This is a retrospective study of SCD patient records. The sample studied included all the patients admitted to a public university hospital over a five-year period (2000–2004) as a result of acute events related to SCD. The hospital, located in Rio de Janeiro City in south-east Brazil, forms part of the national health system (Sistema Único de Saúde) and is a reference centre for the treatment of adolescents and adults with SCD. Patient records were identified by the diagnoses that appear in the hospital's electronic records. The patient records selected were those bearing D57 codes (sickle cell disease), according to the International Classification of Diseases – Tenth Revision. After examining the records, those of patients not admitted during the study period or with sickle-cell trait were excluded. All non-elective admissions for acute events, in which patients remained in the hospital for at least 24 hours, were included. Patients who died during the first hospitalisation were excluded. A form and an instruction manual were developed to guide extraction of information on the patients and their admissions from the patient records. All readmissions of each patient were considered for analysis, regardless of the time elapsed between them.
Time to readmission or to censoring by death or end of the observation period was modeled considering demographic and clinical covariates. Demographic covariates included were age, gender, ethnicity and education. The following independent clinical covariates were evaluated: age, phenotype, vaso-occlusive crisis, acute thoracic syndrome, bacterial infection, chronic renal failure (CRF), length of hospital stay (in days), use of opiates and number of packed red blood cell transfusions. The clinical covariates were selected on the basis of clinical criteria considered relevant in SCD patient admissions.
The statistical approach was based on extensions of Cox's proportional hazard model. Two models were fitted, one marginal and the other conditional or of random effects [11]. The Cox survival model is defined as a semi-parametric model because the baseline hazard function is treated non-parametrically, not assuming any specific probability distribution, while a parametric form is assumed for the covariate effects. It is also called a proportional hazards model as the ratio of hazard rates of any two individuals is proportional and constant over time. However, when an individual can experience multiple events, the dependency between events within individual has to be taken into account, and extensions of this model have been proposed.
Among the various options, the Andersen-Gill (AG) model, also known as the independent increment model was chosen. Independence here means that each event – "event" being each patient's hospital admissions – is not conditioned by previous ones, i.e. by previous admissions. That is, after each admission has occurred, the risk of admission returns to the prior situation, depending only on the characteristics of the individuals involved. The events share the same covariates, but are independent of one another.
Mathematically, the definition of the AG model is similar to the classic Cox proportional hazard model:
where H
it
indicates whether the subject i is at the time t at observation and/or risk (= 1) or not (= 0), λ
o
(t) is an arbitrary baseline hazard rate which remains unspecified, β is the vector of parameters associate to the x vector of covariates. The main difference between this model and the classical Cox model is the construction of the risk set – individuals at risk in each observed time – and the estimation of robust standard errors in the presence of repeated events. In the AG model, as soon as the observed event ends, the individuals is back to the same "at risk" situation.
Another approach to deal with multiple events is via random effects models. Suppose a random variable z
i
representing an unknown random effect, related to each patient, with unit mean and variance ε
i
. Large values of ε
i
reflect a greater degree of heterogeneity among patients. The form often assumed for the hazard rate is
The model holds two standard assumptions: (i) observations within each patient share a common frailty effect over time, (ii) conditional on an chosen parametric distribution – in this case a gamma distribution – the times to events are assumed to be independent.
The intra-individual correlation is thus corrected, and by means of this random effect, a set of unmeasured individual characteristics, which could be described as an individual "frailty", is estimated. Here, differently from the marginal models, the point estimate can be substantially altered, depending on the magnitude of this random effect.
Schoenfeld residuals were used to evaluate the assumptions of proportionality [12]. The explanatory power of the models was estimated using an R2 indicator, calculated as 1 minus double the likelihood ratio between zero and the proposed model [11]. All the models were fitted using R software and the "survival" library [13].
The study was approved by the Research Ethics Committee of the National School of Public Health, Oswaldo Cruz Foundation (Rio de Janeiro).