DISTRIBUTION FUNCTIONS, MEAN SQUARED ERRORS, AND CONFIDENCE LIMITS FOR RIDGE REGRESSION ESTIMATORS

A large number of ridge regression estimators have been proposed and used with little knowledge of their true distributions. Because of this lack of knowledge, these estimators cannot be used to test hypotheses or to form confidence intervals.^ This paper presents a basic technique for deriving the exact distribution functions for a class of generalized ridge estimators. The technique is applied to five prominent generalized ridge estimators. Graphs of the resulting distribution functions are presented. The actual behavior of these estimators is found to be considerably different than the behavior which is generally assumed for ridge estimators.^ This paper also uses the derived distributions to examine the mean squared error properties of the estimators. A technique for developing confidence intervals based on the generalized ridge estimators is also presented. ^

Robust effect sizes and their confidence intervals for group difference between trajectories in hierarchical linear growth model

Hierarchical linear growth model (HLGM), as a flexible and powerful analytic method, has played an increased important role in psychology, public health and medical sciences in recent decades. Mostly, researchers who conduct HLGM are interested in the treatment effect on individual trajectories, which can be indicated by the cross-level interaction effects. However, the statistical hypothesis test for the effect of cross-level interaction in HLGM only show us whether there is a significant group difference in the average rate of change, rate of acceleration or higher polynomial effect; it fails to convey information about the magnitude of the difference between the group trajectories at specific time point. Thus, reporting and interpreting effect sizes have been increased emphases in HLGM in recent years, due to the limitations and increased criticisms for statistical hypothesis testing. However, most researchers fail to report these model-implied effect sizes for group trajectories comparison and their corresponding confidence intervals in HLGM analysis, since lack of appropriate and standard functions to estimate effect sizes associated with the model-implied difference between grouping trajectories in HLGM, and also lack of computing packages in the popular statistical software to automatically calculate them. ^ The present project is the first to establish the appropriate computing functions to assess the standard difference between grouping trajectories in HLGM. We proposed the two functions to estimate effect sizes on model-based grouping trajectories difference at specific time, we also suggested the robust effect sizes to reduce the bias of estimated effect sizes. Then, we applied the proposed functions to estimate the population effect sizes (d ) and robust effect sizes (du) on the cross-level interaction in HLGM by using the three simulated datasets, and also we compared the three methods of constructing confidence intervals around d and du recommended the best one for application. At the end, we constructed 95% confidence intervals with the suitable method for the effect sizes what we obtained with the three simulated datasets. ^ The effect sizes between grouping trajectories for the three simulated longitudinal datasets indicated that even though the statistical hypothesis test shows no significant difference between grouping trajectories, effect sizes between these grouping trajectories can still be large at some time points. Therefore, effect sizes between grouping trajectories in HLGM analysis provide us additional and meaningful information to assess group effect on individual trajectories. In addition, we also compared the three methods to construct 95% confident intervals around corresponding effect sizes in this project, which handled with the uncertainty of effect sizes to population parameter. We suggested the noncentral t-distribution based method when the assumptions held, and the bootstrap bias-corrected and accelerated method when the assumptions are not met.^

Description and analysis of variation in dairy product and concomitant nutrient purchases between Hispanic and non -Hispanic households using a panel of shoppers' purchase records

The purpose of this research was development of a method of estimating nutrient availability in populations as approximated by supermarket purchase records. Demographic information describing 12,516 panel households was obtained from a marketing and advertising program operated by H. E. Butt Grocery Company of San Antonio, Texas. A non-probability sample of 2,161 households meeting expenditure criteria was selected and all purchases of dairy products for this sample of households were organized into a database constructed to facilitate the retrieval, aggregation, and analysis of dairy product purchases and their nutrient contents. Two hypotheses were tested: (1) no difference would be found between Hispanic and non-Hispanic purchases of dairy product categories during the study period and (2) no difference would be found between Hispanic and non-Hispanic purchases of nutrients contained in those dairy products during the thirteen-week study period.^ Food purchase records were used to estimate nutrient exposure on a weekly, per capita basis for Hispanic and non-Hispanic households by linking some 40,000 dairy purchase Universal Product code (UPC) numbers with food composition values contained in USDA Handbook 8-1. Results of this study suggest Hispanic sample households consistently purchased fewer dairy products than did non-Hispanic sample households and consequently had fewer nutrients available from dairy purchases. While weekly expenditures for dairy products among the sample households remained relatively constant during the study period, shifts in the types of dairy products purchased were observed. The effect of ethnicity on dairy product and nutrient purchases was significant over the thirteen-week period. A database consisting of customer, household, and purchase information can be developed to successfully associate food item UPC numbers with a standard reference of food composition to estimate nutrient availability in a population over extended periods of time. ^