The `effectsize`

package contains function to convert
among indices of effect size. This can be useful for meta-analyses, or
any comparison between different types of statistical analyses.

## Converting Between *d* and *r*

The most basic conversion is between *r* values, a measure of
standardized association between two continuous measures, and *d*
values (such as Cohen’s *d*), a measure of standardized
differences between two groups / conditions.

Let’s looks at some (simulated) data:

```
library(effectsize)
data("hardlyworking")
head(hardlyworking)
```

```
> salary xtra_hours n_comps age seniority is_senior
> 1 19745 4.16 1 32 3 FALSE
> 2 11302 1.62 0 34 3 FALSE
> 3 20636 1.19 3 33 5 TRUE
> 4 23047 7.19 1 35 3 FALSE
> 5 27342 11.26 0 33 4 FALSE
> 6 25657 3.63 2 30 5 TRUE
```

We can compute Cohen’s *d* between the two groups:

`cohens_d(salary ~ is_senior, data = hardlyworking)`

```
> Cohen's d | 95% CI
> --------------------------
> -0.72 | [-0.90, -0.53]
>
> - Estimated using pooled SD.
```

But we can also compute a point-biserial correlation, which is
Pearson’s *r* when treating the 2-level `is_senior`

variable as a numeric binary variable:

`correlation::cor_test(hardlyworking, "salary", "is_senior")`

```
> Parameter1 | Parameter2 | r | 95% CI | t(498) | p
> ------------------------------------------------------------------
> salary | is_senior | 0.34 | [0.26, 0.41] | 7.95 | < .001***
>
> Observations: 500
```

But what if we only have summary statistics? Say, we only have \(d=-0.72\) and we want to know what the
*r* would have been? We can approximate *r* using the
following formula (Borenstein et al.
2009):

\[
r \approx \frac{d}{\sqrt{d^2 + 4}}
\] And indeed, if we use `d_to_r()`

, we get a pretty
decent approximation:

`d_to_r(-0.72)`

`> [1] -0.339`

(Which also works in the other way, with `r_to_d(0.12)`

gives 0.723)

As we can see, these are rough approximations, but they can be useful when we don’t have the raw data on hand.

### In multiple regression

Although not exactly a classic Cohen’s d, we can also approximate a
partial-*d* value (that is, the standardized difference between
two groups / conditions, with variance from other predictors partilled
out). For example:

```
fit <- lm(salary ~ is_senior + xtra_hours, data = hardlyworking)
parameters::model_parameters(fit)
```

```
> Parameter | Coefficient | SE | 95% CI | t(497) | p
> -----------------------------------------------------------------------------
> (Intercept) | 14258.87 | 238.71 | [13789.86, 14727.87] | 59.73 | < .001
> is seniorTRUE | 1683.65 | 316.85 | [ 1061.12, 2306.17] | 5.31 | < .001
> xtra hours | 1257.75 | 40.33 | [ 1178.51, 1336.99] | 31.19 | < .001
```

```
>
> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
> using a Wald t-distribution approximation.
```

```
# A couple of ways to get partial-d:
1683.65 / sigma(fit)
```

`> [1] 0.495`

`t_to_d(5.31, df_error = 497)[[1]]`

`> [1] 0.476`

We can convert these semi-*d* values to *r* values, but
in this case these represent the *partial* correlation:

`t_to_r(5.31, df_error = 497)`

```
> r | 95% CI
> -------------------
> 0.23 | [0.15, 0.31]
```

```
correlation::correlation(hardlyworking[, c("salary", "xtra_hours", "is_senior")],
include_factors = TRUE,
partial = TRUE
)[2, ]
```

```
> # Correlation Matrix (pearson-method)
>
> Parameter1 | Parameter2 | r | 95% CI | t(498) | p
> ------------------------------------------------------------------
> salary | is_senior | 0.23 | [0.15, 0.31] | 5.32 | < .001***
>
> p-value adjustment method: Holm (1979)
> Observations: 500
```

```
# all close to:
d_to_r(0.47)
```

`> [1] 0.229`

## Converting Between *OR* and *d*

In binomial regression (more specifically in logistic regression), Odds ratios (OR) are themselves measures of effect size; they indicate the expected change in the odds of a some event.

In some fields, it is common to dichotomize outcomes in order to be
able to analyze them with logistic models. For example, if the outcome
is the count of white blood cells, it can be more useful (medically) to
predict the crossing of the threshold rather than the raw count itself.
And so, where some scientists would maybe analyze the above data with a
*t*-test and present Cohen’s *d*, others might analyze it
with a logistic regression model on the dichotomized outcome, and
present OR. So the question can be asked: given such a OR, what would
Cohen’s *d* have been?

Fortunately, there is a formula to approximate this (Sánchez-Meca, Marı́n-Martı́nez, and Chacón-Moscoso 2003):

\[ d = log(OR) \times \frac{\sqrt{3}}{\pi} \]

which is implemented in the `oddsratio_to_d()`

function.

Let’s give it a try:

```
# 1. Set a threshold
thresh <- 22500
# 2. dichotomize the outcome
hardlyworking$salary_high <- hardlyworking$salary < thresh
# 3. Fit a logistic regression:
fit <- glm(salary_high ~ is_senior,
data = hardlyworking,
family = binomial()
)
parameters::model_parameters(fit)
```

```
> Parameter | Log-Odds | SE | 95% CI | z | p
> -----------------------------------------------------------------
> (Intercept) | 1.55 | 0.16 | [ 1.25, 1.87] | 9.86 | < .001
> is seniorTRUE | -1.22 | 0.21 | [-1.63, -0.82] | -5.86 | < .001
```

```
>
> Uncertainty intervals (profile-likelihood) and p-values (two-tailed)
> computed using a Wald z-distribution approximation.
```

```
>
> The model has a log- or logit-link. Consider using `exponentiate =
> TRUE` to interpret coefficients as ratios.
```

```
# Convert log(OR) (the coefficient) to d
oddsratio_to_d(-1.22, log = TRUE)
```

`> [1] -0.673`

## References

*Introduction to Meta-Analysis*, 45–49.

*Psychological Methods*8 (4): 448.