The sample processing in clinical laboratories requires adequate sample collection for the following tests and avoiding false negatives and false positives in the results. When the sample does not meet the established requirements, it is discarded and becomes a defective product from the perspective of statistical quality control.

Discarded samples generate waste of clinical laboratory resources and can cause user dissatisfaction because a new sampling is required. Therefore, it is essential to implement programs to monitor and control the sampling effectiveness to prevent errors in clinical tests and make decisions to improve the health service.

The data of this article are from a clinical laboratory in Colombia, with continuous monitoring for one year. The interesting variable is the number of clinical samples discarded per week, with a total monitoring time of 50 weeks. Thus, the monitoring parameter (p) in the control charts is the proportion of samples discarded in each subgroup (week).

The diagnosis of process data overdispersion is performed using the Jones and Govindaraju [17] and Heimann [18] methods.

Jones and Govindaraju [17] proposed a graphical method based on the variance ratio test (VRT), comparing the observed variation in a data sample and the expected variation of the corresponding binomial distribution. Let d _i be considered a number of nonconforming units in i subgroups of size n that follow a binomial distribution with parameter p (proportion of the nonconforming units), then:

follows a normal distribution with mean and variance :

where p - is the average proportion of nonconforming units:

The procedure of this method begins by transforming data using (1) and then constructing the normal probability plot of the transformed data and its fitting line. Subsequently, the actual variation of the process is estimated as the distance on the y _i axis that intercepts with the scores +1 y -1 of Z in the fitting line. If the actual variation is greater than the expected variation, estimated as 1.5/√n, it is concluded that the data are overdispersed and it is not possible to state that they follow a binomial distribution.

On the other hand, Heimann [18] proposes a method based on decomposing the total variance and its representation as the sum of the sampling variance and the process variance. The sampling variance represents the difference between the estimate of the probability of producing nonconforming units (from the sample) and the actual value (from the process). The underlying process variance represents the variation in the probability of producing nonconforming units.

Thus, the sampling variance is determined by:

Similarly, let p i be the proportion of nonconforming units for each subgroup i, with a total of m subgroups, the total variance is determined as follows:

and the underlying process variance results from:

Then, r is defined as the variance ratio, between the total variance and the sampling variance:

If r > 1.357, it is concluded that there is extra variability in the data, greater than a binomial distribution implies, and it is not appropriate to choose a p-chart for process monitoring and control.

Various control chart types are developed and compared in order to select the most suitable one for monitoring overdispersed processes in clinical laboratories. Therefore, we develop the p-chart with variable limits, normalized p-chart (Z), moving range chart (X-MR), p’-chart of Laney, and the chart proposed by Goedhart and Woodall.

The construction procedures of the first three control charts are not described in this article since they are widely known and extensively detailed in textbooks on statistical quality control [10] [19]. The construction procedures for the last two control charts are detailed below.

Table 1 shows the data collected from monitoring the total number of samples collected and the number of samples discarded per week in the clinical laboratory under study. The average proportion of nonconforming units, is equal to 0.05. This relatively low value, regarding the standards set by the laboratory, could lead to preliminary inferences that the process has large sample sizes or a low number of nonconforming units per subgroup.

Table 1

Week	Total samples	Nonconforming Samples	Week	Total samples	Nonconforming Samples
1	1467	105	26	1591	34
2	1789	68	27	1726	112
3	1345	140	28	1279	25
4	2347	217	29	2430	132
5	1734	61	30	275	6
6	2378	158	31	1784	96
7	1893	80	32	2694	74
8	2992	260	33	3583	101
9	1935	81	34	6945	276
10	1967	87	35	2895	184
11	1524	98	36	7492	351
12	2592	91	37	1764	155
13	3890	254	38	3790	226
14	1469	79	39	2578	101
15	2693	67	40	3895	114
16	2936	72	41	4895	415
17	1798	66	42	2654	78
18	2348	109	43	4569	62
19	1633	67	44	2697	200
20	1376	80	45	2589	143
21	1234	88	46	2478	62
22	1357	98	47	2737	49
23	1594	158	48	1839	129
24	3789	238	49	1426	84
25	1598	135	50	1925	75
			TOTAL	124208	6241

Source: The authors.

Table 1 shows that both the sample sizes and the number of nonconforming units per subgroup have a high variation. However, the average sample variance is 0.00002559, a relatively low value. This requires a detailed analysis of the data dispersion, as detailed in the next step.

Fig. 1 shows the normal probability plot of the data transformed by applying the Jones and Govindaraju method, described in section 2.2. According to the results, the actual process variation is equal to 0.1073 and the expected variation is equal to 0.0905. Since the actual variance is greater than the expected variance, it is possible to state that the process data are overdispersed.

Figure 1 Source: The authors.

Moreover, applying the Heimann method, it is obtained that = 0.000026 and = 0.000537. Therefore, the variance ratio r is equal to 20.65. Since r > 1.357, it can be stated that the data have a higher variation than expected for a binomial distribution.

Finally, it is possible to state that the highest variation corresponds to the underlying process variation. Data overdispersion is identified in both diagnostic methods, therefore, it is not appropriate to choose a traditional p-chart for process monitoring and control.

In order to compare performance and select the most appropriate control chart(s) for monitoring overdispersed processes in clinical laboratories, we developed the p-chart with variable limits (Fig. 2), the normalized p-chart (Fig. 3), X-MR chart (Fig. 4), p’-chart of Laney (Fig. 5) and the p-chart of Goedhart and Woodall (Fig. 6). The control charts were made using Microsoft Excel^©.

Figure 2 Source: The authors.

Figure 3 Source: The authors.

Figure 4 Source: The authors.

Figure 5 Source: The authors.

Figure 6 Source: The authors.

Traditional control charts have limitations for monitoring overdispersed data due to the assumptions required for their construction [21]. The p-chart with variable limits and the normalized p-chart assume a binomial distribution of the data and that the mean remains constant over time.

The above two assumptions are not satisfied in overdispersed data. Therefore, many points fall outside the control limits, as shown in Figs. 2 and 3. Because of that, it is not appropriate to use either of these two control charts for monitoring overdispersed processes since the data assumptions are not satisfied, and type 1 errors may occur.

According to [17], the acceptable solution for decades to monitor overdispersed data was the development of the X-MR chart (Fig. 4). Although this control chart also assumes a binomial distribution of the data, it incorporates compensation for the inter-sample variation by considering the average moving range as a factor for calculating the control limits. However, it does not consider inter-sample variation and presents constant control limits.

The p'-chart of Laney (Fig. 5) and the p-chart of Goedhart and Woodall (Fig. 6) overcome the limitations of previous control charts and have variable control limits that adjust depending on inter-sample and intra-sample variation. As mentioned in section 2.3, the main difference between these two control charts lies in the effect of inter-sample and intra-sample variations in calculating the control limits. While the p'-chart of Laney has a multiplicative approach, the p-chart of Goedhart and Woodall has an additive approach.

The latter is shown in Fig. 7. The multiplicative effect of the variances causes higher variability in the width of the control limits in the Laney control chart. The maximum range of variation of the control limits, obtained as the difference between the extreme values of the control limits in each control chart, is higher in the p'-chart of Laney (0.255) than in the p-chart of Goedhart and Woodall (0.129). The p-chart of Goedhart and Woodall has control limits that do not overestimate the effect of variances and therefore have lower variability over time.

Figure 7 Source: The authors.

Finally, Table 2 shows a comparative summary of the control charts analyzed based on the results of this work. The decision on the control chart adequate to monitor overdispersed processes depends on fulfilling two main conditions: (i) The control chart must be applicable without data distributional, and stability of the process mean assumptions, and (ii) The control chart must consider both intra-sample and inter-sample variances.

Table 2

Control chart	Applicable without data distributional assumptions?	Consider both intra-sample and inter-sample variances?	Adequate for overdispersed data monitoring?
p-chart	No	No	No
Z-chart	No	No	No
X-MR chart	No	No	No
p´-Laney	Yes	Yes	Yes
p-Goedhart & Woodall	Yes	Yes	Yes

Source: The authors.