The Relationship Between Systemic Lupus Erythematosus and Osteoporosis Based on Different Ethnic Groups: a Two-Sample Mendelian Randomization Analysis

The previous observational studies could not overcome the effects of confounding variables and reverse causality. We aimed to determine whether there is a causal relationship between systemic lupus erythematosus and osteoporosis in East Asian and European populations, respectively, by two-sample Mendelian Randomization analysis. We obtained and downloaded data from publicly available genome-wide association study databases and analyses for East Asian and European populations, including systemic lupus erythematosus (SLE), osteoporosis (OP), multisite bone mineral density (BMD), and OP with fracture. After screening for instrumental single-nucleotide polymorphisms (SNPs) significantly correlated to SLE, the inverse-variance weighted (IVW) method was used for calculating the ratio and 95% confidence interval, besides utilizing MR-Egger, weighted median, and weighted mode to assess the robustness of the primary outcome. Moreover, multiple analyses, including MR-PRESSO, MR-Egger intercept, Cochran's Q test, as well as "leave-one-out" sensitivity, were used for evaluating horizontal pleiotropy, heterogeneity, and stability. Finally, we exchanged exposure and outcome and performed a reverse MR analysis. IVW (OR = 1.05, 95% CI = 1.01-1.09, P = 0.009) indicated a significant positive correlation between genetically predicted SLE and OP in East Asians. Furthermore, neither heterogeneity nor horizontal pleiotropy was observed. In Europe, there was no significant genetically predicted causal relation between SLE and OP. Bi-directional MR analysis showed no reverse causality between SLE and OP. In the East Asian population, genetically predicted SLE may have had a positive causal relationship with OP. In Europe, there is insufficient evidence for a potential causal relation between SLE and OP or BMD and fracture, and the correlations currently observed may be attributed to a variety of confounder variables.

Effect of outliers on estimators and tests in covariance structure analysis

A small proportion of outliers can distort the results based on classical procedures in covariance structure analysis. We look at the quantitative effect of outliers on estimators and test statistics based on normal theory maximum likelihood and the asymptotically distribution-free procedures. Even if a proposed structure is correct for the majority of the data in a sample, a small proportion of outliers leads to biased estimators and significant test statistics. An especially unfortunate consequence is that the power to reject a model can be made arbitrarily--but misleadingly--large by inclusion of outliers in an analysis.

Robust transformation with applications to structural equation modelling

Data sets in social and behavioural sciences are seldom normal. Influential cases or outliers can lead to inappropriate solutions and problematic conclusions in structural equation modelling. By giving a proper weight to each case, the influence of outliers on a robust procedure can be minimized. We propose using a robust procedure as a transformation technique, generating a new data matrix that can be analysed by a variety of multivariate methods. Mardia's multivariate skewness and kurtosis statistics are used to measure the effect of the transformation in achieving approximate normality. Since the transformation makes the data approximately normal, applying a classical normal theory based procedure to the transformed data gives more efficient parameter estimates. Three procedures for parameter evaluation and model testing are discussed. Six examples illustrate the various aspects with the robust transformation.

Structural Equation Modeling with Small Samples: Test Statistics

Structural equation modeling is a well-known technique for studying relationships among multivariate data. In practice, high dimensional nonnormal data with small to medium sample sizes are very common, and large sample theory, on which almost all modeling statistics are based, cannot be invoked for model evaluation with test statistics. The most natural method for nonnormal data, the asymptotically distribution free procedure, is not defined when the sample size is less than the number of nonduplicated elements in the sample covariance. Since normal theory maximum likelihood estimation remains defined for intermediate to small sample size, it may be invoked but with the probable consequence of distorted performance in model evaluation. This article studies the small sample behavior of several test statistics that are based on maximum likelihood estimator, but are designed to perform better with nonnormal data. We aim to identify statistics that work reasonably well for a range of small sample sizes and distribution conditions. Monte Carlo results indicate that Yuan and Bentler's recently proposed F-statistic performs satisfactorily.

Normal theory based test statistics in structural equation modelling

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.

Robust mean and covariance structure analysis

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.

Test of linear trend in eigenvalues of a covariance matrix with application to data analysis

Principal component analysis and factor analysis are the most widely used tools for dimension reduction in data analysis. Both methods require some good criterion to judge the number of dimensions to be kept. The classical method focuses on testing the equality of eigenvalues. As real data hardly have this property, practitioners turn to some ad hoc criterion in judging the dimensionality of their data. One such popular method, the 'scree test' or 'scree plot' as described in many texts and statistical programs, is based on the trend in eigenvalues of sample covariance (correlation) matrix. The principal components or common factors corresponding to eigenvalues which exhibit a slow linear decrease are discarded in further data analysis. This paper develops a formal statistical test for the 'scree plot'. A special case of this test is the classical test for equality of eigenvalues which has been suggested in several texts as the criterion to decide the number of principal components to retain. Comparisons between equality of eigenvalues and the slow linear decrease in eigenvalues on some classical examples support the hypothesis of slow linear decrease. A physical background to such a phenomenon is also suggested.