- Research Article
- Open Access
- Open Peer Review
Simulation of complex data structures for planning of studies with focus on biomarker comparison
- Andreas Schulz^{1, 3}Email authorView ORCID ID profile,
- Daniela Zöller^{2},
- Stefan Nickels^{6},
- Manfred E. Beutel^{7},
- Maria Blettner^{2},
- Philipp S. Wild^{1, 3, 4, 5} and
- Harald Binder^{8}
https://doi.org/10.1186/s12874-017-0364-y
© The Author(s) 2017
- Received: 20 July 2016
- Accepted: 24 May 2017
- Published: 13 June 2017
Abstract
Background
There are a growing number of observational studies that do not only focus on single biomarkers for predicting an outcome event, but address questions in a multivariable setting. For example, when quantifying the added value of new biomarkers in addition to established risk factors, the aim might be to rank several new markers with respect to their prediction performance. This makes it important to consider the marker correlation structure for planning such a study. Because of the complexity, a simulation approach may be required to adequately assess sample size or other aspects, such as the choice of a performance measure.
Methods
In a simulation study based on real data, we investigated how to generate covariates with realistic distributions and what generating model should be used for the outcome, aiming to determine the least amount of information and complexity needed to obtain realistic results. As a basis for the simulation a large epidemiological cohort study, the Gutenberg Health Study was used. The added value of markers was quantified and ranked in subsampling data sets of this population data, and simulation approaches were judged by the quality of the ranking. One of the evaluated approaches, the random forest, requires original data at the individual level. Therefore, also the effect of the size of a pilot study for random forest based simulation was investigated.
Results
We found that simple logistic regression models failed to adequately generate realistic data, even with extensions such as interaction terms or non-linear effects. The random forest approach was seen to be more appropriate for simulation of complex data structures. Pilot studies starting at about 250 observations were seen to provide a reasonable level of information for this approach.
Conclusions
We advise to avoid oversimplified regression models for simulation, in particular when focusing on multivariable research questions. More generally, a simulation should be based on real data for adequately reflecting complex observational data structures, such as found in epidemiological cohort studies.
Keywords
- Simulation
- Planning
- Biomarker
- Risk prediction
- Added value
- Random forest
- Brier score
- AUC
- Nagelkerke- R ^{2}
Background
When planning a new, potentially large cohort study, simulations can help to judge aspects such as sample size or choice of a statistical approach that might be needed for adequately investigating the effect of biomarkers for an outcome of interest. In particular, such simulation studies allow to take potentially complex correlation structures, covariate distributions and potentially non-linear effects or interactions into account when investigating several biomarkers and known risk factors simultaneously. Therefore, a simulation study may be more adequate for example to assess the sample size needed than using probably oversimplifying sample size formulas. A simulation might also be useful beyond sample size planning, e.g. for picking good measures for biomarker performance. Since a simulation study could also be based on oversimplifying assumptions, at the risk of answers that are not better than, e.g., closed-form sample size formulas, it is of prime importance to use a data generating model of adequate complexity, reflecting realistic data structure. Naturally, this entails the danger of requiring a large amount of information about the population and covariate structure of interest, or introducing a considerable number of simulation parameters that cannot be selected adequately. In the following, this work focusses specifically on how to generate correlated covariates with realistic distribution and on what generating model should be used for a simulated outcome to deal with a complex multivariable research questions, exemplarily considering the task of ranking biomarkers with respect to their added value.
There is a lack of literature on sample size or power calculation methods with consideration of correlated multidimensional covariate data in a regression model and as well on a simulation methodology for this setting. Most of the established methods are using sample size formulas for a regression model with only one covariate [1, 2]. Schmoor et al. [3] proposed a sample size formula for a prognostic problem, where additionally a second correlated factor is considered. This requires knowledge about the joint distribution of these two factors and is restricted to a Cox proportional hazards model with one predictor of interest. Jinks et al. [4] derived a formula for a multivariable prognostic model based on the overall prognostic ability, where the prognostic ability is quantified by the measure of discrimination D and sample size calculation is based on the significance of the D value. Comparable approaches can be applied with the overall discrimination ability (AUC) [5] of the model, for which a sample size calculation can be derived as well [6, 7]. Other authors discuss methods for the problem, where the number of predictors is larger than the actual number of samples. This introduces a selection problem, where informative predictors have to be identified in a mixture of informative and non-informative predictors. De Valpine et al. [8] gave a two-step method, where in the first step a simulation is used to reproduce the selection process of informative predictors and in a second step an approximation method for a linear discriminant problem is used. A similar two step approach was developed by Dobbin et al. [9] but with another methodology. A further approach with a variable selection step was proposed in Götte et al. [10], where the sample size determination is focused on the prediction accuracy instead of power. Unfortunately, most of these approaches are based on uncorrelated variables. Binder et al. [11] investigated different scenarios with a small or large amount of information, different covariates distributions and non-linear functional forms of relationship. The simulation revealed the importance of aspects like covariates distributions or functional form and demonstrated the impact. However, the primary aim of that work was not the planning of new studies but comparing approaches for modeling of non-linear effects. Therefore, the present work specifically investigates the degree of complexity that may be required for a realistic simulation study and techniques to use for an adequate generating model.
We exemplarily consider settings with a binary outcome, which are frequently found in observational data for biomedical research questions considering disease risks. However, most aspects of the simulation method can be easily applied to a continuous outcome as well. As candidate technique for generating simulated covariate distributions, i.e. biomarkers and established predictors, two approaches were compared. Drawing from multivariate normal distributions or additionally transformation according to a known empirical distribution to mimic this distribution as exactly as possible. For generating the clinical outcome, standard linear models and extensions via non-linear terms and interactions were used. As a non-parametric approach the random forest model was considered, which requires individual data as basis. As gold standard, repeated sampling from a large population-based, epidemiological cohort, the Gutenberg Health Study (GHS) was used, and judged how close simulated data based on aggregate information, such as odds ratios or correlation matrices (which might be found e.g. in the literature), agree with the gold standard. The use of a pilot study [12] as basis for simulation was also considered, and the effect of the pilot study size was investigated. As a measure to assess the performance of simulation compared to the defined gold standard, the ranking of biomarkers based on simulated data and based on repeated draws from the population data, were compared. For ranking biomarkers according to their added value, the difference in Brier score, the increase in AUC and the difference in pseudo- R ^{2} were considered as added value measures.
In “Population sample” section the GHS study and the exemplary biomarkers and endpoints to be used for investigating simulation approaches are introduced. Concerning the latter, the overall simulation structure is presented in “The general simulation structure” and discuss simulation of covariates in “Covariate matrix” section, and different approaches for generating a simulated phenotype in “Clinical response generating” section. Different measures for added value are discussed in “Quantifying added value” section and the simulation quality criterion in “Reference ranking” section. The population results are presented in “Population sample resultsPopulation sam-ple results” section. Results on different strategies for simulation are presented in “Reference mean ranks” and in “Comparison of simulation approaches” sections. “Pilot sample size” section specifically investigates different pilot study sizes. Concluding remarks are given in “Discussion and Conclusions” sections.
Methods
Population sample
As an application example, the Gutenberg Health Study (GHS [13]) sample was used. The GHS is a population-based prospective, observational, single-center cohort study from Germany at the University Medical Center in Mainz. With the first 5000 participants enrolled from April 2007 to October 2008, it is so far a cross sectional, large, population based sample. The primary aim of the GHS study is to evaluate and improve cardiovascular risk prediction. The participants are aged between 35 and 74 years with nearly equal proportion of men and women. The sample was taken from the population in Mainz and Mainz-Bingen area in Germany. The whole study sample includes 15010 individuals. The analysis was restricted on the first 5000 enrolled participants due to the fact that the measurement of the biomarkers of interest was only accomplished for this subsample. After quality control and data cleaning using the complete case principle for the variables of interest, a sample with 4519 individuals remains. Most missing values occurred in the outcome. Missing values were randomly distributed and resulted mainly from logistical problems. Specifically, the binary variable functional cardiac disorder (FCD [14]) was used as medical outcome. The focus of this work was on a binary outcome, because this approach is commonly used in medicine and plays a more important role in risk prediction than continuous traits do. A basic prediction model for this event was defined as a simple model with sex, age and body-mass-index as covariates. This basic model was extended with different biomarkers. One biomarker was added at a time, and the improvement in prediction was evaluated with three different added value methods described in Quantifying added value. The following biomarkers of interest were selected in advance: MR-proADM, Nt-proBNP, hs-CRP, CT-proAVP and MR-proANP. The results in GHS sample are presented in more detail in 8.
The general simulation structure
Generating artificial data
Covariate matrix
For a successful simulation of a realistic sample, covariate data need to be simulated in a sufficient way. One of the major aspects for the simulation is the correlation structure of the covariate matrix. Not only the correlation between markers and basic model covariates is important, the correlation between markers themselves has a key role. Consequently, the whole correlation matrix including all markers of interest and known risk factors must be taken into account. A natural way to simulate the distribution of covariates while simultaneously considering a correlation structure is to use a multivariate normal distribution based on the correlation matrix, location and dispersion of real data. Where a predefined covariance matrix can be given to determine the variance as well as the correlation structure of generated data [20], this covariance matrix can be easily obtained from the population data. Since the dichotomous variable sex was included into the model, all random covariate data were generated sex-specifically and pooled afterwards. As all covariates of interest, except sex, were nearly normally distributed or log-transformed to approximate the normal distribution. Though the method covers the correlation structure of the original data well, variables are rarely exact normally distributed in a real data set. A more exact method to mimic the real distribution is to generate the covariate matrix with multivariate normal distribution and to take the corresponding quantiles from the empirical distribution of a real data set. It requires more information, but reflects even small deviations from the normal distribution if present, without destroying the correlation structure. By using this method, artificial data with a correct correlation structure can be generated that perfectly mimic the true distribution. This can be seen as the most realistic and sophisticated simulation method of the covariate data. In the simulation of this work, both approaches and the benefit of these additional efforts were explored.
Clinical response generating
For the generation of the clinical response, the relationship between outcome and the covariate matrix has to be taken into account as accurate as possible. For this purpose, prediction models for the outcome were fitted in population data using different modeling approaches. These prediction models are then used to predict the probabilities of the event given the artificial, simulated covariate data. To generate the outcome for the simulation data set, random numbers were generated from the binomial distribution given the predicted probabilities. Even if the biomarker comparison is made by a simple logistic regression model, more complex models can be used for the simulation of the relationship. Additionally, in terms of comparison of markers it is essential that the association to the outcome is simulated considering all markers simultaneously. For that purpose, the prediction models for the binary endpoint in the population data were fitted using all markers and covariates in one model. For the simulation, a set of different models with rising complexity were selected. The simplest approach would be the modeling of the relationship in a linear way with a logistic regression (GLM). Additional interaction effects can be taken in account, which leads to a GLM model with interaction terms (GLM+I). For this model, a set of the six strongest pairwise interaction effects with a stepwise bidirectional selection method based on AIC (Akaike information criterion) [21] was selected and added to the GLM model. Following interactions were selected automatically: sex with age, CT-proAVP with age, Nt-proBNP with MR-proANP and interactions of MR-proADM with BMI, Nt-proBNP, CT-proAVP and MR-proANP. Where even not significant biomarkers were included in interactions and seem to improve the global model. The resulting information could be important because the interactions between the markers cannot be detected in a prediction model including only a single marker, but this may influence the overall results of the simulation. If the relationship between the outcome and the biomarkers is in reality a non-linear relationship, a non-linear modeling would be more appropriate. Generalized additive model with smoothing splines (GAM) [22] was used to model complex non-linear relationships if present, but interactions are omitted in this approach. To cover non-linear relationships and complex interaction structures simultaneously, a more complex model could be necessary. One possibility is to use the random forest model (RF) [23], based on classification trees (CART) [24–26]. For the random forest models, no pruning step was performed, so all trees were maximal grown trees. The number of trees was set to 1000 and the number of variables randomly sampled as candidates at each split was \(\left \lceil \sqrt {p}\right \rceil \) [27], where p is the number of predictors available. This method is described in more details in the next section 8. In this work only both methods to generate the covariate data with the GLM and random forest approaches are presented. For all other approaches, only covariate data simulated based on the empirical distribution are presented as this leads to better simulation results.
Random Forest approach
- 1.
Select randomly a total of N ^{∗} individuals from the original training sample, with replacement. This leads to a bootstrap sample. Repeat this procedure B times.
- 2.
In each bootstrap sample, grow an unpruned classification tree. The tree is constructed by recursively splitting data into two distinct sub-samples. At each node, randomly select mtry predictors from the total m predictor variables. Choose the best split from among the mtry predictors by minimize the Gini-index as a measure of node purity.
- 3.
For calculation of predicted probability, each new individual is dropped down a tree until its terminal node. The majority voting for an event status in this terminal node is determined. The probability estimate is then the average of majority votes over all trees.
Gold standard
Quantifying added value
For comparing the predictive strength of biomarkers, the concept of added value that describes the prediction performance of a model was chosen. This can be measured with several different established measurements. Three of them without intend to have a complete list of existing measures were selected. The first one, the Brier score [31, 32], measures the mean squared difference between the predicted probability and the actual outcome. It takes values between zero and one, since this is the largest possible difference between a predicted probability and a binary outcome. The lower the Brier score, the better the prediction performance. The Brier score is defined for a binary outcome as \(BS=\frac {1}{n}\sum _{i=1}^{n}(p_{i}-y_{i})^{2}\), where p is the predicted probability, n is the sample size and y is the actual, observed outcome. The second common measure is the area under the curve AUC [5, 7, 32–34] from Receiver Operating Characteristic (ROC) methodology which quantifies the discrimination ability. It can be interpreted as the probability that a randomly selected subject with an event will be ranked higher in terms of predicted probability than a randomly selected subject without an event. One possible definition of AUC is given by \(AUC=\frac {1}{n_{1}n_{0}}\left (\sum _{i=1}^{n}(rank(p_{i})y_{i}) - \frac {n_{1}^{2} + n_{1}}{2}\right)\), where n _{1} is number of events and n _{0} is the number of non-events. Third, the coefficient of determination R ^{2}, in this case for a binary outcome the generalization of R ^{2} for generalized linear models from Nagelkerke [35] was used. Nagelkerke R ^{2} coefficient is scaled to a minimum of 0 for no determination and a maximum of 1 for perfect determination. The definition of R ^{2} with log-likelihood is \(\frac {1}{e^{(-2LL_{0}/n)}-1} \left (e^{((-2LL_{1} + 2LL_{0})/n))}-1\right)\) where the L L _{0} is the log-likelihood from the null model only with the intercept term and L L _{1} is the log-likelihood from the model of interest.
Reference ranking
As a criterion for simulation success, the relative ranking of biomarkers was used. It reflects the biomarker comparison study aims in a direct and intuitive way and allows the comparison of the results from different added value measures. Therefore, the top three markers were ranked by each added value measure separately. The simulation is restricted to the top three markers because the others have very small to non-existent effects, see 8. As the basic prediction model is used as a reference for all markers, ranking based on the added value measure itself or on the difference of it lead to the same ranking. To get reference rankings, a resampling method on population data was used, in this case the target criterion is the mean rank of markers. To access the mean rank, 10000 bootstrap samples from population data were drawn. By bootstrapping with real data the distribution and correlation structure of population data is considered in a natural way. For each bootstrap sample, the added value measures were calculated and compared between each biomarker. This leads to a specific rank in each bootstrap sample for each of the top three markers. The mean rank of a marker from all bootstrap samples is used as the reference. This reference has to be replicated in the artificial data in the simulation to ensure the applicability of the simulation approach. The resulting reference rankings are shown in 8. Additionally, the similarity of absolute added values from simulation results to reference values as a more specific criterion was examined. A good consistency in absolute values would demonstrate even better simulation accuracy.
Results
Population sample results
To ensure a stable and well specified model for risk prediction, model diagnostics were carried out in the population data for all models, including the basic model, the extended model with one biomarker and a full model with all markers simultaneously. Model diagnostic covers calibration, influential observations and collinearity. The event frequency was 26.7% (n=1205 o f N=4519), so even in a full model there were 150 events per covariate. The overall calibration was good in all models as the mean predicted probability ranged between 26.02% and 26.87%. The calibration in subgroups of prediction was similar good in all models and only led to a significant Hosmer-Lemeshow test [37] (p=0.042) in one model, which shouldn’t be over-interpreted with this large sample size. By using the Cook’s distance [38, 39], no strong influential observations were detected. The Cook’s distanced is based on a single case deletion statistic which quantifies the changes of estimates after removing a single observation.
The variance inflation factor (VIF) [40, 41] was used to examine the collinearity. Values of the VIF greater than 5 were considered as problematic. In the single extended models with only one additional marker, no VIF values over 2 were observed. In the full model with all biomarkers incorporated simultaneously, the maximum VIF was 2.6 for MR-proANP. In summary, the basic model was stable and well calibrated; there were no collinearity problems and no strong outliers or influential observations after the log-transformation of markers.
Logistic regression models: results from population data
Marker | OR per-SD | Lower 95%CI | Upper 95%CI | p-value | Brier score diff. | Increase in AUC | R ^{2} difference |
---|---|---|---|---|---|---|---|
MR-proADM | 1.18 | 1.08 | 1.30 | 0.00035 | -0.00048 | 0.00232 | 0.00349 |
Nt-proBNP | 1.15 | 1.05 | 1.25 | 0.0017 | -0.00033 | 0.00151 | 0.00272 |
hs-CRP | 1.08 | 1.00 | 1.17 | 0.044 | -0.00024 | 0.00084 | 0.00111 |
CT-proAVP | 1.04 | 0.96 | 1.12 | 0.36 | 0.00000 | 0.00014 | 0.00022 |
MR-proANP | 0.99 | 0.91 | 1.07 | 0.75 | 0.00000 | -0.00004 | 0.00003 |
Reference mean ranks
Mean ranks
Ranking measure | MR-proADM | Nt-proBNP | hs-CRP |
---|---|---|---|
Brier score difference | 1.57 | 1.99 | 2.43 |
Increase in AUC | 1.38 | 1.96 | 2.66 |
R ^{2} difference | 1.45 | 1.86 | 2.69 |
The mean rank for MR-proADM ranges from 1.38 using AUC to 1.57 using the Brier score. Using the R ^{2} resulted in a mean rank of 1.45, which lies somewhere in between. Consequently, the AUC shows a stronger ability to separate the top marker than the Brier score or the R ^{2} in the population data. For Nt-proBNP, Brier score and AUC were about 2, the R ^{2} provided a little smaller mean rank of 1.86 and thus a little higher ranking. The mean rank of the third marker is not important, because it’s already completely determined with the first two ranking positions. It is not exactly clear how the differences can be explained. One possibility could be the fact that different measures represent different aspects of prediction like discrimination or calibration. Another source could be small deviations from the assumptions like linearity or normal distribution, with a heterogeneous effect on the different measurements. The second explanation would make it even more important to cover these aspects in the simulation.
Comparison of simulation approaches
Mean ranks criterion
Absolute values criterion
Pilot sample size
Discussion
In this work we investigated the necessary tools to perform simulation of realistic data with complex structure, e.g. to plan a new study for biomarker comparison or for other subject matter questions corresponding to a multivariable analysis. Except for one specific performance measure (the Nagelkerke R ^{2}), our results indicate a good simulation should be based on individual level pilot data, as the detailed information required for an adequate simulation cannot be extracted from typical aggregate results given in prior publications. Artificial data from theoretical distributions fail to represent the real situation properly and can lead to wrong results. Even small violations of assumptions, which are often negligible in other situations, could be essential for simulation results. Based on a comparison of four different approaches for generating the outcome, the random forest approach (using covariate data drawn from the empirical distribution) seemed to be the most successful. These results are also valid if one takes into account the variation of the rank, which was comparable between the different approaches. Unfortunately, this is the most complex model considered and as all machine learning methods it requires more data [42]. It leads us to the conclusion that non-linear relationship and interaction are playing an important role. Although our reference sample had no clear non-linear relationships, there may be small but nevertheless relevant deviations from linearity. For the analysis of a sample it is common to use a linear model even for a relationship that may be not perfectly linear in order to avoid overfitting, but for a simulation setting is seems to be rather crucial to also consider small deviations from linearity. As the basic model was extended only with one marker at a time, interactions between biomarkers would often be missed in an analysis, but they are playing an important role for the marker ranking in simulations. We think the appropriate modeling of the relationship is a more important aspect than the distribution of covariate data, as long as the correlation structure was adequately reflected. This last part is achieved in a straightforward manner by using covariate data drawn from the empirical distribution, although this requires more information. Nevertheless, we think that this effort is worthwhile as it results in better simulation results and does not need to be adapted if the distribution is non-normal. Additionally, the random forest approach does also need the information on the individual level and thus using the empirical distribution is no additional expenditure in this case. One known disadvantage of random forest approach is the black box nature of the model, but as our aim is to perform a data based simulation, there is no need for a good interpretability of generating model. The model must only adequately reflect the data structure and deliver realistic simulated outcomes. Another aspect to be considered when using the random forest approach is the appropriate choice of tuning parameters, such as the tree size. The difficulty is to choose these parameters in a way that ensures sufficient degrees of freedom, but avoids the overfitting at the same time. The tree size determines the interaction order of the model, so trees with only one split, so called stumps, result in a model without interactions. Each additional split increases the order of possible interactions. This parameter should not be too small; otherwise the model fails to reproduce the interactions in the data. In case of simulations, the choice of adequate tree size is not very critical as long as the model has sufficient degrees of freedom, because a certain amount of overfitting is even desirable. The model must not be generalizable for new data, on the contrary, it must simulate the data with their special characteristics such as random noise as well as possible. Consequently, the pruning of the trees might not be as important in this case. Instead of recommending an added value measure, we recommend to include the choice of the added value measure into the planning phase. For example, a better separation between the markers was possible when using IAUC, but the Brier score difference had a better precision in results. Only the Nagelkerke R ^{2} difference is not recommended as the results could not be recovered well within a simulation study. Another aspect that should be taken into account when choosing the measure is the sample size of a pilot study or interim sample study. With the right choice of the measure, the required sample size could be reduced. From this point of view, the Brier score difference can be recommended. We used the improvement of prediction in nested models with adding a new single marker, but this concept could be used as well for improvement with several new markers simultaneously. This could be important when investigating weak markers, like genetic SNP markers for example [43], where interaction between markers could play an even more important role.
In addition to the scenarios investigated here, other data analysis scenarios might be considered, e.g. a comparison of two sets of markers or a set of markers with one single marker. We expect that our results also transfer to such other multivariable settings to some extent, as the strategy presented here ensures that the correlation structure with exact covariate distribution is considered as well as the relationship to event with interactions and non-linearity. Naturally, our results may strongly depend on the specific data source, the Gutenberg Health Study (GHS), and the specific markers and outcome considered in our investigation, but we expect that the level of complexity seen there is not unusual and needs to be anticipated when setting up a large cohort study.
Conclusions
Generalizing from the present results, we would not recommend potentially oversimplified regression models for representing the relationship between markers and the outcome when simulating complex data. It seems that more flexible approaches, such as random forest, may be more appropriate for adequate simulation of complex multivariate data. This better takes into account important aspects such as non-linear relationships or interactions and therefore provides more adequate results. Yet, these methods require information on individual level and thereby a pilot study or other preliminary data sources. Additionally, the results of the simulation emphasize that some not readily apparent properties of the underlying data structure can affect (the performance of) marker identification. This also is an important lesson for other situations where realistic data structure needs to be simulated.
Declarations
Acknowledgements
We thank all study participants for their willingness to provide data for this research project and we are indebted to all coworkers for their enthusiastic commitment.
Funding
The Gutenberg Health Study is funded through the government of Rhineland-Palatinate (“Stiftung Rheinland-Pfalz für Innovation”, contract AZ 961-386261/733), the research programs “Wissen schafft Zukunft” and “Center for Translational Vascular Biology (CTVB)” of the Johannes Gutenberg-University of Mainz, and its contract with Boehringer Ingelheim and PHILIPS Medical Systems, including an unrestricted grant for the Gutenberg Health Study. PSW is funded by the Federal Ministry of Education and Research (BMBF 01EO1503) and he is PI of the German Center for Cardiovascular Research (DZHK).
Availability of data and materials
Data sharing is not applicable as the used data is a part of a big observational cohort study which has not yet been completed.
Authors’ contributions
AS carried out the simulation study and drafted the manuscript. HB and PSW supervised the project. DZ revised the simulation procedure and statistical methodology. MB, MEB and SN contributed critical revision and discussion. All authors read and approved the final manuscript.
Authors’ information
This publication is a part of the doctoral thesis of Andreas Schulz.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
All individuals signed informed consent before participating. The study protocol and study documents were approved by the local ethics committee of the Medical Chamber of Rhineland-Palatinate, Germany (reference no. 837.020.07; original vote: 22.3.2007, latest update: 20.10.2015).
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References
- Vaeth M, Skovlund E. A simple approach to power and sample size calculations in logistic regression and Cox regression models. Stat Med. 2004; 23(11):1781–92. doi:10.1002/sim.1753.View ArticlePubMedGoogle Scholar
- Schoenfeld DA. Sample-size formula for the proportional-hazards regression model. Biometrics. 1983; 39(2):499–503. doi:10.2307/2531021.View ArticlePubMedGoogle Scholar
- Schmoor C, Sauerbrei W, Schumacher M. Sample size considerations for the evaluation of prognostic factors in survival analysis. Stat Med. 2000; 19(4):441–52. doi:10.1002/(SICI)1097-0258(20000229).View ArticlePubMedGoogle Scholar
- Jinks RC, Royston P, Parmar MK. Discrimination-based sample size calculations for multivariable prognostic models for time-to-event data. BMC Med Res Methodol. 2015; 15(1):82. doi:10.1186/s12874-015-0078-y.View ArticlePubMedPubMed CentralGoogle Scholar
- Chen W, Samuelson FW, Gallas BD, Kang L, Sahiner B, Petrick N. On the assessment of the added value of new predictive biomarkers. BMC Med Res Methodol. 2013; 13(1):1–9. doi:10.1186/1471-2288-13-98.View ArticleGoogle Scholar
- Obuchowski NA. Computing sample size for receiver operating characteristic studies. Investig Radiol. 1994; 29(2):238–43. doi:10.1097/00004424-199402000-00020.View ArticleGoogle Scholar
- Hanley JA, McNeil BJ. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology. 1982; 143(1):29–36. doi:10.1148/radiology.143.1.7063747.View ArticlePubMedGoogle Scholar
- De Valpine P, Bitter HM, Brown MPS, Heller J. A simulation-approximation approach to sample size planning for high-dimensional classification studies. Biostatistics. 2009; 10(3):424–35. doi:10.1093/biostatistics/kxp001.View ArticlePubMedPubMed CentralGoogle Scholar
- Dobbin KK, Simon RM. Sample size planning for developing classifiers using high-dimensional DNA microarray data. Biostatistics (Oxford, England). 2007; 8(1):101–17. doi:10.1093/biostatistics/kxj036.View ArticleGoogle Scholar
- Götte H, Zwiener I. Sample size planning for survival prediction with focus on high-dimensional data. Stat Med. 2013; 32(5):787–807. doi:10.1002/sim.5550.View ArticlePubMedGoogle Scholar
- Binder H, Sauerbrei W, Royston P. Comparison between splines and fractional polynomials for multivariable model building with continuous covariates: A simulation study with continuous response. Stat Med. 2013; 32(13):2262–77. doi:10.1002/sim.5639.View ArticlePubMedGoogle Scholar
- Thabane L, Ma J, Chu R, Cheng J, Ismaila A, Rios LP, Robson R, Thabane M, Giangregorio L, Goldsmith CH. A tutorial on pilot studies: the what, why and how. BMC Med Res Methodol. 2010; 10(10):1–10.View ArticlePubMedPubMed CentralGoogle Scholar
- Wild PS, Zeller T, Beutel M, Blettner M, Dugi Ka, Lackner KJ, Pfeiffer N, Münzel T, Blankenberg S. [The Gutenberg Health Study]. Bundesgesundheitsblatt, Gesundheitsforschung, Gesundheitsschutz. 2012; 55(6-7):824–9. doi:10.1007/s00103-012-1502-7.View ArticlePubMedGoogle Scholar
- Wild PS, Sinning CR, Roth A, Wilde S, Schnabel RB, Lubos E, Zeller T, Keller T, Lackner KJ, Blettner M, Vasan RS, Münzel TF, Blankenberg S. Distribution and categorization of left ventricular measurements in the general population: results from the population-based gutenberg-heart study. Circ Cardiovasc Imaging. 2010;604–13. doi:.10.1161/CIRCIMAGING.109.911933
- Burton A, Altman DG, Royston P, Holder RL. The design of simulation studies in medical statistics. Stat Med. 2006; 25:4279–92. doi:10.1002/sim.View ArticlePubMedGoogle Scholar
- R Developement Core Team. R: A Language and Environment for Statistical Computing. 2015. doi:10.1007/978-3-540-74686-7. http://www.r-project.org
- Genz A, Bretz F, Miwa T, Mi X, Leisch F, Scheipl F, Hothorn T. mvtnorm: Multivariate Normal and T Distributions. 2016. R package version 1.0-5. http://CRAN.R-project.org/package=mvtnorm. Accessed 15 Apr 2016http://CRAN.R-project.org/package=mvtnorm.
- Genz A, Bretz F. Computation of Multivariate Normal and T Probabilities, 1st: Springer Publishing Company, Incorporated; 2009, pp. 1682–90. doi:10.1007/s13398-014-0173-7.2.
- Liaw A, Wiener M. Classification and regression by randomforest. R news. 2002; 2:18–22. doi:10.1177/154405910408300516.Google Scholar
- Ripley BD. Stochastic Simulation.John Wiley & Sons, Inc.; 1987, p. 98. doi:10.1002/9780470316726.fmatter.
- Akaike H. Information theory and an extension of the maximum likelihood principle In: Parzen E, Tanabe K, Kitagawa G, editors. Selected Papers of Hirotugu Akaike. New York: Springer: 1998. p. 199–213. doi:10.1007/978-1-4612-1694-0/_15.Google Scholar
- Hastie T, Tibshirani R. Generalized additive models. Stat Sci. 1986; 1:297–310. doi:10.1214/ss/1177013604.View ArticleGoogle Scholar
- Breiman L. Random forests. Mach Learn. 2001; 45(1):5–32. doi:10.1023/A:1010933404324.View ArticleGoogle Scholar
- Breiman L, Friedman J, Olshen RA, Stone CJ. Classification and Regression Trees.Taylor & Francis; 1984, p. 368. https://books.google.de/books?id=JwQx-WOmSyQC.
- Breiman L. Consistency For a Simple Model of Random Forests. Technical Report 670, Statistics Department, UC Berkeley. 2004. http://www.stat.berkeley.edu/~breiman.
- Biau G, Devroye L, Lugosi G. Consistency of random forests and other averaging classifiers. J Mach Learn Res. 2008; 9(2008):2015–33. doi:10.1145/1390681.1442799.Google Scholar
- Genuer R, Poggi JM, Tuleau C. Random Forests : some methodological insights. ArXiv e-prints. 2008; 6729:32.Google Scholar
- Efron B. Estimating the error rate of a prediction rule: improvement on cross-validation. J Am Stat Assoc. 1983; 78(382):16. doi:10.1080/01621459.1983.10477973.View ArticleGoogle Scholar
- Kruppa J, Liu Y, Biau G, Kohler M, König IR, Malley JD, Ziegler A. Probability estimation with machine learning methods for dichotomous and multicategory outcome: Theory. Biometrical J. 2014; 56(4):534–63. doi:10.1002/bimj.201300068.View ArticleGoogle Scholar
- Kruppa J. Probability estimation with machine learning methods for dichotomous and multicategory outcome: Applications. Biometrical J. 2014; 56(4):564–83.View ArticleGoogle Scholar
- Brier GW. Verification of forecasts expressed in terms of probability. Mon Weather Rev. 1950; 78(1):1–3. doi:10.1126/science.27.693.594.View ArticleGoogle Scholar
- Gerds TA, Cai T, Schumacher M. The performance of risk prediction models. Biom J Biom Z. 2008; 50(4):457–79. doi:10.1002/bimj.200810443.View ArticleGoogle Scholar
- Obuchowski NA. Receiver operating characteristic curves and their use in radiology. Radiology. 2003; 229(1):3–8. doi:10.1148/radiol.2291010898.View ArticlePubMedGoogle Scholar
- Cook NR. Use and misuse of the receiver operating characteristic curve in risk prediction. Circulation. 2007; 115(7):928–35. doi:10.1161/CIRCULATIONAHA.106.672402.View ArticlePubMedGoogle Scholar
- Nagelkerke NJD. A note on a general definition of the coefficient of determination. Biometrics. 1991; 78(3):691–2. doi:10.1093/biomet/78.3.691.View ArticleGoogle Scholar
- Steyerberg EW, Vickers AJ, Cook NR, Gerds T, Gonen M, Obuchowski N, Pencina MJ, Kattan MW. Assessing the performance of prediction models: a framework for traditional and novel measures. Epidemiology (Cambridge, Mass.) 2010; 21(1):128–38. doi:10.1097/EDE.0b013e3181c30fb2.View ArticleGoogle Scholar
- Hosmer DW, Lemeshow S. Applied Logistic Regression. In: Wiley Series in Probability and Statistics. 2nd ed. vol. 23. no. 1. John Wiley & Sons, Inc.: 2000. p. 375. doi:10.1002/0471722146.
- Cook RD, Weisberg S. Residuals and Influence in Regression.Chapman & Hall; 1982, p. 230. doi:10.2307/1269506. https://books.google.de/books?id=MVSqAAAAIAAJ.
- Williams DA. Generalized linear model diagnostics using the deviance and single case deletions. Appl Stat. 1987; 36(2):181. doi:10.2307/2347550.View ArticleGoogle Scholar
- Belsley D, Kuh E, Welsch R. Detecting and Assessing Collinearity. New York: John Wiley and Sons; 1980, pp. 85–91. doi:10.1002/0471725153.ch3.View ArticleGoogle Scholar
- O’Brien RM. A caution regarding rules of thumb for variance inflation factors. Qual Quant. 2007; 41(5):673–90. doi:10.1007/s11135-006-9018-6.View ArticleGoogle Scholar
- van der Ploeg T, Austin PC, Steyerberg EW. Modern modelling techniques are data hungry: a simulation study for predicting dichotomous endpoints. BMC Med Res Methodol. 2014; 14(1):1–13. doi:10.1186/1471-2288-14-137.View ArticleGoogle Scholar
- Bin RD, Herold T, Boulesteix AL. Added predictive value of omics data: specific issues related to validation illustrated by two case studies. BMC Med Res Methodol. 2014; 14(1):1–23. doi:10.1186/1471-2288-14-117.View ArticleGoogle Scholar