Compute effect size indices for standardized difference between two normal multivariate distributions or between one multivariate distribution and a defined point. This is the standardized effect size for Hotelling's $$T^2$$ test (e.g., DescTools::HotellingsT2Test()). D is computed as:

$$D = \sqrt{(\bar{X}_1-\bar{X}_2-\mu)^T \Sigma_p^{-1} (\bar{X}_1-\bar{X}_2-\mu)}$$

Where $$\bar{X}_i$$ are the column means, $$\Sigma_p$$ is the pooled covariance matrix, and $$\mu$$ is a vector of the null differences for each variable. When there is only one variate, this formula reduces to Cohen's d.

## Usage

mahalanobis_d(
x,
y = NULL,
data = NULL,
pooled_cov = TRUE,
mu = 0,
ci = 0.95,
alternative = "greater",
verbose = TRUE,
...
)

## Arguments

x, y

A data frame or matrix. Any incomplete observations (with NA values) are dropped. x can also be a formula (see details) in which case y is ignored.

data

An optional data frame containing the variables.

pooled_cov

Should equal covariance be assumed? Currently only pooled_cov = TRUE is supported.

mu

A named list/vector of the true difference in means for each variable. Can also be a vector of length 1, which will be recycled.

ci

Confidence Interval (CI) level

alternative

a character string specifying the alternative hypothesis; Controls the type of CI returned: "two.sided" (default, two-sided CI), "greater" or "less" (one-sided CI). Partial matching is allowed (e.g., "g", "l", "two"...). See One-Sided CIs in effectsize_CIs.

verbose

Toggle warnings and messages on or off.

...

Not used.

## Value

A data frame with the Mahalanobis_D and potentially its CI (CI_low and CI_high).

## Details

To specify a x as a formula:

• Two sample case: DV1 + DV2 ~ group or cbind(DV1, DV2) ~ group

• One sample case: DV1 + DV2 ~ 1 or cbind(DV1, DV2) ~ 1

## Confidence (Compatibility) Intervals (CIs)

Unless stated otherwise, confidence (compatibility) intervals (CIs) are estimated using the noncentrality parameter method (also called the "pivot method"). This method finds the noncentrality parameter ("ncp") of a noncentral t, F, or $$\chi^2$$ distribution that places the observed t, F, or $$\chi^2$$ test statistic at the desired probability point of the distribution. For example, if the observed t statistic is 2.0, with 50 degrees of freedom, for which cumulative noncentral t distribution is t = 2.0 the .025 quantile (answer: the noncentral t distribution with ncp = .04)? After estimating these confidence bounds on the ncp, they are converted into the effect size metric to obtain a confidence interval for the effect size (Steiger, 2004).

For additional details on estimation and troubleshooting, see effectsize_CIs.

## CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility) intervals and p values are complementary summaries of parameter uncertainty given the observed data. A dichotomous hypothesis test could be performed with either a CI or a p value. The 100 (1 - $$\alpha$$)% confidence interval contains all of the parameter values for which p > $$\alpha$$ for the current data and model. For example, a 95% confidence interval contains all of the values for which p > .05.

Note that a confidence interval including 0 does not indicate that the null (no effect) is true. Rather, it suggests that the observed data together with the model and its assumptions combined do not provided clear evidence against a parameter value of 0 (same as with any other value in the interval), with the level of this evidence defined by the chosen $$\alpha$$ level (Rafi & Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no effect, additional judgments about what parameter values are "close enough" to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser, 1996).

## Plotting with see

The see package contains relevant plotting functions. See the plotting vignette in the see package.

## References

• Del Giudice, M. (2017). Heterogeneity coefficients for Mahalanobis' D as a multivariate effect size. Multivariate Behavioral Research, 52(2), 216-221.

• Mahalanobis, P. C. (1936). On the generalized distance in statistics. National Institute of Science of India.

• Reiser, B. (2001). Confidence intervals for the Mahalanobis distance. Communications in Statistics-Simulation and Computation, 30(1), 37-45.

stats::mahalanobis(), cov_pooled()

Other standardized differences: cohens_d(), means_ratio(), p_superiority(), rank_biserial(), repeated_measures_d()

## Examples

## Two samples --------------
mtcars_am0 <- subset(mtcars, am == 0,
select = c(mpg, hp, cyl)
)
mtcars_am1 <- subset(mtcars, am == 1,
select = c(mpg, hp, cyl)
)

mahalanobis_d(mtcars_am0, mtcars_am1)
#> Mahalanobis' D |      95% CI
#> ----------------------------
#> 2.14           | [1.22, Inf]
#>
#> - One-sided CIs: upper bound fixed at [Inf].

# Or
mahalanobis_d(mpg + hp + cyl ~ am, data = mtcars)
#> Mahalanobis' D |      95% CI
#> ----------------------------
#> 2.14           | [1.22, Inf]
#>
#> - One-sided CIs: upper bound fixed at [Inf].

mahalanobis_d(mpg + hp + cyl ~ am, data = mtcars, alternative = "two.sided")
#> Mahalanobis' D |       95% CI
#> -----------------------------
#> 2.14           | [1.07, 2.90]

# Different mu:
mahalanobis_d(mpg + hp + cyl ~ am,
data = mtcars,
mu = c(mpg = -4, hp = 15, cyl = 0)
)
#> Mahalanobis' D |      95% CI
#> ----------------------------
#> 1.90           | [1.00, Inf]
#>
#> - Deviation from a difference of 15.5242.
#> - One-sided CIs: upper bound fixed at [Inf].

# D is a multivariate d, so when only 1 variate is provided:
mahalanobis_d(hp ~ am, data = mtcars)
#> Mahalanobis' D |      95% CI
#> ----------------------------
#> 0.49           | [0.00, Inf]
#>
#> - One-sided CIs: upper bound fixed at [Inf].

cohens_d(hp ~ am, data = mtcars)
#> Cohen's d |        95% CI
#> -------------------------
#> 0.49      | [-0.23, 1.21]
#>
#> - Estimated using pooled SD.

# One sample ---------------------------
mahalanobis_d(mtcars[, c("mpg", "hp", "cyl")])
#> Mahalanobis' D |      95% CI
#> ----------------------------
#> 12.59          | [9.49, Inf]
#>
#> - One-sided CIs: upper bound fixed at [Inf].

# Or
mahalanobis_d(mpg + hp + cyl ~ 1,
data = mtcars,
mu = c(mpg = 15, hp = 5, cyl = 3)
)
#> Mahalanobis' D |      95% CI
#> ----------------------------
#> 5.31           | [3.97, Inf]
#>
#> - Deviation from a difference of 16.0935.
#> - One-sided CIs: upper bound fixed at [Inf].