Design and properties of the predictive ratio cusum (PRC) control charts

Figures & data

Table 1. The expected ratio of the variance of the likelihood $f(X|\mathit{\theta })$ over the variance of the marginal $f(X|\mathit{Y},{\alpha }_0,\mathit{\tau }),$ defined in (11[11] $$E_{\mathit{\theta }}\left(\frac{\mathit{Var}(X|\mathit{\theta })}{\mathit{Var}(X|\mathit{Y},{\alpha }_0,\mathit{\tau })}\right)=\rho$$[11] ).

Likelihood	Initial prior	Expected ratio (11[11] $$E_{\mathit{\theta }}\left(\frac{\mathit{Var}(X|\mathit{\theta })}{\mathit{Var}(X|\mathit{Y},{\alpha }_0,\mathit{\tau })}\right)=\rho$$[11] )
$f(·|\mathit{\theta })$	${\pi }_0(\mathit{\theta }|\mathit{\tau })$	ρ
P$(\theta ·s_i)$	G(c, d)	$1-\frac{1}{d+{\alpha }_0{\sum }_{j=1}^{n_0}{s}_j+1}$
Bin$(N_i,\theta )$	Beta(a, b)	$1-\frac{N}{a+b+{\alpha }_0{\sum }_{j=1}^{n_0}{N}_j+N}$
NBin$(r,\theta )$	Beta(a, b)	$1-\frac{r}{b+{\alpha }_0{\sum }_{j=1}^{n_0}{y}_j-1+r}$
G$(\alpha ,\theta )$	G(c, d)	$1-\frac{\alpha }{c+\alpha ·({\alpha }_0{n}_0+1)-1}$
W$(\theta ,\kappa )$	$IG(\alpha ,\beta )$	$1-\frac{{\Gamma }^2\left(1+\frac{1}{\kappa }\right)\left[\Gamma (\alpha +{\alpha }_0{n}_0)\Gamma \left(\alpha +{\alpha }_0{n}_0-\frac{2}{\kappa }\right)-{\Gamma }^2\left(\alpha -\frac{1}{\kappa }\right)\right]}{\Gamma (\alpha +{\alpha }_0{n}_0)\Gamma \left(\alpha +{\alpha }_0{n}_0-\frac{2}{\kappa }\right)\Gamma \left(1+\frac{2}{\kappa }\right)-{\Gamma }^2\left(1+\frac{1}{\kappa }\right){\Gamma }^2\left(\alpha -\frac{1}{\kappa }\right)}$
IG$(\alpha ,\theta )$	G(c, d)	$1-\frac{\alpha -2}{c+\alpha ·{\alpha }_0{n}_0-1+\alpha }$
Pa$(m,\theta )$	G(c, d)	$1-\frac{Va{r}_{\theta }\left(\frac{\theta }{\theta -1}\right)}{E_{\theta }(\frac{\theta }{{(\theta -1)}^2(\theta -2)})+Va{r}_{\theta }\left(\frac{\theta }{\theta -1}\right)}$

Figure 1. Determining the decision threshold h for a predictive ratio cusum (PRC) scheme. A decision is represented by a rhombus, and a rectangle corresponds to an operation after a decision is made.

Figure 2. Predictive ratio cusum (PRC) flowchart 2. A parallelogram corresponds to an input/output information, a decision is represented by a rhombus, and a rectangle denotes an operation after a decision-making. In addition, the rounded rectangles indicate the beginning and end of the process. *For the likelihoods with two unknown parameters and total prior ignorance (i.e., initial reference prior and ${\alpha }_0=0$ in the power prior), we need n = 3 to initiate PRC, while for all other cases, PRC starts right after x₁ becomes available.

Figure 3. The family-wise error rate FWER(k) at each time point $k=2,3,\dots ,50,$ the probability of successful detection, probability of successful detection $\mathit{PSD}(\omega )$ and the truncated conditional expected delay, truncated conditional expected delay $\mathit{tCED}(\omega )$ for shifts at locations $\omega =\{11,26,41\},$ of self-starting cusum SSC, cumulative Bayes factor CBF and PRC, under a reference (CBF_r, PRC_r) or a moderately informative (CBF_mi, PRC_mi) prior for OOC scenarios with misspecified likelihood. All the procedures are set for a mean step change size of $1\sigma$ in data from a standard Normal or an rate increase of 50% in P(1) data.

Figure 4. The $\mathit{PSD}(\omega )$ and $\mathit{tCED}(\omega )$ for shifts at locations $\omega =\{11,26,41\},$ of SSC, CBF and PRC, under a reference (CBF_r, PRC_r) or a moderately informative (CBF_mi, PRC_mi) prior for OOC scenarios with misspecified jumps. All the procedures are set for a mean step change of size $1\sigma$ in data from a standard normal distribution.

Figure 5. The FWER(k) at each time point $k=2,3,\dots ,50,$ $\mathit{PSD}(\omega )$ and $\mathit{tCED}(\omega )$ for shifts at locations $\omega =\{11,26,41\},$ of PRC under two misplaced moderately informative priors in the negative and positive directions ($PR{C}_{-}$ and $PR{C}_{+}$ respectively), along with the and CBF under the same priors ($CB{F}_{-}$ and $CB{F}_{+}$). All the methods are set for detecting a positive mean step change of size $1\sigma$ in normal data.

Figure 6. The $\mathit{PSD}(\omega )$ and $\mathit{tCED}(\omega )$ for shifts at locations $\omega =\{11,26,41\},$ of SSC, CBF, and PRC, under a reference (CBF_r, PRC_r) or a moderately informative (CBF_mi, PRC_mi) prior for OOC scenarios with dependent data. All the procedures are set for a mean step change of size $1\sigma$ for the mean or 50% inflation for the standard deviation in data from a standard normal distribution.

Table 2. The Factor V (%) internal quality control observations of the current $\mathit{X}=(x_1,x_2,\dots ,x_{21})$ data, reported during September 24, 2019–October 8, 2019.

$x_1-x_{11}$	31.0	30.0	32.0	28.0	33.2	33.2	35.1	35.1	33.9	37.9	33.2
$x_{12}-x_{21}$	36.5	33.2	35.1	34.5	36.5	33.2	35.1	37.2	32.6	36.5

Figure 7. Predictive ratio cusum (PRC) for normal data. At the top panel, the data are plotted, while at the lower panel, we provide the PRC control chart, focused on detecting an upward or downward mean step change of one standard deviation size, when we aim a $\mathit{FWER}=10\%$ for 21 observations.

Table 3. Counts of adverse events (x_i) and product exposure (s_i) per million ($i=1,2,\dots ,22$), for each quarter reported during July 1, 1999–December 31, 2004 (see Dong, Hedayat, and Sinha 2008).

Adverse events $(x_1-x_{11})$	1	0	0	0	1	0	3	3	3	2	5
Product exposure $(s_1-s_{11})$	0.206	0.313	0.368	0.678	0.974	0.927	0.814	0.696	0.659	0.775	0.731
Adverse events $(x_{12}-x_{22})$	5	2	4	4	3	4	3	8	3	2	2
Product exposure $(s_{12}-s_{22})$	0.710	0.705	0.754	0.682	0.686	0.763	0.833	0.738	0.741	0.843	0.792

Figure 8. Predictive ratio cusum (PRC) for Poisson data. At the top panel, we plot the counts of adverse events x_i (solid line) and the rate of adverse events per million units $x_i/s_i$ (dashed line). At the lower panel, we provide the PRC control chart, focused on detecting 100% rate inflation and the evidence based limit of $h_J=\log (100)\approx 4.605$ is used. For the fast initial response (FIR)-PRC (dashed line) the parameters $(f,d)=(1/2,3/4)$ were used.

Table 4. The sequence of the data $\mathit{X}=(x_1,x_2,\dots ,x_{40}),$ representing the number of defective per 50 sampled shipping papers.

$x_1-x_{20}$	3	3	1	5	2	1	3	3	3	0	2	1	2	1	4	1	1	0	3	2
$x_{21}-x_{40}$	4	6	0	1	3	2	2	4	2	1	2	4	5	2	4	8	4	4	8	5

Figure 9. Predictive ratio cusum (PRC) for binomial data. At the top panel, we plot the number of defective shipping papers x_i, while at the lower panel, we provide the PRC control chart, focused on detecting 100% odds inflation. The evidence based limit of $h_J=\log (100)\approx 4.605$ is used for the first 30 observations, and then these data are used for the derivation of the decision limit $h_m=4.332,$ setting the average run length ${\mathrm{ARL}}_0=400.$

Dong, Y., A. S. Hedayat, and B. K. Sinha. 2008. Surveillance strategies for detecting change point in incidence rate based on exponentially weighted moving average methods. Journal of the American Statistical Association 103 (482):843–53. doi: 10.1198/016214508000000166.

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