F Tests for Mean and Covariance Structure Analysis

The asymptotically distribution free (ADF) method is often used to estimate parameters or test models without a normal distribution assumption on variables, both in covariance structure analysis and in correlation structure analysis. However, little has been done to study the differences in behaviors of the ADF method in covariance versus correlation structure analysis. The behaviors of 3 test statistics frequently used to evaluate structural equation models with nonnormally distributed variables, χ2 test TAGLS and its small-sample variants TYB and TF(AGLS) were compared. Results showed that the ADF method in correlation structure analysis with test statistic TAGLS performs much better at small sample sizes than the corresponding test for covariance structures. In contrast, test statistics TYB and TF(AGLS) under the same conditions generally perform better with covariance structures than with correlation structures. It is proposed that excessively large and variable condition numbers of weight matrices are a cause of poor behavior of ADF test statistics in small samples, and results showed that these condition numbers are systematically increased with substantial increase in variance as sample size decreases. Implications for research and practice are discussed.

In this paper, we derive the asymptotic properties of estimators obtained from various kinds of loss functions in covariance structure analysis. We first show that the estimators except for OLS-based loss functions have the same asymptotic distribution when the dimension of the covariance matrix, , is fixed and the sample size tends to infinity. Then, focusing on the spherical model, we show that this equivalence does not hold when both and become larger. Specifically, we show that some estimators lose consistency, and even consistent estimators have different asymptotic variances. Among the estimators considered, the maximum likelihood estimator shows the best performance, while the less famous invGLS(ub) estimator performs better than the commonly used GLS estimator. We also demonstrate the validity of the likelihood ratio test for the spherical and diagonal models in a high-dimensional framework.

Covariance structure analysis is used to evaluate hypothesized influences among unmeasured latent and observed variables. As implemented, it is not robust to outliers and bad data. Several robust methods in model fitting and testing are proposed. These include direct estimation of M-estimators of structured parameters and a two-stage procedure based on robust M- and S-estimators of population covariances. The large sample properties of these estimators are obtained. The equivalence between a direct M-estimator and a two-stage estimator based on an M-estimator of population covariance is established when sampling from an elliptical distribution. Two test statistics are presented in judging the adequacy of a hypothesized model; both are asymptotically distribution free if using distribution free weight matrices. So these test statistics possess both finite sample and large sample robustness. The two-stage procedures can be easily adapted into standard software packages by modifying existing asymptotically distribution free procedures. To demonstrate the two-stage procedure, S-estimator and M-estimators under different weight functions are calculated for some real data sets.

In a variety of statistical problems the estimate Θn of a parameter Θ is defined as the root of a generalized estimating equation Gn(Θnγn)=0 where γn is an estimate of a nuisance parameter γ. We give sufficient conditions for the asymptotic normality of #x0398;n defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of #x0398;n will not be influenced by that of γn) is noted. As an example, we consider a covariance structure analysis in which both the population mean and the population fourth-order moment are nuisance parameters. Applications to pseudo maximum likelihood, generalized least squares with estimated weights, and M-estimation with an estimated scale parameter are discussed briefly.

Research studying robustness of maximum likelihood (ML) statistics in covariance structure analysis has concluded that test statistics and standard errors are biased under severe non-normality. An estimation procedure known as asymptotic distribution free (ADF), making no distributional assumption, has been suggested to avoid these biases. Corrections to the normal theory statistics to yield more adequate performance have also been proposed. This study compares the performance of a scaled test statistic and robust standard errors for two models under several non-normal conditions and also compares these with the results from ML and ADF methods. Both ML and ADF test statistics performed rather well in one model and considerably worse in the other. In general, the scaled test statistic seemed to behave better than the ML test statistic and the ADF statistic performed the worst. The robust and ADF standard errors yielded more appropriate estimates of sampling variability than the ML standard errors, which were usually downward biased, in both models under most of the non-normal conditions. ML test statistics and standard errors were found to be quite robust to the violation of the normality assumption when data had either symmetric and platykurtic distributions, or non-symmetric and zero kurtotic distributions.

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.

The corneal back surface is known to add some against the rule astigmatism, with implications in cataract surgery with toric lens implantation. This study aimed to set up and validate a deep learning algorithm to predict corneal back surface power from the corneal front surface power and biometric measures. This study was based on a large dataset of IOLMaster 700measurements from two clinical centres. N=19,553measurements of 19,553 eyes with valid corneal front (CFSPM) and back surface power (CBSPM) data and other biometric measures. After a vector decomposition of CFSPM and CBSPM into equivalent power and projections of astigmatism to the 0°/90° and 45°/135° axes, a multi-output feedforward neural network was derived to predict vector components of CBSPM from CFSPM and other measurements. The predictions were compared with a multivariate linear regression model based on CFSPM components only. After pre-conditioning, a network with two hidden layers each having 12 neurons was derived. The dataset was split into training (70%), validation (15%) and test (15%) subsets. The prediction error (predicted corneal back surface power CBSPP - CBSPM) of the network after training and crossvalidation showed no systematic offset, narrower distributions for CBSPP - CBSPM and no trend error of CBSPP - CBSPM vs. CBSPM for any of the vector components. The multivariate linear model also showed no systematic offset, but broader distributions of the prediction error components and a systematic trend of all vector components vs. CFSPM components. The neural network approach based on CFSPM vector components and other biometric measures outperforms the multivariate linear model in predicting corneal back surface power vector components. Modern biometers can supply all parameters required for this algorithm, enabling reliable predictions for corneal back surface data where direct corneal back surface data are unavailable.

Large sample inference for a multivariate linear model with autocorrelated errors

Reflectance data and relative chlorophyll content for Phyllostachys violascens at leaf scale were measured during the growth period from April, 5th to June, 18th using a portable Analytical Spectral Devices （ASD） field spectrometer and a hand-held Chlorophyll Content Meter （CCM）-200. Correlation analyses were conducted between hyper-spectral vegetation indices and chlorophyll content based on the data. Then individual univariate linear inversion models were developed for chlorophyll content and hyper-spectral vegetation indices, such as red edge indexes GM, Vog3, double difference index DD, modified normalized differential vegetation index mND705, modified simple ratio mSR705, and Red-edge positions （REP）. Also multivariate linear models for selected hyper-spectral vegetation indices and chlorophyll content were tested. Multivariate linear models are designed in two methods, strategy A is based on the 20 Phyllostachys violascens samples, and each data for the sample is the average for all the 14 times. On the contrary, strategy B is based on the data of 14 times, which average the 20 samples for each time. Results over the entire growth period showed （1） significant （P＜0.01） correlations between chlorophyll content and hyper-spectral vegetation indices, GM （r = 0.866 3）, Vog3 （r = 0.927 4 ）, DD （r = 0.880 6）, mND705 （r = 0.917 9）, mSR705 （r = 0.924 9）, and REP （r =0.895 4）. At the end of the growth period, all vegetation indices had a favorable relationship with chlorophyll content, showing as the high correlation coefficients, although some indexes perform bad in most other time periods；（2） Using the univariate linear model, correlation for hyper-spectral vegetation indices and chlorophyll content showed r ＞ 0.85. The multivariate linear models of the six hyper-spectral vegetation indices listed above and chlorophyll content using two strategies, both accurately predicted chlorophyll content of Phyllostachys violascens ［with correlation coefficients between predicted values and measured values that were all above r = 0.89］. The multivariate linear models can be used to predicte chlorophyll content in the leaf of Phyllostachys violascens. Considering the calculate method, strategy B is more fit for the dynamic change of chlorophyll content for Phyllostachys violascens at leaf scale. ［Ch, 4 fig. 2 tab. 45 ref.］

Dosimetric Predictive Models for Urinary and Rectal Toxicities after Prostate Stereotactic Body Radiotherapy Assessed from the Expanded Prostate Cancer Index Composite Patient-reported Quality of Life

This note outlines the development of a multivariate linear regression model to predict peak lightning activity a year in advance. Using cloud-to-ground lightning strike data from the inception of the National Lightning Detection Network in 1985, a predictive model is proposed for the peak monthly strike total for the Pacific Northwest. Six climate indices were evaluated as potential predictors of the lightning. From these, a multivariate linear model was developed, based on the year-in-advance values of two of the indices—The El Nino-Southern Oscillation Index, and the Western North Pacific Monsoon Index. The 2007 prediction of 56,618 strikes was tested against the measured value of 38,591 strikes. The values are in agreement within the expected accuracy of the model. The model prediction represents an improvement relative to the expectation based on the past statistical history of the data and should be most useful in predicting extreme events. The model parameters have been updated to include the 2007 ...

Iron metabolism plays an important role in insulin resistance, and the triglyceride-glucose (TyG) index has been proposed in recent years as a more accessible and cost-effective marker for insulin resistance. This study aims to evaluate the association between iron metabolism markers, including ferritin (FER), transferrin (TRF), and transferrin receptor (TFR), and the TyG index. A total of 6524 Chinese individuals aged between 18 and 75 years were included in this study. Multivariable linear models were used to investigate the association between FER, TRF, and TFR levels, and the TyG index. Further subgroup analyses stratified by age and sex were also performed. There was a positive association between FER and TRF levels and the TyG index in all 3 multivariable linear regression models, regardless of stratification by sex and age. Additionally, TFR was positively associated with the TyG index among females and those aged ≥45 years, but not among males and those aged <45 years. Our findings reveal a positive association between FER and TRF levels and the TyG index in a Chinese population, while the association between TFR levels and the TyG index showed different patterns depending on age and gender.

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.

For the testing problem concerning the coefficients of the multivariate linear functional relationship model, the distribution of a statistic previously proposed by A. P. Basu depends on the unknown covariance matrixV of errors, so limiting its applicability. This article proposes new test statistics with sampling distributions which are independent of the unknown parameters for the cases whereV is either unknown or known only up to a proportionality factor. The exact distributions of the test statistics are also discussed.

The asymptotic distributions of statistics are often used to construct finite sample approximations for their distributions. However statistics which arise in different contexts but are equivalent in law can have different asymptotic distributions. We consider Moran's sum of logarithms of uniform spacings statistic which is equal in law to a special case of Bartlett's test statistic for homoscedasticity. Moran's statistic is asymptotically normal and, by accelerating the convergence, we obtain an accurate approximation for small samples. Also, contrary to expectation, we find that the asymptotic chi-square distribution associated with the homoscedasticity test provides as good a basis for approximating the distribution of Moran's statistic. A closer look at likely alternatives leads to a scaled Beta as a more uniformly accurate approximation than either the normal or the chi-square.