help for bkrosenblatt

Blum, Kiefer and Rosenblatt test of bivariate independence

bkrosenblatt
xvar yvar
[if] [in]

Description

bkrosenblatt provides a test of bivariate independence
due to Blum, Kiefer and Rosenblatt (1961), itself a variant on a test
due to Hoeffding (1948). P-values are computed or indicated following
the work of Mudholkar and Wilding (2003, 2005).

Remarks

Bivariate independence of continuous variables is usually
tested for by focusing on a specific alternative hypothesis of linear or
monotonic relationship. However, Harrell (2001, 2015), among others, stresses
the value of screening data for non-monotonic relationships. More
general omnibus tests have been provided by Hoeffding (1948) and, in
related work, by Blum, Kiefer and Rosenblatt (1961) (hereafter BKR).
For an introductory overview and comparison of such tests, with many
worked empirical examples, see Hollander and Wolfe (1999, Ch.8).
Mudholkar and Wilding (2003, 2005) give further references.

These tests have long remained difficult to implement fully
because of a lack of information on sampling distributions for finite
sample sizes. Hollander and Wolfe (1999, pp.733-734) give detailed
tables for Hoeffding's statistic for n = 5(1)9, thereby extending the
results of Hoeffding (1948) for n = 5, 6, 7. Hollander and Wolfe (1999,
p.735) table the BKR statistic only for the limiting (asymptotic)
distribution. Note that the original table in BKR (1961, p.497) carries
one more decimal place.

Mudholkar and Wilding (2003, 2005) have addressed this problem
with a combination of analytical and simulation work. They show that
the Hoeffding and BKR procedures differ more than is widely recognised
and that the asymptotic distribution for the BKR statistic can be a poor
approximation for many sample sizes. More positively, they detail
procedures for calculating P-values for BKR for all but very small
sample sizes.

bkrosenblatt implements the BKR test. See Mudholkar and
Wilding (2003) for arguments and evidence that this is generally
preferable to competing tests in terms of power against positive
dependence alternatives.

The main idea is best explained with some details. For data
x[i], y[i], i = 1,...,n, count the numbers of values lying in each of
four quadrants of the plane relative to each data point. That is, form
vectors with typical elements

N_1[i] = #(x <= x[i] & y <= y[i])
N_2[i] = #(x > x[i] & y <= y[i])
N_3[i] = #(x <= x[i] & y > y[i])
N_4[i] = #(x > x[i] & y > y[i])

For conditions near independence these vectors will have very
similar averages. The reduction used by BKR is the sum of (N_1 N_4 - N_2 N_3)^2
divided by n^4. (All calculations are performed elementwise.) This
statistic, conventionally denoted n B_n , is a measure of bivariate
dependence, but it is not on a particularly intuitive scale. Mudholkar
and Wilding (2005) show that for moderate sample sizes a power
transformation of n B_n has an approximately normal sampling
distribution, which provides a way of computing a z-score and thus a
P-value. The power h to be used and the mean mu and standard deviation
sigma of the resulting distribution all depend on sample size n:

h = -0.36 + 2.866 * n^-0.775 - 0.683 * exp(-0.244 n)

mu = 4.663 - 1/(0.2137 + 0.00448 n) 15 <= n <= 24
= 3.823 - 1/(0.193 + 0.01662 n^0.8481) n >= 25

sigma = 0.614 - 1 / (1.187 + 0.0328 n)

As h is always negative, the transformation reverses high and
low values. Hence the z-score reported by bkrosenblatt is
calculated as z = (mu - (n B_n )^h) / sigma, so that high positive values
correspond to strong bivariate dependence. Note especially that high
negative values correspond to conditions near independence.

Mudholkar and Wilding (2005) do not suggest approximations for
sample sizes less than 15. However, Mudholkar and Wilding (2003) give
simulation results for a range of sample sizes as low as n = 5. For
sample sizes between 5 and 14 therefore, bkrosenblatt does not
calculate a z-score or P-value, but it does display critical values of
n B_n from the simulations for each sample size. See p.46 of the
2003 paper for a fuller table, while noting the irregularity of the
distribution near n = 5. Bootstrapping may appeal as an alternative, but
best of all is to get more data.

Naturally, you should always look at a scatter plot too and
apply whatever subject-matter knowledge is available to interpretation.

Vignettes

Wassily Hoeffding (1914-1991) was born in Mustamäki,
then in Finland. He studied and worked in Denmark and Germany, gaining a
doctorate from Berlin University, before entering the United States in
1946. Hoeffding attended statistical lectures at Columbia before moving
in 1947 to the University of North Carolina at Chapel Hill, where he
settled. He produced much pioneering work in the area of nonparametric
statistics, including the development of U-statistics. Hoeffding is
remembered as well read in Russian literature and as good-natured, even
in the face of prolonged ill-health and disability. He was a member of
the U.S. National Academy of Sciences.

Julius Rubin Blum (1922-1982) was born in Nuremberg in
Germany and moved to the United States in 1937. His parents perished in
the Holocaust. After war service, he gained degrees in mathematics and
statistics from Berkeley. Blum researched and taught at Indiana
University, Sandia Corporation, the University of New Mexico, the
University of Wisconsin, Milwaukee, the University of Arizona and the
University of California, Davis. He was gregarious and highly
productive and collaborative in probability, ergodic theory and
mathematical statistics, with 86 publications and 34 co-authors.

Jack Carl Kiefer (1924-1981) was born in Cincinnati. He
studied electrical engineering, economics, and statistics at MIT and
Columbia and researched and taught at Cornell and Berkeley. Kiefer was
many-sided: he nearly sought a career in show business, he was a very
serious amateur mycologist and he was active in liberal causes such as
opposition to the Vietnam war and to the oppression of Jews and
dissidents. In statistics he is best known for outstanding work on
optimal design of experiments, but he contributed to several other
areas. Kiefer was a member of the U.S. National Academy of Sciences.

Murray Rosenblatt (1926-2019) was born in New York City. He
gained degrees from City College of New York and Cornell before
researching and teaching at Chicago, Indiana, Brown and (from 1964)
University of California, San Diego. Rosenblatt made outstanding
contributions to several areas of probability and statistics, including
Markov processes and time series. He made key specific contributions to
density estimation, central limit theorems under strong mixing and long
memory processes. Rosenblatt was a member of the U.S. National Academy of
Sciences.

Examples

. sysuse auto, clear

. bkrosenblatt price foreign

Saved results

r(n) number of data pairs n
r(n_B_n) n B_n
r(z) z-score (if n >= 15)
r(P_n_B_n) P-value (if n >= 15)

Author

Nicholas J. Cox, Durham University, U.K.

n.j.cox@durham.ac.uk

References

Blum, J.R., Kiefer, J. and Rosenblatt, M. 1961.
Distribution free tests of independence based on the sample
distribution function.
Annals of Mathematical Statistics 32: 485-498.

Bradley, R. and Davis, R. 2019.
Obituary: Murray Rosenblatt 1926-2019.
IMS Bulletin 48(8): 8-9.

Brillinger, D.R. and Davis, R.A. 2009.
A conversation with Murray Rosenblatt.
Statistical Science 24: 116-140.

Brown, L.D. 1984. The research of Jack Kiefer outside the area of
experimental design.
Annals of Statistics 12: 406-415.

Fisher, N.I. and van Zwet, W.R. 2005.
Wassily Hoeffding 1914-1991.
Biographical Memoirs, National Academy of Sciences
86: 208-227.

Fisher, N.I. and van Zwet, W.R. 2008.
Remembering Wassily Hoeffding.
Statistical Science 23: 536-547.

Harrell, F.E. 2001.
Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis.
New York: Springer.

Harrell, F.E. 2015.
Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis.
Cham: Springer.

Hoeffding, W. 1948.
A non-parametric test of independence.
Annals of Mathematical Statistics 19: 546-557.

Hollander, M. and Wolfe, D.A. 1999.
Nonparametric statistical methods.
New York: Wiley.

The publications and writings of Jack Kiefer.
Annals of Statistics 12: 424-430.

Mudholkar, G.S. and Wilding, G.E. 2003.
On the conventional wisdom regarding two consistent tests of
bivariate dependence.
The Statistician 52: 41-57.

Mudholkar, G.S. and Wilding, G.E. 2005.
Two Wilson-Hilferty type approximations for the null distribution
of the Blum, Kiefer and Rosenblatt test of bivariate independence.
Journal of Statistical Planning and Inference
128: 31-41.

Rosenblatt, M. and Samaniego, F.J. 1985. Julius R. Blum 1922-1982.
Annals of Statistics 13: 1-9.

Sacks, J. 1984. Jack Carl Kiefer 1924-1981.
Annals of Statistics 12: 403-405.

Sen, P.K. 1997.
Hoeffding, Wassily. In Johnson, N.L. and Kotz, S. (eds)
Leading personalities in statistical sciences.
New York: John Wiley, 118-122.

Sun, T.C. 1997.
Murray Rosenblatt: his contributions to probability and statistics.
Journal of Theoretical Probability 10: 279-286.

Wynn, H.P. 1984. Jack Kiefer's contributions to experimental design.
Annals of Statistics 12: 416-423.