ci_mahalanobis: Confidence Interval for the Squared Mahalanobis Distance

Description

Computes the squared Mahalanobis distance $D^2$ together with an exact confidence interval for the population squared distance $\Delta^2$, obtained by inverting Hotelling's $T^2$ statistic through its (noncentral) F-distribution as in Reiser (2001). Both the one-sample setting (mean vector against a hypothesized population mean) and the two-sample setting (between-groups distance from discriminant analysis or multivariate group comparison) are supported, and either raw data or a pre-computed $D^2$ with sample sizes can be supplied.

Usage

ci_mahalanobis(
  D2 = NULL,
  group_1 = NULL,
  group_2 = NULL,
  mu_0 = NULL,
  n_1 = NULL,
  n_2 = NULL,
  p = NULL,
  conf_level = 0.95,
  alpha_lower = NULL,
  alpha_upper = NULL,
  ...
)

Value

A one-row data.frame with columns sample_type

("one-sample" or "two-sample"), D2 (point estimate of the squared distance), lower_limit and upper_limit (the confidence limits on the population squared distance $\Delta^2$),

F_value, df_1, df_2, n_1, n_2

(NA in one-sample mode), and p.

Arguments

D2: Optional pre-computed squared Mahalanobis distance. Ignored if group_1 is supplied.
group_1: Optional numeric matrix or data frame for the first sample ($n_1 \times p$). Rows are observations and columns are variables.
group_2: Optional numeric matrix or data frame for the second sample ($n_2 \times p$). When NULL, the function operates in one-sample mode against mu_0.
mu_0: Optional hypothesized population mean for the one-sample case (length-$p$ numeric vector). Defaults to a vector of zeros.
n_1: Sample size for group 1 (required when supplying D2 directly).
n_2: Sample size for group 2 (required for two-sample mode when supplying D2 directly; leave NULL for one-sample mode).
p: Dimensionality (number of variables) when supplying D2 directly.
conf_level: Confidence coverage for a symmetric interval (default 0.95). Set it to NULL to specify the tails directly through alpha_lower and alpha_upper.
alpha_lower, alpha_upper: Optional Type I error rates for the lower and upper tail. To use them, set conf_level = NULL and supply both (an asymmetric or one-sided interval with coverage 1 - alpha_lower - alpha_upper); supplying either alongside a non-NULL conf_level is an error, as in ci_nc_F, to which they are passed.
...: Additional arguments passed to ci_nc_F (for example tol).

Author

Ken Kelley [email protected]

Details

Definition. For a $p$-vector $\mathbf{x}$ drawn from a multivariate normal with mean $\boldsymbol{\mu}$ and covariance $\boldsymbol{\Sigma}$, Mahalanobis's (1936) squared distance from a reference vector $\boldsymbol{\mu}_0$ is $$\Delta^2 = (\boldsymbol{\mu} - \boldsymbol{\mu}_0)^\top \boldsymbol{\Sigma}^{-1} (\boldsymbol{\mu} - \boldsymbol{\mu}_0).$$ In the two-sample case the population distance between groups is $\Delta^2 = (\boldsymbol{\mu}_1 - \boldsymbol{\mu}_2)^\top \boldsymbol{\Sigma}^{-1} (\boldsymbol{\mu}_1 - \boldsymbol{\mu}_2)$, assuming a common covariance. The corresponding sample estimates plug the sample means and the sample (or pooled) covariance into the same quadratic form.

Link to Hotelling's $T^2$. Hotelling's (1931) $T^2$ statistic is $T^2 = n D^2$ (one sample) or $T^2 = \{n_1 n_2 / (n_1 + n_2)\} D^2$ (two samples). Under multivariate normality $$\frac{n_1 + n_2 - p - 1}{(n_1 + n_2 - 2)\,p}\,T^2 \sim F'\!\left(p,\, n_1 + n_2 - p - 1,\; \lambda = \frac{n_1 n_2}{n_1 + n_2}\,\Delta^2\right)$$ in the two-sample case, and analogously $\{(n - p)/[(n-1)p]\}\,T^2 \sim F'(p, n-p, n\Delta^2)$ in the one-sample case (see Anderson, 2003, Section 5.2).

Confidence interval. The CI on $\Delta^2$ is obtained by inverting these distributional results (Reiser, 2001): a CI on the noncentrality parameter $\lambda$ is constructed via ci_nc_F and then mapped back to $\Delta^2$ by $\Delta^2 = \lambda\,(n_1 + n_2)/(n_1 n_2)$ (two sample) or $\Delta^2 = \lambda / n$ (one sample). When the observed $F$ is below the lower-tail critical value of the central F-distribution at the requested confidence level, the lower CI on $\lambda$ (and hence on $\Delta^2$) is clamped to zero, in keeping with the ci_nc_F convention.

Bias. The plug-in estimator $D^2$ is upward biased for $\Delta^2$; the CI from this function is exact for $\Delta^2$ under multivariate normality and reflects the bias structure correctly, but the point estimate reported is the standard plug-in $D^2$.

References

Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (4th ed.). Wiley.

Hotelling, H. (1931). The generalization of Student's ratio. The Annals of Mathematical Statistics, 2(3), 360--378.

Mahalanobis, P. C. (1936). On the generalized distance in statistics. Proceedings of the National Institute of Sciences of India, 2(1), 49--55.

Reiser, B. (2001). Confidence intervals for the Mahalanobis distance. Communications in Statistics--Simulation and Computation, 30(1), 37--45. tools:::Rd_expr_doi("10.1081/SAC-100001856")

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 221--257). Mahwah, NJ: Lawrence Erlbaum.

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1--24. tools:::Rd_expr_doi("10.18637/jss.v020.i08")

Examples

Run this code

# Two-sample distance between the two schools of the Holzinger and
#     Swineford (1939) study on a four-test cognitive battery.
battery <- c("t1_visual_perception", "t2_cubes", "t4_lozenges",
             "t6_paragraph_comprehension")
g1 <- as.matrix(holzinger_swineford[
  holzinger_swineford$school == "Grant-White", battery])
g2 <- as.matrix(holzinger_swineford[
  holzinger_swineford$school == "Pasteur", battery])
ci_mahalanobis(group_1 = g1, group_2 = g2)

# One-sample distance: how far is the Grant-White centroid from a
#     reference vector of (29, 24, 18, 9)?
ci_mahalanobis(group_1 = g1, mu_0 = c(29, 24, 18, 9))

# Pre-computed D^2 (no raw data needed): the two-school distance,
#     reproduced from reported summaries.
ci_mahalanobis(D2 = 0.608, n_1 = 145, n_2 = 156, p = 4)

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