Bioassay Data

5.1.1 Data Description

To compare the profile monitoring schemes between the OAAT and the Delete-All schemes, we analyze the data from DuPont Crop Protection (Woodall, Williams, Birch, and Ferry, 2007).

The data set consists of forty-four weeks (m =44) of in vivo bioassay results run alongside experimental compounds over a one-year time period.

The commercial compound was diluted to eight doses (0.003, 0.009, 0.028, 0.084, 0.25, 0.76, 2.27, and 6.8 (d=8)) and replicated four times at each dose (r = 4) in 96-well microtiter plates for each sampling period i. A spectrophotometer measured the optical density (OD) of the plant organism after the inoculation period.

Let yijk represents the k^th response to the j^th dose at sampling period i, where i = 1,2, …,m , j = 1,2,…,d, and k = 1,2,…, r. For this data set, we have m = 44, d = 8, and r = 4. Both treated and untreated wells were measured for growth inhibition. The percent control (PC) values were calculated using the median OD (M_i) from 96 replications of untreated wells. Williams, Woodall, Ferry, and Birch (2007) let M_i represent the median response of the untreated specimen at sampling period i. Then, the percent control of the chemical for the k^th replication of the j^th dose in sampling period i is calculated as

, 1,..., , 1,..., , 1,.., .

A plot of PCijk values for all m = 44 weeks for one of the standards from the DuPont is given in Figure 27. Because the bioassay data have replications (r = 4), we need to first monitor the variance within profiles. Williams, Woodall, Ferry, and Birch (2007) referred to it as the variance profile monitoring.

Figure 27: Bioassay Data (DuPont Does-Response Data) for all 44 weeks.

5.1.2 Monitoring for Bioassay Data

We analyze the profiles in the bioassay data in Figure 27 and consider the following 4-parameter logistic model which has been used frequently for dose-response studies (Williams, et al. (2007)):

, 1,..., , 1,.., , 1 ( ) ⁱ

i i

ij i ij

ij B i

D A

y A i m j n

x C

− ε

= + + = =

+

(13)

where A is the upper asymptote, _i D is the lower asymptote, _i C is the point where the curve _i reaches halfway between A and _i D , and _i B is a parameter representing the rate of increase or _i decrease from D to _i A in Figure 28. Since the estimators of _i A , _i B , _i C , and _i D are _i correlated, it is more appropriate to account for the correlation among them when testing for unusual values of ˆA , ˆ_i B , ˆ_i C and ˆ_i D . We illustrate the estimated mean profiles for all _i forty-four weeks in Figure 29.

Figure 28: A does-response curve.

Figure 29 : Estimated mean profiles of all 44 weeks in the bioassay data .

We use the RMVE in Equation (7) to monitor dose-response profiles. As a result, both the OAAT and Delete-All schemes remove the same 13^th, 20^th, 21^st, 22^nd, 24^th, 26^th, 32^nd, 34^th, 45^th, 46^th, and 48^th profiles in the end. See Figure 30 (a) gives the control chart of the first iteration and 30(b) shows the result of the last iteration in which the remaining 33 profiles are in control.

(a) (b)

Figure 30: The RMVE chart when monitoring unusual profiles. (a) The 44 dose-response profiles at the first iteration, and (b) the 33 dose-response profiles at the last iteration.

We compute the T₀² statistic according to Equation (11) for the original 44 profiles and find that T₀²=0.992673, very close to the suggested cutoff value 1. The result that both schemes removed the same set of the profiles is in accordance with the result that T is close 1. ₀²

For demonstrating that the OAAT scheme performs better than the Delete-All scheme, we use the variance profile monitoring of Williams, et al. (2007). Since there are 4 replications at every dose j, estimate σ_ij² by

2 2 1

( )

ˆ .

1

ijk ij k

ij ij

PC PC

S r

σ ⁼

−

= =

−

∑

(14)

Following Williams, et al. (2007), we model σˆ_ij² by the model

2 2

0, 1,

(ˆ_ij) ( )_{ij i i} ( )._ij

log σ =log S =θ +θ log x (15)

Figure 31 displays the fitted results of the 44 variance profiles.

Figure 31: Estimated variance profiles of all 44 weeks in the bioassay data.

We use the RMVE statistic in Equation (7) to monitor the variance profiles. The OAAT scheme removes only the 20^th and 45^th weeks in Figure 32(a)(b), and the Delete-All scheme first removes 6^th, 20^th, 22^nd, 24^th, 26^th, and 45^th weeks and then removes 16^th, 19^th, 34^th and 44^th again.

See Figure 33(a)(b)(c), respectively. The OAAT scheme has 42 profiles remaining, and the Delete-All scheme has only 34 profiles left. It is apparent that the Delete-All scheme removes a lot more than the OAAT scheme. It could be a reasonable doubt that the Delete-All scheme picks come false-alarms.

(a) (b)

Figure 32: The RMVE chart when monitoring the remaining variance profiles by the OAAT scheme. (a) The 44 variance profiles, and (b) the 42 variance profiles.

(a) (b)

(c)

Figure 33: The RMVE chart when monitoring the remaining variance profiles by the Delete-All scheme. (a) The 44 initial variance profiles, (b) the 38 variance profiles, after the 1^st iteration, and (c) the remaining 34 variance profiles.

Note that the T value for the variance profiles is 0.2747621, indicating it is likely that some ₀² false-alarms are signaled by the Delete-All scheme.

After removing the profiles that signal out of control in the variance profiles monitoring, we monitor the remaining dose-response profiles. For the 42 remaining profiles, the OAAT scheme removes 13^th, 21^st, 22^nd, 24^th, 26^th, 32^nd, 34^th, 46^th, and 48^th weeks, and see Figure 34(a)(b). For the remaining 34 profiles, Delete-All scheme first removes 13^th, 32^nd, and 48^th weeks, then removes 21^st, and 46^th weeks, see Figure 35(a),(b), and (c), respectively. At the end, the remaining

33 profiles by the OAAT scheme, and the remaining 29 profiles by the Delete-All scheme, are in control now.

(a) (b)

Figure 34: The RMVE chart when monitoring the remaining mean profiles by the OAAT scheme.

(a) The 42 mean profiles, (b) the 33 mean profiles.

(a) (b)

(c)

Figure 35: The RMVE chart when monitoring the remaining mean profiles by the Delete-All scheme. (a) The 34 mean profiles, (b) the 31 mean profiles, and (c) the 29 mean profiles.

Are the additional 6^th, 16^th, 19^th, and 44^th profiles signaled by the Delete-All false-alarm rate?

To see this, we plot 44 profiles in Figure 36 with these four profiles highlighted. The 6^th, 16^th, and 19^th profiles seem fairly normal. But the 44^th profile seems somewhat unusual, and the OAAT scheme does not remove it. The reason might be that 4 parameters A, B, C, and D of this profile are monitored equally weighted by RMVE. Although the parameter B of the 44^th profile is smaller than the other profiles, the parameters A, C, and D are fairly similar to that of other in-control profiles. Thus the difference in B becomes insignificant in the overall measure T_RMVE² . If we want to emphasize a particular feature like the increasing rate represented by B, we may consider put more weights on that component.

Figure 36: The thick lines are the 6^th, 16^th, 19^th, and 44^th profiles.

5.2 Vertical Board Density Data (VDP)