inst/shiny-examples/SCD-effect-sizes/markdown/IRD.md

In jepusto/SingleCaseES: A Calculator for Single-Case Effect Sizes

Definition

The robust improvement rate difference is defined as the robust phi coefficient corresponding to a certain $2 \times 2$ table that is a function of the degree of overlap between the observations each phase (Parker, Vannest, & Davis, 2011).

This effect size does not have a stable parameter definition because its magnitude depends on the number of observations in each phase (Pustejovsky, 2018).

Estimation

Let $y^A_{(1)},y^A_{(2)},...,y^A_{(m)}$ denote the values of the baseline phase data, sorted in increasing order, and let $y^B_{(1)},y^B_{(2)},...,y^B_{(n)}$ denote the values of the sorted treatment phase data. Let $y^A_{(0)} = y^B_{(0)} = -\infty$ and $y^A_{(m + 1)} = y^B_{(n + 1)} = \infty$. For an outcome where increase is desirable, let $\tilde{i}$ and $\tilde{j}$ denote the values that maximize the quantity

$$ \left(i + j\right) I\left(y^A_{(i)} < y^B_{(n + 1 - j)}\right) $$ for $0 \leq i \leq m$ and $0 \leq j \leq n$. For an outcome where decrease is desirable, let $\tilde{i}$ and $\tilde{j}$ instead denote the values that maximize the quantity

$$ \left(i + j\right) I\left(y^A_{(m + 1 - i)} > y^B_{(j)}\right). $$

Now calculate the $2 \times 2$ table

$$ \begin{array}{|c|c|} \hline m - \tilde{i} & \tilde{j} \\ \hline \tilde{i} & n - \tilde{j} \\ \hline \end{array} $$

Parker, Vannest, and Brown (2009) proposed the non-robust improvement rate difference, which is equivalent to the phi coefficient from this table. Parker, Vannest, and Davis (2011) proposed to instead use the robust phi coefficient, which involves modifying the table so that the row- and column-margins are equal. Robust IRD is thus equal to

$$ \text{IRD} = \frac{n - m - \tilde{i} - \tilde{j}}{2 n} - \frac{m + n - \tilde{i} - \tilde{j}}{2 m}. $$

Robust IRD is algebraically related to PAND as

$$ \text{IRD} = 1 - \frac{(m + n)^2}{2mn}\left(1 - \text{PAND}\right). $$

The sampling distribution of robust IRD has not been described, and so standard errors and confidence intervals are not available.

Primary reference

Parker, R. I., Vannest, K. J., & Davis, J. L. (2011). Effect size in single-case research: A review of nine nonoverlap techniques. Behavior Modification, 35(4), 303--22. https://dx.doi.org/10.1177/0145445511399147

Additional references

Parker, R. I., Vannest, K. J., & Brown, L. (2009). The improvement rate difference for single-case research. Exceptional Children, 75(2), 135–150. https://dx.doi.org/10.1177/001440290907500201

Pustejovsky, J. E. (2018). Procedural sensitivities of effect sizes for single-case designs with behavioral outcome. Psychological Methods, forthcoming. https://doi.org/10.1037/met0000179

jepusto/SingleCaseES documentation built on Aug. 21, 2023, 12:08 p.m.