On Graphically Checking Goodness-of-fit of Binary Logistic Regression Models

OBJECTIVES: This paper is concerned with checking goodness-of-fit of binary logistic regression models. For the practitioners of data analysis, the broad classes of procedures for checking goodness-of-fit available in the literature are described. The challenges of model checking in the context of binary logistic regression are reviewed. As a viable solution, a simple graphical procedure for checking goodness-of-fit is proposed. METHODS: The graphical procedure proposed relies on pieces of information available from any logistic analysis; the focus is on combining and presenting these in an informative way. RESULTS: The information gained using this approach is presented with three examples. In the discussion, the proposed method is put into context and compared with other graphical procedures for checking goodness-of-fit of binary logistic models available in the literature. CONCLUSION: A simple graphical method can significantly improve the understanding of any logistic regression analysis and help to prevent faulty conclusions.

Stochastic search for semiparametric linear regression models

This paper introduces and analyzes a stochastic search method for parameter estimation in linear regression models in the spirit of Beran and Millar [Ann. Statist. 15(3) (1987) 1131–1154]. The idea is to generate a random finite subset of a parameter space which will automatically contain points which are very close to an unknown true parameter. The motivation for this procedure comes from recent work of Dümbgen et al. [Ann. Statist. 39(2) (2011) 702–730] on regression models with log-concave error distributions.

The Blinder-Oaxaca decomposition for linear regression models

The counterfactual decomposition technique popularized by Blinder (1973, Journal of Human Resources, 436–455) and Oaxaca (1973, International Economic Review, 693–709) is widely used to study mean outcome differences between groups. For example, the technique is often used to analyze wage gaps by sex or race. This article summarizes the technique and addresses several complications, such as the identification of effects of categorical predictors in the detailed decomposition or the estimation of standard errors. A new command called oaxaca is introduced, and examples illustrating its usage are given.

Detecting outlying studies in meta-regression models using a forward search algorithm.

When considering data from many trials, it is likely that some of them present a markedly different intervention effect or exert an undue influence on the summary results. We develop a forward search algorithm for identifying outlying and influential studies in meta-analysis models. The forward search algorithm starts by fitting the hypothesized model to a small subset of likely outlier-free studies and proceeds by adding studies into the set one-by-one that are determined to be closest to the fitted model of the existing set. As each study is added to the set, plots of estimated parameters and measures of fit are monitored to identify outliers by sharp changes in the forward plots. We apply the proposed outlier detection method to two real data sets; a meta-analysis of 26 studies that examines the effect of writing-to-learn interventions on academic achievement adjusting for three possible effect modifiers, and a meta-analysis of 70 studies that compares a fluoride toothpaste treatment to placebo for preventing dental caries in children. A simple simulated example is used to illustrate the steps of the proposed methodology, and a small-scale simulation study is conducted to evaluate the performance of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.

Development and validation of energy expenditure prediction models based on GT3X accelerometer data in 5- to 9-year-old children

Background: Accelerometry has been established as an objective method that can be used to assess physical activity behavior in large groups. The purpose of the current study was to provide a validated equation to translate accelerometer counts of the triaxial GT3X into energy expenditure in young children. Methods: Thirty-two children aged 5–9 years performed locomotor and play activities that are typical for their age group. Children wore a GT3X accelerometer and their energy expenditure was measured with indirect calorimetry. Twenty-one children were randomly selected to serve as development group. A cubic 2-regression model involving separate equations for locomotor and play activities was developed on the basis of model fit. It was then validated using data of the remaining children and compared with a linear 2-regression model and a linear 1-regression model. Results: All 3 regression models produced strong correlations between predicted and measured MET values. Agreement was acceptable for the cubic model and good for both linear regression approaches. Conclusions: The current linear 1-regression model provides valid estimates of energy expenditure for ActiGraph GT3X data for 5- to 9-year-old children and shows equal or better predictive validity than a cubic or a linear 2-regression model.

ROBREG: Stata module providing robust regression estimators

robreg provides a number of robust estimators for linear regression models. Among them are the high breakdown-point and high efficiency MM-estimator, the Huber and bisquare M-estimator, and the S-estimator, each supporting classic or robust standard errors. Furthermore, basic versions of the LMS/LQS (least median of squares) and LTS (least trimmed squares) estimators are provided. Note that the moremata package, also available from SSC, is required.

Modeling absolute differences in life expectancy with a censored skew-normal regression approach.

Parameter estimates from commonly used multivariable parametric survival regression models do not directly quantify differences in years of life expectancy. Gaussian linear regression models give results in terms of absolute mean differences, but are not appropriate in modeling life expectancy, because in many situations time to death has a negative skewed distribution. A regression approach using a skew-normal distribution would be an alternative to parametric survival models in the modeling of life expectancy, because parameter estimates can be interpreted in terms of survival time differences while allowing for skewness of the distribution. In this paper we show how to use the skew-normal regression so that censored and left-truncated observations are accounted for. With this we model differences in life expectancy using data from the Swiss National Cohort Study and from official life expectancy estimates and compare the results with those derived from commonly used survival regression models. We conclude that a censored skew-normal survival regression approach for left-truncated observations can be used to model differences in life expectancy across covariates of interest.

Blood glucose and prognosis in children with presumed severe malaria: is there a threshold for 'hypoglycaemia'?

OBJECTIVES: Hypoglycaemia (glucose <2.2 mmol/l) is a defining feature of severe malaria, but the significance of other levels of blood glucose has not previously been studied in children with severe malaria. METHODS: A prospective study of 437 consecutive children with presumed severe malaria was conducted in Mali. We defined hypoglycaemia as <2.2 mmol/l, low glycaemia as 2.2-4.4 mmol/l and hyperglycaemia as >8.3 mmol/l. Associations between glycaemia and case fatality were analysed for 418 children using logistic regression models and a receiver operator curve (ROC). RESULTS: There was a significant difference between blood glucose levels in children who died (median 4.6 mmol/l) and survivors (median 7.6 mmol/l, P < 0.001). Case fatality declined from 61.5% of the hypoglycaemic children to 46.2% of those with low glycaemia, 13.4% of those with normal glycaemia and 7.6% of those with hyperglycaemia (P < 0.001). Logistic regression showed an adjusted odds ratio (AOR) of 0.75 (0.64-0.88) for case fatality per 1 mmol/l increase in baseline blood glucose. Compared to a normal blood glucose, hypoglycaemia and low glycaemia both significantly increased the odds of death (AOR 11.87, 2.10-67.00; and 5.21, 1.86-14.63, respectively), whereas hyperglycaemia reduced the odds of death (AOR 0.34, 0.13-0.91). The ROC [area under the curve at 0.753 (95% CI 0.684-0.820)] indicated that glycaemia had a moderate predictive value for death and identified an optimal threshold at glycaemia <6.1 mmol/l, (sensitivity 64.5% and specificity 75.1%). CONCLUSIONS: If there is a threshold of blood glucose which defines a worse prognosis, it is at a higher level than the current definition of 2.2 mmol/l.

Hepatoprotective effect of tumour necrosis factor alpha blockade in psoriatic arthritis: a cross-sectional study

Objective To evaluate the impact of tumour necrosis factor α (TNFα) blockers on the presence of liver fibrosis in patients with rheumatoid arthritis (RA) and psoriatic arthritis (PsA) treated with methotrexate (MTX). Methods Participants were consecutive patients with RA and PsA who had undergone MTX treatment for at least 1 year ± TNF blockade for over 6 months. Liver fibrosis was assessed using non-invasive transient elastography (FibroScan). Regression models were used to compare FibroScan values of patients with RA and patients with PsA receiving TNFα blockers with those who were not. Results FibroScan assessments were performed on 51 patients with RA and 43 patients with PsA. Compared to patients with RA, those with PsA were predominantly young men, received lower cumulative dosages of MTX and exhibited a higher incidence of liver steatosis and hyperlipidaemia. An abnormal result was observed in 7.1% of the anti-TNFα-naïve and in 13% of the anti-TNFα-treated patients in the RA group and in 30% of the anti-TNFα-naïve and 4.3% of the anti-TNFα-treated patients in the PsA group (OR=0.11, 95% CI 0.02 to 0.98). Results of the PsA group were robust when adjusted for baseline characteristics. Conclusion The results suggest a protective effect of TNFα inhibitors against the development of liver fibrosis in patients with PsA.

Stroke, coronary and peripheral artery disease survey on antithrombotic treatment (START IT)

QUESTIONS UNDER STUDY: To determine the perception of primary care physicians regarding the risk of subsequent atherothrombotic events in patients with established cardiovascular (CV) disease, and to correlate this perception with documented antithrombotic therapy. METHODS: In a cross-sectional study of the general practice population in Switzerland, 381 primary care physicians screened 127 040 outpatients during 15 consecutive workdays in 2006. Perception of subsequent atherothrombotic events in patients with established CV disease was assessed using a tick box questionnaire allowing choices between low, moderate, high or very high risk. Logistic regression models were used to determine the relationship between risk perception and antithrombotic treatment. RESULTS: Overall, 13 057 patients (10.4%) were identified as having established CV disease and 48.8% of those were estimated to be at high to very high risk for subsequent atherothrombotic events. Estimated higher risk for subsequent atherothrombotic events was associated with a shift from aspirin monotherapy to clopidogrel, vitamin K antagonist or aspirin plus clopidogrel (p <0.001 for trend). Clopidogrel (12.7% vs 6.8%, p <0.001), vitamin K antagonist (24.5% vs 15.6%, p <0.001) or aspirin plus clopidogrel (10.2% vs 4.2%, p <0.001) were prescribed in patients estimated to be at high to very high risk more often than in those at low to moderate risk. CONCLUSIONS: Perception of primary care physicians regarding risk of subsequent atherothrombotic events varies in patients with CV disease, and as a result antithrombotic therapy is altered in patients with anticipated high to very high risk even though robust evidence and clear guidelines are lacking.

Risk factors promoting hypertensive crises: evidence from a longitudinal study

Background Current knowledge about risk factors promoting hypertensive crisis originates from retrospective data. Therefore, potential risk factors of hypertensive crisis were assessed in a prospective longitudinal study. Methods Eighty-nine patients of the medical outpatient unit at the University Hospital of Bern (Bern, Switzerland) with previously diagnosed hypertension participated in this study. At baseline, 33 potential risk factors were assessed. All patients were followed-up for the outcome of hypertensive crisis. Cox regression models were used to detect relationships between risk factors and hypertensive crisis (defined as acute rise of systolic blood pressure (BP) ≥200mmHg and/or diastolic BP ≥120mmHg). Results The mean duration of follow-up was 1.6 ± 0.3 years (range 1.0–2.4 years). Four patients (4.5%) were lost to follow-up. Thirteen patients (15.3%) experienced hypertensive crisis during follow-up. Several potential risk factors were significantly associated with hypertensive crisis: female sex, higher grades of obesity, the presence of a hypertensive or coronary heart disease, the presence of a somatoform disorder, a higher number of antihypertensive drugs, and nonadherence to medication. As measured by the hazard ratio, nonadherence was the most important factor associated with hypertensive crisis (hazard ratio 5.88, 95% confidence interval 1.59–21.77, P < 0.01). Conclusions This study identified several potential risk factors of hypertensive crisis. Results of this study are consistent with the hypothesis that improvement of medical adherence in antihypertensive therapy would help to prevent hypertensive crises. However, larger studies are needed to assess potential confounding, other risk factors and the possibility of interaction between predictors.

A longitudinal three-center study of dental arch relationship in patients with bilateral cleft lip and palate

OBJECTIVE: To compare and evaluate longitudinally the dental arch relationships from 4.5 to 13.5 years of age with the Bauru-BCLP Yardstick in a large sample of patients with bilateral cleft lip and palate (BCLP). DESIGN: Retrospective longitudinal intercenter outcome study. PATIENTS: Dental casts of 204 consecutive patients with complete BCLP were evaluated at 6, 9, and 12 years of age. All models were identified only by random identification numbers. SETTING: Three cleft palate centers with different treatment protocols. MAIN OUTCOME MEASURES: Dental arch relationships were categorized with the Bauru-BCLP yardstick. Increments for each interval (from 6 to 9 years, 6 to 12 years, and 9 to 12 years) were analyzed by logistic and linear regression models. RESULTS: There were no significant differences in outcome measures between the centers at age 12 or at age 9. At age 6, center B showed significantly better results (p=.027), but this difference diminished as the yardstick score for this group increased over time (linear regression analysis), the difference with the reference category (center C, boys) for the intervals 6 to 12 and 9 to 12 years being 10.4% (p=.041) and 12.9% (p=.009), respectively. CONCLUSIONS: Despite different treatment protocols, dental arch relationships in the three centers were comparable in final scores at age 9 and 12 years. Delaying hard palate closure and employing infant orthopedics did not appear to be advantageous in the long run. Premaxillary osteotomy employed in center B appeared to be associated with less favorable development of the dental arch relationship between 9 and 12 years.

Religion and assisted and non-assisted suicide in Switzerland: National Cohort Study

Background In the 19th century, eminent French sociologist Emile Durkheim found suicide rates to be higher in the Protestant compared with the Catholic cantons of Switzerland. We examined religious affiliation and suicide in modern Switzerland, where assisted suicide is legal. Methods The 2000 census records of 1 722 456 (46.0%) Catholics, 1 565 452 (41.8%) Protestants and 454 397 (12.2%) individuals with no affiliation were linked to mortality records up to December 2005. The association between religious affiliation and suicide, with the Protestant faith serving as the reference category, was examined in Cox regression models. Hazard ratios (HRs) with 95% confidence intervals (CIs) were adjusted for age, marital status, education, type of household, language and degree of urbanization. Results Suicide rates per 100 000 inhabitants were 19.7 in Catholics (1664 suicides), 28.5 in Protestants (2158 suicides) and 39.0 in those with no affiliation (882 suicides). Associations with religion were modified by age and gender (P < 0.0001). Compared with Protestant men aged 35–64 years, HRs (95% CI) for all suicides were 0.80 (0.73–0.88) in Catholic men and 1.09 (0.98–1.22) in men with no affiliation; and 0.60 (0.53–0.67) and 1.96 (1.69–2.27), respectively, in men aged 65–94 years. Corresponding HRs in women aged 35–64 years were 0.90 (0.80–1.03) and 1.46 (1.25–1.72); and 0.67 (0.59–0.77) and 2.63 (2.22–3.12) in women aged 65–94 years. The association was strongest for suicides by poisoning in the 65–94-year-old age group, the majority of which was assisted: HRs were 0.45 (0.35–0.59) for Catholic men and 3.01 (2.37–3.82) for men with no affiliation; 0.44 (0.36–0.55) for Catholic women and 3.14 (2.51–3.94) for women with no affiliation. Conclusions In Switzerland, the protective effect of a religious affiliation appears to be stronger in Catholics than in Protestants, stronger in older than in younger people, stronger in women than in men, and particularly strong for assisted suicides.

Sonographic follow-up of patients with carpal tunnel syndrome undergoing surgical or nonsurgical treatment: prospective cohort study

Purpose To compare changes in the largest cross-sectional area (CSA) of the median nerve in wrists undergoing surgical decompression with changes in wrists undergoing non-surgical treatment of carpal tunnel syndrome (CTS). Methods This study was a prospective cohort study in 55 consecutive patients with 78 wrists with established CTS, including 60 wrists treated with surgical decompression and 18 wrists with non-surgical treatment. A sonographic examination was scheduled before and 4 months after initiation of treatment. We compared changes in CSA of the median nerve between wrists with surgical treatment and wrists with non-surgical treatment using linear regression models. Results Decreases in CSA of the median nerve were more pronounced in wrists with CTS release than in wrists undergoing nonsurgical treatment (difference in means, 1.0 mm2; 95% confidence interval, 0.3–1.8 mm2). Results were robust to the adjustment for age, gender, and neurological severity at baseline. Among wrists with CTS release, those with postoperative CSA of 10 mm2 or less tended to have better clinical outcomes than those with postoperative CSA of greater than 10 mm2 (p=.055). Postoperative sonographic workup in the 3 patients with unfavorable outcome or recurrence identified likely causes for treatment failure in 2 patients. Conclusions In this observational study, surgical decompression was associated with a greater decrease in median nerve CSA than was nonsurgical treatment. Smaller postoperative CSAs may be associated with better clinical outcomes. Additional randomized trials are necessary to determine the optimal treatment strategy in different subgroups of patients with CTS. Type of study/level of evidence Therapeutic III.