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This comes from the non-central $\chi^2$ distribution, where $E[\hat{\chi^2}] = df + \phi ^2 \times N$ => $E[\hat{\phi^2}] = \phi ^2 + df / N$.
This is used in effectsize for:
phi(adjust = TRUE)
cramers_v(adjust = TRUE)
tschuprows_t(adjust = TRUE)
(The latter two also have a weird scaling factor from Bergsma (2013).)
This correction can be applied to all $\phi$-like effect sizes:
cohens_w() - makes the most sense as it applies the same transformation on $\chi^2$ as $\phi$ does.
pearsons_c() - can be seen as a transformed Cohen's w ( $C = \sqrt{W^2 / (W^2 - 1)}$ ) so using an adjusted w would "adjust" C as well.
fei() - same reasoning. Although the additional scaling factor ( $1/min(p_E) - 1$ ) might have to be adjusted in a similar manner as V and T's is. (See next section.)
Some of my thoughts...
Bergsma (2013) suggested changing the scaling factors of V and T in such a way that when (the true) $T=1$, RMSE would be 0 because (regardless of sample size) the estimated T would also be 1.
I'm not sure this is the way to go, because it also means that a sample in which פ=1 will produce an estimate of 1, even when the sample size is arbitrarily small. For example:
From what I can see, small sample bias adjustments almost always shrink the estimate, even when it is perfect (e.g., $R^2_{adj}$, $\omega^2$, $\epsilon^2$). So I think having:
(which uses the regular scaling factor) makes the most sense to me, which will also make it consistent with w for the uniform-binary case, but will make it inconsistent with the adjusted V and T.
The text was updated successfully, but these errors were encountered:
For$\phi$ , the small-sample bias corrected estimate is:
This comes from the non-central$\chi^2$ distribution, where $E[\hat{\chi^2}] = df + \phi ^2 \times N$ => $E[\hat{\phi^2}] = \phi ^2 + df / N$ .
This is used in
effectsize
for:phi(adjust = TRUE)
cramers_v(adjust = TRUE)
tschuprows_t(adjust = TRUE)
(The latter two also have a weird scaling factor from Bergsma (2013).)
This correction can be applied to all$\phi$ -like effect sizes:
cohens_w()
- makes the most sense as it applies the same transformation onpearsons_c()
- can be seen as a transformed Cohen's w (fei()
- same reasoning. Although the additional scaling factor (Some of my thoughts...
Bergsma (2013) suggested changing the scaling factors of V and T in such a way that when (the true)$T=1$ , RMSE would be 0 because (regardless of sample size) the estimated T would also be 1.
To achieve this with פ:
I'm not sure this is the way to go, because it also means that a sample in which פ=1 will produce an estimate of 1, even when the sample size is arbitrarily small. For example:
This is also true for T (by design):
From what I can see, small sample bias adjustments almost always shrink the estimate, even when it is perfect (e.g.,$R^2_{adj}$ , $\omega^2$ , $\epsilon^2$ ). So I think having:
(which uses the regular scaling factor) makes the most sense to me, which will also make it consistent with w for the uniform-binary case, but will make it inconsistent with the adjusted V and T.
The text was updated successfully, but these errors were encountered: