I. Computer Simulation for Estimating Parameters
To test if the Coxian phase-type model can be simulated by directly using a mixture of Poisson process, we fit the continuous positively skewed data on which the research conducted by Marshall[4] was based after simulating their tabular data on LOS of the geriatric patients. As the most adequate model was fitted by a 3-phase Coxian phase-type distribution we simulated the data by a 3-state mixture Poisson process with
the probability density function expressed as follows.
f(t) = π1× θ1e−θ1t+ π2× θ2e−θ2t+ π3× θ3e−θ3t, (π1+ π2+ π3 = 1)
We set π1 = 0.46 , π2 = 0.40 , π3 = 0.14 and θ1 = 0.07 , θ2 = 0.05 , θ3 = 0.02 . The data set in Marshall’s study indicated the LOS ranged from 0 to 350 days, with a mean of 23 days and a median of 12 days. The simulated data shows the LOS ranged from 0 to 358 days, with 23 days and a median of 14 days (Figure 5-1), which was very close to their original empirical findings.
The Coxian phase-type distribution was fitted to the simulated data by using SAS software. SAS implements an optimization function with the method of maximum likelihood estimation (MLE) given the formulation of the log-likelihood function for different kinds of phase-type distribution. We used the Newton-Raphson algorithm and the minimum Bayesian information criterion (BIC) to decide the most parsimonious
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model. In Table 5-1, the estimated parameters in the one or two phase Coxian phase-type model were close between the original results and our simulated data.
However, when the number of phase increased there was a larger discrepancy. The results of simulation suggest while the hidden phases increased the heterogeneity across different phases could not be captured by a mixture of Poisson process.
II. Compliance with colonoscopy from positive FIT of Taiwan nationwide
colorectal cancer screening program
Univariate Analyses and Multivariate Analyses for the Hurdle model
In order to identify factors associated the non-compliance for colonoscopy and those affecting WT for undergoing colonoscopy, we used the hurdle model to deal with these two problems simultaneously.
The hurdle part is to identify which factors might influence subject not to take colonoscopic exam and the non-hurdle part is to identify which factors would affect WT for colonoscopy among attendees complying with colonoscopy. As shown in Table 5-2, the effects of gender on both parts of model were lacking of statistical significance.
Compared with the age group of 50-54, the older age groups had higher odds of refusing to receive colonoscopy whereas the complier after they underwent colonoscopy exam, the effect of age on WT became not statistically significant. In geographic area,
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those residing in Eastern Taiwan or offshore islands had the highest odds of non-compliance and had the longest WT for colonoscopy if they actually received colonoscopy exam; those dwelling in Northern Taiwan had the lowest odds of non-compliance and had the shorter WT for colonoscopy than those dwelling in Southern or Eastern Taiwan or offshore islands. Those who attended screening at public health centers had the lowest odds of non-compliance and had the short WT for colonoscopy; those who attended screening at local clinic had the highest odds of non-compliance and had the longest WT for colonoscopy. Attendees living in secondary urban or undergoing screening at inaugural period or being detected at subsequent screening had the lowest odds of non-compliance and had the shortest WT for colonoscopy.
Before fitting the multivariate model, the model selection was done and shown in Table 5-3. Because the change in the structure of screening program during the year from 2009 to 2010 might results in heterogeneity between the inaugural period and rolling out period, we evaluated the interaction between factors and periods of screening program. The results of model selection reveal that the hurdle part included seven baseline characteristics and interaction of periods of screening program between geographic areas and type of screening units, and the non-hurdle part contained six baseline characteristics (excluding gender effect) and interaction of periods of screening
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program between geographic areas, type of screening units and urbanization levels. As presented in Table 5-4, multivariate analysis in hurdle part found female, older people, those who lived less urbanized area and those were detected at prevalent screen had higher chance of not complying with colonoscopy. During inaugural period, attendees of eastern Taiwanese or offshore islands had a highest odds of not complying with colonoscopy (OR: 1.51, 95% CI: 1.36-1.66) compared with the northern attendee, but in rolling out period, middle Taiwanese had the highest odds of not complying with colonoscopy (OR: 1.08 95% CI: 1.06-1.10). Although screening at public health centers had the lowest odds in both periods, screening at hospital had the highest odds (OR:
2.54, 95% CI: 2.39-2.69) in inaugural period but decreased during the rolling out period (OR: 1.08, 95% CI 1.06-1.10) and screening at local clinic (OR: 1.79, 95% CI:
1.74-1.84) had the highest odds in rolling out period. When taking the non-hurdle part into account, the results presented in Table 5-5 show attendees who aged between 65 and 69 years had the longest WT for colonoscopy if they actually complied with colonoscopy, but three other age groups had not much difference. Those detected at subsequent screen had shorter WT for colonoscopy than prevalent screen. During inaugural period, attendees living in middle Taiwan (RR: 1.14, 95% CI: 1.07-1.20) or main urban or undergoing screening at public health centers (RR: 1.22, 95% CI:
0.99-1.46) had the shortest WT for colonoscopy. During the rolling out period, those
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who lived in middle Taiwan (RR: 1.13, 95% CI: 1.09-1.16) or secondary urban area (RR:
1.07, 95% CI: 1.06-1.09) or undergoing screening at hospital (RR: 1.14, 95% CI:
1.12-1.15) had the shortest WT. It indicates the similar trend in geographic areas in both periods, and the estimates of RR of northern people increased from 1.03 to 1.12.
Queue Hurdle Coxian Phase-type model
As we had already known there was heterogeneity between the inaugural period (2004-2009) and the rolling out period (2010-2013), so we analyzed these two separately. In the current thesis, we only considered the modelling with the Coxian phase-type model using the data on the inaugural period. The continuous data are positively skewed with a long tail, representing those few attendees who had not received colonoscopic exam for an extremely long WT (Figure 5-2) that justifies the WT had better be modelled by the Coxian phase-type distribution. To decide the most appropriate model, we still used the minimum BIC to determine. In Table 5-6, we found the Queue hurdle 2-phase Coxian phase-type model was the most suitable model due to minimum BIC score and could be classified as short waiting (step-by-step) phase and long waiting (shilly-shally) phase. It can be clearly seen that the 3-phase Coxian phase-type model had higher BIC value and also showed the identifiability problem between the referral rate from the moderate waiting phase (µ2) and the transition rate
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from the moderate waiting phase to long waiting phase (λ2). We observed that regardless of numbers of WT phases in the model, all of them indicate the same arrival rate equal to 0.00021 per person-days and the probability of not complying with colonoscopy was 0.2647. It is very interesting to note that the referral rate was five times greater in the short waiting phase than the long waiting phase. Around 15%
subjects were in a dilemma to be referred to undergo colonoscopy so as to be trapped in long waiting phase. In Queue hurdle 2-phase Coxian phase-type model, Table 5-7 shows that the mean WT in short waiting phase and the mean WT in long waiting phase was 32 days and 169 days, respectively. The marginal mean WT was 35 days.
Assuming covariates would affect referral rate and the transition rate, we used the coefficients estimated from the non-hurdle part of the hurdle regression model as a new
covariate (score):
scorei = −3.7554 + 0.0217 × age50−54+ 0.0181 × age55−59
+0.0198 × age60−64+ 0.0321 × areanorth+ 0.1276 × areamid +0.0770 × areasouth− 0.2038 × unithospital
+0.2006 × unitpublic− 0.0227 × urbansecondary
−0.0567 × urbanrural+ 0.0364 × subsequent
All of these significant covariates transformed into a new continuous covariate, so we could reduce parameters to be estimated from 11 to 1. We also made it become a
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binary outcome according to the cutoff of median value
G = �1 , if scorei> −3.4981 0 , if scorei< −3.4981
Using the proportional hazards regression form to compare referral rate of higher score
group and lower score group gave the following expression:
µ1 = µ01exp (γ × G) µ2 = µ02exp (γ × G) λ1 = λ01exp (γ × G)
We fitted 1- and 2-phase model to determine which model was more suitable to explain data. In Table 5-8, the result of 1-phase model shows the higher score, the faster referral rate for colonoscopy (P < 0.001). The mean WT was 38 days in lower score group and 32 days in higher score group. In the 2-phase models, risk score might have the impact on the transition rate from short waiting phase to undergoing colonoscopy (µ1), or from long waiting phase to undergoing colonoscopy (µ2), or from short waiting phase to long waiting phase (λ1). Table 5-9 indicates the model with score related to 𝜇𝜇1 was the most appropriate model. In addition, the 2-phase model was better than 1-phase model as well. The mean WT in short waiting phase were 36 days and 30 days corresponding to low score group and high score group, separately. In longer waiting phase, the mean WT was 167 days among these two groups. The marginal mean WT was 38 days in low
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score group and 32 days in high score group.
According to this model with score related to the referral rate, we could predict its transition probabilities at different times. In Figure 5-3, the probability of staying in short waiting phase, P11, declined over time and those with lower score had longer WT in short waiting phase than higher score given the same probability of staying in short
waiting phase. The transition probability from short waiting phase to long waiting phase, P12, was pretty small and no difference between these two groups. The transition
probability of undergoing colonoscopy, P13, increased over time, because patients would receive colonoscopic exam eventually. Under the same transition probability to undergo colonoscopy, low score group had longer WT than high score group.
III. Application II : Hospitalization of colorectal cancer patients
There were 178 CRC patients in Shin Kong Wu Ho-su Memorial Hospital (SKH) between 1999 and 2013. The variables of interest include the patients’ LOS (recorded in days), measured from the day of admission of a patient until they have been discharged.
The continuous data are positively skewed with a long tail, representing those few patients who have remained in hospital for an extremely long time (Figure 5-4). The LOS ranges from 1 to 215 days, with a mean of 13.8 days and a median of 7 days.
The Coxian phase-type distribution was fitted to the LOS data. Table 5-10 displays
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the fitted parameters and the BIC score for each of the model under consideration. It could be found that the most suitable number of phases was 3, because it had the minimum BIC score, which was 1157. A 3-phase model could be classified as short-stay, medium-stay and longer-stay care because if patient had the serious condition, he/she would have poor resistance to infections. Therefore, the severer condition CRC patients had, the longer patients stayed in the hospital (Figure 5-5). The absorbing rate of discharge for short-stay was five times than that of medium-stay and long-stay. The transition from short-stay to medium stay was five times that from medium-stay to long-stay. The expected LOS is displayed in Table 5-11. In short-stay phase, the expected LOS was 10 days whereas both medium-stay and longer-stay phases were 49 days. The marginal expected LOS was 14 days.
As we found that the transition rate from medium stay to discharge or death and the transition rate from longer stay to discharge or death were very close we therefore tested this current 3-phase model against a new model assuming these two transition rates were equal. Table 5-12 shows the original 3-phase model was better because of the smaller AIC. Therefore, we still kept two transition rates distinct although they were close.
To make sure if there still existed a better model, we attempted to fit another model.
We could find that discharge type 1, 3 and 5 had shorter LOS and could be regarded as
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discharge due to recovery. Discharge type 4, 6 and A had longer LOS and could be regarded as discharge due to severer condition. These patients had aggravated condition or even death. Therefore, we divided discharge types into recovery and death and we then used Coxian phase-type model with two absorbing states to fit the data. At first, we needed to determine how many phases would be appropriate, so we were also based on BIC score to decide. As shown in Table 5-13, 3-phases model was the most suitable case.
However, if we only focused on 3-phases Coxian phase-type models, it indicates the model with one absorbing state was still better than that with two absorbing states due to the smaller BIC score. As a result, we reckon the 3-phases Coxian phase-type model with one absorbing state was the most appropriate model.
Coxian phase-type models with covariates
After confirming the 3-phase Coxian phase-type model was the most suitable one to fit the data, we wonder if the transition rate would be influenced by covariates so that the number of phase could be reduced in the model. As a result, we used the 2-phase Coxian phase-type model to explore this issue and applied the proportional hazards form. We assumed that gender or age would affect transition rate in the following five scenarios:
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(1) From short-stay to longer-stay
λ1 = λ01exp(β1× genderFemale)
or
λ1 = λ01exp(β1× age60−74+ β2× age≤60)
(2) From short-stay to absorbing state (Death/Discharge) µ1 = µ01exp(β1× genderFemale)
or
µ1 = µ01exp(β1× age60−74+ β2× age≤60)
(3) From longer-stay to absorbing state
µ2 = µ02exp(β1× genderFemale)
or
µ2 = µ02exp(β1× age60−74+ β2× age≤60)
(4) From short-stay to absorbing state and from longer-stay to absorbing state µ1 = µ01exp(β1× genderFemale)
µ2 = µ02exp(β2× genderFemale)
or
µ1 = µ01exp(β1× age60−74+ β2× age≤60) µ2 = µ02exp(β3× age60−74+ β4× age≤60)
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(5) Joint effect on 3 transition paths
µ1 = µ01exp(β1× genderFemale) µ2 = µ02exp(β2× genderFemale) λ1 = λ01exp(β3× genderFemale)
or
µ1 = µ01exp(β1× age60−74+ β2× age≤60) µ2 = µ02exp(β3× age60−74+ β4× age≤60) λ1 = λ01exp(β5× age60−74+ β6× age≤60)
Firstly, we took gender into account. Distributions of gender list in Table 5-14.The results of these five situations are shown in Table 5-15. Based on their corresponding BIC scores, it indicates that the model having gender effect on transition rate from short-stay to absorbing state was the most appropriate model. It also shows that male would discharge or die earlier than female. The mean LOS in short-stay was 9 and 12 days corresponding to male and female, separately, and both were 53 days in longer-stay.
According to this 2-phase Coxian phase-type model with gender effect on the transition rate from short-stay to absorbing state, we could predict its transition probabilities at different times. In Figure 5-6, the probability of staying in short-stay state, P11, declined over time and female had longer LOS in short-stay than male given the same probability of staying in short-stay state. The transition probability from short-stay to longer-stay,
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P12, was pretty small, but female still had higher transition probability than male under
the same LOS. The transition probability from short-stay to absorbing state, P13, increased over time, because patients would discharge or die eventually. Under the same transition probability from short-stay to absorbing state, female had longer LOS than male.
Secondly, we took age into account. Distributions of age list in Table 5-16. The results of these five scenarios are shown in Table 5-17. Based on their corresponding BIC scores, it indicates that the model having age effect on transition rate from longer-stay to absorbing state was the most appropriate model. It also shows that the elderly would discharge or die earlier than the young. All of their mean LOS in short-stay was 9 days and the mean LOS in longer-stay was 18, 19 and 80 days corresponding to those aged above 75, 60-74 and below 60, separately. According to this 2-phase Coxian phase-type model with age effect on transition rate from longer-stay to absorbing state, we could also predict its transition probabilities at different times. In Figure 5-7, there was no age effect on the probability of staying in short-stay state and we also found that those aged at 60-74 or above 74 were not different irrespective of the transition probabilities from short-stay to longer-stay or from short-stay to absorbing state. The transition probability from short-stay to longer-stay in patients aged 60 years or below had higher transition probability than the other two groups given the same
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LOS. Under the same transition probability from short-stay to absorbing state, those aged below 60 also had longer LOS.
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