Confidence Interval for the Squared Mahalanobis Distance

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,
  ...
)

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 conf_limits_ncf, to which they are passed.
...: Additional arguments passed to conf_limits_ncf (for example tol).

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.

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 conf_limits_ncf 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 conf_limits_ncf 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. 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. doi:10.18637/jss.v020.i08

Author

Ken Kelley kkelley@nd.edu

Examples

# 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)
#>  sample_type D2    lower_limit upper_limit F_value df_1 df_2 n_1 n_2 p
#>  two-sample  0.608 0.259       0.975       11.3    4    296  145 156 4
#> 
#> Confidence level: 95%

# 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))
#>  sample_type D2    lower_limit upper_limit F_value df_1 df_2 n_1 n_2  p
#>  one-sample  0.289 0.11        0.474       10.2    4    141  145 <NA> 4
#> 
#> Confidence level: 95%

# 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)
#>  sample_type D2    lower_limit upper_limit F_value df_1 df_2 n_1 n_2 p
#>  two-sample  0.608 0.259       0.974       11.3    4    296  145 156 4
#> 
#> Confidence level: 95%