We illustrate the proposed methods using data collected by the nationwide DPCR (Munk-Jørgensen and Mortensen, 1997
). The DPCR has monitored contacts with psychiatric institutions throughout Denmark since 1969 and has included all contacts in emergency rooms and outpatient clinics since 1995. Each admission record includes date of admission, date of discharge, one main discharge diagnosis and up to 3 auxiliary discharge diagnoses. The data used in our analysis is from a cohort of 286 individuals with first contact with Danish psychiatric services during the period from April 1, 1970 to December 31, 1970 with a diagnosis of schizophrenia. For our cohort, 63% were male, 20% were married, and the median onset age of schizophrenia was 26 with a range from 14 to 88. We are interested in assessing the effects of onset age, gender, and marital status on the risk of rehospitalization due to a schizophrenic episode. Although previous literature (Olesen and Mortensen, 2002
; Olesen and Parner, 2006
) shows that the readmission risk seems to be stable over time, to avoid potential long-term pattern changes in recurrent gap times, we set the maximum follow-up time for each person to be 3 years. Forty percent of these individuals (171 out of 286) did not experience another hospitalization during the 3 year follow-up. On average, each patient experienced 1.7 hospitalizations after the initial contact.
We apply the nonparametric trend test proposed by Wang and Chen (2000)
to the recurrent gap times of the study cohort and find that there is no significant change in the distributional pattern of gap times. Univariate Cox regression using the method by Huang and Chen (2003)
shows that 3 risk factors, age of schizophrenia onset (P
< 0.001), male (P
= 0.09), and unmarried (P
= 0.02) are associated with a shorter gap time between repeated hospitalizations. When all 3 variables are included in the Cox model, the effects of onset age remains highly significant (P
< 0.001), while the effects of gender (P
= 0.85) and marital status (P
= 0.47) are insignificant. The multivariate Cox model estimates that a person who has onset at 20 has almost a 25% higher instantaneous risk of rehospitalization than a person who has an onset at 30, adjusting for gender and marital status. Further analysis reveals that both gender and marital status were highly correlated with onset age. The median onset age was 24 and 34 for males and females, and was 23 and 38 for unmarried and married patients of the studied cohort. Hence, in the remaining analysis, we will only focus on the effect of onset age on the distribution of recurrent gap times.
To check the assumption of proportional hazards for the candidate model with age at onset, denoted by Z, as the only covariate, we plot the standardized pseudo-score process Wph*(t) from the fitted model. shows the observed standardized pseudo-score process versus follow-up time since last hospitalization along with approximating processes simulated from the null hypothesis of a correct model fit. The observed process seems to be typical of the null hypothesis simulations, and the proportional hazards assumption fails to be rejected with an estimated P value 0.29 that is based on 1000 simulated processes.
Fig. 3. Diagnostic plots for the DPCR data. Upper panel: plots of the standardized pseudo-score process Wph*(t) for checking the proportional hazards assumption. The covariates in the fitted models are (a) Z, (b) , and (c) log(Z). Lower panel: plots of the partial-mean (more ...)
Next, we plot the partial-mean process Wz*
) and apply the supremum test to check the linear functional form of Z
. displays the observed partial-mean process along with 20 approximating processes generated from the null distribution. In addition to the linear Cox model, we also consider 2 different functional forms,
), of Z
. The corresponding plots are given by . The estimated P
values for the 3 models are 0.42, 0.71, and 0.77, respectively. These results suggest that the linear form of onset age, although reasonable, may not fit as well as the proportional hazards models with either
) as the covariate.
Plots of standardized pseudo-score process for the other 2 candidate models are also obtained to examine the assumption of proportional hazards. As shown in , the estimated p
values for the models with
) as the only covariate are 0.50 and 0.75, respectively. Based on these results, the logarithm transformation of the schizophrenia onset age seems to be more appropriate than the other functional forms considered in this section. The pseudo-partial likelihood method estimates the coefficient of logZ
to be − 1.04 (SE 0.21) in the Cox model.