Interpretation problems of the partial correlation with nonnormally distributed variables

In manufacturing processes, a vector of multiple responses is often monitored to assess if the process is in-control. A multivariate Shewhart control chart is one method of monitoring the mean vector. If the chi-square statistic exceeds an upper control limit (UCL), then an out-of-control signal occurs. The average run length (ARL) is used to determine the UCL value. ARLs have been estimated under an assumption of a multivariate normal (MVN) assumption. This research explores the sensitivity of ARLs when the MVN assumption is incorrect. ARLs for data from Multivariate t, lognormal, uniform, and beta distributions are estimated and compared to ARLs under the MVN assumption.

Monitoring and improving manufacturing processes involves identifying, investigating and eliminating problems responsible for inefficiencies in production operations. While statistical process control tools, such as control charts, are available for process monitoring at the operational level, methods for evaluating system performance from more strategic and tactical levels are limited. The traditional control charts that monitor a single process parameter at a time may not be appropriate in situations where interrelationships among various system measures exist. Although multivariate process control techniques allow for simultaneous monitoring of several process parameters, they require assumptions of independence and multivariate normality of data. In addition, their application has mostly been at an operational level. In order to assist managers in monitoring and improving manufacturing system performance, this paper proposes an individual control chart that monitors an integrated performance index generated from a non-parametric method, which effectively considers multiple performance measures and the relationships between them. The primary advantages of this method are that a single integrated measure can be monitored, does not require assumptions of independence and multivariate normality of data, and allows for the integration of decision-maker's input when the system measures that are monitored have unequal importance.

Abstract. We investigated the influence of meteorological variability on the interannual variation of springtime boundary layer ozone over Japan during 1981–2005 by multiyear simulations with the Models-3 Community Multiscale Air Quality (CMAQ) modeling system and the Regional Emission Inventory in Asia (REAS). CMAQ/REAS generally reproduced the observed interannual variability of springtime ozone over Japan, showing year-to-year variations larger than the annual rate of increase of the long-term trend. We then analyzed the influence of the interannual variation of meteorology in simulated results by using the fixed emissions for 2000 and meteorological fields for each year. As a reference parameter, we calculated the area-weighted surface pressure anomaly over the Pacific Ocean east of Japan. When the anomaly has a large negative value, polluted air masses from continental Asia tend to be transported directly to Japan by westerly winds. In contrast, when the anomaly has a large positive value, influence of the outflow from continental Asia tends to be small because the westerly components of wind fields around Japan are comparatively weak. Instead, southerly winds are relatively strong and transport clean air masses from the Pacific Ocean to Japan. Consequently, springtime ozone over Japan is higher (lower) than in ordinary years when the anomaly has a large negative (positive) value. In general, the interannual variation of springtime ozone over Japan is sensitive to the outflow from continental Asia. We also found some correlation between springtime ozone over Japan and the El Niño-Southern Oscillation, indicating that higher and lower springtime ozone over Japan are related to La Niña and El Niño, respectively. Differences in the meridional displacement and diversity of cyclone tracks near Japan between El Niño and La Niña years may be responsible for interannual variations in the springtime boundary layer ozone over Japan.

This article presents the determination of the weights of socio-economic factors affecting the volume of passenger transport on individual railway lines located in the Łodź region. Passenger transport on these lines is operated by the carrier Łodź Agglomeration Railway. The transport data containing the number of transported passengers and the transport work done in 2017 were used for the analysis. A partial correlation coefficient was used to obtain values of weights of the analyzed factors. During the calculation of correlation coefficients, transport gauges were used as explanatory variables (dependent variables). The socio-economic factors influencing the volume of rail passenger transport on individual lines were taken into account. To calculate the partial correlation coefficient, a correlation matrix, containing interrelations between all analyzed variables, was used. During calculating the weights of particular factors affecting the volume of rail passenger transport on individual lines located in the Łodź region for each socio-economic factor, the absolute value of partial correlation coefficients was summed for the number of trains and the number of transported passengers. Then the summed value was divided by the number of analyzed gauges describing the volume of transport. It allowed to determine the weights of individual factors affecting the volume of passenger transport on railway lines in the Łodź region. The obtained weights can be used in the future during the multi-criteria analysis of railway lines in the Łodź region. These weights allow to avoid the subjective assessments of experts, that have been often used in previous analyzes.

Most traditional multivariate control charts for monitoring multiple process quality variables are designed and evaluated under the assumption that the process data follow a multivariate normal distribution with known covariance matrix. However, in many practical situations, this assumption of multivariate normality may not hold or may be difficult to verify. Recently, nonparametric Shewhart-type control charts that are based on the notion of simplicial data depth have been proposed for monitoring the locations and scales of multiple process variables. Although these Shewhart charts have potential for addressing the hard problem of multivariate nonparametric quality control, there exist some important unresolved issues with these charts. This paper investigates the effect of the size of the reference sample and the size of the subgroups on the statistical properties of Shewhart-type control charts that are based on simplicial data depth. The sizes of the reference sample and subgroups are the primary factors which determine whether these charts can be designed to achieve sufficiently large in-control average run lengths forKeywordsControl ChartExponentially Weighted Moving AverageStatistical Process ControlQuality TechnologyFederal Aviation AdministrationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Analysis of healthcare utilization, such as hospitalization duration and medical costs, is crucial for policymakers and doctors in experimental and epidemiological investigations. Herein, we examine the healthcare utilization data of patients with systemic lupus erythematosus (SLE). The characteristics of the SLE data were measured over a 10-year period with outliers. Multivariate linear models with multivariate normal error distributions are commonly used to evaluate long series of multivariate longitudinal data. However, when there are outliers or heavy tails in the data, such as those based on healthcare utilization, the assumption of multivariate normality may be too strong, resulting in biased estimates. To address this, we propose multivariate t-linear models (MTLMs) with an autoregressive moving-average (ARMA) covariance matrix. Modeling the covariance matrix for multivariate longitudinal data is difficult since the covariance matrix is high dimensional and must be positive-definite. To address these, we employ a modified ARMA Cholesky decomposition and hypersphere decomposition. Several simulation studies are conducted to demonstrate the performance, robustness, and flexibility of the proposed models. The proposed MTLMs with ARMA structured covariance matrix are applied to analyze the healthcare utilization data of patients with SLE.

It is difficult to detect interactions between continuous variables in field research using moderated multiple regression (MMR). One reason is that multivariate normality, which occurs in field research but not experimental research, suppresses the residual variance of interaction terms. SPSS, SAS, and Matlab programs that expose the extent of this problem for any given data set are provided. Ironically, when significant interactions are found in field research using MMR, this is because the assumption of multivariate normality has been violated, thus rendering the significance tests potentially invalid. The programs presented in this article compute more trustworthy significance levels via data randomization.

Consider a portfolio consisting of quantities x = (x 1, x 2, ⋯, x n )′ of assets 1,2,⋯, n with time t values v = (v 1, v 2, ⋯ v n )′.2 Then the change in the price of the portfolio, V, over the next interval Δt is given by $$ \Delta v = \sum\limits_{t = 1}^n {x_t \Delta v_t ,} $$ (1.1) where Δ, (Δ V) denotes the change in the value of asset (the portfolio) over the interval t to t + Δt. The value-at-risk of the portfolio {bdx} for some defined probability level α, is defined as the level of loss, ΔV(α), such that the probability that ΔV ≤ ΔV is equal to α. When the joint distribution of the change in asset values can be taken as multivariate normal with a known mean and variance, the calculation of Δ V is straightforward. In many cases, however, and particularly when some of the assets are options, the assumption of multivariate normality will be inappropriate, even when appropriate for the underlying rates and prices.

This paper deals with an analysis of covariance model for analyzing monotone samples from a factorial design with repeated measurements under the assumption of multivariate normality with an ante-dependence structure. The likelihood-based procedures are valid when the missing data are missing either at random or completely at random (Rubin, 1976). The likelihood ratio test of a general linear hypothesis on location parameters is derived. The adjusted means of the location parameters along with their dispersion matrix are also derived. A numerical example is given to illustrate the procedures. The factorization property of the likelihood is also used in estimating intermediate missing values, if any.

Even though data sets in psychology are seldom normal, the statistics used to evaluate covariance structure models are typically based on the assumption of multivariate normality. Consequently, many conclusions based on normal theory methods are suspect. In this paper, we develop test statistics that can be correctly applied to the normal theory maximum likelihood estimator. We propose three new asymptotically distribution-free (ADF) test statistics that technically must yield improved behaviour in samples of realistic size, and use Monte Carlo methods to study their actual finite sample behaviour. Results indicate that there exists an ADF test statistic that also performs quite well in finite sample situations. Our analysis shows that various forms of ADF test statistics are sensitive to model degrees of freedom rather than to model complexity. A new index is proposed for evaluating whether a rescaled statistic will be robust. Recommendations are given regarding the application of each test statistic.

For multivariate analysis with p variables the problem that often arises is the ambiguous nature of the correlation or covariance matrix. When p is moderately or very large it is generally difficult to identify the true nature of relationship among the variables as well as observations from the covariance or correlation matrix. Under such situations a very common way to simplify the matter is to reduce the dimension by considering only those variables (actual or derived) which are truly responsible for the overall variation. Important and useful dimension reduction techniques are Principal Component Analysis (PCA), Factor Analysis, Multidimensional Scaling, Independent Component Analysis (ICA), etc. Among them PCA is the most popular one. One may look at this method in three different ways. It may be considered as a method of transforming correlated variables into uncorrelated one or a method of finding linear combinations with relatively small or large variability or a tool for data reduction. The third criterion is more data oriented. In PCA primarily it is not necessary to make any assumption regarding the underlying multivariate distribution but if we are interested in some inference problems related to PCA then assumption of multivariate normality is necessary. The eigen values and eigen vectors of the covariance or correlation matrix are the main contributors of a PCA. The eigen vectors determine the directions of maximum variability whereas the eigen values specify the variances. In practice, decisions regarding the quality of the principal component approximation should be made on the basis of eigen value–eigen vector pairs.

Approximations to the distributions of goodness-of-fit indexes in structural equation modeling are derived with the assumption of multivariate normality and slight misspecification of models. The fit indexes considered in this article are Joreskog and Sorbom's goodness-of-fit index (GFI) and the adjusted GFI, McDonald's absolute GFI, Steiger and Lind's root mean squared error of approximation, Steiger's Γ1 and Γ2, Bentler and Bonett's normed fit index, Bollen's incremental fit index and ρ1, Tucker and Lewis's index ρ2, and Bentler's fit index (McDonald and Marsh's relative noncentrality index). An approximation to the asymptotic covariance matrix for the fit indexes is derived by using the delta method. Furthermore, approximations to the densities of the fit indexes are obtained from the transformations of the asymptotically noncentral chi-square distributed variable. A simulation is carried out to confirm the accuracy of the approximations.

When the assumption of multivariate normality is violated or when a discrepancy function other than (normal theory) maximum likelihood is used in structural equation models, the null distribution of the test statistic may not be χ2 distributed. Most existing methods to approximate this distribution only match up to 2 moments. In this article, we propose 2 additional approximation methods: a scaled F distribution that matches 3 moments simultaneously and a direct Monte Carlo–based weighted sum of i.i.d. χ2 variates. We also conduct comprehensive simulation studies to compare the new and existing methods for both maximum likelihood and nonmaximum likelihood discrepancy functions and to separately evaluate the effect of sampling uncertainty in the estimated weights of the weighted sum on the performance of the approximation methods.

Wiggle matching is an important and powerful technique in radiocarbon dating that can be used to improve the precision of calendar age estimates. All radiocarbon determinations require calibration to provide calendar age estimates. This calibration is achieved by comparing the determinations against a calibration curve $\mu ( \cdot )$ to calculate the probability the sample arises from any particular calendar age t. Wiggle matching involves the calibration of a set of radiocarbon determinations taken from samples with known separations between their calendar ages. Since the calendar age separations between samples are known, all the calendar ages are known functions of one particular age, ${T_1}$ — commonly the most recent calendar age. Dating the sequence then reduces to considering $p({T_1} = {t_1}|data)$, the probability of the calendar age ${t_1}$ given the set of radiocarbon determinations. In previous work, a Bayesian approach has been used to derive a nice formula for this quantity under the assumption we have independent pointwise estimates of the calibration curve $\mu (t)$. In this paper, we derive a generalization of this formula showing how to incorporate covariance information from the calibration curve under an assumption of multivariate normality.

Estimation of intraclass correlation when family sizes are unequal