uva transplant project

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LINEAR STATISTICAL MODELS

SYS 4021

Project 3:

Design Improvements for

the University of Virginia Transplant

Center

Donald E. BROWN Laura BARNES

[email protected] [email protected]

Summary

This study considers the number of kidney and liver transplants at UVA and comes up with an evaluation for

these organic transplants with the MCV and Duke center overall and in different ethnic group especially for

minorities. UVA has the smallest trend on the number of kidney transplants overall and in non-white group

as compared to the two centers over the period 1988 2012. The t-test shows that there is a difference

between the number of transplants at UVA and the other centers at 5% level. The 95% bootstrap confidence

interval of the mean difference also indicates that I can reject the null hypothesis of mean difference is zero.

Time series linear model is constructed to predict the mean difference between two centers in 2013.The

results from Bootstrap and Monte-Carlo simulation reveals that the 95% prediction confidence interval does

not contain zero, meaning that there is a difference between the prediction numbers of kidney transplants.The negative confidence interval tells that the predicted number of kidney transplants at UVA overall and for

non-whites in 2013 is less than the predicted number of kidney transplants at MCV and Duke. This suggests

UVA to do better at recruiting people overall and at recruiting people from other ethnicities. For liver

transplants, it is hard to conclude that building the new Roanoke center in 2005 has increased the number of

liver transplants at UVA. Linear model and Poisson model to model the number of liver transplants show

contradict results. Based on linear model with time series, the p-value of Roanoke variable is 0.014 and is

less than 0.05. With this model, I can reject the null hypothesis at 5% and conclude that building the

Roanoke center has increased the number of liver transplants. Meanwhile based on Poisson model with time

series, Roanoke variable does not affect the number of liver transplants and is not significant at 5% level.

This suggests UVA to do more research on liver transplants and it may be interesting to collect data at UVA

C-ville and UVA Roanoke center.

Honor pledge: On my honor, I pledge that I am the sole author of this paper and I have accurately cited all

help and references used in its completion.

Imran A. Khan

December 7, 13


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1. Problem Description

1.1.

Situation

Organ transplantation replaces diseased or damaged organs with functioning organs from either deceased or

living donors. The complexity of these procedures requires highly skilled teams of physicians, nurses, and

support staff as well as facilities for the surgery and recovery. According to research from the United States,the need for organ donation has become a growing concern over the last decade as the gap between organ

donors and those awaiting transplants widens [4].

The University of Virginia has conducted organ transplantation for more than 30 years and now provides

services for kidney, pancreas, liver, islet, heart, and lung transplantation [2]. The UVA Health System is

consistently ranked as one of the Top 100 Hospitals in America [5]. The availability of first-rate

transplantation services is a component of these rankings. The Transplant Center desires to continue to

increase the number of transplants in all categories but needs guidance on how to achieve this goal [2].

Organ transplantation processes have five primary steps [2]:

1. Referral from a primary care physician;

2. Determination of eligibility and placement on a waiting list;

3. Matching of donor organ with the patient;

4. Acceptance of the organ by the transplant center;

5. Transplantation surgery and recovery.

Figure 1 displays the total number of kidney transplants and donors from the 11 geographic regions. It shows

that the number of transplants is always higher than the number of donors. The center with the most recent

kidney transplants is MCV and the center with the least kidney transplants is UVA and UNC in the last few

years. Duke has done more kidney transplants in 20002005 than all other centers.

Figure 1.Plot Kidney Transplants


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Table 1 shows that MCV center has the highest number of kidney transplants in a single year while UVA

and UNC center has the lowest number of kidney transplants in a single year. On average, DUKE center has

significantly more kidney transplants than other centers.

Table 1. Summary statistics on number of kidney transplants by center/region

Statistics UVA UNV MCV Duke R11Donor

Most kidney transplants 107 90 137 121 1337

Least kidney transplants 24 24 29 43 456

Kidney transplants in 2012 68 90 136 74 1304

Average per year 65.8 54.64 75.64 81.76 956.52

Standard error of the mean 4.16 3.3 7.24 4.88 57.08

MCV appears to have the largest trend in kidney transplants and UVA appears to have the smallest trend in

kidney transplants over the period 1988 2012. UVA and Duke have nearly parallel trend in kidney

transplants over the period of time (Figure 2).

Figure 2.Kidney transplants trend over the period of time by centers

Taking into account the ethnic group, I can observe that UVA performs kidney transplants better for white

people and MCV performs better for minorities or non-whites (Figure 3). This means that UVA center has to

improve its performance of kidney transplants for non-whites. Meanwhile, Duke center performs better in

the beginning of the year for white people but then at the end of the year more kidney transplants for non-

white people than for white people. Comparing the number of kidney transplants of non-whites between the

other centers, I can see that UVA has the poorest performance with the smallest trend and MCV has the

greatest performance with the largest trend (Figure 4).


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Figure 3.Plot of kidney transplants by ethnic group for each center

Figure 4.Plot of kidney transplants by center for each ethnic group

Figure 5. Scatterplot matrix for kidney transplants between centers and region


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Figure 5 shows that the total number of kidney transplants, donors, and the number of kidney transplants in

each center has a strong positive correlation. This can be easily explained as all these numbers have been

increasing over the last 10 years.

Unlike kidney transplants, there are more liver donors than liver transplants as shown in Figure 6. In general,

the four centers fluctuate between decreasing and increasing over the year. UVA reaches the highest liver

transplants in 2009 but then it drops in 2010. In addition, all centers perform well for liver transplants ofwhite people than non-white people (Figure 7 & 8).

Figure 6.Plot of liver transplants

Figure 7.Plot of liver transplants by ethnic group for each center


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Figure 8.Plot of liver transplants by center for each ethnic group

Table 2 summarizes the statistics on number of liver transplants. On average, UVA has more liver

transplants than the other centers. Also, it has the highest liver transplants in 2012.

Table 2. Summary statistics on number of liver transplants

Statistics UVA UNV MCV Duke

Most liver transplants 87 73 66 67

Least liver transplants 1 0 16 11

Liver transplants in 2012 68 31 60 67

Average per year 46.52 38.68 45.24 38.08

Standard error of the mean 4.575 4.398 2.815 2.542

Based on the above situation, I can see that MCV and Duke do better job than UVA for the overall kidney

transplants. Also, UVA has the poorest performance of kidney transplants of non-whites as compared to the

other centers. This motivates me to compare kidney transplants at UVA with MCV, since MCV has the

largest trend or a stable increase overall. Also, MCV has similarity in demographics as UVA, so MCV is a

good choice to compare the kidney transplants overall and in non-white group. In addition, it is also my

interest to compare between UVA and Duke since they both have similar trend over the period of time.

There is no need to compare the number of liver transplants between UVA and the other centers since UVA

has shown a good performance over the period of time. But it should be noted that UVA built a new center in

Roanoke in 2005. As shown in Figure 6, I see that the number of liver transplants at UVA tends to increase a

lot from 2005. So it is my interest to see if building the new center has increased the number of liver

transplants.


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1.2.

Goal

The aim of this study is to come up with a new design that could potentially increase the number of kidney

and liver transplants at UVA. In terms of kidney transplants, the goal is to figure out on how to increase or

improve the number of transplants overall and in non-white group by comparing to MCV and Duke centers.

In terms of liver transplants, the goal is to analyze the efficiency in increasing the number of transplants

when the new center Roanoke was built in 2005.

1.3.

Metrics

For kidney transplants analysis, the difference between the number of kidney transplants at UVA and MCV

and at UVA and Duke are used as a response variable in linear regression model. For liver transplants

analysis, the number of liver transplants at UVA is used as a response variable in linear regression model.

Adjusted R2, AIC, and MSE are considered to compare the performance between the fitted models. In

addition to liver transplant analysis, Poisson model is also considered. The Diagnostic plots (Normal Q-Q

plot, Residuals vs. Fitted, Residuals vs. Leverage, Scale-Location), Autocorrelation function (ACF), and

Partial Autocorrelation function (PACF) are used to investigate if autoregressive term is needed in the

model. Bootstrapping method is also applied to estimate the regression parameters with B=2000 as the

number of bootstrap samples. This includes constructing 95% percentile and BCa confidence interval.

I use significance level of 5% for the analysis throughout this study. If the confidence level (p-value) is less

than 0.05, then my (null) hypothesis is rejected in favor of the alternative. Alternatively, if p-value is greater

than 0.05, then my null hypothesis should not be rejected.

1.4.

Hypotheses

Hypothesis 1:

H0: There is no difference between number of kidney transplants at UVA and MCV in 2013

H1: There is a difference between number of kidney transplants at UVA and MCV in 2013

Hypothesis 2:

H0: There is no difference between number of non-white kidney transplants at UVA and MCV

H1: There is a difference between number of non-white kidney transplants at UVA and MCV

Hypothesis 3:

H0: There is no difference between number of kidney transplants at UVA and Duke in 2013

H1: There is a difference between number of kidney transplants at UVA and Duke in 2013

Hypothesis 4:H0: There is no difference between number of non-white kidney transplants at UVA and Duke

H1: There is a difference between number of non-white kidney transplants at UVA and Duke

Hypothesis 5:

H0: Building the Roanake center does not increase the number of liver transplants at UVA

H1: Bulding the Roanake center increases the number of liver transplants at UVA


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2. Approach

2.1.

Data

The data for this study comes from the Organ Procurement and Transplantation (OPTN) [1] for 4 transplant

centers and two region US, region 11 (Kentucky, North Carolina, South Carolina, Tennessee and Virginia).

Each of regions has number of transplants performed at center/ region, background of the patients (age andethnic group) There are 15 databases related to organ transplant data in csv (comma-separated values)

format, i.e. USdonor.csv, UStransplant.csv, R11donor.csv, R11xplant.csv, UVAxplant.csv, Dukexplant.csv,

MCVxplant.csv, UNC.csv, MCVage.csv, MCVethnic.csv, R11age.csv, R11ethnic.csv, UVAage.csv, and

UVAethnic.csv.

In total, there are 26 observations in each data showing the number of organ transplants from 1988 to 2013.

In this study, I only consider the number of kidney and liver transplants at both region and center. The series

of number kidney and transplants over the last 26 years are tabulated in Table A1 and A2 (Appendix A).

There are no missing values in the data but it should be noted the 26thobservation (2013) is not included in

the analysis since the data are incomplete for that year. For liver transplant data at UVA center, there is one

additional variable, i.e. binary variable, indicates before (0) and after (1) building the Roanoke center.

2.2.

Analysis

2.2.1.

Kidney Transplants Analysis

To predict the difference of kidney transplant between UVA and the other centers in 2013, linear regression

model is built by using R software. Before I start with model building, I make a time series plot of kidney

transplants for each center and also a time series plot of the difference of kidney transplant at UVA and the

other centers (MCV and Duke). A classical paired t-test is also performed to see if there is a difference of

kidney transplants from 1988 to 2012. This test works best on normally distributed data, thus if the

assumption of normality is violated, bootstrapping method is then applied.

In order to reject my hypotheses related to kidney transplants, I build 4 linear regression models by using

r11donor variable to control for Region 11:

Model 1: ( )= + 11_ +

Model 2: ( )= + 11_ +

Model 3: ( )= + 11_ +

Model 4: ( )= + 11_ +

where:

- is the number of kidney transplants at UVA,

- is the number of kidney transplants at MAC,

- is the number of kidney transplants at Duke,

- is the number of kidney transplants of non-whites at UVA,


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- is the number of kidney transplants of non-whites at MCV,

- is the number of kidney transplants of non-whites at Duke,

- 11_ is the number of kidney transplants at region 11, and

-

11_ is the number of kidney transplants of non-whites at region 11.

The stages of model building are as follows:

1.

Fit model 14 and check the residuals of the fitted model whether they are correlated or not by looking

at ACF and PACF plot. Also, observe graphical diagnostic plots to investigate how well the regression

assumptions are satisfied.

2. Determine the number of autoregressive (AR) terms needed to model the residuals by examining the

AIC plot.

3. Fit time series regression models, i.e. model 1 4 after accounting AR terms in the model, and then

examine diagnostic plots and ACF and PACF plots of the residuals of the time series regression models.

4.

Asses the fitted model before and after accounting AR terms by using adjusted R2, AIC, and MSE

criteria.

5.

Apply bootstrapping method to the time series regression model to estimate standard error and

confidence interval of the parameters.

6. Predict the difference of kidney transplants at UVA and other centers in 2013 from the time series

regression model by utilizing bootstrap and Monte-Carlo simulation approach.

2.2.2.

Liver Transplants Analysis

In order to answer my hypothesis related to liver transplants, I start with performing a classical two sample t-

test to see if there is a difference between the number of liver transplants before (1988 2004) and after

(20052012) building the Roanoke center. A non-parametric t-test is also performed, .i.e. Wilcoxon test, incase normal assumption is violated under t-test. Time series of liver transplants is also considered in this

analysis. The residuals of the time series model is then used in t-test and Wilcoxon test to compare the

number of liver transplants before and after building the Roanoke center. Furthermore, I consider linear

model and Poisson model to test whether the number of liver transplant increases after building the Roanoke

center by controlling for Region 11 in the model.

The stages of building linear model are as follows:

1. Consider a linear model with Region 11 and Roanoke center as predictor variables :

Model 5: = + 11+ + where ={1,0,

.

2. Examine diagnostic plot to check if regression assumptions are satisfied.

3. Examine ACF and PACF plots to check if the residuals series are correlated.


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4. Use AIC plot to determine the number of AR terms needed.

5. Asses the fitted model before and after accounting AR terms by using adjusted R2, AIC, and MSE.

6. Apply bootstrapping method to the time series regression model to estimate standard error and

confidence interval of the coefficient value of Roanoke center.

The stages of building Poisson model are as follows:

1.

Consider a Poisson model with Region 11 as a predictor variable:

Model 6.1: log() = + 11_

2. Examine diagnostic plot to check if regression assumptions are satisfied. Also, examine ACF and PACF

plots to check if the residuals series are correlated.

3. Check the dispersion of the fitted Poisson model. If the dispersion is not close to 1 then consider a Quasi

Poisson model.

4. Consider a (Quasi) Poisson model with time series component:

Model 6.2: log() = + 11_ +5. Examine diagnostic plot and ACF and PACF plot of the fitted model in 4.

6. Add Roanoke variable in the (Quasi) Poisson model with time series:

Model 6.3: log() = + 11_ + + 7. Examine diagnostic plot and ACF and PACF plot of the fitted model in 6.

8. Perform chi square test between the time series (Quasi) Poisson model with and without Roanoke center.

3. Evidence

3.1. Kidney Transplants

Figure 9 shows the series of kidney transplants at UVA, MCV, and Duke from 1988 to 2012 and their

corresponding ACF and PACF plots. The series plot shows non-stationary and the correlogram (ACF) showsthat the series are correlated since lags 1-2 are significant for kidney transplants at UVA, lags 1-5 are

significant for kidney transplants at MCV, and lags 1-3 are significant for kidney transplants at Duke.

Similar result is observed for series of kidney transplants of non-whites as shown in Figure 10: ACF plot

shows sinusoidal decay and PACF plot cuts off after lag 1.


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Figure 9.Kidney transplants series and its corresponding ACF and PACF plots

Figure 10.Kidney transplants series of non-whites and its corresponding ACF and PACF plots


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A classical paired t-test is performed between the number of kidney transplants at UVA and other centers

from 1988 to 2012. The result is summarized in Table 3. On average, the number of kidney transplants at

UVA is smaller than the other centers and thus I have a negative mean difference. The test shows that I can

reject the null hypothesis, meaning there is a difference between the number of kidney transplants of non-

white people at UVA and MCV and at UVA and Duke at 5% level. The p-values are smaller than 0.0001.

But for the overall number of kidney transplants at UVA and MCV, the test shows that the difference is zero

because I do not have a strong evidence to reject the null hypothesis at 5% level (p-value = 0.079). One of

the reason I cannot reject the null hypothesis is that the sample size is very small, only 25 observations. Also,

t-test may not be valid to the data because it is a parametric method that relies on an assumption of normal

distribution of the data. The histogram and the QQ plot as shown in Figure B1 (Appendix B) shows that the

difference between kidney transplants at UVA and other centers are not normal.

Table 3.Paired t-test on number of kidney transplants at UVA and other centers

Meana MeanbMean

Differencet-test DF p-value

95% CI

Lower Upper

Overall kidney transplants

UVAa

vs. MCVb

65.8 75.64 -9.84 -1.834 24 0.079 -20.913 1.233UVAa vs. Dukeb 65.8 81.76 -15.96 -3.828 24 0.001 -24.565 -7.355

Non-whites kidney transplants

UVAa vs. MCVb 17.08 49.88 -32.80 -7.751 24


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Figure 11.Plot of difference between kidney transplants at UVA and other centers

Figure 12 displays the ACF and PACF plot of difference of number of kidney transplants at UVA and MCV

and at UVA and Duke. It shows that the difference series are correlated and it may be necessary to considerautoregressive model.

Figure 12.ACF and PACF plot of difference of number of kidney transplants

Table 5 summarizes the estimate, standard error, and p-value of the regression parameters for Model 1 4.

The overall model is significant at 5% except Model 2. Diagnostic plot indicates that the regression

assumptions are violated. I can observe the model has non-constant variance and lack of fit based on the

residual v.s fitted plot. Also, the residuals do not follow Gaussian based on the QQ plot (Figure 13-16).


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Table 5.Estimate and standard error of linear model with mean difference of kidney transplants as the response

Model 1 Model 2 Model 3 Model 4

Estimate

(se)P-value

Estimate

(se)P-value

Estimate

(se)P-value

Estimate

(se)P-value

43.079(15.789) 0.012-7.042

(15.054)0.644

0.642

(4.446)0.886

-9.281

(4.514)0.051

-0.055

(0.016) 0.002-0.009

(0.015) 0.543-0.262

(0.031)


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Figure 15.Diagnostic plots for model 3

Figure 16.Diagnostic plots for model 4

ACF and PACF plots of residuals for Model 14 show that the series correlated and it may be necessary to

consider AR(1) to model the residuals since the sample PACF plot has insignificant peak after lag 1,

especially for Model 1 and 3 (Figure 17). AIC plot for the different number of lag as shown in Figure 18

indicates that it may be adequate to consider AR(1) for residuals of Model 1 and 3 and AR(4) for residuals of

Model 2. I dont need to consider AR term for residuals of Model 4 since PACF plot show no significant lag.


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Figure 17.ACF and PACF plots of linear model residuals

Figure 18.AIC plot for AR model

My final linear model after accounting AR terms can be re-written as follows:

Model 1: ( )= + 11_+ +

Model 2: ( )= + 11_+ +

Model 3:( )= + 11_ + + +

+ +

Model 4: ( )= + 11_+


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Table 6.Estimate and standard error of linear model after accounting autoregressive terms

Model 1 Model 2 Model 3 Model 4

Estimate

(se)P-value

Estimate

(se)P-value

Estimate

(se)P-value

Estimate

(se)P-value

58.061(12.022)


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Figure 20.Diagnostic plot of model 2 after accounting AR terms

Figure 21.Diagnostic plot of model 3 after accounting AR terms

Figure 22.ACF and PACF plots of residuals of model 1-3 after accounting AR terms

Model 1 Model 2

Model 3


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I also perform bootstrapping method to the time series regression models. I have similar results as obtained

in Table 6. The confidence interval for the regression parameters of Model 1 do not contain zero as shown in

Table 7. This is similar as t-test for the time series regression model with p-values < 0.0001, meaning to

reject both the null hypothesis that the coefficient of r11donor and AR(1) equal to zero.

Table 7.Bootstrap results on time series regression model

Original Bias Std. Error 95% Percentile CI 95% BCa CI

Model 1

* 58.06 -0.0921 10.767 (37.0800, 77.770 ) (36.120, 77.310 )* -0.07 0.0002 0.011 (-0.0906, -0.0500 ) (-0.0913, -0.0505 )* 0.71 0.0007 0.151 ( 0.4172, 1.0109 ) ( 0.4210, 1.0171 )Model 2

* -2.05 0.0399 14.321 (-30.449, 25.506 ) (-31.879, 23.051 )* -0.01 0.0000 0.013 (-0.0384, 0.0124 ) (-0.0398, 0.0119 )* 0.58 0.0050 0.196 ( 0.2057, 0.9814 ) ( 0.2123, 0.9888 )

* -0.28 -0.0068 0.234 (-0.7543, 0.1728 ) (-0.7549, 0.1686 )

* 0.07 0.0089 0.227 (-0.3731, 0.5344 ) (-0.3936, 0.5158 )* -0.47 -0.0047 0.185 (-0.8412, -0.1029 ) (-0.8115, -0.0784 )Model 3

* 0.52 0.0593 4.275 (-7.8556, 9.0951 ) (-7.5756, 9.3440 )* -0.26 -0.0002 0.029 -0.3215, -0.2049 ) (-0.3185, -0.2028 )* 0.32 0.0046 0.200 (-0.0877, 0.7176 ) (-0.0979, 0.7044 )Model 4

* -9.28 0.0439 4.315 (-17.351, -0.625 ) (-16.651, 0.167 )* -0.11 0.0001 0.030 (-0.1672, -0.0522 ) (-0.1672, -0.0521 )

Based on adjusted R2, AIC, MSE criteria, regression model after accounting AR terms performs better than

the model before accounting AR terms. Adjusted R2 is higher for time series regression model. It means

modeling the residuals with autoregressive model has improved the fitted model. AIC and MSE values are

also smaller for model after accounting AR terms (Table 8).

Table 8.Model assessment based on adjusted R2, AIC, and MSE

Before accounting AR terms After accounting AR terms

adj. R2 AIC MSE adj. R2 AIC MSE

Model 1 0.318 229.76 451.50 0.682 204.23 208.17

Model 2 -0.026 227.38 410.44 0.411 183.78 189.95Model 3 0.750 192.77 102.82 0.754 185.88 96.89

Model 4 0.321 193.54 106.02 0.321 193.54 106.02


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My final model in Table 6 can be used to predict the mean difference of kidney transplants in 2013. To do

this, I need to forecast the r11 donors and the model residuals. This is done by forecasting r11donor series in

2013 and then I use this point forecast to predict the time series regression model. Prediction result with

bootstrap method is shown in Table 9. I also use Monte-Carlo simulation for improved CI and the result is

shown in Table 10. Comparing the two results, i.e. bootstrap and simulation prediction, I can see that Monte-

Carlo simulation prediction tends to have wider confidence interval. Also, in general the two methods show

that I am doing better than my estimate already since the predication estimate from bootstrap and simulation

is smaller (Figure 23). The bootstrap and simulation prediction do not show any deviation from normal

distribution (Figure 24-25). The 95% confidence interval for the predicted mean difference of kidney

transplants at UVA and MCV and at UVA and DUKE do not contain zero. This means there is indeed a

difference between the two centers at 5% level. Similar results are also obtained for non-whites kidney

transplants. Therefore, I can reject my null hypotheses of no mean difference in 2013 with p


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Figure 23.Forecast plot

Figure 24.Histogram and QQ-plot of bootstrap prediction


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Figure 25.Histogram and QQ-plot of simulation prediction

3.2.Liver Transplants

Figure 26 plots the number of liver transplants before and after building the Roanoke center. It appears that

there are more liver transplants when the Roanoke center is built. Starting from year 2005, there is an

increasing trend although there is a drop in year 2011. The right panel of Figure 26 shows the distribution of

liver transplants which is non-normal. Table 11 confirms my plot that the mean of liver transplants is higher

(68.13) after the Roanoke center is built. T-test indicates that there is a difference between the number of

liver transplants before and after the Roanoke center (p-value = 0.0013). I also utilize Wilcoxon test because

the sample size is very small and also because the data are not normally distributed (Figure B6, Appendix B).

The result is in line with t-test that I can reject the null hypothesis of the difference is equal to zero at 5%level.

Figure 26.Plot and histogram of number of liver transplants


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Table 11. T-test and Wilcoxon test on number of liver transplants before and after the Roanoke center

Mean before

the Roanoke

center

Mean after the

Roanoke center

T-test Wilcoxon test

t-stat DF p-value W-stat p-value

36.35 68.13 -4.090 12.749 0.0013 12.5 0.0013

ACF plot of liver transplants indicates that the series are correlated and it may be necessary to consider

AR(1) based on PACF plot. AIC plot in Figure B7 (Appendix B) also supports this. The residuals of AR(1)

model is no longer correlated as shown in bottom panel of Figure 27.The residuals are then used to compare

if there is a difference between number of liver transplants before and after building the Roanoke center.

Table 12 reveals that there is no difference between the two since the p-value from both t-test and Wilcoxon

test is greater than 0.05. However, this test may not be valid because it is important to control for Region 11.

Figure 27.ACF and PACF of liver transplants series and residuals of AR(1)

Table 12. T-test and Wilcoxon test on residuals of AR(1) before and after the Roanoke center

T-test Wilcoxon test

t-stat DF p-value W-stat p-value

-1.812 11.750 0.0957 32 0.0523

Linear model for liver transplants with Region 11 and Roanoke variables as predictor show that the

regression assumptions are violated based on diagnostic plot in Figure 16. The residual from the fitted linear

model indicates that the series are correlated. I then consider AR(1) to model the residuals since the PACF

plot cuts off after lag 1 (Figure 29). Accounting AR term in the linear model has improved the fitted model

based on diagnostic plot in Figure 30. The residuals vs. fitted plot shows constant variance and the QQ plot

shows the residuals are approximately normal. Moreover, the ACF and PACF plots indicate the residual

series are no longer correlated and no significant lags in both plot (Figure 31). The estimated parameters and


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its corresponding standard errors for linear model and linear model with time series are summarized Table

13.

Figure 28.Diagnostic plot for linear regression model on number of liver transplants

Figure 29.Plot of residuals from linear regression model and its corresponding ACF and PACF plot

Figure 30.Diagnostic plot for linear regression model with time series on number of liver transplants

Figure 31.Plot of residuals from linear model with time series and its corresponding ACF and PACF plot


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Table 13.Results for linear model and linear model with time series

Linear model (Model 5)Linear model with time

series

Estimate (se) p-value Estimate (se) p-value

31.482 (12.79) 0.022 42.972 (12.23) 0.002 0.012 (0.029) 0.690 -0.011 (0.027) 0.704 26.846 (14.38) 0.075 34.011 (12.62) 0.014 0.428 (0.178) 0.026

Adj. R2 0.391 0.516

AIC 219.9 203.0

MSE 280.7 182.3

Overall modelF-statistic: 8.691 on 2 and 22

DF, p-value: 0.001653

F-statistic: 9.18 on 3 and 20

DF, p-value: 0.0005048

Both models are significant overall at 5% level. Linear regression model with time series performs best

based on adjusted R2, AIC, and MSE criteria. The adjusted R2is larger and AIC is smaller for linear model

after accounting AR term in the model. Also, it fits better because it has decreased the MSE value. My final

linear model for liver transplants can be written as follows:

= 42.9720.01111+ 34.011 + 0.428

Bootstrapping the regression for Roanoke center (*) shows that the confidence interval of the parameterdoes not contain zero (Table 14). This means that I can be 95% confident that there is a difference between

the number of liver transplants before and after the Roanoke center is built. Thus, I can reject my null

hypothesis at 5%, meaning building Roanoke center helps to improve the number of liver transplants at

UVA. The histogram and QQ-plot for the bootstrap estimate of * shows no violation of normality (Figure32).

Table 14.Bootstrap estimate for linear model with time series

Original BiasStd.

Error95% Percentile CI 95% Bca CI

* 42.972 0.128 11.071 (20.08, 63.80 ) (18.71, 62.57 )* -0.011 0.000 0.025 (-0.0582, 0.0408 ) (-0.0552, 0.0431 )* 34.011 -0.120 11.376 (11.30, 56.26 ) (10.28, 55.20 )* 0.428 0.002 0.165 ( 0.1148, 0.7610 ) ( 0.1077, 0.7586 )


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Figure 32.Histogram and QQ-plot of bootstrap estimate of Roanoke center

Considering Poisson model to model the number liver transplants at UVA as a response variable, thedispersion of Model 6.1 is 8.32. Since this number is not close to 1 therefore I use Quasi Poisson model

instead. The diagnostic plot for the residuals of this model shows that the tail distribution lack of fit Gaussian

in normal QQ plot. The residual vs. fitted plot shows non-constant variance (Figure 33). The PACF of

residuals in Figure 34 appears to be insignificant after lag 1 then I consider one autoregressive term in the

Poisson model (Model 6.2). The model looks better now as shown in the diagnostic plot (Figure 35). Also,

the residual series are no longer correlated (Figure 36). This is also true for Model 6.3 where Roanoke center

is included as a predictor variable (Figure 37-38). The comparison between the estimated parameters from

Poisson model, Quasi Poisson model, and Quasi Poisson model with time series are summarized in Table 15.

Figure 33.Diagnostic plot for Quasi Poisson model (Model 6.1)


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Figure 34.Plot of residuals from Quasi Poisson model (Model 6.1) and its corresponding ACF and PACF plot

Figure 35.Diagnostic plot for Quasi Poisson model with time series (Model 6.2)

Figure 36.Plot of residuals from Quasi Poisson model with time series (Model 6.2) and its corresponding ACF and

PACF plot

Figure 37.Diagnostic plot for Quasi Poisson model with time series and Roanoke center (Model 6.3)


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Figure 38.Plot of residuals from Quasi Poisson model with time series and Roanoke center (Model 6.3) and its

corresponding ACF and PACF plot

The model utility test shows that I can reject my null hypothesis and prefer the full model. The Quasi-

Poisson model for Roanoke center (Model 6.3) does not show any significant effect of building Roanoke

center to the response variable. The Roanoke center variable has p-value 0.078 greater than significant level

0.05.

Table 15.Results for (Quasi) Poisson model

Poisson Model (6.1)Quasi Poisson

Model (6.1)

Quasi Poisson

Model with time

series (6.2)

Quasi Poisson Model

with time series and

Roanoke variable (6.3)

Estimate

(se)p-value

Estimate

(se)p-value

Estimate

(se)p-value

Estimate

(se)p-value

3.105(0.085) < 0.00013.105

(0.246)< 0.0001

3.173

(0.197)< 0.0001

3.616

(0.289)< 0.0001

0.001(0.0001) < 0.00010.001

(0.0004)0.003

0.001

(0.0003)0.001

0.00005

(0.001)0.944

0.545(0.293) 0.078

0.083(0.024) 0.003 0.069(0.023) 0.007

Model

utility test

Deviance: 96.195,

DF:1, P-value < 2.2E-

16

Deviance: 96.195,

DF:1, P-

value:0.00674

Deviance: 117.74,

DF:2, P-value


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null hypothesis of building the Roanoke center does not increase the number of liver transplants at UVA at

5% level. This contradicts the result obtained from linear model with time series.

4. Recommendation

It is evidence that modeling the difference of kidney transplants at UVA and MCV and at UVA and Duke

with time series linear regression model fits better than linear model without time series based on adjustedR2, AIC, and MSE criteria. The final model selected is significant overall at 5% level. By using the bootstrap

and Monte Carlo simulation on prediction the time series model for the difference number of kidney

transplants, at level p


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Appendix A

Table A 1.Number of kidney transplants from 1988 to 2013

r11donor UVA MCV Duker11donor UVA MCV Duke

Year W NW W NW W NW W NW

1988 456 24 41 61 254 46 20 4 24 17 39 221989 493 39 47 49 265 42 28 11 20 27 36 13

1990 618 56 57 43 363 53 43 13 22 35 28 15

1991 638 54 34 62 344 57 43 11 14 20 40 22

1992 625 53 38 47 325 62 48 5 14 24 23 24

1993 659 43 43 47 352 83 32 11 15 28 30 17

1994 720 34 37 68 390 73 24 10 13 24 47 21

1995 719 53 29 59 373 72 42 10 15 14 36 23

1996 806 68 38 72 411 81 54 14 11 27 30 42

1997 838 74 53 78 402 94 54 20 18 35 38 40

1998 889 69 53 70 408 91 49 20 21 32 27 431999 937 63 52 70 418 101 46 17 17 35 33 37

2000 949 55 64 86 451 93 44 11 25 39 42 44

2001 1005 58 68 111 432 111 45 12 22 46 67 44

2002 1049 63 82 96 456 109 50 13 33 49 53 43

2003 1105 68 99 121 434 126 52 16 34 65 69 52

2004 1166 73 96 119 501 135 52 21 27 69 60 59

2005 1284 94 112 100 549 167 73 21 26 86 51 49

2006 1337 107 107 95 653 203 73 33 38 69 42 53

2007 1268 102 106 84 611 209 75 27 36 70 40 44

2008 1237 103 108 108 615 206 74 29 32 76 43 65

2009 1281 79 137 105 606 251 55 24 43 94 41 64

2010 1255 69 130 104 608 254 45 24 35 95 37 67

2011 1275 76 124 115 600 251 52 24 48 76 43 72

2012 1304 68 136 74 687 219 42 26 41 95 41 33

2013 791 44 87 56 401 155 26 18 24 63 23 33W: number of kidney transplants of white people

NW: number of kidney transplants of non-white people


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Table A 2.Number of liver transplants from 1988 to 2013

Year R11donor UVA MCV Duke

1988 148 1 21 11

1989 182 17 18 22

1990 253 54 16 31

1991 261 51 27 34

1992 270 36 31 33

1993 344 66 37 21

1994 386 62 33 38

1995 372 54 39 37

1996 433 37 66 37

1997 439 24 60 32

1998 498 23 53 48

1999 510 23 60 25

2000 550 37 45 34

2001 550 40 46 36

2002 571 29 46 38

2003 569 28 57 35

2004 643 36 57 41

2005 757 40 54 41

2006 884 58 55 46

2007 833 83 47 30

2008 838 86 54 44

2009 883 87 55 57

2010 819 78 48 51

2011 781 45 46 63

2012 811 68 60 67

2013 523 44 36 38


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Appendix B

Figure B 1.Histogram and QQ-Plot on mean difference of kidney transplants at UVA and other centers

Figure B 2.Histogram and QQ-plot of bootstrap estimate of UVA - MCV

Figure B 3.Histogram and QQ-plot of bootstrap estimate of UVA - Duke


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Figure B 4.Histogram and QQ-plot of bootstrap estimate of UVA - MCV of non-whites

Figure B 5.Histogram and QQ-plot of bootstrap estimate of UVA - Duke of non-whites

Figure B 6.Histogram and QQ plot of number of liver transplants before and after the Roanoke center


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Figure B 7.AIC plot for AR model

uva transplant project

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