for application in a multiple case design with a substantial number of measurements, technically at least 20 to 25, but preferably 60 or more. PAND is defined as 100% minus the percentage of data points that need to be removed from either phase in order to ensure nonoverlap between the phases. Several approaches have been suggested to calculate PAND, leading to potentially different outcomes. In their 2007 paper, Parker and colleagues present an algorithm for computing PAND. The algorithm involves sorting the scores of a time series, including the associated phases, and comparing the resulting phase order with the original phase order using a contingency table. To account for ties, the algorithm includes a randomization process where ties are randomly assigned to one of the two phases. Consequently, executing the algorithm multiple times could yield different results. It is important to note that this algorithm does not produce the same results as the PAND definition provided earlier in the same paper. However, it offers the advantage of allowing the calculation of an effect size measure phi, and the application of statistical tests for frequency distributions. Pustejovsky (2019) presented a mathematical formulation of Parker's original definition for comparing two phases of a single case: $$PAND = \frac{1}{m+n}max\{(i+j)I(y^A_{i}<y^B_{n+1-j}\}$$ This formulation provides accurate results for PAND, but the original definition has the drawback of an unknown distribution under the null hypothesis, making a statistical test difficult. The pand() function enables the calculation of PAND using both methods. The first approach (method = "sort") follows the algorithm described above, with the exclusion of randomization before sorting to avoid ambiguity. It calculates a phi measure and provides the results of a chi-squared test and a Fisher exact test. The second approach (method = "minimum") applies the aforementioned formula. The code of this function is based on the code of the SingleCaseES package (function calc_PAND). For a multiple case design, overlaps are calculated for each case, summed, and then divided by the total number of measurements. No statistical test is conducted for this method.

Functions

print(sc_pand): Print results
export(sc_pand): Export results as html table (see export())

References

Parker, R. I., Hagan-Burke, S., & Vannest, K. (2007). Percentage of All Non-Overlapping Data (PAND): An Alternative to PND. The Journal of Special Education, 40, 194-204.

Parker, R. I., & Vannest, K. (2009). An Improved Effect Size for Single-Case Research: Nonoverlap of All Pairs. Behavior Therapy, 40, 357-367.

Pustejovsky, J. E. (2019). Procedural sensitivities of effect sizes for single-case designs with directly observed behavioral outcome measures. Psychological Methods, 24(2), 217-235. https://doi.org/10.1037/met0000179

Pustejovsky JE, Chen M, Swan DM (2023). SingleCaseES: A Calculator for Single-Case Effect Sizes. R package version 0.7.1.9999, https://jepusto.github.io/SingleCaseES/.

Author

Juergen Wilbert

Examples

## REplication of the Parker et al. 2007 example
pand(Parker2007)
#> Percentage of all non-overlapping data
#> 
#> Method: sort 
#> 
#> PAND = 85.7%
#> Φ =  0.713  ; Φ² =  0.508 
#> 
#> 28 measurements (13 Phase A, 15 Phase B) in 3 cases
#> Overlapping data: n = 4 ; percentage = 14.3 
#> 
#> 2 x 2 Matrix of percentages
#>          A    B total
#> A     39.3  7.1  46.4
#> B      7.1 46.4  53.6
#> total 46.4 53.6 100.0
#> 
#> 2 x 2 Matrix of counts
#>        A  B total
#> A     11  2    13
#> B      2 13    15
#> total 13 15    28
#> 
#> 
#> Chi-Squared test:
#> X² = 14.227, df = 1, p = 0.000 
#> 
#> Fisher exact test:
#> Odds ratio = 29.007, p = 0.000 

## Calculate the PAND with an expected decrease of phase B scores
cubs <- scdf(c(20,22,24,17,21,13,10,9,20,9,18), B_start = 5)
pand(cubs, decreasing = TRUE)
#> Percentage of all non-overlapping data
#> 
#> Method: sort 
#> 
#> PAND = 63.6%
#> Φ =  0.214  ; Φ² =  0.046 
#> 
#> 11 measurements (4 Phase A, 7 Phase B) in 1 cases
#> Overlapping data: n = 4 ; percentage = 36.4 
#> 
#> 2 x 2 Matrix of percentages
#>          A    B total
#> A     18.2 18.2  36.4
#> B     18.2 45.5  63.6
#> total 36.4 63.6 100.0
#> 
#> 2 x 2 Matrix of counts
#>       A B total
#> A     2 2     4
#> B     2 5     7
#> total 4 7    11
#> 
#> 
#> Chi-Squared test:
#> X² = 0.505, df = 1, p = 0.477 
#> 
#> Fisher exact test:
#> Odds ratio = 2.288, p = 0.576