Monitoring of three-phase variations in the mortality of COVID-19 pandemic using control charts: where does Pakistan stand?

BACKGROUND: At the end of December 2019, the world in general and Wuhan, the industrial hub of China, in particular, experienced the COVID-19 pandemic. Massive increment of cases and deaths occurred in China and 209 countries in Europe, America, Australia, Asia and Pakistan. Pakistan was first hit by COVID-19 when a case was reported in Karachi on 26 February 2020. Several methods were presented to model the death rate due to the COVID-19 pandemic and to forecast the pinnacle of reported deaths. Still, these methods were not used in identifying the first day when Pakistan enters or exits the early exponential growth phase. OBJECTIVE: The present study intends to monitor variations in deaths and identify the growth phases such as pre-growth, growth, and post-growth phases in Pakistan due to the COVID-19 pandemic. METHODS: New approaches are needed that display the death patterns and signal an alarming situation so that corrective actions can be taken before the condition worsens. To meet this purpose, secondary data on daily reported deaths due to the COVID-19 pandemic have been considered, and the $c$ and exponentially weighted moving average (EWMA) control charts are used To meet this purpose, secondary data on daily reported deaths in Pakistan due to the COVID-19 pandemic have been considered. The $ c$ and exponentially weighted moving average (EWMA) control charts have been used for monitoring variations. RESULTS: The chart shows that Pakistan switches from the pre-growth to the growth phase on 31 March 2020. The EWMA chart demonstrates that Pakistan remains in the growth phase from 31 March 2020 to 17 August 2020, with some indications signaling a decrease in deaths. It is found that Pakistan moved to a post-growth phase for a brief period from 27 July 2020 to 28 July 2020. Pakistan switches to re-growth phase with an alarm on 31/7/2020, right after the short-term post-growth phase. The number of deaths starts decreasing in August in that Pakistan may approach the post-growth phase shortly. CONCLUSION: This amalgamation of control charts illustrates a systematic implementation of the charts for government leaders and forefront medical teams to facilitate the rapid detection of daily reported deaths due to COVID-19. Besides government and public health officials, it is also the public's responsibility to follow the enforced standard operating procedures as a temporary remedy of this pandemic in ensuring public safety while awaiting a suitable vaccine to be discovered.

A side-sensitive synthetic chart for the multivariate coefficient of variation.

Control charts for the coefficient of variations (γ) are receiving increasing attention as it is able to monitor the stability in the ratio of the standard deviation (σ) over the mean (µ), unlike conventional charts that monitor the µ and/or σ separately. A side-sensitive synthetic (SS) chart for monitoring γ was recently developed for univariate processes. The chart outperforms the non-side-sensitive synthetic (NSS) γ chart. However, the SS chart monitoring γ for multivariate processes cannot be found. Thus, a SS chart for multivariate processes is proposed in this paper. A SS chart for multivariate processes is important as multiple quality characteristic that are correlated with each other are frequently encountered in practical scenarios. Based on numerical examples, the side-sensitivity feature that is included in the multivariate synthetic γ chart significantly improves the sensitivity of the chart based on the run length performance, particularly in detecting small shifts (τ), and for small sample size (n), as well as a large number of variables (p) and in-control γ (γ0). The multivariate SS chart also significantly outperforms the Shewhart γ chart, and marginally outperforms the Multivariate Exponentially Weighted Moving Average (MEWMA) γ chart when shift sizes are moderate and large. To show its implementation, the proposed multivariate SS chart is adopted to monitor investment risks.

Multivariate process dispersion monitoring without subgrouping.

The memory-type adaptive and non-adaptive control charts are among the best control charts for detecting small-to-moderate changes in the process parameter(s). In this paper, we propose the Crosier CUSUM (CCUSUM), EWMA, adaptive CCUSUM (ACCUSUM) and adaptive EWMA (AEWMA) charts for efficiently monitoring the changes in the covariance matrix of a multivariate normal process without subgrouping. Using extensive Monte Carlo simulations, the length characteristics of these control charts are computed. It turns out that the ACCUSUM and AEWMA charts perform uniformly and substantially better than the CCUSUM and EWMA charts when detecting a range of shift sizes in the covariance matrix. Moreover, the AEWMA chart outperforms the ACCUSUM chart. A real dataset is used to explain the implementation of the proposed control charts.

Economic-statistical design of synthetic np chart with estimated process parameter.

The economic-statistical design of the synthetic np chart with estimated process parameter is presented in this study. The effect of process parameter estimation on the expected cost of the synthetic np chart is investigated with the imposed statistical constraints. The minimum number of preliminary subgroups is determined where an almost similar expected cost to the known process parameter case is desired for the given cost model parameters. However, the available number of preliminary subgroups in practice is usually limited, especially when the number of preliminary subgroups is large. Consequently, the optimal chart parameters of the synthetic np chart are computed by considering the practical number of preliminary subgroups in which the cost function is minimized. This leads to a lower expected cost compared to that of adopting the optimal chart parameter corresponding to the known process parameter case.

Optimal designs of the median run length based double sampling X chart for minimizing the average sample size.

Designs of the double sampling (DS) X chart are traditionally based on the average run length (ARL) criterion. However, the shape of the run length distribution changes with the process mean shifts, ranging from highly skewed when the process is in-control to almost symmetric when the mean shift is large. Therefore, we show that the ARL is a complicated performance measure and that the median run length (MRL) is a more meaningful measure to depend on. This is because the MRL provides an intuitive and a fair representation of the central tendency, especially for the rightly skewed run length distribution. Since the DS X chart can effectively reduce the sample size without reducing the statistical efficiency, this paper proposes two optimal designs of the MRL-based DS X chart, for minimizing (i) the in-control average sample size (ASS) and (ii) both the in-control and out-of-control ASSs. Comparisons with the optimal MRL-based EWMA X and Shewhart X charts demonstrate the superiority of the proposed optimal MRL-based DS X chart, as the latter requires a smaller sample size on the average while maintaining the same detection speed as the two former charts. An example involving the added potassium sorbate in a yoghurt manufacturing process is used to illustrate the effectiveness of the proposed MRL-based DS X chart in reducing the sample size needed.