<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0121-7488</journal-id>
<journal-title><![CDATA[Ciencia en Desarrollo]]></journal-title>
<abbrev-journal-title><![CDATA[Ciencia en Desarrollo]]></abbrev-journal-title>
<issn>0121-7488</issn>
<publisher>
<publisher-name><![CDATA[Universidad Pedagógica y Tecnológica de Colombia]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0121-74882022000100079</article-id>
<article-id pub-id-type="doi">10.19053/01217488.v13.n1.2022.13655</article-id>
<title-group>
<article-title xml:lang="es"><![CDATA[Comparación de algunos métodos para estimar el modelo de riesgos proporcionales de Cox para datos con censura a intervalo]]></article-title>
<article-title xml:lang="en"><![CDATA[Comparison of Some Methods to Estimate the Cox Proportional Hazards Model for Interval-Censored Data]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Bustos-Giraldo]]></surname>
<given-names><![CDATA[Olga A.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Jaramillo-Elorza]]></surname>
<given-names><![CDATA[Mario C.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Lopera-Gómez]]></surname>
<given-names><![CDATA[Carlos M.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad Nacional de Colombia  ]]></institution>
<addr-line><![CDATA[Medellín ]]></addr-line>
<country>Colombia</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidad Nacional de Colombia  ]]></institution>
<addr-line><![CDATA[Medellín ]]></addr-line>
<country>Colombia</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Universidad Nacional de Colombia  ]]></institution>
<addr-line><![CDATA[Medellín ]]></addr-line>
<country>Colombia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>06</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>06</month>
<year>2022</year>
</pub-date>
<volume>13</volume>
<numero>1</numero>
<fpage>79</fpage>
<lpage>92</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0121-74882022000100079&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0121-74882022000100079&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0121-74882022000100079&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen Los datos con censura a intervalo son comunes en varias áreas del conocimiento, tales como: epidemiología, finanzas, demografía, medicina, entre otras. Ocurren cuando el evento de interés, el tiempo de falla, no se observa exactamente, sino que se encuentra dentro de algún intervalo del tiempo de observación. Con frecuencia en esta situación se realiza una imputación de los datos que no se conocen exactamente. Algunos de los métodos de imputación múltiple propuestos en la literatura son el algoritmo PMDA (Poor Man's Data Augmentation) y el algoritmo ANDA (Asymptotic Normal Data Augmentation), los cuales permiten estimar los parámetros del modelo de riesgos proporcionales de Cox utilizando métodos clásicos de estimación. También existen métodos alternativos para realizar estas estimaciones, como el algoritmo ICM (Iterative Convex Minorant) y un enfoque Bayesiano, que no realizan imputación de los datos con censura a intervalo. En este trabajo se realizó una comparación vía simulación del desempeño de los estimadores de los parámetros del modelo de Cox producidos por cada uno de los métodos anteriormente mencionados. Los resultados evidenciaron que en términos generales los métodos ICM y el enfoque Bayesiano presentan valores de probabilidad de cobertura más altos y errores cuadráticos medios más bajos, además al aumentar el tamaño de la muestra estos valores mejoran notablemente comparados con los métodos PMDA y ANDA. En estos últimos no se evidenciaron diferencias considerables entre los resultados. Finalmente, se realizó una aplicación con datos reales asociados a un estudio de mastitis en ganado lechero.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract Interval censored data is common in several areas of knowledge, such as: epidemiology, finance, demography, medicine, among others. They occur when the event of interest, the failure time, is not observed exactly, but is within some interval of the observation time. Often in this situation an imputation is made of the data that is not exactly known. Some methods of multiple imputation proposed in the literature are the PMDA (Poor Man's Data Augmentation) algorithm and the ANDA (Asymptotic Normal Data Augmentation) algorithm, which allow estimating the parameters of the Cox proportional hazards model using classical estimation methods. There are also alternative methods to make these estimations such as the ICM (Iterative Convex Minorant) algorithm and a Bayesian approach, which do not impute the data with interval censoring. In this work, a comparison was made via simulation of the performance of the estimators of the Cox model parameters produced by each of the aforementioned methods. The results showed that in general terms the ICM methods and the Bayesian approach present higher coverage probability values and lower mean square errors, in addition when increasing the sample size these values significantly improve compared to the PMDA and ANDA multiple imputation methods. In the latter, there were no significant differences between the results. Finally, an application was made with real data associated with a study of mastitis in milk cattle.]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[Métodos de imputación múltiple]]></kwd>
<kwd lng="es"><![CDATA[Censura a intervalo]]></kwd>
<kwd lng="es"><![CDATA[Enfoque bayesiano]]></kwd>
<kwd lng="es"><![CDATA[Algoritmo ICM (Iterative Convex Minorant)]]></kwd>
<kwd lng="es"><![CDATA[Modelo de riesgos proporcionales de Cox]]></kwd>
<kwd lng="en"><![CDATA[Multiple imputation methods]]></kwd>
<kwd lng="en"><![CDATA[Interval-censored]]></kwd>
<kwd lng="en"><![CDATA[Bayesian approach]]></kwd>
<kwd lng="en"><![CDATA[ICM (Iterative Convex Minorant) algorithm]]></kwd>
<kwd lng="en"><![CDATA[Cox proportional hazards model]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<label>[1]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lin]]></surname>
<given-names><![CDATA[X]]></given-names>
</name>
<name>
<surname><![CDATA[Cai]]></surname>
<given-names><![CDATA[B]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Zhang]]></surname>
<given-names><![CDATA[Z]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A Bayesian proportional hazards model for general interval-censored data]]></article-title>
<source><![CDATA[Lifetime Data Analysis]]></source>
<year>2015</year>
<volume>21</volume>
<page-range>470-90</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>[2]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Strapasson]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A simulation study to compare imputation methods to handle grouped survival data]]></article-title>
<source><![CDATA[Revista Brasileira de Biometría]]></source>
<year>2009</year>
<volume>27</volume>
<page-range>210-24</page-range></nlm-citation>
</ref>
<ref id="B3">
<label>[3]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Hsu]]></surname>
<given-names><![CDATA[C. H.]]></given-names>
</name>
<name>
<surname><![CDATA[Taylor]]></surname>
<given-names><![CDATA[J. M.]]></given-names>
</name>
<name>
<surname><![CDATA[S]]></surname>
<given-names><![CDATA[Murray]]></given-names>
</name>
<name>
<surname><![CDATA[Commenges]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Survival analysis using auxiliary variables via non-parametric multiple imputation]]></article-title>
<source><![CDATA[Statistics in Medicine]]></source>
<year>2006</year>
<volume>25</volume>
<page-range>3503-17</page-range></nlm-citation>
</ref>
<ref id="B4">
<label>[4]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Jaramillo-Elorza]]></surname>
<given-names><![CDATA[M. C.]]></given-names>
</name>
<name>
<surname><![CDATA[Lopera-Gómez]]></surname>
<given-names><![CDATA[C. M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Estudio del efecto de la imputación de fallas en la estimación de la curva de supervivencia bajo censura a intervalo]]></article-title>
<source><![CDATA[Ciencia en Desarrollo]]></source>
<year>2017</year>
<volume>8</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>21-8</page-range></nlm-citation>
</ref>
<ref id="B5">
<label>[5]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pan]]></surname>
<given-names><![CDATA[W]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A multiple imputation approach to Cox regression with interval-censored data]]></article-title>
<source><![CDATA[Biometrics]]></source>
<year>2000</year>
<volume>56</volume>
<page-range>199-203</page-range></nlm-citation>
</ref>
<ref id="B6">
<label>[6]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Groeneboom]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Wellner]]></surname>
<given-names><![CDATA[J. A.]]></given-names>
</name>
</person-group>
<source><![CDATA[Information Bounds and Nonparametric Maximum Likelihood Estimation]]></source>
<year>1992</year>
<publisher-name><![CDATA[Springer]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B7">
<label>[7]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Pan]]></surname>
<given-names><![CDATA[W]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Extending the iterative convex mino-rant algorithm to the Cox model for interval-censored data]]></article-title>
<source><![CDATA[Journal of Computational and Graphical Statistics]]></source>
<year>1999</year>
<volume>8</volume>
<page-range>109-20</page-range></nlm-citation>
</ref>
<ref id="B8">
<label>[8]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Huang]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
<name>
<surname><![CDATA[Wellner]]></surname>
<given-names><![CDATA[J. A.]]></given-names>
</name>
</person-group>
<source><![CDATA[Interval censored survival data: A review of recent progress]]></source>
<year>1997</year>
<publisher-name><![CDATA[Springer]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B9">
<label>[9]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Sun]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
</person-group>
<source><![CDATA[Interval censoring. Encyclopedia of Biostatistics]]></source>
<year>1998</year>
<publisher-name><![CDATA[Wiley]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B10">
<label>[10]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cox]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Oakes]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Regression models and life tables]]></article-title>
<source><![CDATA[Journal of the Royal Statistical Society]]></source>
<year>1972</year>
<volume>34</volume>
<page-range>187-220</page-range></nlm-citation>
</ref>
<ref id="B11">
<label>[11]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Mangasarian]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Algorithms of nonlinear mathematical programming]]></article-title>
<source><![CDATA[CS730 Course Notes, Computer Sciences Department]]></source>
<year>1996</year>
<publisher-loc><![CDATA[Madison, WI ]]></publisher-loc>
<publisher-name><![CDATA[University of Wisconsin]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B12">
<label>[12]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bertsekas]]></surname>
<given-names><![CDATA[D. P.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Projected newton methods for optimization problems with simple constraints]]></article-title>
<source><![CDATA[SIAM Journal on Control and Optimization]]></source>
<year>1982</year>
<volume>20</volume>
<page-range>221-46</page-range></nlm-citation>
</ref>
<ref id="B13">
<label>[13]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Finkelstein]]></surname>
<given-names><![CDATA[D. M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A proportional hazards model for interval-censored failure time data]]></article-title>
<source><![CDATA[Biometrics]]></source>
<year>1986</year>
<volume>42</volume>
<page-range>845-54</page-range></nlm-citation>
</ref>
<ref id="B14">
<label>[14]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Sun]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
</person-group>
<source><![CDATA[The Statistical Analysis of Interval- Censored Failure Time Data]]></source>
<year>2007</year>
<publisher-name><![CDATA[Springer]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B15">
<label>[15]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ramsay]]></surname>
<given-names><![CDATA[J. O.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Monotone regression splines in action]]></article-title>
<source><![CDATA[Statistical Science]]></source>
<year>1988</year>
<volume>3</volume>
<page-range>425-61</page-range></nlm-citation>
</ref>
<ref id="B16">
<label>[16]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Joly]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Commenges]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Letenneur]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A penalized likelihood approach for arbitrarily censored and truncated data: Application to age-specific incidence of dementia]]></article-title>
<source><![CDATA[Biometrics]]></source>
<year>1998</year>
<volume>54</volume>
<page-range>185-94</page-range></nlm-citation>
</ref>
<ref id="B17">
<label>[17]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cai]]></surname>
<given-names><![CDATA[B]]></given-names>
</name>
<name>
<surname><![CDATA[Lin]]></surname>
<given-names><![CDATA[X]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Bayesian proportional hazards model for current status data with monotone splines]]></article-title>
<source><![CDATA[Computational Statistics and Data Analysis]]></source>
<year>2011</year>
<volume>55</volume>
<page-range>2644-51</page-range></nlm-citation>
</ref>
<ref id="B18">
<label>[18]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lin]]></surname>
<given-names><![CDATA[X]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A semiparametric probit model for case 2 interval-censored failure time data]]></article-title>
<source><![CDATA[Statistics in Medicine]]></source>
<year>2010</year>
<volume>29</volume>
<page-range>972-81</page-range></nlm-citation>
</ref>
<ref id="B19">
<label>[19]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lin]]></surname>
<given-names><![CDATA[X]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[L.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Bayesian proportional odds models for analyzing current status data: Univariate, clustered, and multivariate]]></article-title>
<source><![CDATA[Communications in Statistics-Simulation and Computation]]></source>
<year>2011</year>
<volume>40</volume>
<page-range>1171-81</page-range></nlm-citation>
</ref>
<ref id="B20">
<label>[20]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Dunson]]></surname>
<given-names><![CDATA[D. B.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Semiparametric Bayes proportional odds models for current status data with underreporting]]></article-title>
<source><![CDATA[Biometrics]]></source>
<year>2011</year>
<volume>67</volume>
<page-range>1111-8</page-range></nlm-citation>
</ref>
<ref id="B21">
<label>[21]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Gilks]]></surname>
<given-names><![CDATA[W. R.]]></given-names>
</name>
<name>
<surname><![CDATA[Wild]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Adaptive rejection sampling for Gibbs sampling]]></article-title>
<source><![CDATA[Applied Statistics]]></source>
<year>1992</year>
<volume>41</volume>
<page-range>337-48</page-range></nlm-citation>
</ref>
<ref id="B22">
<label>[22]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Tanner]]></surname>
<given-names><![CDATA[M. A.]]></given-names>
</name>
<name>
<surname><![CDATA[Wong]]></surname>
<given-names><![CDATA[W. H.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The calculation of posterior distributions by data augmentation]]></article-title>
<source><![CDATA[Journal of the American Statistical Association]]></source>
<year>1987</year>
<volume>82</volume>
<page-range>528-40</page-range></nlm-citation>
</ref>
<ref id="B23">
<label>[23]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Wei]]></surname>
<given-names><![CDATA[G. C.]]></given-names>
</name>
<name>
<surname><![CDATA[Tanner]]></surname>
<given-names><![CDATA[M. A.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Applications of multiple imputation to the analysis of censored regression data]]></article-title>
<source><![CDATA[Biometrics]]></source>
<year>1991</year>
<volume>47</volume>
<page-range>1297-309</page-range></nlm-citation>
</ref>
<ref id="B24">
<label>[24]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Goethals]]></surname>
<given-names><![CDATA[K]]></given-names>
</name>
<name>
<surname><![CDATA[Ampe]]></surname>
<given-names><![CDATA[B]]></given-names>
</name>
<name>
<surname><![CDATA[Berkvens]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Laevens]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Janssen]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Duchateau]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Modeling interval-censored, clustered cow udder quarter infection times through the shared gamma frailty model]]></article-title>
<source><![CDATA[Journal of Agricultural, Biological, and Environmental Statistics]]></source>
<year>2009</year>
<volume>14</volume>
<page-range>1-14</page-range></nlm-citation>
</ref>
<ref id="B25">
<label>[25]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Weller]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
<name>
<surname><![CDATA[Saran]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
<name>
<surname><![CDATA[Zeliger]]></surname>
<given-names><![CDATA[Y]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Genetic and environmental relationships among somatic cell count, bacterial infection, and clinical mastitis]]></article-title>
<source><![CDATA[Journal of Dairy Science,]]></source>
<year>1992</year>
<volume>75</volume>
<page-range>2532-40</page-range></nlm-citation>
</ref>
<ref id="B26">
<label>[26]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Adkinson]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
<name>
<surname><![CDATA[Ingawa]]></surname>
<given-names><![CDATA[K]]></given-names>
</name>
<name>
<surname><![CDATA[Blouin]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Nickerson]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Distribution of clinical mastitis among quarters of the bovine udder]]></article-title>
<source><![CDATA[Journal of Dairy Science]]></source>
<year>1993</year>
<volume>76</volume>
<page-range>3453-9</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
