Recent advances in information technologies are generating a growth in the amount of available biomedical data. In this paper, we studied the impact sample size may have on the categorisation of a continuous predictor variable in a logistic regression setting. Two different approaches to categorise predictor variables were compared.MINECO: MTM2011-28285-C02-01, MTM2013-40941-P, MTM2014-55966-P.
Basque Government: IT620-13.
University of the Basque Country UPV/EHU: UFI11/52
Agrupamento INBIOMED from DXPCTSUG-FEDER unha maneira de facer Europa (2012/273)

Arostegui, I.

Barrio, I.

Rodríguez-Álvarez, M.X.

English

Barrio I.

Arostegui I.

Rodríguez-Álvarez M.X.

Name not available

Sample size impact on the categorisation of continuous variables in clinical prediction

BCAM's Institutional Repository Data

BIOSTATNET Workshop on Biomedical (Big) DataSample size impact on the categorisation ofcontinuous variables in clinical predictionIrantzu Barrio, Inmaculada Arostegui, and Maŕıa-Xosé Rodŕıguez-ÁlvarezDepartamento de M.A. y Estad́ıstica e I.O. Universidad del Páıs Vasco UPV/EHUirantzu.barrio@ehu.eusDepartamento de M.A. y Estad́ıstica e I.O. Universidad del Páıs Vasco UPV/EHUBCAM- Basque Center for Applied Mathematicsinmaculada.arostegui@ehu.eusDepartamento de Estad́ıstica e I.O and Biomedical Research Centre (CINBIO). Universidade de Vigomxrodriguez@uvigo.esAbstract: Recent advances in information technologies are generating a growth inthe amount of available biomedical data. In this paper, we studied the impact sample sizemay have on the categorisation of a continuous predictor variable in a logistic regressionsetting. Two different approaches to categorise predictor variables were compared.1 MotivationRecent advances in information technologies are generating a growth in the amount ofavailable biomedical information and data, what is known as Big Data. This fact makesthat the data available in some biomedical research studies is getting larger in the lastyears. The collection and analysis of this data allows improving the quality and efficiencyof health care services and enhance the quality and longevity of life [1]. Furthermore, thedevelopment of prediction models to estimate the risk of developing a particular diseaseare nowadays relevant in the decision making process [2], with a significant growth in thenumber of predictive models developed in the last years. When developing predictionmodels to be used in clinical practice, categorised versions of continuous predictor vari-ables are commonly used by clinical researchers [3]. It is possible that research studieswith a big amount of data may require or use a categorisation of the continuous predictorvariables and hence, we think it is necessary to evaluate the impact the sample size mayhave on the selection of the cut points to categorise the predictor variable.Recently two different methodologies have been proposed to categorise a continuouspredictor variable in a logistic regression setting [4, 5]. The first approach is based on agraphical display using generalised additive models (GAM [6]) with P-spline smoothersto determine the relationship between the continuous predictor and the outcome. Thesecond approach proposes to select the optimal cut points based on the maximisation ofthe area under the ROC curve (AUC) of the logistic regression model for the categorisedvariable. When developing a prediction model to predict the risk of poor evolutionof patients with chronic obstructive pulmonary disease (COPD) in the IRYSS-COPDstudy [7], we categorised the predictor variable respiratory rate into three categoriesusing both methods. The same categorisation proposal was obtained with both methods,being 20 and 24 the cut points. In this case, the sample size we had was of 1350 patients.However, we wondered whether same results would be obtained with higher sample sizes.Therefore, this question motivated the work presented in this paper, where the aim is toResearch Perspectives CRM Barcelona, Fall 2015, vol. 7, in Trends in MathematicsSpringer-Birkhuser, Basel12 Sample size impact on the categorisation of continuous variables in clinical predictionstudy how sample is related to the location of the cut points to categorise a predictorvariable in a logistic regression setting.2 MethodsIn this section, we briefly describe the two methods considered for the categorisationof a continuous predictor variable. Let us assume that there is a dichotomous responsevariable Y and a continuous predictor variable X which we wish to categorise in a logisticregression setting.Categorisation proposal based on GAM with P-spline smoothers:Barrio et al. (2013) [4] proposed a methodology to categorise a continuous predictorvariable which consists of creating at least one average-risk category along with high- andlow-risk categories based on a GAM with P-spline smoothers. Let logit(p) = β0 + f(X)be the logistic GAM for X where p = P (Y = 1|X) and f() is the smooth function ofthe GAM regression model. The average-risk category is created by building an intervalaround the point x0 ∈ X for which f(x0) = 0. Lets denote π̂0 = logit−1(β0 + f(x0)) =logit−1(β0) and(π̂0inf , π̂0sup)its 95% confidence interval. The average-risk category(x0inf , x0sup)is obtained as f−1(logit(π̂0inf )− β0) = x0inf and f−1(logit(π̂0sup)− β0) =x0sup . Thus, this categorisation proposal considers at least three categories. In a contextin which only two categories are considered, Hin et al. (1999) [8] proposed to dichotomisea continuous variable with x0 as the optimal cut point.Optimal categorisation based on the maximisation of the AUC:Given k = 2 the number of cut points set for categorising X in 3 intervals, lets denote asXcat the categorised variable taking values from 0 to 2. Barrio et al. (2015) [5] proposedthat the vector of 2 cut points v = (x1, x2) which maximises the AUC of the logisticregression model P (Y = 1|Xcat) = logit−1(β0 + β11{Xcat=1} + β21{Xcat=2}), is thus thevector of the optimal cut points. In general, this method allows to search for any possiblenumber of cut points, nevertheless in order to compare both method we will focus onk = 2 number of cut points.For ease of notation and interpretation we will refer to these two approaches as the“GAM approach” and the “AUC approach”, respectively.3 Simulation studyA simulation study was performed under known theoretical conditions that verify lineareffects in the logistic regression model. We used this setting to evaluate the performanceof the GAM approach and the AUC approach when different sample sizes were used.Scenarios and set upThe predictor variable X was simulated from a normal distribution separately in each ofthe populations defined by the outcome (Y = 0 and Y = 1). Specifically, we consideredX|(Y = 0) ' N(0, 1) and X|(Y = 1) ' N(1.5, 1). When the aim is to maximisethe AUC, the theoretical location of cut points to categorise the predictor variable isknown [9], as well as the AUC associated with the corresponding categorical covariate.BIOSTATNET Workshop on Biomedical (Big) Data 3The simulations were performed for different sample sizes assuming the same number ofindividuals in Y = 0 and Y = 1. As far as the number of cut points is concerned, k = 2were considered. When using the AUC approach, we considered the Genetic algorithmto estimate the optimal number of cut points.ResultsFigure 1 depicts the boxplot of the estimated optimal cut points over 200 simulated datasets for the different sample sizes and each of the categorisation approaches considered.Different results were obtained when the sample size was increased with the two cate-gorisation proposals considered. The AUC approach identified the optimal cut pointsfor any sample size considered (see Figure 1(a)). Under this scenario the theoretical cutpoints were 0.227 and 1.274.Table 1. Numerical results obtained over 200 simulated data sets whenthe GAM approach was used to estimate the cut points. Mean, standarddeviation and median for the estimated x0 point together with the meanand standard deviation for the average-risk category values are reported.SampleSizex0Average-risk categoryLow limit Upper limitmean (sd) median mean (sd) mean (sd)100 0.751 (0.198) 0.759 0.395 (0.183) 1.099 (0.181)500 0.760 (0.087) 0.752 0.605 (0.085) 0.915 (0.087)1000 0.754 (0.050) 0.754 0.645 (0.048) 0.863 (0.054)1500 0.755 (0.046) 0.755 0.665 (0.046) 0.845 (0.048)3000 0.749 (0.036) 0.749 0.685 (0.036) 0.813 (0.037)5000 0.751 (0.026) 0.748 0.701 (0.026) 0.800 (0.027)10000 0.751 (0.019) 0.751 0.716 (0.019) 0.786 (0.019)On the other hand, the cut points obtained with the GAM approach differed morefrom those theoretical cut points as the sample size was increased (see Figure 1(b) ). Infact, for larger sample sizes, the average-risk category obtained converged to the point x0for which f(x0) = 0 which turned out to be close to the theoretical cut point for k = 1,i.e, 0.773. This results can be seen in Table 1, where the numerical results obtained withthe GAM approach are shown.4 ConclusionsTo summarise, we have seen that the sample size has an impact on the categorisationof a continuous predictor variable. For a large sample size, the GAM approach leadsto a very narrow average-risk category which can be interpreted as a unique cut point,being thus equivalent to the proposal of Hin et al. (1999), this is, a dichotomisation ofthe continuous variable. On the other hand, the AUC approach performs satisfactorilyin large sample sizes when looking for two cut points, i.e. categorising the predictorvariable into three categories. In general, as long as it is feasible, we recommend theuse of the AUC approach. Otherwise, we should take into account that for large samplesizes the GAM approach does not provide an optimal categorisation when the goal is to4 Sample size impact on the categorisation of continuous variables in clinical prediction(a) AUC Approach(b) GAM ApproachFigure 1. Boxplot of the estimated cut points for k = 2, based on 200simulated data sets for different sample sizes (N = 100, N = 500, N =1000, N = 1500, N = 3000, N = 5000 and N = 10000). From top tobottom: results obtained with the AUC approach and the GAM approach,respectively. True cut points are represented with a dashed line which arev2 = (0.227, 1.274) for k = 2 and v1 = (0.773) for k = 1.categorise the predictor variable into three categories.Funding: this study was supported by grants IT620-13, MTM2011-28285-C02-01, UFI11/52,MTM2013-40941-P, MTM2014-55966-P and Agrupamento INBIOMED from DXPCTSUG-FEDER unha maneira de facer Europa (2012/273).References[1] Costa FF. Big data in biomedicine. Drug Discov Today 2014; 19: 433-440.[2] Steyerberg EW. Clinical prediction models. A practical approach to development, validation, andupdating. New York: Springer, 2009.[3] Turner E, Dobson J and Pocock J. Categorisation of continuous risk factors in epidemiologicalpublications: a survey of current practice. Epidemiol Perspect Innov 2010; 7: 9.BIOSTATNET Workshop on Biomedical (Big) Data 5[4] Barrio I, Arostegui I, Quintana JM, et al. Use of generalised additive models to categorise continuousvariables in clinical prediction. BMC Med Res Methodol 2013; 13: 83.[5] Barrio I, Arostegui I, Rodŕıguez-Álvarez MX and Quintana JM. A new approach to categorisingcontinuous variables in prediction models: Proposal and validation. Stat Methods Med Res (in press).[6] Hastie T and Tibshirani R. Generalized additive models. London: Chapman & Hall, 1990.[7] Quintana JM, Esteban C, Barrio I, et al. The IRYSS-COPD appropriateness study: objectives,methodology, and description of the prospective cohort BMC Health Serv Res 2011; 11:322.[8] Hin LY, Lau TK, Rogers MS, et al. Dichotomization of continuous measurements using generalizedadditive modelling - application in predicting intrapartum caesarean delivery. Stat Med 1999; 18:1101-1110.[9] Tsuruta H and Bax L. Polychotomization of continuous variables in regression models based on theoverall C index. BMC Med Inform Decis Mak 2006; 6: 41.

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Sample size impact on the categorisation of continuous variables in clinical prediction

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