Use correction for small-sample bias in all Chisq effect size

For $\phi$, the small-sample bias corrected estimate is:

$$
\widetilde{\phi} = \sqrt{\phi^2 - \frac{df}{N-1}}
$$

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.

To achieve this with פ:

$$
\widetilde{פ} = \sqrt{\frac{\widetilde{\phi^2}}{\frac{1}{min(p_E)} - 1 - \frac{k-1}{n-1}}}
$$

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:

O <- c(2, 0)
E <- c(0.35, 0.65)

res <- chisq.test(O, p = E, correct = FALSE)

chisq <- unname(res$statistic)
df <- unname(res$parameter)
N <- sum(O)

phi2_adj <- chisq / N - df / (N - 1)

# adjusted Fei
sqrt(phi2_adj / 
       (1 / min(E) - 1 - df / (N - 1)))
#> [1] 1

# unadjusted Fei
effectsize::fei(O, p = E, ci = NULL)
#> Fei 
#> ----
#> 1.00
#> 
#> - Adjusted for uniform expected probabilities.

This is also true for T (by design):

mat <- diag(2)
mat[1,1] <- 2
mat
#>      [,1] [,2]
#> [1,]    2    0
#> [2,]    0    1

effectsize::tschuprows_t(mat, ci = NULL)
#> Tschuprow's T (adj.)
#> --------------------
#> 1.00

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:

$$
\widetilde{פ} = \sqrt{\frac{\widetilde{\phi^2}}{\frac{1}{min(p_E)} - 1}}
$$

(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.