Ordinal Ridge Regression with Categorical Predictors

In multi-category response models categories are often ordered. In case of ordinal response models, the usual likelihood approach becomes unstable with ill-conditioned predictor space or when the number of parameters to be estimated is large relative to the sample size. The likelihood estimates do not exist when the number of observations is less than the number of parameters. The same problem arises if constraints on the order of intercept values are not met during the iterative fitting procedure. Proportional odds models are most commonly used for ordinal responses. In this paper penalized likelihood with quadratic penalty is used to address these issues with a special focus on proportional odds models. To avoid large differences between two parameter values corresponding to the consecutive categories of an ordinal predictor, the differences between the parameters of two adjacent categories should be penalized. The considered penalized likelihood function penalizes the parameter estimates or differences between the parameters estimates according to the type of predictors. Mean squared error for parameter estimates, deviance of fitted probabilities and prediction error for ridge regression are compared with usual likelihood estimates in a simulation study and an application

Ridge Estimation for Multinomial Logit Models with Symmetric Side Constraints

In multinomial logit models, the identifiability of parameter estimates is typically obtained by side constraints that specify one of the response categories as reference category. When parameters are penalized, shrinkage of estimates should not depend on the reference category. In this paper we investigate ridge regression for the multinomial logit model with symmetric side constraints, which yields parameter estimates that are independent of the reference category. In simulation studies the results are compared with the usual maximum likelihood estimates and an application to real data is given

Regularized Proportional Odds Models

The proportional odds model is commonly used in regression analysis to predict the outcome for an ordinal response variable. The maximum likelihood approach becomes unstable or even fails in small samples with relatively large number of predictors. The ML estimates also do not exist with complete separation in the data. An estimation method is developed to address these problems with MLE. The proposed method uses pseudo observations to regularize the observed responses by sharpening them so that they become close to the underlying probabilities. The estimates can be computed easily with all commonly used statistical packages supporting the fitting of proportional odds models with weights. Estimates are compared with MLE in a simulation study and two real life data sets

Proportional Odds Models with High-dimensional Data Structure

The proportional odds model (POM) is the most widely used model when the response has ordered categories. In the case of high-dimensional predictor structure the common maximum likelihood approach typically fails when all predictors are included. A boosting technique pomBoost is proposed that fits the model by implicitly selecting the influential predictors. The approach distinguishes between metric and categorical predictors. In the case of categorical predictors, where each predictor relates to a set of parameters, the objective is to select simultaneously all the associated parameters. In addition the approach distinguishes between nominal and ordinal predictors. In the case of ordinal predictors, the proposed technique uses the ordering of the ordinal predictors by penalizing the difference between the parameters of adjacent categories. The technique has also a provision to consider some mandatory predictors (if any) which must be part of the final sparse model. The performance of the proposed boosting algorithm is evaluated in a simulation study and applications with respect to mean squared error and prediction error. Hit rates and false alarm rates are used to judge the performance of pomBoost for selection of the relevant predictors

Multinomial Logit Models with Implicit Variable Selection

Multinomial logit models which are most commonly used for the modeling of unordered multi-category responses are typically restricted to the use of few predictors. In the high-dimensional case maximum likelihood estimates frequently do not exist. In this paper we are developing a boosting technique called multinomBoost that performs variable selection and fits the multinomial logit model also when predictors are high-dimensional. Since in multicategory models the effect of one predictor variable is represented by several parameters one has to distinguish between variable selection and parameter selection. A special feature of the approach is that, in contrast to existing approaches, it selects variables not parameters. The method can distinguish between mandatory predictors and optional predictors. Moreover, it adapts to metric, binary, nominal and ordinal predictors. Regularization within the algorithm allows to include nominal and ordinal variables which have many categories. In the case of ordinal predictors the order information is used. The performance of the boosting technique with respect to mean squared error, prediction error and the identification of relevant variables is investigated in a simulation study. For two real life data sets the results are also compared with the Lasso approach which selects parameters

Multiple imputation with compatibility for high-dimensional data

Multiple Imputation (MI) is always challenging in high dimensional settings. The imputation model with some selected number of predictors can be incompatible with the analysis model leading to inconsistent and biased estimates. Although compatibility in such cases may not be achieved, but one can obtain consistent and unbiased estimates using a semi-compatible imputation model. We propose to relax the lasso penalty for selecting a large set of variables (at most n). The substantive model that also uses some formal variable selection procedure in high-dimensional structures is then expected to be nested in this imputation model. The resulting imputation model will be semi-compatible with high probability. The likelihood estimates can be unstable and can face the convergence issues as the number of variables becomes nearly as large as the sample size. To address these issues, we further propose to use a ridge penalty for obtaining the posterior distribution of the parameters based on the observed data. The proposed technique is compared with the standard MI software and MI techniques available for high-dimensional data in simulation studies and a real life dataset. Our results exhibit the superiority of the proposed approach to the existing MI approaches while addressing the compatibility issue

Modeling the Factors Associated with Incomplete Immunization among Children

Immunization is a precautionary measure that helps to stop diseases before their occurrence. Vaccine-preventable diseases are a primary cause of death among children under the age of five in many developing nations. The purpose of this study is to investigate the immunization status and associated demographic characteristics among children aged 12-23 months in Punjab, Pakistan. The study used the data from the Multiple Indicator Cluster Survey (MICS) for Punjab, Pakistan. Data were collected from caregivers using interviewer-administered questionnaires. To summarize the data, descriptive statistics are computed, and logistic regression is used to identify the significant factors that are responsible for complete immunization among the children in Punjab. Odds ratios, 95% CI, and Chi-square statistics were computed to identify the factors associated with no or partial immunization. The prevalence of complete immunization coverage was 89.1%. Women in the rich wealth quantile had the highest odds of completing the immunization for their children (AOR = 2.314; 95% CI: 1.642-3.261) compared to those who are poor. Those in rural areas were more likely to fully vaccinate their children (AOR = 1.54; 95% CI: 1.232-1.925) compared to those in urban areas. Those in the highest level of the educational group (AOR = 2.639; 95% CI: 1.800-3.87) are more likely to complete vaccination for their children compared to those with no formal education. However, female children are less likely to complete immunization compared to male children (AOR = 0.813; 95% CI: 0.687-0.963). The immunization status of children shows a significant association with maternal education, wealth status, and area of residence

Adjunctive Platelet-Rich Plasma (PRP) in Infrabony Regenerative Treatment: A Systematic Review and RCT's Meta-Analysis

Background and Objective. The purpose of this study was to highlight the clinical performance of platelet-rich plasma (PRP) used as an adjunctive tool for regeneration in infrabony periodontal defects using different biomaterials or performing different surgical flap approaches. Comparative evaluation of main clinical outcomes as probing pocket depth reduction, clinical attachment gain, and recession reduction with and without the use of PRP has been analysed. Materials and Methods. According to the focused question, an electronic and hand searching has been performed up to December 2016. From a batch of 73 articles, the selection strategy and Jadad quality assessment led us to include 15 studies for the meta-analysis. Results. Despite the high heterogeneity found and the lack of complete data regarding the selected clinical outcomes, a comparative analysis has been possible by the categorization of used biomaterials and surgical flap approaches. This method led us to observe the best performance of grafts with the use of adjunctive PRP in CAL gain and PPD reduction. No difference has been outlined with a specific surgical flap. Conclusion: s. Although PRP is considered a cheap and patient's derived growth factor, the not conclusive data reported would suggest that its use in addition to bone substitutes could be of some clinical benefit in the regenerative treatment of infrabony defects. Clinical Relevance. This systematic review was intended to sort out the huge controversial debate in the field about the possible use of PRP in regenerative surgery in infrabony defect. The clinical relevance of using blood-borne growth factors to conventional procedures is effective as these could determine a better performance and outcomes despite the surgical approach adopted and limit the use of additional biomaterials for the blood clot stabilization

Modeling and forecasting exchange rate dynamics in Pakistan using ARCH family of models

The main objective of this paper is to provide an exclusive understanding about the theoretical and empirical working of the GARCH class of models as well as to exploit the potential gains in modeling conditional variance, once it is confirmed that conditional mean model errors present time varying volatility. Another objective is to search the best time series model among autoregressive moving average (ARMA), autoregressive conditional heteroscedasticity (ARCH), generalized autoregressive conditional heteroscedasticity (GARCH), and exponential generalized autoregressive conditional heteroscedasticity (EGARCH) to give best prediction of exchange rates. The data used in present study consists of monthly exchange rates of Pakistan for the period ranging from July 1981 to May 2010 obtained from the State Bank of Pakistan. GARCH (1,2) is found to be best to remove the persistence in volatility while EGARCH(1,2) successfully overcome the leverage effect in the exchange rate returns under study.