Order selection tests with multiply-imputed data.

We develop nonparametric tests for the null hypothesis that a function has a prescribed form, to apply to data sets with missing observations. Omnibus nonparametric tests do not need to specify a particular alternative parametric form, and have power against a large range of alternatives, the order selection tests that we study are one example. We extend such order selection tests to be applicable in the context of missing data. In particular, we consider likelihood-based order selection tests for multiply- imputed data. A simulation study and data analysis illustrate the performance of the tests. A model selection method in the style of Akaike's information criterion for multiply imputed datasets results along the same lines.Akaike information criterion; Hypothesis test; Multiple imputation; lack-of-fit test; Missing data; Omnibus test; Order selection;

Multiple imputation in Cox regression when there are time-varying effects of covariates.

In Cox regression, it is important to test the proportional hazards assumption and sometimes of interest in itself to study time-varying effects (TVEs) of covariates. TVEs can be investigated with log hazard ratios modelled as a function of time. Missing data on covariates are common and multiple imputation is a popular approach to handling this to avoid the potential bias and efficiency loss resulting from a "complete-case" analysis. Two multiple imputation methods have been proposed for when the substantive model is a Cox proportional hazards regression: an approximate method (Imputing missing covariate values for the Cox model in Statistics in Medicine (2009) by White and Royston) and a substantive-model-compatible method (Multiple imputation of covariates by fully conditional specification: accommodating the substantive model in Statistical Methods in Medical Research (2015) by Bartlett et al). At present, neither accommodates TVEs of covariates. We extend them to do so for a general form for the TVEs and give specific details for TVEs modelled using restricted cubic splines. Simulation studies assess the performance of the methods under several underlying shapes for TVEs. Our proposed methods give approximately unbiased TVE estimates for binary covariates with missing data, but for continuous covariates, the substantive-model-compatible method performs better. The methods also give approximately correct type I errors in the test for proportional hazards when there is no TVE and gain power to detect TVEs relative to complete-case analysis. Ignoring TVEs at the imputation stage results in biased TVE estimates, incorrect type I errors, and substantial loss of power in detecting TVEs. We also propose a multivariable TVE model selection algorithm. The methods are illustrated using data from the Rotterdam Breast Cancer Study. R code is provided

Handling protest responses in contingent valuation surveys

OBJECTIVES: Protest responses, whereby respondents refuse to state the value they place on the health gain, are commonly encountered in contingent valuation (CV) studies, and they tend to be excluded from analyses. Such an approach will be biased if protesters differ from non-protesters on characteristics that predict their responses. The Heckman selection model has been commonly used to adjust for protesters, but its underlying assumptions may be implausible in this context. We present a multiple imputation (MI) approach to appropriately address protest responses in CV studies, and compare it with the Heckman selection model. METHODS: This study exploits data from the multinational EuroVaQ study, which surveyed respondents' willingness-to-pay (WTP) for a Quality Adjusted Life Year (QALY). Here, our simulation study assesses the relative performance of MI and Heckman selection models across different realistic settings grounded in the EuroVaQ study, including scenarios with different proportions of missing data and non-response mechanisms. We then illustrate the methods in the EuroVaQ study for estimating mean WTP for a QALY gain. RESULTS: We find that MI provides lower bias and mean squared error compared with the Heckman approach across all considered scenarios. The simulations suggest that the Heckman approach can lead to considerable underestimation or overestimation of mean WTP due to violations in the normality assumption, even after log-transforming the WTP responses. The case study illustrates that protesters are associated with a lower mean WTP for a QALY gain compared with non-protesters, but that the results differ according to method for handling protesters. CONCLUSIONS: MI is an appropriate method for addressing protest responses in CV studies

Comparison of techniques for handling missing covariate data within prognostic modelling studies: a simulation study

Background: There is no consensus on the most appropriate approach to handle missing covariate data within prognostic modelling studies. Therefore a simulation study was performed to assess the effects of different missing data techniques on the performance of a prognostic model. Methods: Datasets were generated to resemble the skewed distributions seen in a motivating breast cancer example. Multivariate missing data were imposed on four covariates using four different mechanisms; missing completely at random (MCAR), missing at random (MAR), missing not at random (MNAR) and a combination of all three mechanisms. Five amounts of incomplete cases from 5% to 75% were considered. Complete case analysis (CC), single imputation (SI) and five multiple imputation (MI) techniques available within the R statistical software were investigated: a) data augmentation (DA) approach assuming a multivariate normal distribution, b) DA assuming a general location model, c) regression switching imputation, d) regression switching with predictive mean matching (MICE-PMM) and e) flexible additive imputation models. A Cox proportional hazards model was fitted and appropriate estimates for the regression coefficients and model performance measures were obtained. Results: Performing a CC analysis produced unbiased regression estimates, but inflated standard errors, which affected the significance of the covariates in the model with 25% or more missingness. Using SI, underestimated the variability; resulting in poor coverage even with 10% missingness. Of the MI approaches, applying MICE-PMM produced, in general, the least biased estimates and better coverage for the incomplete covariates and better model performance for all mechanisms. However, this MI approach still produced biased regression coefficient estimates for the incomplete skewed continuous covariates when 50% or more cases had missing data imposed with a MCAR, MAR or combined mechanism. When the missingness depended on the incomplete covariates, i.e. MNAR, estimates were biased with more than 10% incomplete cases for all MI approaches. Conclusion: The results from this simulation study suggest that performing MICE-PMM may be the preferred MI approach provided that less than 50% of the cases have missing data and the missing data are not MNAR

Peroxiredoxin 4, a novel circulating biomarker for oxidative stress and the risk of incident cardiovascular disease and all-cause mortality

BACKGROUND: Oxidative stress has been suggested to play a key role in the development of cardiovascular disease (CVD). The aim of our study was to investigate the associations of serum peroxiredoxin 4 (Prx4), a hydrogen peroxide-degrading peroxidase, with incident CVD and all-cause mortality. We subsequently examined the incremental value of Prx4 for the risk prediction of CVD compared with the Framingham risk score (FRS). METHODS AND RESULTS: We performed Cox regression analyses in 8141 participants without history of CVD (aged 28 to 75 years; women 52.6%) from the Prevention of Renal and Vascular End-stage Disease (PREVEND) study in Groningen, The Netherlands. Serum Prx4 was measured by an immunoluminometric assay in baseline samples. Main outcomes were: (1) incident CVD events or CVD mortality and (2) all-cause mortality during a median follow-up of 10.5 years. In total, 708 participants (7.8%) developed CVD events or CVD mortality, and 517 participants (6.3%) died. Baseline serum Prx4 levels were significantly higher in participants with incident CVD events or CVD mortality and in those who died than in participants who remained free of outcomes (both P<0.001). In multivariable models with adjustment for Framingham risk factors, hazard ratios were 1.16 (95% CI 1.06 to 1.27, P<0.001) for incident CVD events or CVD mortality and 1.17 (95% CI 1.06 to 1.29, P=0.003) for all-cause mortality per doubling of Prx4 levels. After the addition of Prx4 to the FRS, the net reclassification improvement was 2.7% (P=0.01) using 10-year risk categories of CVD. CONCLUSIONS: Elevated serum Prx4 levels are associated with a significantly higher risk of incident CVD events or CVD mortality and all-cause mortality after adjustment for clinical risk factors. The addition of Prx4 to the FRS marginally improved risk prediction of future CVD

The East German Wage Structure after Transition

We extend the literature on transition economies' wage structures by investigating the returns to tenure and experience. This study applies recent panel data and estimation approaches that control for hitherto neglected biases. We compare the life cycle structure in East and West German wages for fulltime employed men in the private sector. The patterns in the returns to seniority are similar for the two regional labor markets. The returns to experience lag behind in the East German labor market, even almost 20 years after unification. The results are robust when only individuals are considered who started their labor market career in the market economy and they hold across skill groups.wage structure, life cycle earnings, returns to tenure, returns to experience