Converting Between d, r, and OR

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 simulate some data:

set.seed(1)
data <- bayestestR::simulate_difference(
  n = 10,
  d = 0.2,
  names = c("Group", "Outcome")
)

>    Group Outcome
> 1      0  -1.280
> 2      0  -0.597
> 3      0  -0.100
> 4      0   0.397
> 5      0   1.080
> 6      1  -1.080
> 7      1  -0.397
> 8      1   0.100
> 9      1   0.597
> 10     1   1.280

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

cohens_d(Outcome ~ Group, data = data)

> Cohen's d |        95% CI
> -------------------------
> -0.22     | [-1.46, 1.03]
> 
> - Estimated using pooled SD.

But we can also treat the 2-level group variable as a numeric variable, and compute Pearson’s r:

correlation::correlation(data, include_factors = TRUE)[2, ]

> # Correlation Matrix (pearson-method)
> 
> Parameter1 | Parameter2 |     r |        95% CI |  t(8) |      p
> ----------------------------------------------------------------
> Group.0    |    Outcome | -0.12 | [-0.70, 0.55] | -0.35 | > .999
> 
> p-value adjustment method: Holm (1979)
> Observations: 10

But what if we only have summary statistics? Say, we only have \(d=-0.37\) 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(-0.31)

> [1] -0.153

(Which also works in the other way, with r_to_d(0.17) gives -0.345)

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

fit <- lm(mpg ~ am + hp, data = mtcars)

parameters::model_parameters(fit)

> Parameter   | Coefficient |       SE |         95% CI | t(29) |      p
> ----------------------------------------------------------------------
> (Intercept) |       26.58 |     1.43 | [23.67, 29.50] | 18.65 | < .001
> am          |        5.28 |     1.08 | [ 3.07,  7.48] |  4.89 | < .001
> hp          |       -0.06 | 7.86e-03 | [-0.07, -0.04] | -7.50 | < .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:
5.28 / sigma(fit)

> [1] 1.81

t_to_d(4.89, df_error = 29)[[1]]

> [1] 1.82

t_to_r(4.89, df_error = 29)

> r    |       95% CI
> -------------------
> 0.67 | [0.43, 0.80]

correlation::correlation(mtcars[, c("mpg", "am", "hp")], partial = TRUE)[1, ]

> # Correlation Matrix (pearson-method)
> 
> Parameter1 | Parameter2 |    r |       95% CI | t(30) |         p
> -----------------------------------------------------------------
> mpg        |         am | 0.67 | [0.42, 0.83] |  4.97 | < .001***
> 
> p-value adjustment method: Holm (1979)
> Observations: 32

# all close to:
d_to_r(1.81)

> [1] 0.671

From Odds ratios

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 <- 0

# 2. dichotomize the outcome
data$Outcome_binom <- data$Outcome < thresh

# 3. Fit a logistic regression:
fit <- glm(Outcome_binom ~ Group,
  data = data,
  family = binomial()
)

parameters::model_parameters(fit)

> Parameter   | Log-Odds |   SE |        95% CI |     z |     p
> -------------------------------------------------------------
> (Intercept) |     0.41 | 0.91 | [-1.39, 2.43] |  0.44 | 0.657
> Group [1]   |    -0.81 | 1.29 | [-3.54, 1.70] | -0.63 | 0.530

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

# Convert log(OR) (the coefficient) to d
oddsratio_to_d(-0.81, log = TRUE)

> [1] -0.447

OR <- 3.5
baserate <- 0.85

oddsratio_to_riskratio(OR, baserate)

> [1] 1.12

OR <- 3.5
baserate <- 0.04

oddsratio_to_riskratio(OR, baserate)

> [1] 3.18