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. ^

Selection bias and the cross-validation of regression models for prediction

Strategies are compared for the development of a linear regression model with stochastic (multivariate normal) regressor variables and the subsequent assessment of its predictive ability. Bias and mean squared error of four estimators of predictive performance are evaluated in simulated samples of 32 population correlation matrices. Models including all of the available predictors are compared with those obtained using selected subsets. The subset selection procedures investigated include two stopping rules, C$\sb{\rm p}$ and S$\sb{\rm p}$, each combined with an 'all possible subsets' or 'forward selection' of variables. The estimators of performance utilized include parametric (MSEP$\sb{\rm m}$) and non-parametric (PRESS) assessments in the entire sample, and two data splitting estimates restricted to a random or balanced (Snee's DUPLEX) 'validation' half sample. The simulations were performed as a designed experiment, with population correlation matrices representing a broad range of data structures.^ The techniques examined for subset selection do not generally result in improved predictions relative to the full model. Approaches using 'forward selection' result in slightly smaller prediction errors and less biased estimators of predictive accuracy than 'all possible subsets' approaches but no differences are detected between the performances of C$\sb{\rm p}$ and S$\sb{\rm p}$. In every case, prediction errors of models obtained by subset selection in either of the half splits exceed those obtained using all predictors and the entire sample.^ Only the random split estimator is conditionally (on $\\beta$) unbiased, however MSEP$\sb{\rm m}$ is unbiased on average and PRESS is nearly so in unselected (fixed form) models. When subset selection techniques are used, MSEP$\sb{\rm m}$ and PRESS always underestimate prediction errors, by as much as 27 percent (on average) in small samples. Despite their bias, the mean squared errors (MSE) of these estimators are at least 30 percent less than that of the unbiased random split estimator. The DUPLEX split estimator suffers from large MSE as well as bias, and seems of little value within the context of stochastic regressor variables.^ To maximize predictive accuracy while retaining a reliable estimate of that accuracy, it is recommended that the entire sample be used for model development, and a leave-one-out statistic (e.g. PRESS) be used for assessment. ^

Effect of nonnormal random components on level and power in group-randomized trials

The use of group-randomized trials is particularly widespread in the evaluation of health care, educational, and screening strategies. Group-randomized trials represent a subset of a larger class of designs often labeled nested, hierarchical, or multilevel and are characterized by the randomization of intact social units or groups, rather than individuals. The application of random effects models to group-randomized trials requires the specification of fixed and random components of the model. The underlying assumption is usually that these random components are normally distributed. This research is intended to determine if the Type I error rate and power are affected when the assumption of normality for the random component representing the group effect is violated. ^ In this study, simulated data are used to examine the Type I error rate, power, bias and mean squared error of the estimates of the fixed effect and the observed intraclass correlation coefficient (ICC) when the random component representing the group effect possess distributions with non-normal characteristics, such as heavy tails or severe skewness. The simulated data are generated with various characteristics (e.g. number of schools per condition, number of students per school, and several within school ICCs) observed in most small, school-based, group-randomized trials. The analysis is carried out using SAS PROC MIXED, Version 6.12, with random effects specified in a random statement and restricted maximum likelihood (REML) estimation specified. The results from the non-normally distributed data are compared to the results obtained from the analysis of data with similar design characteristics but normally distributed random effects. ^ The results suggest that the violation of the normality assumption for the group component by a skewed or heavy-tailed distribution does not appear to influence the estimation of the fixed effect, Type I error, and power. Negative biases were detected when estimating the sample ICC and dramatically increased in magnitude as the true ICC increased. These biases were not as pronounced when the true ICC was within the range observed in most group-randomized trials (i.e. 0.00 to 0.05). The normally distributed group effect also resulted in bias ICC estimates when the true ICC was greater than 0.05. However, this may be a result of higher correlation within the data. ^

THE IMPROVEMENT REGION IN RIDGE REGRESSION

One of the difficulties in the practical application of ridge regression is that, for a given data set, it is unknown whether a selected ridge estimator has smaller squared error than the least squares estimator. The concept of the improvement region is defined, and a technique is developed which obtains approximate confidence intervals for the value of ridge k which produces the maximum reduction in mean squared error. Two simulation experiments were conducted to investigate how accurate these approximate confidence intervals might be. ^

Inappropriate imaging utilization in Texas: Geographic variation and predictive factors

Objective::Describe and understand regional differences and associated multilevel factors (patient, provider and regional) to inappropriate utilization of advance imaging tests in the privately insured population of Texas. Methods: We analyzed Blue Cross Blue Shield of Texas claims dataset to study the advance imaging utilization during 2008-2010 in the PPO/PPO+ plans. We used three of CMS "Hospital Outpatient Quality Reporting" imaging efficiency measures. These included ordering MRI for low back pain without prior conservative management (OP-8) and utilization of combined with and without contrast abdominal CT (OP-10) and thorax CT (OP-11). Means and variation by hospital referral regions (HRR) in Texas were measured and a multilevel logistic regression for being a provider with high values for any the three OP measures was used in the analysis. We also analyzed OP-8 at the individual level. A multilevel logistic regression was used to identify predictive factors for having an inappropriate MRI for low back pain. Results: Mean OP-8 for Texas providers was 37.89%, OP-10 was 29.94% and OP-11 was 9.24%. Variation was higher for CT measure. And certain HRRs were consistently above the mean. Hospital providers had higher odds of high OP-8 values (OP-8: OR, 1.34; CI, 1.12-1.60) but had smaller odds of having high OP-10 and OP-11 values (OP-10: OR, 0.15; CI, 0.12-0.18; OP-11: OR, 0.43; CI, 0.34-0.53). Providers with the highest volume of imaging studies performed, were less likely to have high OP-8 measures (OP-8: OR, 0.58; CI, 0.48-0.70) but more likely to perform combined thoracic CT scans (OP-11: OR, 1.62; CI, 1.34-1.95). Males had higher odds of inappropriate MRI (OR, 1.21; CI, 1.16-1.26). Pattern of care in the six months prior to the MRI event was significantly associated with having an inappropriate MRI. Conclusion::We identified a significant variation in advance imaging utilization across Texas. Type of facility was associated with measure performance, but the associations differ according to the type of study. Last, certain individual characteristics such as gender, age and pattern of care were found to be predictors of inappropriate MRIs.^

PREDICTIVE VARIABLES FOR WEIGHT CHANGE 2 YEARS FOLLOWING PARTICIPATION IN A WEIGHT LOSS INTERVENTION PROGRAM

A historical prospective study was designed to assess the man weight status of subjects who participated in a behavioral weight reduction program in 1983 and to determine whether there was an association between the dependent variable weight change and any of 31 independent variables after a 2 year follow-up period. Data was obtained by abstracting the subjects records and from a follow-up questionnaire administered 2 years following program participation. Five hundred nine subjects (386 females and 123 males) of 1460 subjects who participated in the program, completed and returned the questionnaire. Results showed that mean weight was significantly different (p < 0.001) between the measurement at baseline and after a 2 year follow-up period. The mean weight loss of the group was 5.8 pounds, 10.7 pounds for males and 4.2 pounds for females after a 2 year follow-up period. A total of 63.9% of the group, 69.9% of males and 61.9% of females were still below their initial weight after the 2 year follow-up period. Sixteen of the 31 variables assessed utilizing bivariate analyses were found to be significantly (p (LESSTHEQ) 0.05) associated with weight change after a 2 year follow-up period. These variables were then entered into a multivariate linear regression model. A total of 37.9% of the variance of the dependent variable, weight change, was accounted for by all 16 variables. Eight of these variables were found to be significantly (p (LESSTHEQ) 0.05) predictive of weight change in the stepwise multivariate process accounting for 37.1% of the variance. These variables included: Two baseline variables (percent over ideal body weight at enrollment and occupation) and six follow-up variables (feeling in control of eating habits, percent of body weight lost during treatment, frequency of weight measurement, physical activity, eating in response to emotions, and number of pounds of weight gain needed to resume a diet). It was concluded that a greater amount of emphasis should be placed on the six follow-up variables by clinicians involved in the treatment of obesity, and by the subjects themselves to enhance their chances of success at long-term weight loss. ^

Comparing the factorial structure, invariance, and predictive validity of transtheoretical model constructs for alcohol use across restricted and unrestricted settings

With substance abuse treatment expanding in prisons and jails, understanding how behavior change interacts with a restricted setting becomes more essential. The Transtheoretical Model (TTM) has been used to understand intentional behavior change in unrestricted settings, however, evidence indicates restrictive settings can affect the measurement and structure of the TTM constructs. The present study examined data from problem drinkers at baseline and end-of-treatment from three studies: (1) Project CARE (n = 187) recruited inmates from a large county jail; (2) Project Check-In (n = 116) recruited inmates from a state prison; (3) Project MATCH, a large multi-site alcohol study had two recruitment arms, aftercare (n = 724 pre-treatment and 650 post-treatment) and outpatient (n = 912 pre-treatment and 844 post-treatment). The analyses were conducted using cross-sectional data to test for non-invariance of measures of the TTM constructs: readiness, confidence, temptation, and processes of change (Structural Equation Modeling, SEM) across restricted and unrestricted settings. Two restricted (jail and aftercare) and one unrestricted group (outpatient) entering treatment and one restricted (prison) and two unrestricted groups (aftercare and outpatient) at end-of-treatment were contrasted. In addition TTM end-of-treatment profiles were tested as predictors of 12 month drinking outcomes (Profile Analysis). Although SEM did not indicate structural differences in the overall TTM construct model across setting types, there were factor structure differences on the confidence and temptation constructs at pre-treatment and in the factor structure of the behavioral processes at the end-of-treatment. For pre-treatment temptation and confidence, differences were found in the social situations factor loadings and in the variance for the confidence and temptation latent factors. For the end-of-treatment behavioral processes, differences across the restricted and unrestricted settings were identified in the counter-conditioning and stimulus control factor loadings. The TTM end-of-treatment profiles were not predictive of drinking outcomes in the prison sample. Both pre and post-treatment differences in structure across setting types involved constructs operationalized with behaviors that are limited for those in restricted settings. These studies suggest the TTM is a viable model for explicating addictive behavior change in restricted settings but calls for modification of subscale items that refer to specific behaviors and caution in interpreting the mean differences across setting types for problem drinkers. ^

Mean corpuscular volume as a marker for adherence to antiretroviral therapy among HIV infected children and adolescents in Uganda

Mean corpuscular volume, which is an inexpensive and widely available measure to assess, increases in HIV infected individuals receiving zidovudine and stavudine raising the hypothesis that it could be used as a surrogate for adherence.^ The aim of this study was to examine the association between mean corpuscular volume and adherence to antiretroviral therapy among HIV infected children and adolescents aged 0–19 years in Uganda as well as the extent to which changes in mean corpuscular volume predict adherence as determined by virologic suppression.^ The investigator retrospectively reviewed and analyzed secondary data of 158 HIV infected children and adolescents aged 0–19 years who initiated antiretroviral therapy under an observational cohort at the Baylor College of Medicine Children's Foundation - Uganda. Viral suppression was used as the gold standard for monitoring adherence and defined as viral load of < 400 copies/ml at 24 and 48 weeks. ^ Patients were at least 48 weeks on therapy, age 0.2–18.4 years, 54.4% female, 82.3% on zidovudine based regimen, 92% WHO stage III at initiation of therapy, median pre therapy MCV 80.6 fl (70.3–98.3 fl), median CD4% 10.2% (0.3%–28.0%), and mean pre therapy viral load 407,712.9 ± 270,413.9 copies/ml. For both 24 and 48 weeks of antiretroviral therapy, patients with viral suppression had a greater mean percentage change in mean corpuscular volume (15.1% ± 8.4 vs. 11.1% ± 7.8 and 2.3% ± 13.2 vs. -2.7% ± 10.5 respectively). The mean percentage change in mean corpuscular volume was greater in the first 24 weeks of therapy for patients with and without viral suppression (15.1% ± 8.4 vs. 2.3% ± 13.2 and 11.1% ± 7.8 vs. -2.7% ± 10.5 respectively). In the multivariate logistic regression model, percentage change in mean corpuscular volume ≥ 20% was significantly associated with viral suppression (adjusted OR 4.0; CI 1.2–13.3; p value 0.02). The ability of percentage changes in MCV to correctly identify children and adolescents with viral suppression was higher at a cut off of ≥ 20% (90.7%; sensitivity, 31.7%) than at ≥ 9% (82.9%; sensitivity, 78.9%). Negative predictive value was lower at ≥ 20% change (25%; specificity, 84.8%) than at ≥ 9% change (33.3%; specificity, 39.4%).^ Mean corpuscular volume is a useful marker of adherence among children and adolescents with viral suppression. ^