Missing data is a common issue that can cause problems in estimation and inference in biomedical, epidemiological and social research. Multiple imputation is an increasingly popular approach for handling missing data. In case of a large number of covariates with missing data, existing multiple imputation software packages may not work properly and often produce errors. We propose a multiple imputation algorithm called mispr based on sequential penalized regression models. Each variable with missing values is assumed to have a different distributional form and is imputed with its own imputation model using the ridge penalty. In the case of a large number of predictors with respect to the sample size, the use of a quadratic penalty guarantees unique estimates for the parameters and leads to better predictions than the usual Maximum Likelihood Estimation (MLE), with a good compromise between bias and variance. As a result, the proposed algorithm performs well and provides imputed values that are better even for a large number of covariates with small samples. The results are compared with the existing software packages mice, VIM and Amelia in simulation studies. The missing at random mechanism was the main assumption in the simulation study. The imputation performance of the proposed algorithm is evaluated with mean squared imputation error and mean absolute imputation error. The mean squared error ($\hat{\mathbf{\beta }}$ ), parameter estimates with their standard errors and confidence intervals are also computed to compare the performance in the regression context. The proposed algorithm is observed to be a good competitor to the existing algorithms, with smaller mean squared imputation error, mean absolute imputation error and mean squared error. The algorithm’s performance becomes considerably better than that of the existing algorithms with increasing number of covariates, especially when the number of predictors is close to or even greater than the sample size. Two real-life datasets are also used to examine the performance of the proposed algorithm using simulations.

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 1223 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: 1.6423.261) compared to those who are poor. Those in rural areas were more likely to fully vaccinate their children (AOR=1.54; 95 CI: 1.2321.925) compared to those in urban areas. Those in the highest level of the educational group (AOR=2.639; 95 CI: 1.8003.87) 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: 0.6870.963). The immunization status of children shows a significant association with maternal education, wealth status, and area of residence.

In multi-category response models, categories are often ordered. In the 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 constraint on the order of intercept values is not met during the iterative procedure. Proportional odds models (POMs) 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 POMs. 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 parameter 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.

Osteoporosis is characterized by low bone mineral density leading to enhanced bone fragility and a consequent increase in fracture risk. The focus of this case‐control study was to identify significant socioeconomic risk factors of osteoporosis in Pakistani women and examine how the risk increases for different levels of risk factors. A case‐control study was conducted from November 2018 to August 2019 in two main hospitals in Faisalabad, Pakistan. Multiple logistic regression was used to explore the significant risk factors of osteoporosis and how the risk increases in cases (cases = 120) as compared to the control group (controls = 120) in the presence of these risk factors. The mean age ± standard deviation for cases and controls was 59.62 ± 10.75 and 54.27 ± 10.09, respectively. The minimum and maximum ages were 36 and 80 years, respectively. In addition to age, bone fracture, family history, regular physical activity, family size, use of meat, type of birth, breastfeeding, premature menopause, loss of appetite, and use of anticoagulants were significant risk factors with p‐values less than 0.05. The risk prediction model with significant risk factors was a good fit with a p‐value of 0.28, corresponding to the Hosmer–Lemeshow test value (χ2 = 9.78). This parsimonious model with Cox–Snell R2 = 0.50 (with a maximum value = 0.75) and Nagelkerke R2 = 0.66 showed an AUC of 0.924 as compared to the full model with all risk factors under study that exhibited an AUC of 0.949.

Summary: The proportional odds model 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 techniquepomBoostis proposed to fit 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) that 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 ofpomBoostfor selection of the relevant predictors.