Methods Of Competing Risks Flexible Parametric Modeling For Estimation .
Nouri et al. BMC Medical Research -z(2020) 20:17RESEARCH ARTICLEOpen AccessMethods of competing risks flexibleparametric modeling for estimation of therisk of the first disease among HIV infectedmenSahar Nouri1, Mahmood Mahmoudi1*, Kazem Mohammad1, Mohammad Ali Mansournia1, Mahdi Yaseri1 andNoori Akhtar-Danesh2,3AbstractBackground: Patients infected with the Human Immunodeficiency Virus (HIV) are susceptible to many diseases. Inthese patients, the occurrence of one disease alters the chance of contracting another. Under such circumstances,methods for competing risks are required. Recently, competing risks analyses in the scope of flexible parametricmodels have risen to address this requirement. These lesser-known analyses have considerable advantages overconventional methods.Methods: Using data from Multi Centre AIDS Cohort Study (MACS), this paper reviews and applies methods ofcompeting risks flexible parametric models to analyze the risk of the first disease (AIDS or non-AIDS) among HIVinfected patients. We compared two alternative subdistribution hazard flexible parametric models (SDHFPM1 andSDHFPM2) with the Fine & Gray model. To make a complete inference, we performed cause-specific hazard flexibleparametric models for each event separately as well.Results: Both SDHFPM1 and SDHFPM2 provided consistent results regarding the magnitude of coefficients and riskestimations compared with estimations obtained from the Fine & Gray model, However, competing risks flexibleparametric models provided more efficient and smoother estimations for the baseline risks of the first disease. Wefound that age at HIV diagnosis indirectly affected the risk of AIDS as the first event by increasing the number ofpatients who experience a non-AIDS disease prior to AIDS among 40 years. Other significant covariates had directeffects on the risks of AIDS and non-AIDS.Discussion: The choice of an appropriate model depends on the research goals and computational challenges. TheSDHFPM1 models each event separately and requires calculating censoring weights which is time-consuming. Incontrast, SDHFPM2 models all events simultaneously and is more appropriate for large datasets, however, when thefocus is on one particular event SDHFPM1 is more preferable.Keywords: Competing risks, Flexible parametric models, Multicenter AIDS cohort study, Hazard function,Subdistribution Hazard function, Risk* Correspondence: mahmoodim@tums.ac.ir1Department of Epidemiology and Biostatistics, School of public health,Tehran University of Medical Sciences, Tehran, IranFull list of author information is available at the end of the article The Author(s). 2020 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication o/1.0/) applies to the data made available in this article, unless otherwise stated.
Nouri et al. BMC Medical Research Methodology(2020) 20:17BackgroundThere are a variety of possible subsequent outcomesother than Acquired Immunodeficiency Syndrome(AIDS) among HIV-infected patients [1–3]. When ourprimary focus is on the occurrence of each of theseevents separately in the presence of each other, theproblem of competing risks arises [4, 5]. Competingrisks occur when subjects can experience one or moreevents, which ‘compete’ with each other and occurrenceof one event may prohibit observing the other events ormodify their chances of occurrence. With the occurrence of non-AIDS diseases or death prior to AIDS, thetrue survival time for developing AIDS would be unobservable (e.g., tuberculosis, a lung infection, can expedite progression to AIDS). In general, among HIVinfected patients, diseases can affect each other throughimmune system suppression. Treatment of one diseasemay reduce the chance of developing another disease[6–8]. One solution to assess an unbiased estimate ofthe risk of the event of interest is restricting the analysis to competing events where the follow up for a patient ends at the onset of the first event, and notfocusing on multiple events in a patient [9–11]. In thisscenario, traditional survival analysis presumes thatonly one event can occur is not valid and this calls forspecial methods for competing risks, [5, 12]. Usingwell-known Kaplan-Meier method, which considerseach patient experiencing a competing event as censoring at the time of its occurrence, undermines animportant assumption underlying this estimator: independent censoring. However, the subject who was censored due to failing in a competing event will neverexperience the event of interest as the first event andconsidering competing events to be independent censorings leads to biases and overestimates the probabilityof failure [5, 12, 13]. An appropriate measure to evaluate the probability of failure within a competing riskframework is the cumulative incidence function (CIF)[14–16]. It is defined as the probability of failure fromthe event of interest in an interval of time from the beginning of the study until a particular time when it isquite clear how many subjects experienced competingevents.To evaluate the effects of covariates in a competingrisks analysis, model-based approaches on two important measures —the cause-specific hazard and subdistribution hazard— are more common. Thedifference between these two measures is in their risksets. In the cause-specific hazard, patients who experienced a competing event will be excluded from therisk set for the event of interest and considered ascensored. Modeling on cause-specific hazards providescausal effects of covariates on the hazard of the eventof interest in a counterfactual world, where there arePage 2 of 15no competing events and patients can only fail fromthe event of interest [10, 11, 14, 16]. In contrast, inthe cause-specific subdistribution hazard a patient experiencing a competing event will remain in the riskset until the end of the study because he can neverexperience the event of interest as the initial event.Modeling on subdistribution hazard creates the effectof covariates on the probability of failing in the eventof interest at the presence of competing events whichis a more realistic depiction of subjects who followthe event of interest and takes into account thechance of failing from other causes prior to the eventof interest [5, 16–18]. The choice of modeling hazardor subdistribution hazard depends on the researchgoals. For etiological goals, to investigate the effect ofcovariates on the occurrence of a particular event,cause-specific hazard models are suitable. In contrast,for prognosis questions, to know what fraction ofsubjects at a specified time are at risk of experiencingthe event of interest as the first event, consideringthe fact that they can experience a competing eventas the initial event, subdistribution hazard models arepreferable [5, 11, 12]. The two most popular modelsbased on cause-specific hazards and subdistributionhazards are Cox and Fine & Gray regression modelsrespectively [19, 20].There are alternative approaches in modeling ofcompeting risks based on the decomposition of theCIF (i.e., the joint distribution of event times and typesof events) known as mixture and vertical models. Mixture models factorize the CIF as the product of themarginal distribution of event type and the conditionaldistribution of failure time given the type of the event.A multinomial regression model is used to assess theeffects of covariates on the type of event and parametric or semi-parametric hazard regression models areused to evaluate their effects on the conditional failuretimes [21–24]. In contrast, vertical models decomposethe joint distribution of event times and types of eventsas the product of the marginal distribution of time offailure (ignoring the type of event) and conditional distribution of event type given the time of failure, whichprovides relative hazards [25]. With that said, thesemodels have some difficulties in their estimations, andinterpretation of their results and are not practicallywell-developed [26].One important issue with Cox or Fine & Graymodels is that the baseline hazard or subdistributionhazard functions remain unspecified and are not estimated parametrically [4, 9, 20]. Their use of partiallikelihood functions and inferences are limited to therelative rate of the two hazards or subdistributionhazards. To capture the shape of baseline functionsmore accurately, the use of parametric models in
Nouri et al. BMC Medical Research Methodology(2020) 20:17competing risks modeling, and generally for survivalanalysis, are preferred. The ordinary forms of parametric models (e.g., Weibull and log-logistic have aconstraint of linear association between transformations of the survival function and log time [27]. Thenthe flexibility of the models to fit adequately on thedata is limited. So, bias estimations and inaccuratepredictions result. In recent years, the application offlexible parametric models in the competing risksmodeling for both cause-specific hazard and subdistribution hazard approaches have been proposed [28–30]. Flexible parametric models are an extension ofparametric models and can be defined on a wide classof different scales (e.g., hazard scale, odds scale orprobit). They model a transformation of baseline survival functions on the log time scale using NaturalCubic Splines (NCSs) instead of linear functions andhave substantial capabilities in assessing parametricestimates of the absolute measure of hazard or subdistribution hazard of the event of interest at eachtime point [31, 32].The primary aim of this paper is to apply two alternative subdistribution hazard flexible parametricmodels to the HIV-infected men population and compare these models with the Fine & Gray model as astandard model in competing risks analysis. We haveidentified the competing risks as the first event—AIDS, non-AIDS, and death prior to AIDS or nonAIDS diseases— in an HIV-infected male populationand evaluated the covariates that are associated withthe risks of these outcomes. Furthermore, we implemented a cause-specific hazard flexible parametricmodel to investigate the direct (causal) and indirect(noncausal) effects of covariates on the risk of thecompeting events. In the next section, we present a description of the Multicenter AIDS Cohort Study(MACS). In the third section, the association of the cumulative incidence function with the hazard and subhazard functions is explained. In the fourth section,the Competing risks flexible parametric models(CRFPMs) for multiple types of events are reviewed. Inthe fifth section, the CRFPMs are applied to the MACSdataset and predictions for the risks are obtained. Thelast two sections contain results and conclusions.Study descriptionStudy population and patient selectionMulti Centre AIDS Cohort Study (MACS) is a 30–year prospective study of HIV infection among homosexual or bisexual men (HBM) who were 18 years orolder with no prior AIDS-defining illness. MACSbegan in 1984 at four US sites; Chicago, Illinois;Baltimore, Maryland; Pittsburgh, Pennsylvania; andLos Angeles, California. It has multiple patientPage 3 of 15recruitments. The first recruitment, which consists of4954 HIV-infected and uninfected HBM was conducted in 1984–1985. In 1987–1990, recruitment wasreopened and 668 HBM were enrolled. During 2001–2003 another 1350 HBM were enrolled. AnotherMACS expansion commenced at the beginning of2010 and 371 HBM were recruited until April 2014.HIV-related symptoms, demographic characteristics,blood specimens, and behavioral history at each 6month follow-up visit were collected. Among 7232HBM, seroconverter patients were selected and patients with a non-AIDS disease before seroconversionwere excluded from the study. This analysis includes674 seroconverter or prevalent patients with a knownvisit of seroconversion.Outcomes and covariatesStudy outcomes were determined as the time durationfrom seroconversion to the occurrence of the firstevent. The midpoint of the last negative and the firstpositive visits was used as the time of seroconversion.The primary event was the occurrence of AIDS without evidence of a non-AIDS disease before. The secondary event was time to a non-AIDS disease priorto AIDS. Non-AIDSs included the following diseases:kidney, liver, cardiovascular, cerebrovascular diseases;lung infection, bacteremia, septicemia; malignancies,neurologic; cancers —all cancers excluding Kaposisarcoma, lymphoma, and invasive cervical cancer.Since the occurrence of death precludes observingAIDS or non-AIDS diseases, we considered unrelateddeath as the third outcome. These types of deathsmay occur for reasons unrelated to AIDS or nonAIDS diseases (e.g., cerebral artery occlusion). Otherpatients who were lost to follow-up or did not experience any failure event at the end of the study (i.e.,April 2014) were censored. Such variables as MACSrecruitment calendar years, age at seroconversiontime, laboratory results including the number of positive CD4 cell counts, CD8 cell counts, white bloodcells, red blood cells and platelets at baseline wereconsidered based on the expert knowledge and previous studies on HIV/AIDS [6, 7, 33, 34]. Measurements obtained at the first positive visit are referredto as ‘baseline’. We used categorical covariates insteadof continuous to have perceptible clinical interpretations. The cut points were determined based on clinical considerations and previous studies on the MACSdata [33–38]. Sparse groups were integrated with adjacent categories. The study follow-up time was restricted to 15 years from HIV positive diagnosis toexclude non-AIDS diseases or causes not related toHIV infection or death related to aging.
Nouri et al. BMC Medical Research Methodology(2020) 20:17Page 4 of 15Relations of risk with Hazard and subhazardfunctionsCause-specific CIF is a measure of absolute risk and defined as the probability of failure from kth (k 1 K)cause by time t while being at risk of failing from othercompeting events [9]. So, we have:ZF k ðt Þ ¼ pðT t; event ¼ k Þ ¼0thcsk ðuÞ:S ðuÞduð1ÞIn the above equation, Fk(t) is the CIF function, T isthe survival time i.e. the minimum of the true survivaltime and censoring time, hcsk ðuÞ is the cause-specific hazard functions at time u t. S is the overall survival function and is defined as.K RPu cs SðuÞ ¼ expf 0 hk ðvÞdvg:k¼1Equation (1) implies that the risk of the event of interest is the combination of its hazard (hcsk ) and the chancethat competing events get to the event of interest tooccur as the first event (S). An increase in the hazard ofcompeting events will lower the risk occurrence of theevent of interest as the first event by decreasing theoverall survival function. In other words, in a competingrisks framework, a covariate can increase the risk of theevent of interest directly via increasing the hazard of theevent or indirectly by decreasing the hazards of competing events. So, a discrepancy between the hazard andrisk of the event of interest may be observed. The magnitude of this discrepancy depends on the severity ofcompeting risks. The stronger the competing risk, thegreater the discrepancy [10, 12, 14]. The cause-specifichazard is the instantaneous failure rate from a particularevent among patients who did not experience any priorcompeting events and has the form of hcsk ðt Þ ¼ LimΔt 0 pðt T t þ Δt; event ¼ kjT t Þ:ΔtIn contrast, the cause-specific subdistribution hazardis defined as hsdk ðt Þ ¼ LimΔt 0 pðt T t þ Δt; event ¼ kjT t ðT t K k ÞÞ:ΔtAlthough this definition of the risk set is not practically meaningful because the patient does not actually remain in the risk set to experience the event of interest asthe first event; however, this can lead to the followingdirect relationship between subdistribution hazard functions and corresponding cause-specific CIFs: Z t hsdðuÞduF k ðt Þ ¼ 1 exp k0ð2ÞSo, modeling on cause-specific subdistribution hazardwill show the associations between covariates on thecause-specific CIF. The subdistribution hazard-basedmodels use the information that the occurrence of competing events gives about the event of interest. That is,the event of interest never occurs as the first event if acompeting event has already occurred. So, whenever acompeting event occurs the chance of occurrence of theevent of interest as the first event will be reduced. However, to investigate whether an association is direct or indirect the effect of the covariate on all cause-specifichazards should be assessed.ModelsCause-specific Hazard flexible parametric modelsThe CSHFPM performs a flexible parametric model foreach type of event separately considering competingevents as censoring. It regresses the cause-specific logcumulative hazard function on a Natural Cubic Splines(NCSs) function of the log of time that islnH csk ðtjX k Þ ¼ ln H 0k ðt Þ þ X k βk ¼ NCS k lnt; γ k ; d 0k þ X k βkð3ÞWhere Hcs(t X) is the cause-specific cumulative hazard function for event k with matrix Xk of covariates attime t, H0k (t) is the cause-specific baseline cumulativehazard function, βk is a vector of covariates coefficientsand NCSk{lnt; γk, d0k} is a natural cubic spline functionof ln(t) with d0 knots and parameters γ for event k.The number and position of knots in the spline function determine the complexity of the baseline cumulative hazard function. However, sensitivity analysisshowed that they have little effect on the model fittingand there is no need for optimal selection to have agood fit [31]. Model [3] has the capability of carryingtime dependent effects of covariates, for handling nonproportional hazards (non-PH) form, easily through incorporating the interactions of covariates with splinefunctions [32].Instead of separate CSHFPMs, Hinchliffe and Lambert introduced a unified CSHFPM on stacked databased on the Lunn-McNeil approach [28, 39]. In thestacked data, each patient has one row of observations for each particular competing event to have theopportunity of failing in that event. For each eventtype, an indicator variable would be added to thedata. An additional indicator variable also would be
Nouri et al. BMC Medical Research Methodology(2020) 20:17Page 5 of 15created to identify the type of event for each patient.The unified CSHFPM is fitted for all competing eventssimultaneously and parameters related to each competing event are jointly estimated [40]. Different baseline hazard functions are considered for each cause offailure. This approach has the capability of considering shared covariate effects for all competing events.If knot positions in the unified model are the same asthose in separate CSHFPMs, covariate estimations ofthis model would be equivalent to those obtainedfrom separate models [40, 41].Subdistribution Hazard flexible parametric models(SDHFPM)In the SDHFPM, the effects of covariates directly modelon the cause-specific log cumulative subdistributionhazard functions using NCSs as follows:lnH sdk ðtjX k Þ ¼ ln ð ln ð1 F k ðtjX k ÞÞÞ ¼ NCS k lnt; γ k ; d 0k þ X k βkð4ÞLambert et al. proposed a parametric version of theFine & Gray model [29]. Patients who experienced acompeting event are kept in the risk set and have thechance of being censored before the end of the followup. The survival time is defined as the minimum ofcensoring time and true survival time. So, the censoringprobabilities should be calculated and incorporate inthe likelihood function for obtaining an unbiased estimation of the subdistribution hazard for the event ofinterest [40]. Fine and Gray calculated the censoringprobabilities non-parametrically and used inverse probabilities of censoring weights in the partial likelihoodfunction [20]. In the Lambert model, the censoring distribution is estimated parametrically using FPMs. Theyused a weighted likelihood where weights are conditional probability of not being censored after experiencing a competing event which is time dependentbecause the censoring probability grows over time. Inaddition, they extended the Geskus approach to be ableto estimate SHDFPM using standard software of FPMs[42]. To achieve this, the follow up time after competing event would be split into a number of intervals andtime dependent weights are applied to each interval.Then, standard packages for FPMs can be applied forthe event of interest [29]. Lambert showed that there isno need to have a very fine number of splits and thebias of estimation is negligible. Like the Fine & Graymodel, the Lambert model is fitted for each event ofinterest separately [29, 41].As an alternative method, Mozumder et al. introduced a unified likelihood function to obtain a directestimate of all cause-specific CIFs simultaneouslyusing FPMs [30]. This model is also on the logcumulative subdistribution hazards scale (Eq. (4)). Inthis method, however, instead of using the censoringweights, the likelihood function is directly constructed based on subdistribution hazards and CIFs[30, 43]. We hereafter refer to the former subdistribution hazard model as SDHFPM1 and the latter asSDHFPM2.Statistical analysisWe performed a Complete Case Analysis (CCA), 629of 674 patients, and assumed the data were MissedCompletely at Random (MCAR). However, MultipleImputations (MI) using the multivariate normalregression method with 10 imputed data sets formissing data were performed to explore the sensitivityof the inferences to departures from MCARassumptions.The possibility of a reduction in the number of initial set of covariates, generated based on the expert(background) knowledge, was explored. We usedtranscan function in R, which transforms covariateswhile imputing missing values of them [44]. The transformation for each covariate is conducted using canonical variates in a way that the covariate has themaximum correlation with the optimum linear combinations of other covariates. The number of knots forbaseline hazard and subhazard functions were determined using the Akaike Information Criterion (AIC)statistic. We used main effects of covariates andstarted with five degrees of freedom and identified thecomplexity of models based on the lowest value ofAIC. Internal and boundary Knots for NCSs of eachcompeting event separately located in their equallyspaced centiles and the first and last event times respectively. After determining the optimal number ofknots for each model, a forward stepwise regressionmethod was performed to build final models [45].However, we did not consider variable selection onCD4 cell count due to its important role in HIV/AIDSstudies. In SDHFPM1, for patients experiencing a competing event the time was split every .1-year. The timedependent censoring weights were estimated throughfitting an FPM on the initial set of covariates, withthree degrees of freedom. The CSHFPMs were conducted for each event separately.All significant effects were detected using the Likelihood Ratio (LR) tests at a 5% level. The findingsare summarized with hazard and subdistribution hazard ratios and 95% Confidence Intervals (CI). Statistical analyses were performed in STATA (StataCorp.2017. Stata Statistical Software: Release 15. College
(2020) 20:17Nouri et al. BMC Medical Research MethodologyPage 6 of 15Table 1 Baseline Characteristics of Seroconverter HBM in the MACS DataVariables*MACS recruitmentN 6741984–85 & 1987–902001–3 & 2010Age at diagnosisN 667 40patientsAIDS(N 267)Non-AIDS(N 156)Death(N 26)14(58.33) 40194(29.09)60(22.64)62(39.74)10(41.67)Baseline CD4, per μlN 629 2)47(18.88)16(10.74)6(31.58) 500476(75.68)180(72.29)124(83.22)13(68.42)Baseline CD8, per μlN 629 7(53.58)129(51.81)79(53.02)12(63.16) 1000201(31.96)75(30.12)53(35.57)3(15.79)Baseline WBC, per μlN 663 5000191(28.81)78(29.43)39(25.32)10(43.48) e RBC, 10 , per μlN 667 4.590(13.49)25(9.43)31(19.87)1(4.17) 4.5577(86.51)240(90.57)125(80.13)23(95.83)Baseline platelets, 103, per μlN 660 58)127(47.92)90(59.21)8(34.78) ces between the number of patients and the sum of the AIDS, Non-AIDS and Death columns indicate the number of censored patientsStation, TX: StataCorp LLC.) and R (Version 3.5.2)[46].ResultsThe descriptive statistics for patients in the study arereported in Table 1. Patients who were diagnosed withAIDS as the first event tended to be younger, havemore prevalence in 1984–85 and 1987–90recruitments, lower CD4, higher CD8, and higher RBCin comparison with patients who were diagnosed withnon-AIDS diseases as the first event. The results oftranscan function showed that none of the covariatesselected by the expert (background) knowledge hadstrong correlations with the others (refer toAdditional file 1: Table S1). The AICs and BICs in allthree CSHFPM, SDHFPM1 and SDHFPM2 with 3, 1, 1knot(s) for AIDS, nonAIDS, and death respectively, hadminimum values. For all CRFPMs, Interactions betweenthe age with ln(t) on non-AIDS diseases was statisticallysignificant (LR test P-values .05, .021 and .024, forCSHFPM, SDHFPM1 and SDHFPM2, respectively).More complicated time dependent effects of age hadbeen assessed. However, ln (t) had the lowest AIC andBIC among them. Table 2 presents the results of multivariable SDHFPM1, SDHFPM2, and Fine & Gray model,which show the effects of covariates on the subdistribution hazards or equivalently on the risk of the first observed event. There was a high agreement betweenestimated subhazard ratios obtained from Fine & Grayand two SDHFPMs. The results of performing CSHFPMalso show the effects of covariates on the hazard ofeach event separately (refer to Table 2). For simplicity,we refer to 1984–85 and 1987–90 MACS recruitmentsas period 1 and 2001–3 and 2010 recruitments asperiod 2. The rest of this section is concentrated on theresults of SDHFPM1 and CSHFPM. However, due to thefair agreement between models, the SDHFPM2 wouldhave the same interpretation. The interpretation of subdistribution hazard ratios is not often appealing. Itshould be noted that their magnitudes are not
(2020) 20:17Nouri et al. BMC Medical Research MethodologyPage 7 of 15Table 2 Cause-Specific and Subdistribution Hazard Ratios Estimated from the Cause-Specific Hazard (CSHFPM) and Cause-SpecificSubdistribution Hazard models (SDHFPM1, SDHFPM2 and Fine and Gray)EventVariableAIDSMACS recruitmentSDHFPM1SDHFPM2Fine&Gray ModelCSHFPMSDHRSDHRSDHRCSHR(95% CI)(95% CI)(95% CI)(95% CI)1984–85 & ��3 & 1).039(.005–.28)*LR test .0001 .0001 40ReferenceReferenceReferenceReference 64–1.18)LR test.017.016 3501.83 1.83)1.38(1.00–1.91)1.33(.96–1.84) 500ReferenceReferenceReferenceReferenceLR test.016.0381984–85 & ��3 & 5.35)2.45(1.51–3.99)LR test .0001 .0001 40ReferenceReferenceReferenceReference 68)3.72(2.34–5.90)LR test .0001 .0001 1.11).57(.29–1.12).68(.35–1.32) 500ReferenceReferenceReferenceReferenceLR test.19.42 40ReferenceReferenceReferenceReference 04)3.71(1.34–10.28)LR test.037.021.001Age at diagnosis.38Baseline CD4Non-AIDS.016MACS recruitment.0009Age at diagnosis .0001Baseline CD4Death.33Age at diagnosis.012Baseline CD4 erence 500ReferenceReferenceLR test.17.13.18*LR test is the Likelihood Ratio test for evaluating the effect of each covariate in the multivariable SDHFPM1, SDHFPM2, and CSHFPMequivalent to the effect of covariates on the risk of theevent of interest. However, they contain informationabout the significance and direction of the effects of covariates on the risk of the event of interest [47]. The results of analyses on the multiply imputed data sets werealmost identical to CCA (refer to Appendix).Associations of MACS recruitments with the risk of AIDSprior to non-AIDS diseases, non-AIDS prior to AIDS andunrelated deathThe results showed that the hazard of AIDS occurrence in period 2 declined to about 96% compared toperiod 1 (P-value .002). In addition, the hazard of
Nouri et al. BMC Medical Research Methodology(2020) 20:17non-AIDS diseases among patients of period 2 was2.45 times that of period 1 (P-value .002). By increasing the hazard of non-AIDS diseases in period 2,fewer patients remained at risk to experience AIDSprior to non-AIDS diseases compared to period 1.Consequently, the risk of observing AIDS as the firstevent was lower in period 2 compared to period 1 (Pvalue .0001). In a similar way, the risk of observinga non-AIDS disease prior to AIDS increased in period2 compared to period 1 (P-value .0001). Therefore,there were direct and indirect associations betweenthe MACS recruitments and the risk of AIDS andnon-AIDS. MACS recruitments were not associatedwith the hazard and risk of unrelated death.Associations of age at HIV diagnosis with the risk of AIDSprior to non-AIDS diseases, non-AIDS prior to AIDS
flexible parametric models in the competing risks modeling for both cause-specific hazard and subdistri-bution hazard approaches have been proposed [28- 30]. Flexible parametric models are an extension of parametric models and can be defined on a wide class of different scales (e.g., hazard scale, odds scale or probit).
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Outline Introduction to competing risks Kaplan-Meier for cause-specific survival Cumulative Incidence Function Cause-specific hazards; conditional probability
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