PERFORMANCE EVALUATION OF CONVENTIONAL EXPONENTIALLY WEIGHTED MOVING AVERAGE (EWMA) AND P-VALUE CUMULATIVE SUM (CUSUM) CONTROL CHART

This paper is aimed at comparing the performances of the conventional Exponentially Weighted Moving Average (EWMA) and p-value Cumulative Sum (CUSUM) control chart. These charts were applied in monitoring the outbreak of pulmonary tuberculosis in Delta State University Teaching Hospital (DELSUTH), Oghara for a period of eighty four (84) calendar months. Line chart and histogram were plotted to test for stationary and normality of the data. Autocorrelation plot was also used to study the randomness of the data. The results of the control charts show that conventional EWMA chart detects shifts faster in monitoring process mean than the p-value CUSUM control chart.


INTRODUCTION
Control charts are classified in view of whether to use or not to use the historical values of control statistic. Shewhart control charts are applied based on the information of the process contained in the present observations only and it disregards any information given by the entire series of points [1]. Hence, Shewhart control chart is categorized as control chart that has no memory. Therefore, Shewhart control chart has been established to be less susceptible in detecting smaller shifts, mainly smaller than 1.5 times the standard deviation [2]. When smaller shifts are of significance, an alternative to Shewhart control chart is Cumulative Sum (CUSUM) control chart and the Exponentially Weighted Moving Average (EWMA) control chart [3]. These charts are based on reminiscence and perform better than Shewhart chart in detecting smaller shifts. The information from the past observations is cumulated up to the recent sample and then the decision about the mean process is taken [4]. Also for CUSUM chart, the EWMA chart is better in detecting small shifts in the process mean. These charts are used for monitoring the mean of a process on the basis that samples are taken from the process at set times ( i.e, shifts, hours, days, weeks, months, year).
The measurements of these samples at a certain time comprise a subgroup [5]. The EWMA chart depends on the specification of target value and a reliable approximate of the standard deviation. For this reason, it is better to adopt the moving average chart after process control has been established. The exponentially weighted moving average (EWMA) control chart scheme was first introduced by Roberts in 1959, which is a good substitute to Shewhart chart when one is interested in a small process shift [6]. In a situation of hypothesis testing, early testing practice make decisions using the idea of rejection and acceptance region [7]. A null hypothesis will be rejected when the observed value of the associated test statistic falls within the rejection region. This usual way of hypothesis testing has therefore been replaced by the pvalue since the p-value approach cannot just make a decision about the hypotheses, but as well tell us how strong the data in the observed values that is against the null hypothesis [8]. Motivated by this p-value method in hypothesis testing, the p-value approach is used to design a CUSUM chart also. By p-value approach, for a set control chart, the incontrol (IC) distribution of control charting statistic is initially estimated [9]. Subsequently, at a given point, the p-value equivalent to the observed value of such charting statistic is then obtained. If the p-value is smaller than a specified significance level, the chart indicates a process distributional shift [10]. In contrast to conventional control charts using control limits, the p-value method has an advantage that at a known time point, even if a shift in the process is not detected, the p-value can give us a quantitative measure of the possibility of a potential shift, so that the subsequent sampling interval can be adjusted properly. All control charts using p-value approach have the same format, such that the vertical axis always ranges between 0 and1, denoting the p-values, and only one control limit corresponds to the significance level, therefore making the charts more suitable to use. The pvalue computation of the charting statistic of the control chart was discussed in [11]. When the in-control process distribution is assumed to be normal with a known variance, [12] gave an approximate formula for the in control distribution of the statistic of the CUSUM chart.
[8] studied the rate in multistage process mean monitoring, wherein p-value calculation of the charting statistics were applied in different stages of the process monitoring. Nevertheless, comparison of EWMA and pvale CUSUM is still not much available in literature.

METHODS / CALCULATIONS Cumulative Sum (CUSUM) Chart
The main feature of the CUSUM control chart scheme is that it cumulates the difference between observed value and a set target value, μ a . The cumulative sum of deviations from the target value, μ a given by ∑ (1) which is plotted on a chart or tabulated in order to detect an upward or downward shift (change) from target. The tabular CUSUM statistic (2) (3) is used to discover an increase and a decrease in the process mean shift, where µ a is the acceptable process mean. The process is said to be out-of-control if where H is the decision interval [13].

DETERMINING THE VALUE OF K AND H
Parameter K is known as the reference value for the CUSUM control chart scheme and is chosen to be between acceptable process mean µ a , which the CUSUM scheme is said to detect quickly. [14] concludes that for a control scheme designed to detect a specific mean shift of , a value of be used, gave the formula for determining K as (4) where is the variation from the acceptable process mean. This reference value is also known as the reference value for hypothesis testing.
(5) Subsequent to the selection of K, the decision interval H is obtained from extracted table from CUSUM Average Run Length (ARL) without or with First Initial Response (FIR). The value of H should give appropriately large ARL when the process is at its desired Acceptable Quality Level (AQL) and appropriately small ARL when the process has shifted to undesired Reject-able Quality Level (RQL). Usually the FIR feature is used to ascertain a faster detection in case of start-up problems after a control action. An alarm is signaled whenever the CUSUM statistic exceeds H.

THE p-VALUE CUSUM APPROACH
A null hypothesis would be rejected when the observed value of the related test statistic falls within the rejection region [2]. This conventional way of hypothesis testing has been replaced by the p-value approach, because the p-value method cannot only make a decision about the hypotheses, but also tell us the evidence in the observed data is against the null hypothesis. By the p-value approach in hypothesis testing, suggestion is made in designing a control charting scheme using the p-value approach. By the p-value approach, for a known control chart, the in-control (IC) distribution of the charting statistic is computed first. Subsequently, at a given time point, the p-value corresponding with the observed value of the charting statistic is then obtained. If the p-value is less than a pre-specified level, then the chart signals a process shift [15,16].

ASSESSMENT OF THE PERFORMANCE OF p-VALUE CUSUM CHART
Several authors have studied CUSUM control scheme on the basis of Average Run Length (ARL) computation. Instead of using ARL at a specified quality level which is the average number of samples taken before an out of control signal is detected, the P-value is used. The chart gives a signal of mean shift when where h is a control limit (Decision interval) chosen to achieve a prespecified IC ARL (denoted as ARL0) value. Instead of comparing the charting statistic with the control limit (Decision interval) value h; here, computing the p-value corresponding to the value of and thereafter comparing the p-value with a pre-specified significance level α for making a decision whether the process is outof-control (OC) are used. Hence, there is a need to find the in-control distribution first. Some researchers provided an approximation formula for this IC distribution in cases when the IC process distribution is normal with a known variance [17,18]. At a given time point and a given allowance constant k, generate the Phase II observations and compute the reference value. This process is then repeated several times (e.g., a million times), and the experimental distribution of can be determined by the computed values. For a given observed value of , denoted as , the corresponding p-value is then computed by: and (6) where is the CUSUM values greater than zero.

THE EXPONENTIAL WEIGHTED MOVING AVERAGE (EWMA)
The Exponential Weighted Moving Average (EWMA) chart is used for monitoring process by dividing the data such that it gives less weight to older data as samples are taken and give a more weight to most recent data. It is also effective in detecting smaller shifts.
The EWMA charting procedure is used to monitor the rate of occurrence of uncommon events where the time between successive occurrences is exponentially distributed. This procedure can be used extensively in time series forecasting and modeling, [19]. The EWMA for individual value may be defined as: (7) where and Assuming the following notations: = Exponentially Weighted Moving Average, = EWMA weight parameter (0 < w < 1), = Process mean, = j th measure of i th subgroup, with = standard deviation of the process, = Sample size of i th group, q = σlimit, ̅ = Mean of measurement in i th subgroup. If , then subgroup mean trim down to the observation in the group. If a known value for µ is specified, , otherwise . The previous equation can be rewritten as: (8) which states the current EWMA plus the weighted error in the prediction of the current mean based on the previous mean. The EWMA for the i th subgroup can be written as: (9) expresses the EWMA as weighted average of historical subgroup means where the weight reduces exponentially as more weight is assigned to recent subgroup mean.

COMPUTATION OF CONTROL LIMIT
The fact that the weights decrease exponentially is the reason it is called an exponentially weighted moving average control chart. As increases, the control limits approach constant values. If the observations are said to be independent with common variance , then the variance is: ( ∑ ) ∑ (13) On replacing with , when and are known, the EWMA chart is build up by plotting versus the sampled number i with control limits at: (14) and (15)

DISCUSSION OF RESULTS AND APPLICATION
The set of data used in this study are secondary data on the pulmonary tuberculosis cases in DELSUTH, which consist of the monthly record from 2012 to 2018 as shown in the appendix. Statistical software, MINITAB 17 was implored to carry out the statistical analysis and plotting of control charts while anygeth software was used in obtaining the target values (k) and reference values (h). The findings from the study and application are presented below.

TEST FOR STATIONARY AND NORMALITY OF THE DATASET.
This study uses Augmented Dickey Fuller test to test for stationary of the dataset in order to ascertain the validity and the result is shown in Table 1 and Histogram chart is to ascertain Non-normality if it is not bell shape as shown in Figure 1.  1: Histogram of the pulmonary tuberculosis The histogram above is not normally distributed since it has long tail to the right (i.e., positively skewed).

AUTO-CORRELATION
The study considers the Autocorrelation chart to test for randomness and the significance of the lags. Any lag that is above the line dot is random and significant as shown in Figure 2 and Figure 3.  Figure 2 shows that lag 0, 1, 2,3, 5and 7are random and significant The Partial Autocorrelation Function (PACF) chart in Figure 3 shows that lag 1 and 2 are random and significant.

CUSUM CHART
The idea of CUSUM chart design for the data on pulmonary tuberculosis cases in DELSUTH is detecting the first shift when the process is actually in control given that and as shown in Figures 4 and 5.

Figure 4:
The CUSUM chart for = 1.5 The first shift is detected at month 6 in the chart of Figure 4 with ARL = 6. This indicates that at the sixth month, the tuberculosis control system was out of control in DELSUTH. Therefore, the management system needs to intensify more efforts to bring the process in control so that the outbreak can be under control. The first shift is detected at month 12 in the chart of Figure 5 with ARL = 12. This indicates that at the twelfth month, the tuberculosis control system was out of control in DELSUTH. Therefore, the management system needs to intensify more efforts to bring the system in control so that the outbreak can be under control.
Generally, it is observed that there were shift in the process mean for the various values of , and the smaller the value of , the faster the shift is detected.

p-VALUE FOR THE STATISTICAL PROCESS CONTROL
The p-Value CUSUM method is also used to monitor the outbreak of tuberculosis data used in this study in order to compare the performances of the two schemes. In designing p-value CUSUM chart, the dataset is conditioned to be stationary at p-value < 0.05 and also be in statistical control as shown in Figures 6 and 7.

Figure 6:
The p-value CUSUM chart for = 1.5 The first shift is detected at month 7 in the chart of Figure 6 with the ARL = 7

CONCLUSIONS
The paper concluded that the early shift was detected when the is small in p-value CUSUM. Also, the conventional EWMA control chart is better in detecting early shift in the mean process than p-value CUSUM and; from the observation, it is noticed that there was always a high outbreak of Tuberculosis in month 7 which corresponds to the month of July for p-value CUSUM and January for EWMA control chart.