Gamma effect size powered by the middle non-zero quantile absolute deviation

In previous posts, I covered the concept of the gamma effect size. It’s a nonparametric effect size which is consistent with Cohen’s d under the normal distribution. However, the original definition has drawbacks: this statistic becomes zero if half of the sample elements are equal to each other. Last time, I suggested) a workaround for this problem: we can replace the median absolute deviation by the quantile absolute deviation. Unfortunately, this trick requires parameter tuning: we should choose a proper quantile position to make this approach work. Today I want to suggest a strategy that provides a way to make a generic choice: we can use the middle non-zero quantile absolute deviation.

Recall

First of all, let’s recall the general equation for the gamma effect size for the \(p^\textrm{th}\) quantile:

\[\gamma_p = \frac{Q_p(y) - Q_p(x)}{\operatorname{PMAD}_{xy}} \]

where \(Q_p\) is a quantile estimator of the \(p^\textrm{th}\) quantile, \(\operatorname{PMAD}_{xy}\) is the pooled median absolute deviation:

\[\operatorname{PMAD}_{xy} = \sqrt{\frac{(n_x - 1) \operatorname{MAD}^2_x + (n_y - 1) \operatorname{MAD}^2_y}{n_x + n_y - 2}}, \]

\(\operatorname{MAD}_x\) and \(\operatorname{MAD}_y\) are the median absolute deviations of \(x\) and \(y\):

\[\operatorname{MAD}_x = C_{n_x} \cdot Q_{0.5}(|x_i - Q_{0.5}(x)|), \quad \operatorname{MAD}_y = C_{n_y} \cdot Q_{0.5}(|y_i - Q_{0.5}(y)|), \]

\(C_{n_x}\) and \(C_{n_y}\) are consistency constants that makes \(\operatorname{MAD}\) a consistent estimator for the standard deviation estimation. They can be chosen based on the used quantile estimators:

• Constants for the traditional quantile estimator
• Constants for the Harrell-Davis quantile estimator
• Constants for the trimmed Harrell-Davis quantile estimator

QAD instead of MAD

The \(\operatorname{MAD}\) approach has a severe drawback: if half of the sample elements equal to the \(p^\textrm{th}\) quantile, \(\operatorname{MAD}\) becomes zero. Thereby, we can’t use the gamma effect size to compare quantile values.

The problem can be solved using the Quantile Absolute Deviation(QAD) around the given quantile:

\[\operatorname{QAD}_x(p, q) = C_n \cdot Q_q(|x_i - Q_p(x)|) \]

It’s easy to see that the \(\operatorname{MAD}\) is just a special case of \(\operatorname{QAD}\):

\[\operatorname{MAD}_x = \operatorname{QAD}_x(0.5, 0.5). \]

By analogy with \(\operatorname{MAD}\), we can define the pooled quantile absolute deviation \(\operatorname{PQAD}_{xy}\):

\[\operatorname{PQAD}_{xy}(p, q) = \sqrt{\frac{ (n_x - 1) \operatorname{QAD}^2_x(p, q) + (n_y - 1) \operatorname{QAD}^2_y(p, q)}{n_x + n_y - 2}}, \]

The only problem with the approach is that we have to define \(q\).

MNZQAD instead of QAD

In my previous post, I suggested the idea of the middle non-zero quantile absolute deviation:

\[\operatorname{MNZQAD(x, p)} = \operatorname{QAD(x, p, q_m)}, \]

\[q_m = \frac{q_0 + 1}{2}, \quad q_0 = \frac{\max(k - 1, 0)}{n - 1}, \quad k = \sum_{i=1}^n \mathbf{1}_{Q(x, p)}(x_i), \]

where \(\mathbf{1}\) is the indicator function:

\[\mathbf{1}_U(u) = \begin{cases} 1 & \textrm{if}\quad u = U,\\ 0 & \textrm{if}\quad u \neq U. \end{cases} \]

Thus, we peek the middle \(q\) values across all \(q\) values that gives non-zero \(\operatorname{QAD}\):

We can also define a pooled version of \(\operatorname{MNZQAD}\):

\[\operatorname{PMNZQAD}_{xy}(p) = \sqrt{\frac{ (n_x - 1) \operatorname{MNZQAD}^2_x(p) + (n_y - 1) \operatorname{MNZQAD}^2_y(p)}{n_x + n_y - 2}}, \]

With this enchantment, the gamma effect size is always defined for samples with non-zero ranges.