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A Risk-Adjusted Multi-Attribute Cumulative Sum Control Scheme in HealthCare Systems S. N. Shojaei1, S. T. A. Niaki1 1
Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran (
[email protected])
ABSTRACT- Hospitals increasingly use control charts to monitor clinical processes and their outcomes. In medical context, control charts should have stable performance when different patients with different levels of risk enter the hospital. In order to monitor multi-attribute medical processes, we propose a new control chart with entities having different levels of risk. First, risk-adjusted multivariate cumulative sum control chart (RA-MCUSUM) is developed. Then, simulation experiences are performed to demonstrate the application and to evaluate its performance in terms of in-control average run length (ARL0) stability with the one of a standard MCUSUM chart. The results show that while the standard MCUSUM shows a sensitive performance, the RA-MCUSM has a robust performance when different entities with different levels of risk enter the system. Keywords – CUSUM, multi-attribute, risk-adjusted
I.
INTRODUCTION
Hospitals increasingly use control charts to monitor clinical processes and outcomes. Similar to vital signs and laboratory indices, which are used to trace patients‟ health, control charts can be used to monitor the performance of surgical and nursing care. In all these techniques, abnormal values indicate a possible deterioration requiring intervention. These control charts can be a powerful tool to stabilize the processes of surgical care and improve the post-operative outcomes of processes and patients‟ satisfaction. [1, 2] When monitoring in the medical context, it is often necessary to account for heterogeneity in patient conditions which lead to different prior levels of risk at the time of treatment (for example, rate of newly developed ulcers can be affected by diabetes mellitus in nursing care [3] ). Hence, when monitoring the adverse outcomes of medical procedures, the variation of the baseline risk must be taken into account by using a riskadjusted (RA) chart. Implementing RA charts will eliminate unwanted false alarms and decrease the response time to changes. So, the notion of risk adjustment was introduced into the monitoring processes, which means that the adjustments to the type of procedure
and patient conditions (risk factors) are collectively referred to as case-mix [4].
A. Risk Adjusted Control Charts Many previous studies have proposed RA control charts for monitoring medical procedures. Woodall [5] and Cook et al. [6] reviewed several studies in the field of RA control charts; among which, Cook et al.[7] suggested a risk-adjusted P-chart. Steiner et al. [8], Grigg and Spiegelhalter [9], Biswas and Kalbfleisch [10], and Steiner and Jones [11] designed different RA control charts to be used in medical applications. The proposed charts are based either on the cumulative sum (CUSUM) chart [8,10] or the exponentially weighted moving average (EWMA) chart [9,11]. In addition, they are all developed to monitor one attribute and the RA-CUSUM and RAEWMA charts are more sensitive to small and moderate procedure shifts than the RA-P-control chart [6]. B. Multi-Attribute Control Charts There are many processes, in which quality characteristics cannot be measured numerically. In these cases, each inspected item is usually classified as either conforming or nonconforming to specify the quality characteristic. These quality characteristics are referred to as attributes. The most common attribute control charts are the p- and np-charts (for binomially distributed processes) and the c and u control charts (for Poisson distributed processes). The traditional attribute control charts such as the control chart for fraction nonconforming (p-chart) and the control chart for the number of defects or non-conformities (c chart) are onedimensional or univariate. This means that one quality characteristic is taken into account to assess the overall quality of a product[12]. Nevertheless, in many cases, the quality of a product depends on more than one correlated quality attribute. In such cases, the common practice is to monitor each attribute with a separate univariate chart. However, this can lead to misleading results in terms of type I error when the attributes are highly correlated. As a result, it would be necessary to use a multivariate control chart, which considers a correlation structure between the attributes. Besides, it is more practical to use a single multi-attribute control scheme than several uni-attribute ones. In addition, multivariate control procedures are much more preferable compared to separate univariate
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ones. In fact, the multivariate control charts have the ability to take into consideration the relationship between the variables or attributes and provide more sensitive control than that provided by the univariate control charts[13]. Patel [14] proposed a control chart for multivariate attribute data and suggested a Hotelling-type chart for observations from multivariate binomial or multivariate Poisson distribution. Yu et al. [15] presented a step-bystep procedure for both the construction of the SPRT( sequential probability ratio test) under the estimated probability density function of the linguistic data and the control limits of their proposed chart. The purpose of this paper is to develop a multiattribute RA control scheme based on CUSUM chart and analyze its performance when different entities enter the system. In the next section, the concept of a RA chart is first introduced. Then, uni-attribute and multi-attribute CUSUM-based RA control charts are proposed. In Section III, simulation experiments are designed to evaluate the performance of the proposed method. . The results are discussed in Section IV, and the conclusion is delivered in Section V.
system odds are monitored. However, in standard control charts the process is monitored based on . In other words, monitoring of medical procedures can be formulated by testing the hypothesis: { where
(4)
is the number of attributes being monitored.
B. Uni-attribute Risk-adjusted CUSUM Control Chart Steiner et al. [8] originally proposed RA CUSUM. Let be the number of patients whose outcomes are monitored. RA-CUSUM involves monitoring (
to
)
(5)
where and is the sample weight assigned patient defined as: [
] (6)
II. METHODOLOGY
{
A. Concept of Risk Adjusted Control Schemes Risk adjustment accounts for patients‟ prior risk factors when assessing treatment risk. In the context of side effects of surgical operation, let be the failure probability of event for the patient. This probability is a function of the patient‟s covariates. There are different approaches to approximate . For instance, in cardiac surgery, a logistic regression model shown in Eq. (1) is used to estimate patients‟ risk [16]: (
)
(1)
where are scores of the patient and parameters and can be estimated using a historical data set. In medical context, the odds of attribute failure, consists of patient risk and system odds. The relationship is as follows:
[
]
In (6), indicates the outcome for patient (i.e., whether the patient died after a surgery in 30 days or not). Although non-negative weights are assumed in CUSUM, some other weight schemes are suggested in [17, 18]. If , the medical process is diagnosed to be in out-of-control state. The control limit ( ) can be estimated using Markov chain or simulation.[8] C. Multi-attribute Risk-adjusted CUSUM Control chart The RA-CUSUM scheme introduced in the last section is extended in this paper to a multi-attribute one, called thereafter RA-MCUSUM. In multivariate control charts, instead of one weight, there is a vector of weights for each patient ( ). The elements of this vector are shown in (7) and the vector itself is given in (8). Having an initial zero-vector in (9) and using (10) cumulatively, Equation (11) calculates which is being monitored. if , then the medical process is diagnosed to be in outof-control state:
(2) (
[
(3) [
where is the probability of occurrence of attribute. is the risk of patient and is the odds of system‟s failure for attribute. In risk adjusted control charts,
{
)
] )
( (
)
(
)
]
(7)
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200 0.5-0.5 risk
(8) ]
0.2-0.2 risk
(9) [ ]
{ {
} }
(10) (11)
In-control average run lenghth(ARL0)
[
180 160 0.8-0.8 risk 140 120 100 80 60 40 20
III. RESULTS In medical systems, patients with different levels of risk enter a medical procedure. The procedure is assumed to follow a multivariate Bernoulli distribution with two correlated attributes. The parameters of this distribution are estimated using (3). To assess the performance of the proposed RA-MCUSUM scheme, we first simulate the medical system with three groups of entities (low, medium and high). Then, both the standard and the riskadjusted control charts are implemented to monitor the system and to observe ARL0. Consider a multivariate Bernoulli process with two correlated attributes. In-control odds of the system are assumed equal to one and patients can have a low risk of (0.2), medium risk of (0.5), and high risk of (0.8) in their attributes. Equations (2) and (3) calculate the odds of failure. For example, if a low-risk patient in attributes 1 and 2 enters to an in-control process (odds of system=1), then the odds of failure is 0.25. The same can be done for the second attribute. To simulate the system, simulation experiments are first performed in order to estimate the covariance matrix . Then, the medical system is simulated and the RAMCUSUM is implemented to observe its performance in terms of in-control ARL stability and to compare it with the one of a standard MCUSUM. ARL0 stability accounts for robust performance of a RA chart. The results are presented in Table I. In addition, these results are illustrated in Fig. [1] and fig. [2].
0 1 9 17 25 33 41 49 57 65 73 81 89 97
where n is the number of attitudes which are monitored, t is number of patient and ∑ is covariance matrix of the random vector . In this paper, we estimate ∑ h with simulation. In addition, to avoid negative weights, a constant parameter K is added to (7).
control limit (h) Fig 1. ARL0 of Standard MCUSUM control chart based on different control limits with various level of risk for entities
IV. DISCUSSION In this paper, we set the „in-control ARL‟ when the patients have the medium risk level. After that, we change the patients risk level and check the stability of ARL0. The results in Table I and fig. 1 and 2 indicate that when using RA-MCUSUM, the monitoring system remains robust for all entities. In other words, no matter which level of risk a patient has, the control chart demonstrate almost a unique performance. However, the standard MCUSUM is sensitive to the entities. For instance, in a system with (in-control state), when the patients has higher levels of risk, ARL0 is decreased. V. CONCLUSION In this paper, a RA CUSUM-based multi-attribute scheme was first proposed to monitor medical treatment processes; when different patients with different risk levels enter the system. The RA control chart, monitors the odds of system for different attributes simultaneously. This control chart uses weight for every attribute. Then, simulation experiments were designed to compare the performance of the proposed scheme in terms of incontrol average run length stability to the one of a standard MCUSUM. The results showed that the RA MCUSUM chart remains robust when different kinds of
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TABLE I ARL0 of risk adjusted and standard control charts when the risk of entities changes
h
ARL0 0.80.8
ARL0 0.20.2
36
11
10
12
8
42
43
12
13
14
9
46
51
14
15
17
10
48
57
16
17
18
11
55
65
18
19
21
12
57
74
21
20
Standard
24
14
63
RA
81
22
23
Standard
26
15
68
RA
92
26
25
Standard
31
17
81
RA
99
27
27
34
82
18
ARL0 0.5-0.5 RA 10 Standard RA
entities enter the system. Further, the control chart had much more stable performance in comparison with the standard MCUSUM, which is sensitive to the risk level of entities entering the system. Developing multi-stage risk adjusted control charts, which monitors one attribute in different stages and designing multivariate-multistage risk adjusted are the future areas for research.
12 Standard RA 14 Standard RA
ACKNOWLEDGMENT The authors wish to thank Seyyed Ali Shojaee and Morteza Davari for their thoughtful and detailed suggestions to improve this paper.
16 Standard RA
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18 Standard RA
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20
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26 Standard
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30
In-control average run lenghth(ARL0)
0.5-0.5 risk 25
0.2-0.2 risk 0.8-0.8 risk
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20 [7]
15 [8]
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1 11 21 31 41 51 61 71 81 91
0 Control limit(h) Fig 2. ARL0 of RA- MCUSUM control chart in different control limit and with different level of risk for entities
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