Sample Partial Correlation Research Articles

Permutation Tests for Linear Models

Several approximate permutation tests have been proposed for tests of partial regression coefficients in a linear model based on sample partial correlations. This paper begins with an explanation and notation for an exact test. It then compares the distributions of the test statistics under the various permutation methods proposed, and shows that the partial correlations under permutation are asymptotically jointly normal with means 0 and variances 1. The method of Freedman & Lane (1983) is found to have asymptotic correlation 1 with the exact test, and the other methods are found to have smaller correlations with this test. Under local alternatives the critical values of all the approximate permutation tests converge to the same constant, so they all have the same asymptotic power. Simulations demonstrate these theoretical results.

Comparison of correlation coeffìcients when conditional independence constraints between variables are present

Partial correlations are the natural interaction terms to be associated with the edges of the independence graph of a multivariate normal distribution. Two edges of the graph can thus be compared by comparing the corresponding partial correlations. The comparison of dependent correlations is a well known problem in statistics but, when the variables satisfy some conditional independence relations, the maximum likelihood estimates of partial correlations are different from the sample partial correlations, so that classical results no longer bold. In this paper we analyze to what extent the classical test statistics can be applied. We show that maximum likelihood estimates of partial correlations are more efficient than sample partial correlations. Furthermore we show that the conditional independence structure of the model can be used to turn the comparison of dependent correlations into the comparison of independent parameters.

Confidence Intervals for Partial Rank Correlations

Abstract A partial rank correlation coefficient T(n) XY|Z , based on comparing pairs for which the values of the conditioning variable follow each other in a numerical ordering, is studied. Simulation results show that it is possible to obtain reasonably good confidence intervals by estimating σ 2/(1 − τ 2 XY|Z ). The advantage of using the coefficient T(n) XY|Z is that it is always clear what this coefficient measures, in contrast to, for example, Pearson's, Spearman's, or Kendall's partial correlation coefficients, which can give values far from 0 even in cases of conditional independence. The main disadvantage is that the asymptotic efficiency relative to the sample partial correlation coefficient (in the case of trivariate normal variables) is never higher than .33. Coefficients like T(n) XY|Z have been studied by Goodman and by Quade.

Confidence Intervals for Partial Rank Correlations

Abstract A partial rank correlation coefficient T(n) XY|Z , based on comparing pairs for which the values of the conditioning variable follow each other in a numerical ordering, is studied. Simulation results show that it is possible to obtain reasonably good confidence intervals by estimating σ 2/(1 − τ 2 XY|Z ). The advantage of using the coefficient T(n) XY|Z is that it is always clear what this coefficient measures, in contrast to, for example, Pearson's, Spearman's, or Kendall's partial correlation coefficients, which can give values far from 0 even in cases of conditional independence. The main disadvantage is that the asymptotic efficiency relative to the sample partial correlation coefficient (in the case of trivariate normal variables) is never higher than .33. Coefficients like T(n) XY|Z have been studied by Goodman and by Quade.

Performance analysis of complex lattice processors for signal detection

Analytical performance results are derived for a partial-correlation-coefficient-based signal detector. Although the result is based on large sample theory, computer simulations show that it also provides sufficiently accurate approximations for small or moderate samples. The joint distribution function of complex sample partial correlation coefficients is derived to prove the results will be found useful in many other situations as well. >

From Bareiss' algorithm to the stable computation of partial correlations

This paper presents a new algorithm for the stable computation of sample partial correlation coefficients. We start by generalizing Bareiss' algorithm for the solution of linear systems of equations with (non-symmetric) Toeplitz coefficient matrix to matrices that are not Toeplitz, and show that it computes their LU and UL factorizations. For symmetric positive-definite matrices B, the normalized version of Bareiss' algorithm is the Hyperbolic Cholesky algorithm, which computes the upper and lower triangular Cholesky factors U and L of B by means of 2×2 hyperbolic rotations. If B is a convariance matrix, certain partial correlation coefficients may be obtained as the geometric means of the multiplier pairs in Bareiss' algorithm or as the hyperbolic tangents of the hyperbolic rotations. The partial correlation coefficients for a given data matrix A may suffer from serious loss of numerical accuracy when computed from the covariance matrix B = A T A. While the data flow graph of Bareiss' algorithm suggests a method, due to Cybenko, for computing partial correlation coefficients directly from the columns of A, the graph of Hyperbolic Cholesky suggests an alternative that requires fewer operations nd exhibits a higher degree of parallelism. In that method one computes the QR decomposition A = QU of the data matrix and applies orthogonal rotations to transform U to L; the sines of the rotation angles constitute a set of partial correlation coefficients. We extend this approach to determine arbitrary partial correlation coefficients at little extra computational cost.

Asymptotic second-order properties of sample partial correlations

The asymptotic behavior of the sample partial correlations (PARCORs) is studied in the case of a general stationary process. An explicit formula for PARCOR's second-order moments is provided. The analysis presented extends to more general processes as shown using previously reported results on the statistics of sample partial correlations of autoregressive (AR) and mixed autoregressive moving-average (ARMA) processes. This extension should be useful as the processes encountered in applications are not exactly AR or ARMA.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Time Series Model Identification and Quantile Spectral Analysis

Abstract This paper describes research in progress whose goal is to apply techniques of Quantile Data Analysis to interpret non-parametric functions such as the spectral density, sample correlations, and sample partial correlations, which are used to analyze time series. We propose to plot identification-quantile versions of these functions. The aim is to identify for an observed time series its memory type, and type of parametric time domain models that fit it. A measure of memory is the index δ of regular variation of a spectral density; estimators are proposed for δ. Identification quantile scatter diagrams and IQQ plots are defined as other illustrations of quantile domain statistical data analysis.

Ante-dependence Analysis of an Ordered Set of Variables

For a set of variables in a given order, $s$th ante-dependence will be said to obtain if each one of the variables, given at least $s$ immediate antecedent variables in the order, is independent of all further preceding variables. If the number of variables is $p$, ante-dependence is of some order between 0 and $p - 1.$ 0th ante-dependence and $(p - 1)$st ante-dependence are equivalent to complete independence and to completely arbitrary patterns of dependence, respectively, and are defined irrespective of the ordering of the variables. 1st to $(p - 2)$nd ante-dependence are defined in terms of a specific order only. If $X_1, X_2, \cdots, X_p$ are multivariate normal, $s$th ante-dependence is equivalent to each $X_i$, given $X_{i - 1}, X_{i - 2}, \cdots, X_{i - s}, \cdots, X_{i - s - z}$, being uncorrelated with $X_{i - s - z - 1}, X_{i - s - z - 2}, \cdots, X_2, X_1$ for any non-negative $z$. In other words, the partial correlation of $X_i$ and $X_{i - s - z - k}$, given all the variables $X_{i - 1}, X_{i - 2}, \cdots, X_{i - s - z}$, is zero for all $i, k$ and $z$. The hypothesis that the covariance matrix is such that all the above partial correlations vanish will be denoted by $D_s (s = 0, 1, \cdots, p - 1)$, so that, for the multivariate normal distribution, $D_s$ denotes the hypothesis of $s$th ante-dependence. It is shown that for any set of ordered variables, normal or otherwise, $D_s$ is equivalent to the following correspondence between the regression equations of $X_i$ on all other variables, and on $X_{i - s}, X_{i - s + 1}, \cdots, X_{i = 1}, X_{i + 1}, \cdots, X_{i + s}$ only: the multiple correlations are equal, and the regression coefficients of $X_{i - s}, X_{i - s + 1}, \cdots, X_{i + s}$ are equal in both equations, all other coefficients in the former equation being zero. It is also equivalent to the $(p - s)(p - s - 1)/2$ elements in the upper right (and also lower left) corner of the inverse covariance matrix being zero. Indeed, any null hypothesis on a set of elements of the inverse covariance matrix may be formulated, and tested, as a hypothesis $D_s$ if the variables can be so ordered as to put the zero elements in the upper right and lower left corners of the inverse. Maximum likelihood estimates are derived under $D_s$ for the normal case. Likelihood ratio tests of any one $D_s$ against any other follow immediately and may be expressed in terms of the sample partial correlations. Exact distributions are not investigated, but for large samples $\chi^2$ approximations are available. Thus a sequence of tests of $D_{p - 2}$ under $D_{p - 1}, D_{p - 3}$ under $D_{p - 2}, \cdots, D_0$ under $D_1$, is obtained which, in effect, forms a breakdown of the large sample test of independence, $D_0$, under the general alternative, $D_{p - 1}$. The assumptions of ante-dependence are clearly analogous to those of Markov processes and autoregressive schemes for time series; the motivation for the study and application of these models is also similar. The present model is more general in that it relaxes the usual autoregression assumptions of equal variances and, more crucial, of equal correlations between all pairs of equidistant variables (distance being meant in terms of the order of the set or time series). This greater generality requires analysis of a sample of observations for the study of ante-dependence, whereas for autoregressive schemes there are methods of analysis based on a single observation of the time series. The ante-dependence models can be generalized to the case of several variables at each stage of the ordering. This would be analogous to the study of multiple time-series. $s$-ante-dependent sets of variables may be generated by $s$ successive summations of independent variables. This may be relevant for some applications of such models. Ante-dependence models might be applicable to observations ordered in time or otherwise. Observations on growth of organisms up to each of several ages could be analyzed in such a manner. Where growth is recorded on several dimensions, e.g., height and weight, the analysis would proceed in terms of the multidimensional generalization of the model. Other possible fields of application include batteries of psychological tests increasing in complexity, and data on the successive location of travelling objects. A study of some such applications is now under way.