Usual Correlation Research Articles

A simple graphic method to assess correlation

This study explores the correlation between two variables and to demonstrate a simple graphic method to assess their degree of correlation. Following the lead of early English biometricians, it has been tacitly assumed that the studied variables develop in the same direction: when variable A’s measurements are higher from one object to another, the measurements of variable B, also are higher. The customary measure of co-relation relies on a least squares fitted trend line, then assuming that the trend is more real than, and has priority over the individually recorded data. The situation changes when measurements of variables develop in opposite directions: The very first data set I used to perform a correlation analysis was a study of student grades achieved and the percentage of their having missed classes: the more a student was absent from class, the lower were his achieved grades. In that situation the accepted model of correlation analysis – the mathematically fitted straight line and the squared distance of each student’s record from that line - was not appropriate. The usual correlation coefficient contradicted visual evidence of those data because the model underlying that situation treats the individual data as having more reality value than the general trend, but not as deviations or errors. The visual appearance, the graph of that situation, resembles a rectangular triangle, formed by the horizontal and vertical axis as its catheters, and the hypotenuse formed by a line through and representing the highest data points. This image justifies the expression “Triangular correlation”.

An Optimal Decision Population Code that Accounts for Correlated Variability Unambiguously Predicts a Subject’s Choice

Decisions emerge from the concerted activity of neuronal populations distributed across brain circuits. However, the analytical tools best suited to decode decision signals from neuronal populations remain unknown. Here we show that knowledge of correlated variability between pairs of cortical neurons allows perfect decoding of decisions from population firing rates. We recorded pairs of neurons from secondary somatosensory (S2) and premotor (PM) cortices while monkeys reported the presence or absence of a tactile stimulus. We found that while populations of S2 and sensory-like PM neurons are only partially correlated with behavior, those PM neurons active during a delay period preceding the motor report predict unequivocally the animal's decision report. Thus, a population rate code that optimally reveals a subject's perceptual decisions can be implemented just by knowing the correlations of PM neurons representing decision variables.

Measuring linkage disequilibrium by the partial correlation coefficient

Measuring linkage disequilibrium by the partial correlation coefficient

The Gaussian rank correlation estimator: robustness properties

The Gaussian rank correlation equals the usual correlation coefficient computed from the normal scores of the data. Although its influence function is unbounded, it still has attractive robustness properties. In particular, its breakdown point is above 12%. Moreover, the estimator is consistent and asymptotically efficient at the normal distribution. The correlation matrix obtained from pairwise Gaussian rank correlations is always positive semidefinite, and very easy to compute, also in high dimensions. We compare the properties of the Gaussian rank correlation with the popular Kendall and Spearman correlation measures. A simulation study confirms the good efficiency and robustness properties of the Gaussian rank correlation. In the empirical application, we show how it can be used for multivariate outlier detection based on robust principal component analysis.

The Gaussian Rank Correlation Estimator: Robustness Properties

The Gaussian rank correlation equals the usual correlation coefficient computed from the normal scores of the data. Although its influence function is unbounded, it still has attractive robustness properties. In particular, its breakdown point is above 12%. Moreover, the estimator is consistent and asymptotically efficient at the normal distribution. The correlation matrix based on the Gaussian rank correlation is always positive semidefinite, and very easy to compute, also in high dimensions. A simulation study confirms the good efficiency and robustness properties of the proposed estimator with respect to the popular Kendall and Spearman correlation measures. In the empirical application, we show how it can be used for multivariate outlier detection based on robust principal component analysis.

The bivariate statistical analysis of environmental (compositional) data

Environmental sciences usually deal with compositional (closed) data. Whenever the concentration of chemical elements is measured, the data will be closed, i.e. the relevant information is contained in the ratios between the variables rather than in the data values reported for the variables. Data closure has severe consequences for statistical data analysis. Most classical statistical methods are based on the usual Euclidean geometry — compositional data, however, do not plot into Euclidean space because they have their own geometry which is not linear but curved in the Euclidean sense. This has severe consequences for bivariate statistical analysis: correlation coefficients computed in the traditional way are likely to be misleading, and the information contained in scatterplots must be used and interpreted differently from sets of non-compositional data. As a solution, the ilr transformation applied to a variable pair can be used to display the relationship and to compute a measure of stability. This paper discusses how this measure is related to the usual correlation coefficient and how it can be used and interpreted. Moreover, recommendations are provided for how the scatterplot can still be used, and which alternatives exist for displaying the relationship between two variables.

Difference correlation can distinguish deterministic chaos from 1/fα-type colored noise

Distinguishing deterministic chaos from colored noise with the power-law spectra or random fractal sequences (fractional Brownian motions) is one of the important problems in chaotic time series analysis. In this paper, we describe a simple method for solving this problem, which seems easier than the other algorithms that have already been proposed. In order to show how well our procedure works, first we apply a nonlinear prediction to time series data, produced from both nonlinear dynamical systems and stochastic systems with the power-law spectra. Next, we evaluate the prediction performance by calculating two kinds of correlation coefficients between actual time series and predicted time series, which are called a conventional correlation coefficient and a difference correlation coefficient. The conventional correlation coefficient is a usual correlation coefficient between actual time series and predicted time series, and the difference correlation coefficient is between first-difference time series obtained from actual time series and predicted time series. When the one-step-ahead nonlinear prediction is applied to deterministic chaos without observational noise, not only conventional correlation coefficients but also difference correlation coefficients are very high values, namely, the coefficients take values almost 1.0 even if the number of data points is small. On the other hand, in the case of 1/${\mathrm{f}}^{\mathrm{\ensuremath{\alpha}}}$-type colored noise, although conventional correlation coefficients are relatively high values, difference correlation coefficients turn out to be low values, even though the scaling exponent of the power spectrum \ensuremath{\alpha} is large. This difference between conventional correlation and difference correlation can be a good criterion for distinguishing deterministic chaos from colored noise with the power-law spectra. Finally, several real time series data are analyzed in order to confirm the applicability of the proposed method.

The percentage bend correlation coefficient

A well-known result is that the usual correlation coefficient,ρ, is highly nonrobust: very slight changes in only one of the marginal distributions can alterρ by a substantial amount. There are a variety of methods for correcting this problem. This paper identifies one particular method which is useful in psychometrics and provides a simple test for independence. It is not recommended that the new test replace the usual test ofH 0:ρ = 0, but the new test has important advantages over the usual test in terms of both Type I errors and power.

Estimating Winsorized correlations in a univariate or bivariate random effects model

A well‐known result is that slight departures from normality can have a large effect on the usual correlation coefficient rendering the magnitude of the correlation difficult to interpret and potentially misleading. In the context of a random effects model, which is the focus of attention in this paper, this means that effect size, as measured by the intraclass correlation, might be small due to outliers or heavy‐tailed distributions rather than a lack of differences among the groups being compared. Similarly, a large intraclass correlation might be due to trivial shifts away from normality which would become small if an adjustment for non‐normality were made. Moreover, this problem has to do with the effects of non‐normality on population parameters, not just statistics, so problems can arise even with large sample sizes. This follows almost immediately from results in Tukey (1960), and it is briefly illustrated here. One approach to this problem is to use a Winsorized analogue of the intraclass correlation. This paper suggests three ways the Winsorized intraclass correlation might be estimated and compares them via simulations. A bivariate generalization of the random effects model is also considered, and two methods of estimating the group‐level correlation are described and compared. Alternatives to Winsorization are also discussed.

Some results on a Winsorized correlation coefficient

Recent investigations indicate that psychometric measures often have distributions with very heavy tails, and that outliers are quite common. It has long been known that in terms of both power and Type I errors, even slight departures from normality can have serious consequences, and it is fairly evident that there are problems when using conventional measures of effect size, as is briefly illustrated in this paper. Similar problems arise when dealing with the usual correlation coefficient. Robust correlation coefficients have already been proposed that reflect the linear relationship between two random variables, but many have the unfortunate property of not always being equal to 0 under independence. One exception is the Winsorized correlation. The primary goal in this paper is to suggest a simple method for testing the hypothesis that the Winsorized correlation is equal to zero. A minor goal is to describe a formal definition of the Winsorized correlation coefficient in terms of Winsorized expected values.

Correlation Curves: Measures of Association as Functions of Covariate Values

For experiments where the strength of association between a response variable $Y$ and a covariate $X$ is different over different regions of values for the covariate $X$, we propose local nonparametric dependence functions which measure the strength of association between $Y$ and $X$ as a function of $X = x$. Our dependence functions are extensions of Galton's idea of strength of co-relation from the bivariate normal case to the nonparametric case. In particular, a dependence function is obtained by expressing the usual Galton-Pearson correlation coefficient in terms of the regression line slope $\beta$ and the residual variance $\sigma^2$ and then replacing $\beta$ and $\sigma^2$ by a nonparametric regression slope $\beta(x)$ and a nonparametric residual variance $\sigma^2(x) = \operatorname{var}(Y \mid x)$, respectively. Our local dependence functions are standardized nonparametric regression curves which provide universal scale-free measures of the strength of the relationship between variables in nonlinear models. They share most of the properties of the correlation coefficient and they reduce to the usual correlation coefficient in the bivariate normal case. For this reason we call them correlation curves. We show that, in a certain sense, they quantify Lehmann's notion of regression dependence. Finally, the correlation curve concept is illustrated using data from a study of the relationship between cholesterol levels $x$ and triglyceride concentrations $y$ of heart patients.

Information gain and a general measure of correlation

SUMMARY Given a parametric model of dependence between two random quantities, X and Y, the notion of information gain can be used to define a measure of correlation. This definition of correlation generalizes both the usual product-moment correlation coeffi- cient for the bivariate normal model and the multiple correlation coefficient in the standard linear regression model. The use of this information-based correlation in a descriptive statistical analysis is examined and several examples are given. If the dependence between two random quantities, X and Y, is modelled parametri- cally, then the concept of information gain can be used to define a measure of correlation. This correlation coefficient can appear in two possible contexts, depending on whether one models the joint distribution of X and Y, or just the conditional distribution of Y given X. An important motivating feature for this information-based correlation coefficient is the fact that it generalizes both the usual product-moment correlation coefficient for the bivariate normal model and the usual multiple correlation coefficient for the standard multiple regression model with normal errors. Our intuition is well developed for these usual correlation coefficients, and hopefully our intuition will still be applicable for-the information-based correlation in more general modelling situations of parametric dependence. Further, since our correlation coefficient is based on information gain, we might hope to extend our intuition to interpret the information gain in any statistical modelling situation where we want to assess how much better a more complicated model is than a simpler model. The concept of information gain for general statistical models is described in ? 2. This concept is then used to define an information-based measure of correlation; the joint case is covered in ? 3 and the conditional case in ? 4. Estimation of the correlation coefficient is carried out by estimating the corresponding information gain; see ?? 5-7. The use of information gain for the purpose of model choice in a descriptive statistical analysis is discussed in ? 8 and a comparison between our approach and Akaike's information criterion is given in ?9. Some examples of the use of this information-based correlation are given in ?? 10 and 11.

Inequalities for Distributions with Given Marginals

An ordering on discrete bivariate distributions formalizing the notion of concordance is defined and shown to be equivalent to stochastic ordering of distribution functions with identical marginals. Furthermore, for this ordering, $\int\varphi dH$ is shown to be $H$-monotone for all superadditive functions $\varphi$, generalizing earlier results of Hoeffding, Frechet, Lehmann and others. The usual correlation coefficient, Kendall's $\tau$ and Spearman's $\rho$ are shown to be monotone functions of $H$. That $\int\varphi dH$ is $H$-monotone holds for distributions on $\mathbb{R}^n$ with fixed $(n - 1)$-dimensional marginals for any $\varphi$ with nonnegative finite differences of order $n$. Some related results are obtained. Stochastic ordering is preserved under certain transformations, e.g., convolutions. A distribution on $\mathbb{R}^\infty$ is constructed, making $\max(X_1,\cdots, X_n)$ stochastically largest for all $n$ when $X_i$ have given one-dimensional distributions, generalizing a result of Robbins. Finally an ordering for doubly stochastic matrices is proposed.

Asymmetric measures of association, closed data, and multivariate analysis

The association between constant-sum variables Xiand Xjexpressed as percentages can be calculated as a product-moment correlation between Xiand Xj/(100 − Xi) and a correlation between Xjand Xi/(100 − Xj). An asymmetric, square matrix may be formed from these coefficients, and multivariate analysis performed by two methods: singular value decomposition and canonical decomposition. Either analysis avoids problems in the interpretation of correlation coefficients determined from closed arrays, and provides information about dependencies among the variables beyond that obtained from the usual correlation coefficient between Xiand Xj.Two examples show the canonical decomposition to have the greater usefulness.

On the yielding efficiency of Japanese rice varieties. : Stem length.

1. As one of the series of the writer's reports concerning the yielding efficiency of Japanese rice varieties, the data on the length of stem of paddy rice plants in Japan are reported. 2. In all the agronomic varieties of the paddy rice of Japan studied, the minimum of stem length is 44 cm, and thg maximum 140 cm, the mode being 92 cm, the mean 91.21 cm, and the σ 10.49 cm. The distribution is a very typical monomodal symmetric curve. 3. The distribution curve of the stem length of rice varieties in each prefecture presents a typical monomodal symmetric one. The minimum of the mean is 80 cm in Akita, 81 cm in Aomori, 82 cm in Fukusima, etc. The maximum is 100 cm in Gumma, 98 om in Nara, 96 cm in Yamagata, etc. 4. A close negative correlations were found between the stem length and number of stems per tubo, the days from seeding to heading, and the latitude coefficient etc., while a close positive correlations were made clear betwen the stem length and the yield per tan, prefectural average yield per tan, and the mean temperature, etc. 5. From 1931 to 1940, a diminution of the stem length occurred according to the straightline drawn by means of R=-0.6114y+97.49 cm. 6. The positive yearly correlations were found in each prefectures between the stem length and yield per tan, - the mean temperature from April to November, - the mean temperature of July. Negative correlation were made clear betwen the stem length and the heading date, the number of stems, the amount of rain fall from April to November, and the sun shine from April to November. 7. The number of stems shows a closer correlation with the yield, than that between the length of stems with the same, if considered by the usual correlation coefficient. But the stem length shows a closer correlation with the yield, when considered from the least square equation of the yearly tendency. 8. Sembon-asahi and Norin no.1, showing the mean stem lengths 73 cm and 79 cm respectively, are the representative short stem varieties, while Watarifune and Kokuryomiyako, the mean stem length of which are 102 cm and 104 cm respectively, are the standard long stem varieties. 9. By the deviation from the mean method which I proposed in order to present the relative of a given variety compared to the mean of the population of the varieties concerned, the deviation from the mean value of each variety is calculated. The values are as follows. Sembon-asahi - 17.0 cm, Norin no.1 -13.3 cm etc. (shorter varieties) Watarifune +7.0 cm, Kokuryomiyako +12.3 cm etc. (longer varieties). The tendency deviation from the mean method coincide with the tendency of absolute numbers. 10. We can calculate the theoretical length of the stem of any variety at any locality by adding or substituting the deviation from the mean value from the local stem length. 11. Among the varieties of rice plants, the correlations betwen the stem length and the yield per tan, the number of head per plant, and the number of head per tubo are negative and significant, while the correlation between the absolute stem length and the deviation from the mean value of the stem length is positive and distinct.

On the partial correlation ratio

(1) In a paper communicated to the Royal Society in 1903 I gave very briefly in a footnote the properties of the correlation ratio . These properties were discussed more at length in my memoir, “On the General Theory of Skew Correlation and Non-linear Regression,” published in 1905. The two papers dealt only with the total correlation ratio , or the relation between two variates without consideration of any other correlated variates. The introduction of the correlation ratio enabled the measure of the relationship between two variates to be expressed by a single number, measuring its total intensity, in cases where the regression line was of any form. The ratio passed into the usual correlation coefficient when the regression line became straight. This correlation ratio has been generally accepted by statisticians as a useful measure of relationship in cases of skew correlation and non-linear regression. Shortly after the appearance of the above memoirs I generalised this coefficient in a manner comparable with the generalisation of the coefficient of correlation, namely, by the definitions of the multiple correlation ratio and of the partial correlation ratio . These ratios correspond to the multiple correlation coefficient and the partial correlation coefficient in multiple linear regression. Their importance is very considerable, as they enable us to measure the intensity of association between two variates when other correlated variates are considered as constant without any assumption that the regression is linear, still less that the frequencies follow the normal (or Laplace-Gaussian) surface. I had not intended to discuss the results of the present paper before the probable errors had been provided, but the recent revival of interest in skew regression, and its fundamental importance in all higher statistical inquiry, justifies, at least, the publication of those formulæ which are fundamental to the subject. (2) I deal first with the problem of three variates, although the extension to any number is not hard to make.