compare_cor $statistic not z-score

[Lakens]

compare_cor() reports unstandardized numerator as z-statistic

Package: TOSTER
Function: compare_cor() with method = "fisher" and alternative = "equivalence"

---

Description

When running an equivalence test via compare_cor(), the reported z statistic
in the output is not the standardized test statistic. Instead, it is the raw
numerator (diff - bound) before dividing by the standard error. The p-value
is computed correctly (it does use zlo/z_se internally), but the labelled
statistic is misleading and inconsistent with what a z-statistic conventionally means.

---

Expected behavior

The reported z should be the standardized test statistic:

z = (diff - bound) / SE

A reader using the reported z to verify or recompute the p-value via pnorm()
should get the same p-value shown in the output.

---

Actual behavior

The reported z is the unstandardized numerator zlo = diff - min(znull).
Looking at the source:

zlo = diff - min(znull)
plo = p_from_z(zlo/z_se, alternative = "greater")  # p-value uses zlo/z_se
...
z = zlo   # but reported z stores only zlo, not zlo/z_se

---

Reproducible example

library(TOSTER)

result <- compare_cor(
  r1 = 0.6, df1 = 18,
  r2 = 0.8, df2 = 23,
  null = 0.4,
  method = "fisher",
  alternative = "equivalence"
)

result$statistic
#> z = 0.018184   ← this is diff - bound, not a z-statistic

# Manually computing the correct z-statistic:
z1   <- atanh(0.6); z2 <- atanh(0.8)
diff <- z1 - z2                              # -0.4055
SE   <- sqrt(1/17 + 1/22)                   #  0.3229
bound_z <- atanh(0.4)                        #  0.4236 (TOSTER converts null via atanh)

(diff - (-bound_z)) / SE                     # 0.0562 ← actual z-statistic

# Verify p-value matches the reported one:
1 - pnorm((diff - (-bound_z)) / SE)          # 0.4775 ✓
1 - pnorm(result$statistic)                  # 0.4928 ✗  (wrong if you use reported z)

---

Additional context

• The p-value itself is correct — this is purely a reporting/labelling issue.
• It becomes practically misleading if a user tries to verify the p-value from
  the reported z, or reports the z-statistic in a paper, as it has no standard
  interpretation (it is in the units of Fisher-z differences, not a
  standardized normal deviate).
• Separately: null is passed through rho_to_z() internally, so null = 0.4
  is treated as r = 0.4 → z = 0.4236. This may also be worth documenting more
  explicitly, as users setting bounds "in Fisher-z units" will get slightly
  different bounds than intended.

[arcaldwell49]

@Lakens good catch. working on a fix now.