In confirmatory factor analysis CFAthe use of maximum likelihood ML assumes that the observed indicators follow a continuous and multivariate normal distribution, which is not appropriate for ordinal observed variables. Diagonally weighted least squares WLSMVon the other hand, is specifically designed for ordinal data.
Although WLSMV makes no distributional assumptions about the observed variables, a normal latent distribution underlying each observed categorical variable is instead assumed. A Monte Carlo simulation was carried out to compare the effects of different configurations of latent response distributions, numbers of categories, and sample sizes on model parameter estimates, standard errors, and chi-square test statistics in a correlated two-factor model.
In the social and behavioral sciences, researchers often employ Likert-type scale items to operationalize unobserved constructs e.
Confirmatory factor analysis CFA has been widely used as evidence of construct validity in theory-based instrument construction. A confirmatory factor-analytic model takes into account the differences between the true and observed scores by including pertinent error variances as model parameters in a structural equation modeling framework.
The most common method used to estimate parameters in CFA models is maximum likelihood MLbecause of its attractive statistical properties i. When this assumption is considered tenable, ML maximizes the likelihood of the observed data to obtain parameter estimates. However, ML is not, strictly speaking, appropriate for ordinal variables.
When the normality assumption is not deemed empirically tenable, the use of ML may not only reduce the precision and accuracy of the model parameter estimates, but may also result in misleading conclusions drawn from empirical data.
Robust ML has been widely introduced into CFA models when continuous observed variables slightly or moderately deviate from normality. WLSMV, on the other hand, is specifically designed for categorical observed data e. Although WLSMV makes no distributional assumptions about observed variables, a normal latent distribution underlying each observed categorical variable is instead assumed. As compared to ML estimation, a robust ML approach is less dependent on the assumption of multivariate normal distribution.
When the normality assumption about observed variables does not hold, and robust ML is implemented, parameter estimates are still obtained using the asymptotically unbiased ML estimator, but standard errors and chi-square test statistics are statistically corrected to enhance the robustness of ML against departures from normality in the forms of skewness, kurtosis, or both. The corrected standard error estimates are calculated by taking the square roots of the diagonal elements of the above estimated asymptotic covariance matrix.
The robust corrections applied to the chi-square statistic vary slightly across different current software programs. Note that a mean- and variance-adjusted chi-square statistic i.
Robust Maximum Likelihood Estimation
Weighted least squares is generally referred to as the asymptotically distribution-free estimator when data are continuous but nonnormal and a consistent estimate of the asymptotic covariance matrix of sample-based variances and covariances is used Browne, However, neither the assumption of normality nor the continuity property is clearly met by observed variables that are measured on an ordinal scale.
An estimated polychoric correlation captures the linear relationship between two normal, latent response variables. However, as the number of observed variables and response categories increases, the weight matrix grows rapidly in size. The diagonal weight matrix prevents software programs from engaging in extensive computations and encountering numerical problems in model estimation.
It is worth noting that the aim of statistical corrections to standard errors in WLSMV is to compensate for the loss of efficiency when the full weight matrix is not calculated, and the mean and variance adjustments for test statistics in WLSMV are targeted to make the shapes of the test statistics be approximately close to the reference chi-square distribution with the associated degrees of freedom.
Simulation studies have investigated the properties of different estimation methods, typically reporting on the relative performance e. A literature review of Monte Carlo simulation studies carrying out ordinal confirmatory factor-analytic models was conducted across several high-impact journals e. The empirical findings, using ML and WLS and their statistical corrections, can be briefly summarized below.Mplus provides both Bayesian and frequentist inference.
Posterior distributions can be monitored by trace and autocorrelation plots. Convergence can be monitored by the Gelman-Rubin potential scaling reduction using parallel computing in multiple MCMC chains. Posterior predictive checks are provided. Frequentist analysis uses maximum likelihood and weighted least squares estimators. Mplus provides maximum likelihood estimation for all models. With censored and categorical outcomes, an alternative weighted least squares estimator is also available.
For all types of outcomes, robust estimation of standard errors and robust chi-square tests of model fit are provided. These procedures take into account non-normality of outcomes and non-independence of observations due to cluster sampling.
Robust standard errors are computed using the sandwich estimator.
Robust chi-square tests of model fit are computed using mean and mean and variance adjustments as well as a likelihood-based approach. Bootstrap standard errors are available for most models. Linear and non-linear parameter constraints are allowed. With maximum likelihood estimation and categorical outcomes, models with continuous latent variables and missing data for dependent variables require numerical integration in the computations.
The numerical integration is carried out with or without adaptive quadrature in combination with rectangular integration, Gauss-Hermite integration, or Monte Carlo integration.
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Email Address. Sign In. Access provided by: anon Sign Out. First, the maximum-power-voltage-based control scheme and direct maximum power control scheme are introduced for the maximum power point tracking MPPT. By using T-S fuzzy model representation, the two MPPT control schemes are formulated to an output tracking control problem in a unified form. Then, the T-S fuzzy observer and controller are developed to achieve asymptotic MPPT for uncertain PV power systems, where the controller and observer gains are able to be separately solved from novel linear matrix inequality formulation.
The Robust Maximum Principle
Furthermore, the MPPT robustness is also discussed in the presence of rapidly changing atmosphere and external disturbances. Different from the traditional MPPT approaches, the proposed T-S fuzzy controller does not require searching the maximum power operational point and using coordinate transformation. All the buck, boost, or buck-boost converters can be used to achieve MPPT in the same control method. Finally, the satisfactory performance is shown from the simulation and experimental results.
Robust Regression | Stata Annotated Output
Date of Publication: 06 January DOI: Need Help?In many applications, statistical estimators serve to derive conclusions from data, for example, in finance, medical decision making, and clinical trials.
However, the conclusions are typically dependent on uncertainties in the data. We use robust optimization principles to provide robust maximum likelihood estimators that are protected against data errors. Both types of input data errors are considered: a the adversarial type, modeled using the notion of uncertainty sets, and b the probabilistic type, modeled by distributions. We provide efficient local and global search algorithms to compute the robust estimators and discuss them in detail for the case of multivariate normally distributed data.
The estimator performance is demonstrated on two applications.
First, using computer simulations, we demonstrate that the proposed estimators are robust against both types of data uncertainty and provide more accurate estimates compared with classical estimators, which degrade significantly, when errors are encountered. We establish a range of uncertainty sizes for which robust estimators are superior. Second, we analyze deviations in cancer radiation therapy planning. When analyzing a large set of past clinical treatment data, robust estimators lead to more reliable decisions when applied to a large set of past treatment plans.
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In addition, deregulation of the power industry and pressure from climate change introduced new sources of uncertainties on the generation side of the system. Generation expansion and retirement become highly uncertain as well.
Some of the uncertainties do not have probability distributions, making it difficult to use stochastic programming. Techniques like robust optimization that do not require a probability distribution became desirable. To address these challenges, we study two optimization criteria for the transmission expansion planning problem under the robust optimization paradigm, where the maximum cost and maximum regret of the expansion plan over all uncertainties are minimized, respectively.
With these models, our objective is to make planning decisions that are robust against all scenarios. We use a two-layer algorithm to solve the resulting tri-level optimization problems. Then, in our case studies, we compare the performance of the minimax cost approach and the minimax regret approach under different characterizations of uncertainties. Article :. Date of Publication: 08 April DOI: Need Help?Ordinary least squares OLS regression is an extremely useful, easily interpretable statistical method.
However, it is not perfect. When running an OLS regression, you want to be aware of its sensitivity to outliers. Robust regression offers an alternative to OLS regression that is less sensitive to outliers and still defines a linear relationship between the outcome and the predictors.
Note that robust regression does not address leverage. This page shows an example of robust regression analysis in Stata with footnotes explaining the output. We will use the crime data set. We will drop the observation for Washington, D. To determine if a robust regression model would be appropriate, OLS regression is a good starting point. After running the regression, postestimation graphing techniques and an examination of the model residuals can be implemented to determine if there are any points in the data that might influence the regression results disproportionately.
The commands for an OLS regression, predicting crime with poverty and singleand a postestimation graph appear below. Details for interpreting this graph and other methods for detecting high influence points can be found in the Robust Regression Data Analysis Example. We will be interested in the residuals from this regression when looking at our robust regression, so we have added a predict command and generated a variable containing the absolute value of the OLS residuals.
The same model can be run as a robust regression. From this model, weights are assigned to records according to the absolute difference between the predicted and actual values the absolute residual. The records with small absolute residuals are weighted more heavily than the records with large absolute residuals.
Then, another regression is run using these newly assigned weights, and then new weights are generated from this regression. This process of regressing and reweighting is iterated until the differences in weights before and after a regression is sufficiently close to zero. For a detailed illustration of this process, see Chapter Six of Regression with Graphics.
The Stata command for robust regression is rreg. The model portion of the command is identical to an OLS regression: outcome variable followed by predictors. We have added gen weight to the command so that we will be able to examine the final weights used in the model.
Huber iteration — These are iterations in which Huber weightings are implemented.Please cite us if you use the software. This Scaler removes the median and scales the data according to the quantile range defaults to IQR: Interquartile Range.
The IQR is the range between the 1st quartile 25th quantile and the 3rd quartile 75th quantile. Centering and scaling happen independently on each feature by computing the relevant statistics on the samples in the training set.
Median and interquartile range are then stored to be used on later data using the transform method. Standardization of a dataset is a common requirement for many machine learning estimators.
Typically this is done by removing the mean and scaling to unit variance. In such cases, the median and the interquartile range often give better results. Read more in the User Guide. If True, center the data before scaling. This will cause transform to raise an exception when attempted on sparse matrices, because centering them entails building a dense matrix which in common use cases is likely to be too large to fit in memory.
Default: If False, try to avoid a copy and do inplace scaling instead. This is not guaranteed to always work inplace; e. The data used to compute the median and quantiles used for later scaling along the features axis. If True, will return the parameters for this estimator and contained subobjects that are estimators. The method works on simple estimators as well as on nested objects such as pipelines.
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