overlap provides a table with some of the most important overlap indices for each case of an scdf. For calculating overlap indicators it is important to know if a decrease or an increase of values is expected between phases. By default overlap assumes an increase in values. If the argument decreasing = TRUE is set, calculation will be based on the assumption of decreasing values.

Overlap measures refer to a comparison of two phases within a single-case data-set. By default, overlap compares the first to the second phase.

8.1.1 Select and recombine phases

The select_phases function call

select_phases(data, A, B)

The select_phases() function is needed if you like to compare specific phases or even like to combine several phases. select_phases() is designed to work within a pipe structure. So the first argument is an scdf and it returns an scdf.

scdf %>%select_phases(A =1, B =3) %>% ...

select_phases() has the arguments A and B. Each argument takes a vector with the names or the numbers of the phases to be selected. If you want to compare the first to the third phase you can set select_phases(scdf, 1,3). If the phases of your case are named ‘A’, ‘B’, and ‘C’ you could alternatively set select_phases(scdf, "A","C"). It is also possible to compare a combination of several cases against a combination of other phases. Each of the two list-elements could contain more than one phase which are concatenated with the c command. For example if you have an ABAB-Design and like to compare the two A-phases against the two B-phases select_phases(scdf, c(1,3), c(2,4) ) will do the trick.

(As an alternative approach you can set the phases argument within the overlap() function. This argument takes a list with two elements where the first element defines the phases for the A-phase and the second argument the phases for the B-phase.)

Standardized mean differences can be calculated in various ways. They refer to the difference in the means of two phases. The smd function provides an overview of the most common parameters for each single-case:

smd(exampleAB_score)

Standardized mean differences
Christiano Lionel Neymar
mA 2.70 3.10 2.30
mB 15.35 15.35 15.60
sdA 1.42 1.59 1.49
sdB 2.13 1.60 2.19
sd cohen 1.81 1.60 1.87
sd hedges 1.93 1.60 1.99
Glass' delta 8.92 7.68 8.90
Hedges' g 6.54 7.67 6.68
Hedges' g correction 6.37 7.46 6.50
Hedges' g durlak correction 6.15 7.21 6.28
Cohen's d 6.98 7.67 7.10

The percentage of non-overlapping data (PND) effect size measure was described by Scruggs, Mastropieri, & Casto (1987) . It is the percentage of all data-points of the second phase of a single-case study exceeding the maximum value of the first phase. In case you have a study where you expect a decrease of values in the second phase, PND is calculated as the percentage of data-point of the second phase below the minimum of the first phase.

The function pnd provides the PND for each case as well as the mean of all PNDs of that scdf. When you expect decreasing values set decreasing = TRUE. When there are more than two phases or phases are not named A and B, use the phases argument as described at the beginning of this chapter.

pnd(exampleAB)

Percent Non-Overlapping Data
Case PND Total Exceeds
Johanna 100% 15 15
Karolina 86.67% 15 13
Anja 93.33% 15 14
Mean : 93.33 %

The pem function returns the percentage of phase B data exceeding the phase A median. Additionally, a binomial test against a 50/50 distribution is computed. Different measures of central tendency can be addressed for alternative analyses.

pem(exampleAB)

Percent Exceeding the Median
PEM positives total binom.p
Johanna 100 15 15 0
Karolina 100 15 15 0
Anja 100 15 15 0
Alternative hypothesis: true probability > 50%

8.5 Percentage exceeding the regression trend (PET)

The pet function provides the percentage of phase B data points exceeding the prediction based on the phase A trend. A binomial test against a 50/50 distribution is computed. Furthermore, the percentage of phase B data points exceeding the upper (or lower) 95 percent confidence interval of the predicted progress is computed.

pet(exampleAB)

Percent Exceeding the Trend
N cases = 3
PET binom.p PET CI
Johanna 100.000 0 86.667
Karolina 93.333 0 0.000
Anja 100.000 0 100.000
Binom.test: alternative hypothesis: true probability > 50%
PET CI: Percent of values greater than upper 95% confidence threshold (greater 1.645*se above predicted value)

The pand function calculates the percentage of all non-overlapping data (Richard I. Parker, Hagan-Burke, & Vannest, 2007), an index to quantify a level increase (or decrease) in performance after the onset of an intervention. The argument correction = TRUE makes pand use a frequency matrix, which is corrected for ties. A tie is counted as the half of a measurement in both phases. Set correction = FALSE to use the uncorrected matrix, which is not recommended.

pand(exampleAB)

Percentage of all non-overlapping data
PAND = 93.3 %
Φ = 0.822 ; Φ² = 0.676
Number of cases: 3
Total measurements: 60 (in phase A: 15; in phase B: 45)
n overlapping data per case: 0, 2, 2
Total overlapping data: n = 4 ; percentage = 6.7
2 x 2 Matrix of proportions
% expected
A B total
% A 21.7 3.3 25
real B 3.3 71.7 75
total 25 75
2 x 2 Matrix of counts
expected
A B total
A 13 2 15
real B 2 43 45
total 15 45
Note. Matrix is corrected for ties
Correlation based analysis:
z = 6.316, p = 0.000, τ = 0.822

PAND indicates nonoverlap between phase A and B data (like PND), but uses all data and is therefore not based on one single (probably unrepresentative) datapoint. Furthermore, PAND allows the comparison of real and expected associations (Chi-square test) and estimation of the effect size Phi, which equals Pearsons r for dichotomous data. Thus, phi-Square is the amount of explained variance. The original procedure for computing PAND does not account for ambivalent datapoints (ties). The newer NAP overcomes this problem and has better precision-power (Richard I. Parker, Vannest, & Davis, 2011a).

The nap function calculates the nonoverlap of all pairs (Richard I. Parker & Vannest, 2009). NAP summarizes the overlap between all pairs of phase A and phase B data points. If an increase of phase B scores is expected, a non-overlapping pair has a higher phase B data point. The NAP equals number of pairs showing no overlap / number of pairs. Because NAP can only take values between 50 and 100 percent, a rescaled and therefore more intuitive NAP (0-100%) is also displayed. NAP is equivalent to the the U-test and Wilcox rank sum test. Thus, a Wilcox test is conducted and reported for each case.

nap(exampleAB)

Nonoverlap of All Pairs
Case NAP Rescaled Pairs Positives Ties W p
Johanna 100 100 75 75 0 0.0 0.00062
Karolina 97 93 75 72 1 2.5 0.00129
Anja 98 96 75 73 1 1.5 0.00095

The Tau-U statistic has been proposed by Richard I. Parker, Vannest, Davis, & Sauber (2011b) and is one of the more broadly used approach for reporting effect sizes of single case data. Unfortunately, various and ambiguous implementations of Tau-U exist (Brossart, Laird, & Armstrong, 2018; Pustejovsky, 2016). The tau_u function tries to cover several of these implementation. It takes an scdf and returns Tau-U calculations for each single-case within that file. Additionally, an overall Tau-U value is calculated for all cases based on a meta-analysis.

8.8.1 Variations of Tau-U

Several arguments an be set to define how Tau-U should be calculated. By setting the argument method = "parker", Tau-U is calculated as described in Richard I. Parker et al. (2011b). This procedure could lead to Tau-U values above 1 and below -1 which are difficult to interpret. method = "complete, which is the default, applies a correction that keeps the values within the -1 to 1 range and should be more appropriate. In the original method proposed by Richard I. Parker et al. (2011b) data, calculations are based on Kendall’s Tau A which does not correct for ties. Alternatively, Kendall’s Tau B has a correction for Tau in the presence of ties. The tau_method can be set to decide on the tau method to use "a" for Kendall’s Tau A and "b"` for Kendall’s Tau B.

Here is an example with setting that reconstruct the values from the original example in Richard I. Parker, Vannest, Davis, & Sauber (2011c) :

Tau-U
Method: parker
Applied Kendall's Tau-a
95% CIs for tau are reported.
CI method:
Case: Case1
Tau CI lower CI upper SD_S Z p
A vs. B 0.80 0.29 0.96 8.16 1.96 .05
A vs. B - Trend A 0.65 -0.02 0.92 9.59 1.36 .18
A vs. B + Trend B 0.77 0.21 0.95 9.59 2.40 <.05
A vs. B + Trend B - Trend A 0.56 -0.17 0.89 9.59 2.09 <.05

Tau-U
Method: parker
Applied Kendall's Tau-b
95% CIs for tau are reported.
CI method:
Case: Johanna
Tau CI lower CI upper SD_S Z p
A vs. B 1.00 NaN NaN 22.91 3.27 <.001
A vs. B - Trend A 0.59 0.20 0.82 23.26 3.22 <.001
A vs. B + Trend B 0.79 0.53 0.91 30.53 4.75 <.001
A vs. B + Trend B - Trend A 0.77 0.49 0.90 30.81 4.71 <.001

Another online calculator created by Rumen Manolov is available at https://manolov.shinyapps.io/Overlap/. It uses an R code developed by Kevin Tarlow to calculate Tau-U. This setting will replicate the results of this approach:

Tau-U
Method: complete
Applied Kendall's Tau-a
95% CIs for tau are reported.
CI method:
Case: Johanna
Tau CI lower CI upper SD_S Z p
A vs. B 1.00 NaN NaN 22.91 3.27 <.001
A vs. B - Trend A 0.88 0.72 0.95 30.82 2.43 <.05
A vs. B + Trend B 0.81 0.56 0.92 30.82 4.70 <.001
A vs. B + Trend B - Trend A 0.76 0.48 0.90 30.82 4.70 <.001

The standard return of the tau_u function does not display all calculations. If you like to have more details, apply the print function with the additional argument complete = TRUE.

tau_u(exampleAB$Johanna) %>%print(complete =TRUE)

Tau-U
Method: complete
Applied Kendall's Tau-b
95% CIs for tau are reported.
CI method:
Case: Johanna
pairs pos neg ties S D Tau CI lower
A vs. B 75 75 0 0 75 75.00 1.00 NaN
Trend A 10 5 5 0 0 10.00 0.00 -0.88
Trend B 105 87 17 1 70 104.50 0.67 0.24
A vs. B - Trend A 85 80 5 0 75 126.75 0.59 0.20
A vs. B + Trend B 180 162 17 1 145 184.45 0.79 0.53
A vs. B + Trend B - Trend A 190 167 22 1 145 189.50 0.77 0.49
CI upper SD_S VAR_S SE_Tau Z p n
A vs. B NaN 22.91 525.00 0.31 3.27 <.001 20
Trend A 0.88 4.08 16.67 NaN 0.00 1.00 5
Trend B 0.88 20.21 408.33 0.19 3.46 <.001 15
A vs. B - Trend A 0.82 23.26 541.22 0.18 3.22 <.001 20
A vs. B + Trend B 0.91 30.53 932.39 0.17 4.75 <.001 20
A vs. B + Trend B - Trend A 0.90 30.81 949.00 0.16 4.71 <.001 20

8.8.2 Meta analyses

Note

The procedure for calculating the meta-analyses has changed with scan version 0.55.7. Please make sure you are using the latest scan version.

If you pass multiple cases to the tau-u function, it will calculate a Tau-U table for each case and an overall calculation via a meta-analysis.

Calculating a Tau-U meta analysis

The calculation of the Tau-U-meta-analyses involves the following steps:

The tau values are Fisher-Z transformed to \(Tau_z\).

The standard error for each transformed value is calculated as either:

The average \(tau_z\) is the mean of \(tau_z\) weighted by \(1 \over se_z^2\)

The standard error of the average \(tau_z\) is \(se_{M_{tau_z}} = \sqrt{\frac{1}{\sum{weights}}}\)(Cooper, Hedges, & Valentine, 2009)

The p value is calculated with a Z-test (form \(Z = \frac{M_{tau_z}}{se_{M_{tau_z}}}\) )

The overall tau value is derived from an inverse-Fisher-Z-transformation.

8.8.3 Confidence intervals

Note

The default method for calculating the confidence interval has changed with scan version 0.55.7. Confidence intervals could have been outside the [-1, 1] in earlier versions. Set ci_method = "s" for a replication of results from scan version 0.55.6 or earlier.

By default, 95% percent intervals are calculated for each tau value. You can specify a different interval with the ci argument (ci = 0.90 will calculate a 90% interval). There are three alternative approaches to calculating the confidence intervals. When ci_method = "z" is set (the default), a general formula for calculating the standard-error of Fisher-Z values is used (Hotelling, 1953). If ci_method = "tau", a specific formula for Fisher-Z transformed tau values is applied (Fieller et al., 1957). Both approaches give similar results. A third approach is derived from the standard deviation of the S statistic^{1}. Set ci_method = "s" for this method. The S method could lead to implausible values blow -1 or above 1. I recommend to use the common “z” method or the accurate “tau” method.

tau_u(exampleAB, ci =0.90, ci_method ="tau")

Tau-U
Method: complete
Applied Kendall's Tau-b
90% CIs for tau are reported.
CI method:
Tau-U meta analyses:
Weight method: z
90% CIs are reported.
Model Tau_U se CI lower CI upper z p
A vs. B 1.00 0.14 1.00 1.00 Inf 0.0e+00
A vs. B - Trend A 0.59 0.14 0.42 0.72 4.8 1.3e-06
A vs. B + Trend B 0.75 0.14 0.63 0.83 6.9 4.4e-12
A vs. B + Trend B - Trend A 0.74 0.14 0.61 0.82 6.7 1.8e-11
Case: Johanna
Tau CI lower CI upper SD_S Z p
A vs. B 1.00 NaN NaN 22.91 3.27 <.001
A vs. B - Trend A 0.59 0.39 0.74 23.26 3.22 <.001
A vs. B + Trend B 0.79 0.66 0.87 30.53 4.75 <.001
A vs. B + Trend B - Trend A 0.77 0.63 0.86 30.81 4.71 <.001
Case: Karolina
Tau CI lower CI upper SD_S Z p
A vs. B 0.94 0.90 0.96 22.91 3.06 <.001
A vs. B - Trend A 0.55 0.34 0.71 23.25 3.01 <.001
A vs. B + Trend B 0.80 0.69 0.88 30.52 4.85 <.001
A vs. B + Trend B - Trend A 0.78 0.65 0.87 30.79 4.81 <.001
Case: Anja
Tau CI lower CI upper SD_S Z p
A vs. B 0.97 0.94 0.98 22.91 3.14 <.001
A vs. B - Trend A 0.62 0.43 0.76 23.21 3.36 <.001
A vs. B + Trend B 0.63 0.43 0.76 30.45 3.74 <.001
A vs. B + Trend B - Trend A 0.64 0.45 0.78 30.71 3.91 <.001

This method has been proposed by Tarlow (2016). The baseline data are checked for a significant autocorrelation (based on Kendalls Tau). If so, a non-parameteric Theil-Sen regression is applied for the baseline data where the dependent values are regressed on the measurement time. The resulting slope information is then used to predict data of the B-phase. The dependent variable is now corrected for this baseline trend and the residuals of the Theil-Sen regression are taken for further calculations. Finally, Kendalls tau is calculated for the dependent variable and the dichotomous phase variable. The function here provides two extensions to this procedure: The alternative Siegel repeated median regression is applied when repeated = TRUE(Siegel, 1982) and a continuity correction is applied when continuity = TRUE (both not the defaults).

Here is a replication of an example provided by Tarlow (2016) :

Baseline corrected tau
Method: Theil-Sen regression
Continuity correction not applied.
Case1 :
tau z p
Baseline autocorrelation -0.75 -2.31 <.05
Uncorrected tau -0.58 -2.98 <.01
Baseline corrected tau 0.69 3.57 <.001
Baseline correction should be applied.

Basically, the reliable change index (rci) depicts if a post-test is above a pre-test value. Based on the reliability of the measurements and the standard-deviation the standard error is calculated. The mean difference between phase-A and phase-B is divided by the standard-error. Several authors proposed refined methods for calculating the rci.

The rci function computes three indices of reliable change (Wise, 2004) and corresponding descriptive statistics.

rci(exampleAB$Johanna, rel =0.8, graph =TRUE)

Reliable Change Index
Mean Difference = 19.53333
Standardized Difference = 1.678301
Descriptives:
n mean SD SE
A-Phase 5 54.60000 2.408319 1.077033
B-Phase 15 74.13333 8.943207 3.999524
Reliability = 0.8
95 % Confidence Intervals:
Lower Upper
A-Phase 52.48905 56.71095
B-Phase 66.29441 81.97226
Reliable Change Indices:
RCI
Jacobson et al. 18.13624
Christensen and Mendoza 12.82426
Hageman and Arrindell 18.49426

Brossart, D. F., Laird, V. C., & Armstrong, T. W. (2018). Interpreting Kendall’s Tau and Tau-U for single-case experimental designs. Cogent Psychology, 5(1), 1–26. https://doi.org/10.1080/23311908.2018.1518687

Fieller, E. C., Hartley, H. O., & Pearson, E. S. (1957). Tests for rank correlation coefficients. Biometrika, 44, 470–481.

Hotelling, H. (1953). New Light on the Correlation Coefficient and its Transforms. Journal of the Royal Statistical Society: Series B (Methodological), 15(2), 193–225. https://doi.org/10.1111/j.2517-6161.1953.tb00135.x

Parker, Richard I., Hagan-Burke, S., & Vannest, K. (2007). Percentage of AllNon-OverlappingData (PAND) AnAlternative to PND. The Journal of Special Education, 40(4), 194–204. Retrieved from http://sed.sagepub.com/content/40/4/194.short

Parker, Richard I., Vannest, K. J., & Davis, J. L. (2011a). Effect Size in Single-CaseResearch: AReview of NineNonoverlapTechniques. Behavior Modification, 35(4), 303–322. https://doi.org/10.1177/0145445511399147

Parker, Richard I., Vannest, K. J., Davis, J. L., & Sauber, S. B. (2011b). Combining Nonoverlap and Trend for Single-CaseResearch: Tau-U. Behavior Therapy, 42(2), 284–299. https://doi.org/10.1016/j.beth.2010.08.006

Parker, Richard I., Vannest, K. J., Davis, J. L., & Sauber, S. B. (2011c). Combining nonoverlap and trend for single-case research: Tau-u. Behavior Therapy, 42(2), 284–299. https://doi.org/10.1016/j.beth.2010.08.006

Scruggs, T. E., Mastropieri, M. A., & Casto, G. (1987). The QuantitativeSynthesis of Single-SubjectResearchMethodology and Validation. Remedial and Special Education, 8(2), 24–33. https://doi.org/10.1177/074193258700800206

Tarlow, K. R. (2016). An Improved Rank Correlation Effect Size Statistic for Single-Case Designs: Baseline Corrected Tau. Behavior Modification, 41(4), 427–467. https://doi.org/10.1177/0145445516676750

Wise, E. A. (2004). Methods for analyzing psychotherapy outcomes: A review of clinical significance, reliable change, and recommendations for future directions. Journal of Personality Assessment, 82(1), 50–59. Retrieved from http://www.tandfonline.com/doi/abs/10.1207/s15327752jpa8201_10

S is the difference between concordant and discordant comparisons in a Kendall’s tau calculation. This is the same statistic used to calculate the p-value.↩︎