Programs for Problems Created by Continuous Variable Distributions in Moderated Multiple Regression

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.

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.

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.

Multivariate normality assumption is often required when employing multivariate statistical models, mixed models, etc. The validity of this assumption is seldom examined due to the confusion of which test should be used. More than 50 tests of multivariate normality have been proposed, but relatively little is known about the performance of these procedures. Mecklin and Nundfrom (2005) indicated no single test is most powerful based on Monte Carlo simulation comparing the performance of 13 tests for 25 alternative distributions. Based on the union intersection principle, Malkovich and Afifi (1973) proposed a test for multivariate normality derived from the univariate normal test proposed by Shapiro and Wilk (1965). Since this test requires finding the extreme value from an infinite set, Malkovich and Afifi (1973) suggested finding an approximate value of the extreme value based on the set of boundaries. However, the construction of this test depends upon the dimension and sample size of the data. This paper implements the Box Muller transformation to find another approximation. The proposed computative procedure is found to maintain the nominal level and achieve higher power as compared to old computative tests.

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.

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.

This study investigates the impact of renewable energy integration on energy efficiency within the China-Pakistan Economic Corridor (CPEC) from 2000 to 2022. Efficient renewable energy utilization is crucial for the CPEC’s sustainable development. We employed multiple linear regression analysis on data from national energy statistics databases to examine the relationship between energy efficiency defined as the ratio of total renewable energy output to input in (TWh) and four renewable energy sources: hydropower, biofuel, solar PV, and geothermal. Our model controlled for potential confounding factors and met assumptions of linearity, multivariate normality, and absence of multicollinearity. Hydropower exhibited a highly significant negative correlation with energy efficiency (-0.632, p < 0.001), with a regression coefficient of -7.642 × 10⁻⁴ (p < 0.001). Similarly, biofuel showed a significant negative correlation (-0.222, p < 0.001) and a coefficient of -9.580 × 10⁻⁴ (p < 0.001). These findings suggest that increased production from these sources is associated with decreased energy efficiency, potentially due to transmission losses or inefficiencies in conversion technologies. In contrast, solar PV (-0.027, p = 0.638) and geothermal (-0.014, p = 0.806) showed no statistically significant relationship with energy efficiency, indicating that at their current scales of deployment, they do not significantly impact overall efficiency. A moderate negative correlation (r=-0.114, p = 0.047) was observed between solar PV and geothermal production, possibly reflecting resource allocation choices within the renewable energy portfolio. These results highlight the need for targeted interventions to improve the efficiency of hydropower and biofuel production and emphasize the potential for future contributions from solar PV and geothermal as their deployment and grid integration expands. The findings contribute to a more nuanced understanding of the complex interplay between renewable energy generation and overall energy efficiency within large-scale development projects.

PurposeThe purpose of this paper is to show that multivariate t-distribution assumption provides a better description of stock return data than multivariate normality assumption.Design/methodology/approachThe EM algorithm is applied to solve the statistical estimation problem almost analytically, and the asymptotic theory is provided for inference.FindingsThe authors find that the multivariate normality assumption is almost always rejected by real stock return data, while the multivariate t-distribution assumption can often be adequate. Conclusions under normality vs under t can be drastically different for estimating expected returns and Jensen’s αs, and for testing asset pricing models.Practical implicationsThe results provide improved estimates of cost of capital and asset moment parameters that are useful for corporate project evaluation and portfolio management.Originality/valueThe authors proposed new procedures that makes it easy to use a multivariate t-distribution, which models well the data, as a simple and viable alternative in practice to examine the robustness of many existing results.

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.

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.

The partial correlation is a commonly used measure for assessing the bivariate correlation of two quantitative variables after eliminating the influence of one or more other variables. The partial correlation is generally interpreted as the correlation that would result if the variables to be eliminated were fixed (not allowed to vary and influence the other variables), which is referred to in the statistical literature as conditional correlation. The present paper demonstrates, by means of theoretical derivations and practical examples, that when the assumption of multivariate normality is violated (e.g., as a result of nonlinear relationships among the variables investigated) the usual interpretation of the partial correlation coefficient will be basically incorrect. In extreme cases the value of the partial correlation coefficient may be strongly positive, close to 1, whereas the conditional correlation may have a large negative value. To solve this problem the paper suggests to partial out a certain function (in most cases the square) of the variables whose effects are to be eliminated if nonlinear relationships are likely to occur.

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.