Model selection for Poisson autoregressive model

Chapter 4 Model specification

4.5 Estimation of parameters

4.5.3 Model selection for Poisson autoregressive model

Akaike’s information criterion (AIC) was used for model selection in the Poisson

autoregressive model ⁷⁶. The last selected model were based on minimizing AIC.

Additionally, the residuals of the models were checked whether there were

autocorrelations by Ljung-Box statistics methods and any patterns by residual plots ⁷⁷. Model selection for Bayesian dynamic linear model was based deviance information criteria (DIC).

69 Chapter 5 Results

5.1 Basic results

A total of 552,233 admissions were registered during the period from 1 January, 1994 to 31 December, 2013. 6,776 (1.2%) admissions were excluded either because information on variables (date of birth or department of admission) were unavailable (n=254) or the admission was indicated for a health check-up or consultations only (n=6522). Therefore, 545,457 admissions from 315,209 patients with 4,210,609 patient-days were available for analysis in this study. Of these, there were 10,117 patients with healthcare-associated infection and 20,221 cultures were identified.

There were 2,965 deaths ascertained within 30 days after the onset of HAIs (Figure 5.1.1). Figure 6 shows the incidence rates of HAI by calendar year. The median length of stay for HAI patients was 35 days (IQR: 22-58 days, range:1-4,033 days). The median length of stay was 5 days (IQR: 3-8) for non-HAI patients. Males accounted for 52.29% of cases (8,450 episodes) with a median age of 68.00 years (IQR: 54.73-76.94); and females accounted for 47.71% of cases (7,711 episodes) and had a median age of 71.94 years (IQR: 60.92-80.25).

The HAIs cohort was systematically collected between 1994 and 2013 that was continued in a 921-bed teaching medical center hospital in Taipei. Total admissions

was about 25,000-28,000 every year. There was no substantial change in the hospitalized admission by year. The healthcare-associated infectious (HAI) cases among those admissions were around 900-1,200 by year (Table 1). The range of HAIs incidence rates of were about 2.86-5.16 per 1000 patient-days for every calendar year.

The trend of overall HAI incidence between 1994 and 2013 showed obvious decline, especially in the last two years (see Table 5.1.1). The detail statistical results were demonstrated in Table 5.1.2. The incidence rate of HAI in male was slight higher compared with female. Regarding the admission department, the highest HAI incidence rate was revealed in Department of Oncology, followed by GI, Pediatric, and Chest. The Incidence rates were different from seasons. The HAI incidence rate in summer was higher than others. The incidence rate was highest for urinary tract infections (1.59 episodes per 1,000 patient-days), followed by bacteremia (1.09), pneumonia (0.72), and surgical site infection (0.45).

In multi-variable Poisson model in evaluating the causes of incidence rate, age was a significant factor. The relative risk was highest in the age group of over 80 years old. The lowest risk age group was between 20 to 29 years old. Male patients had higher relative risk of contracting HAIs than female patients. Emergency

department staying had higher incidence risk of HAIs compared to other departments.

Patients admitted as an emergency had a higher risk of HAIs compared to those admitted from outpatient departments. After adjusting for age, gender, and the department of hospitalization, patient admission type (outpatient or emergency) and sites of infection were still associated with HAIs incidence (Table 5.1.3).

5.2 Decomposition method with Generalized Linear Time-series Model

among selected infection sites and species

Based on Table 5.1.3, we focused on four infection sites and the two species of Gram (-) based on the longitudinal follow-up for the incidence of HAI from 1994 to 2013.

(A) Overall HAI incidence

The longitudinal follow-up for the incidence of HAI from 1994 to 2013 was shown in Figure 5.2.1. The monthly time series of the HAIs count was shown in Figure 5.2.4. Seasonality was noted in the HAIs incidence. The incidence was higher in the summer and spring than that in the winter (p<0.0001) (Table 5.2.1). The relative risk was higher in the summer than in the winter. Figure 5.2.5 is the time series after de-seasonalization of the HAI, and there were five outliers noted in this figure. After

deleting the outliers, there was a significant third degree polynomial trend (Table 5.2.2). The autoregression analysis for the de-seasonalized and de-trend time series revealed lacking of autoregressive order and white noise (Table 5.2.3).

(B) Bacteremia incidence

The monthly time series of the healthcare-associated bacteremia count was shown in Figure 5.2.7. Seasonality was noted. The incidence was higher in the summer and spring than that in the winter (p<0.0001) (Table 5.2.4). The relative risk was 19% higher in the summer than in the winter, and 16% higher in the spring than in the winter. Figure 5.2.8 is the time series after de-seasonalization of the HAI bacteremia, and there were five possible outliers noted in this figure. After deleting the outliers, there was a significant linear trend (Table 5.2.5). The analysis for the de-seasonalized and de-trend time series revealed lacking of autoregressive order (Table 5.2.6).

(C) Pneumonia incidence

The monthly time series of the healthcare-associated pneumonia count was shown in Figure 5.2.10. Although the univariate seasonal variation analysis was not

significant, the incidence tended to be higher in the summer than in the winter (Table 5.2.7). Figure 5.2.11 is the time series after de-seasonalization of the HAI pneumonia, and there were three outliers noted in this figure. After deleting the outliers, there was a significant linear trend (Table 5.2.8). The analysis for the seasonalized and de-trend time series revealed second-order autoregressive pattern (Table 5.2.9).

(D) Surgical site infection incidence (SSI)

The monthly time series of the healthcare-associated SSI count was shown in Figure 5.2.13. Seasonality was noted. The incidence was higher in the summer and spring than that in the winter (p=0.0029) (Table 5.2.10). The relative risk was 20%

higher in the summer than in the winter, and 24% higher in the spring than in the winter. Figure 5.2.14 was the time series after de-seasonalization of the healthcare-associated SSI, and there were five outliers noted in this figure. After deleting the outliers, there was a significant linear trend (p<0.0001) (Table 5.2.11). There was lacking of autoregressive pattern for the de-seasonalized and de-trend time series (Table 5.2.12).

(E) Urinary tract infection incidence (UTI)

The monthly time series of the healthcare-associated UTI count was shown in Figure 5.2.16. Seasonality was noted. The incidence was higher in the summer and spring than that in the winter (p<0.0001) (Table 5.2.13). The relative risk was 16%

higher in the summer than in the winter, and 11% higher in the spring than in the winter. Figure 5.2.17 was the time series after de-seasonalization of the healthcare-associated UTI, and there were four outliers noted in this figure. After deleting the outliers, there was a significant third degree polynomial trend (Table 5.2.14). The analysis for the de-seasonalized and de-trend time series revealed fourth-order autoregressive pattern (Table 5.2.15).

(F) E. coli Bacteremia incidence

The monthly time series of the healthcare-associated E. coli bacteremia count was shown in Figure 5.2.18. Seasonality was not significant (Table 5.2.16). Figure 5.2.19 was the time series after de-seasonalization of the HAI E. coli bacteremia, and there were four outliers noted in this figure. After deleting the outliers, there was a significant quadratic trend (Table 5.2.17). The autoregression analysis for the de-seasonalized and de-trend time series revealed second-order autoregressive pattern (Table 5.2.18).

75 (G) P. aeruginosa Bacteremia incidence

The monthly time series of the healthcare-associated P. aeruginosa bacteremia count was shown in Figure 5.2.20. Seasonal variation was not significant (Table 5.2.19). Figure 5.2.21 is the time series after de-seasonalization of the HAI P.

aeruginosa bacteremia, and there were four outliers noted in this figure. Despite

deleting the outliers, there was no significant trend (Table 5.2.20). The autoregression analysis for the de-seasonalized and de-trend time series revealed lacking of

autoregressive pattern (Table 5.2.21).

Autoregressive integrated moving-average model (ARIMA)

The traditional ARIMA time series analysis for the original overall HAI

incidence revealed an order one moving average after one differencing of the original incidence time series. The model for bacteremia, pneumonia, and surgical site

infections incidence was ARIMA (0,1,1). The ARIMA model for UTI was ARIMA (1.0.2).

In the bacteremia, the E. coli incidence shows no significant autoregressive or moving average order. In P. aeruginosa incidence, both the order two autoregressive

and moving average order were significant, the absolute value of the higher order was small than that of the low order are also shown in (Table 5.2.22, Table 5.2.23).

5.3 Generalized Time-series Model with covariates, seasonality, time trend,

and autoregressive order: A MLE approach

(A) Overall HAI incidence

In the multi-variable analysis, we adjusted covariates of age, gender, and admission department. We also adjusted seasonal factor, third-degree polynomial trend effect and autoregressive order one. Under the assumption of normal

distribution, the result showed negative of Hessian and the model was not stable. Thus, we adjusted age, gender, season, third-degree polynomial trend effect and

autoregressive order one. After adjustment, the incidence of overall incidence had third-degree polynomial trend and autoregressive order one pattern (Table 5.3.1). In the Poisson time series model, it revealed the over-dispersion shown in the deviance.

Thus we applied negative binomial distribution to fit the data, the seasonal factor result became not significant. Although third-degree polynomial trend effect was significant in all three model forms, the regression coefficients were very small (Table 5.3.1-5.3.3).

77 (B) Bacteremia

In the multi-variable analysis of bacteremia incidence, the linear trend was significant after adjusting the age, gender, and seasonal effect. The autoregressive order and seasonal effect was not significant (Table 5.3.4). The seasonal factor was not significant in both Poisson and negative binomial model, either. The linear trend was obvious in three models. The autoregressive order one was significant in Poisson and negative binomial model, but not in the normal distribution model (Table 5.3.4-5.3.6).

(C) Pneumonia

In the multi-variable analysis of pneumonia incidence, the autoregressive order two effect was significant after adjusting the age, gender, trend and seasonal effect.

The autoregressive order was significant, whereas the seasonal and trend effect were not (Table 5.3.7). The negative binomial model results on the seasonal effects, trend, and autoregressive pattern was similar with those in the normal distribution model (Table 5.3.7, Table 5.3.9).

78 (D) Surgical site infections

In the multi-variable analysis of surgical site infection incidence, the linear trend was significant after adjusting the age, gender, and seasonal effect. The autoregressive order and seasonal effect was not significant (Table 5.3.10). In both Poisson and negative binomial model, the results of seasonal effects, linear trend, and

autoregressive order one agreed with those in the normal model (Table 5.3.11, Table 5.3.12).

(E) Urinary tract infections

The seasonal effect was not significant in the urinary tract HAIs. The third-degree polynomial trend was shown in all three distribution model forms (Table 5.3.13, Table 5.3.14, and Table 5.3.15). The autoregressive order three was significant in Poisson and negative binomial model.

(F) E. coli bacteremia

In the multi-variable analysis of E. coli bacteremia incidence, the seasonal effect, trend, and autoregressive order were not significant after adjusting the age and gender (Table 5.3.16). However, in the Poisson and negative binomial model, they revealed quadratic trend effects (Table 5.3.17, Table 5.3.18).

79 (G) P. aeruginosa bacteremia

After adjusting covariates of age and gender, the seasonal effect, trend effect, and autoregressive order were not significant in all three distribution models (Table 5.3.19, Table 5.3.20, and Table 5.3.21).

5.4 Bayesian dynamic linear model

In the Bayesian dynamic time linear model, the model assumed normal

distribution assumption. The age was a dichotomous variable, with a variable of 50-years-old versus those below 50.

(A) Overall HAI incidence

It revealed first order autoregressive pattern after adjusting seasonal effect, trend effect, age and gender. The seasonal and trend effect was not significant. As compared with the result of non-Bayesian time series model, in which the significant third-polynomial trend became insignificant in the Bayesian analysis. The seasonal effect results were similar (Table 5.4.1).

(B) Bacteremia

The first-order autoregressive pattern was still significant after adjusting age,

gender, trend, and seasonal factors. The trend effect was not significant (Table 47).

(C) Pneumonia

There was a significant second-order autoregressive pattern in the Bayesian analysis after adjusting age, gender, and seasonal effect. The trend and seasonal effect were not significant (Table 5.4.3).

(D) Surgical site infections

There was a significant first-order autoregressive pattern shown in the Table 54. In the multi-variable Bayesian analysis, there was no seasonal effect. After adjusting the seasonal effect, the linear trend became significant, though the regression coefficient was very small (Table 5.4.4).

(E) Urinary tract infections

The autoregressive pattern was fourth-order in the urinary tract HAI. The seasonal and trend effect were not significant (Table 5.4.5).

(F) E. coli bacteremia

The seasonal effect was not significant in the model after adjust the age, gender, trend and autocorrelations. It revealed third-order autoregressive pattern and linear trend in the adjusted model (Table 5.4.6).

(G) P. aeruginosa bacteremia

There was a third-order autoregressive pattern and linear trend in the multi-variable Bayesian analysis. The seasonal effect was not significant (Table 5.4.7).

5.5 Bayesian Autoregressive Moving Average model

1. Bayesian autoregressive model (1) HAIs

Table 5.5.1 shows the estimated regression coefficients with Bayesian

autoregressive model incorporating time trend and seasonal variation in the model but without considering any other non time-series covariates for overall HAI. In the model of first order autoregressive, summer, spring, and autumn had higher risk of HAI than winter. The estimated regression coefficients were 0.133 (0.081-0.186), 0.14 (0.082-0.199), 0.092 (0.039-0.146) for spring, summer, and autumn, respectively. The first autoregressive order was statistically significant (0.45(0.46-0.63)). Linear time trend significantly decreased with time. When higher orders of autoregressive terms were considered, the effects of seasons were similar. However, time trend effect became insignificant in the second, third, and fourth order autoregressive model. In Table 5.5.1, fourth-order autoregressive term was not statistically significant.

The trigonometric function was used as an alternative for seasonal variation

(Table 5.5.2). The regression coefficient for the series of autoregressive models was close to their counterparts in Table 5.5.1. The results still support that the fourth-order of autoregressive terms was not suggested.

The results of Bayesian autoregressive model incorporating time trend, seasonal variation, and non time-series covariates (age and gender) for overall HAI were shown in Table 5.5.3. The regressive coefficients for season and time trend effect did

not changed a lot. For the first autoregressive term, the regression coefficient became smaller (0.2750.019) than that in Table 5.5.1 (0.5450.042). Younger patients had

significantly less HAIs than the elder patients aged over 70. The estimated regression coefficients were -2.069 (95% CI: -2.145- -1.991) and -1.195 (95% CI: -1.248- -1.143) for age under 40 and age between 40 and 69, respectively. Male had significantly larger HAI counts than female. The estimated coefficient was 0.086 (95% CI: 0.042-0.131).

This thesis further considers cubic form of time trend in the first order

autoregressive model (Table 5.5.4). Such model did not change much of the estimated

results for age, gender, and season. The regression coefficient of the first

autoregressive became smaller (0.2230.02) than that in Table 5.5.3 (0.2750.019).

Regarding time trend, linear trend became insignificant, but both quadratic and cubic

83 terms were statistically significant.

The cubic time trend model of first-order autoregressive model (Table 5.5.4) was further extended to cubic time trend model of third-order autoregressive model (Table 5.5.5). The effects of age, gender, season, and time trend were similar in the two models. All first-, second-, and third-order of autoregressive were statistically significant (Table 5.5.5). The third-order autoregressive model had smaller DIC (7491.98) than that in the first-order autoregressive model (7599.01). With cubic form of time trend, the autoregressive of first, second, third, and fourth-order were all statistically significant with DIC of 7482.3 (Table 5.5.6).

(2) UTI

The estimated results of the Bayesian autoregressive model incorporating time trend and seasonal variation in the model but without considering any other non time-series covariates for UTI were shown in Table 5.5.7. In the model of first order autoregressive, summer, spring, and autumn had higher risk of UTI than winter. The estimated regression coefficients were 0.135 (0.050-0.220), 0.186 (0.090-0.277), 0.109 (0.022-0.194) for spring, summer, and autumn, respectively. The first

autoregressive order was statistically significant (0.51 (95% CI: 0.42-0.59)). Linear

time trend was positive and statistically significant. When higher orders of

autoregressive terms were concerned, the effects of seasons were similar. However, time trend effect was only significant in the second-order autoregressive model but not in the third and fourth order autoregressive model. The same as overall HAI, the fourth-order autoregressive term was not statistically significant for UTI.

The estimated results of using trigonometric seasonal variation for UTI are shown in Table 5.5.8. The regression coefficient for the series of autoregressive models was similar to their counterparts in Table 5.5.7, except that linear time trend was

consistently insignificant in all the four models in Table 5.5.7. The results still supported that the fourth-order of autoregressive terms was not suggested.

The results of Bayesian autoregressive model incorporating time trend in cubic form, seasonal variation, and non time-series covariates (age and gender) for UTI are shown in Table 5.5.9. The regressive coefficients for seasonal variation did not were similar. The linear time trend became significant. The magnitude of autoregressive terms was half of that in Table 5.5.7. Table 5.5.10 shows the results with fourth-order of autoregressive terms. Younger patients had significantly less UTI than the elder patients aged over 70. The estimated regression coefficients were 2.866 (3.072 -2.674) and -1.526 (-1.644- -1.406) for age under 40 and age between 40 and 69,

respectively. Male had significantly larger UTI counts than female. The estimated coefficient was -0.247 (95% CI: -0.331- -0.161).

(3) E. coli. bacteremia

The estimated results of the Bayesian autoregressive model incorporating time trend and seasonal variation in the model but without considering any other non time-series covariates for E. coli. bacteremia infections are shown in Table 5.5.11. In the model of first order autoregressive, seasons did not have statistically significant effects. The first autoregressive order was statistically significant (0.246 (95% CI:

0.117-0.377)). Linear time trend was insignificant. When higher orders of autoregressive terms were concerned, the effects of seasons and time trend were similar. However, higher order of autoregressive were insignificant.

The estimated results of using trigonometric seasonal variation for E. coli.

bacteremia infection are shown in Table 5.5.12. The regression coefficient for the series of autoregressive models and linear time trend were similar to their counterparts in Table 5.5.11.

The results of Bayesian autoregressive model incorporating time trend in quadratic form, seasonal variation, and non time-series covariates (age and gender)

for E. coli. bacteremia infection are shown in Table 5.5.13. The regressive coefficients for seasonal variation, linear time trend, and first-order autoregressive were similar.

Younger patients had significantly less E. coli. bacteremia infection than the elder patients aged over 70. The estimated regression coefficients were 2.861 (3.716 -2.217) and -1.109 (-1.474- -0.815) for age under 40 and age between 40 and 69, respectively. Male had higher E. coli. bacteremia infection counts than female. The estimated coefficient was 0.106 (95% CI: -0.140-0.349) (p>0.05).

The results with second-order autoregressive terms of Bayesian autoregressive model incorporating time trend in quadratic form, seasonal variation, and non time-series covariates (age and gender) for E. coli. bacteremia infection are shown in Table 5.5.14. The effects for all covariates were similar to those in Table 5.5.12. The second order autoregressive was not significant.

(4) P. aeruginosa bacteremia

The same as observed in E coli bacteremia infection, seasons did not have statistically

significant effects in the model of first order autoregressive. The first autoregressive order was also insignificant (0.079 (95% CI: -0.097-0.258)). Linear time trend was decreasing. When higher orders of autoregressive terms were concerned, the effects of seasons and time trend were similar. All the autoregressive terms were not statistically significant.

The estimated results of using trigonometric seasonal variation for P. aeruginosa bacteremia infection are shown in Table 5.5.16. The regression coefficient for the series of autoregressive models and linear time trend were similar to their counterparts in Table 5.5.15.

The results of Bayesian autoregressive model incorporating time trend in

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