7. Prince (2007) performed a 25????2 fractional factorial in the process of op-

timizing a clinical assay to detect Streptococcus pyogenes with real-time PCR. Optimization of this process would allow hospitals to detect Strep infections in less than 30 minutes with 99% accuracy. The factors he stud-

ied were A=number of S. pyogenes colonies (1 or 4), B=Boiling time (1 min. or 3 min.), C=Centrifuge time (0 min. or 2 min.), D=cycling temper-

ature (60 or 61), E=Cycling time (5/10 sec. or 8/13 sec.). The generators for the design were D = AC and E = BC.

(a) The response data (in standard order) from the eight experiments were:

1.31091, 1.43201, 1.29951, 1.37199, 1.33566, 1.46820, 1.39023, 1.41531.

Calculate the eects and make a half-normal plot of the eects. Identify any eects you believe may be signicant.

(b) Determine the confounding pattern or alias structure for the design.

(c) What are the aliases of the largest eect in absolute value?

(d) Prince performed 8 more experiments according to a foldover (simi-

lar to that shown in Table 6.8) and the resulting data were: 1.31702, 1.38881, 1.32222, 1.36248, 1.33826, 1.32654, 1.32654, 1.34635. Combin-

ing this data with the data from the original 8 runs, calculate the 15 eects including a block eect for the dierences in the two groups of experiments.

(e) What is the dening relationship and alias structure for the complete set of 16 experiments?

(f) Make a half-normal plot of the 15 eects and determine what appears to be signicant from the combined set of data.

(g) Provide an interpretation for all eects and interactions you determine are signicant, and discuss how the conclusions and interpretation after 16 experiments diers from what could have been concluded after the rst 8 experiments.