Multifactor Experimental Designs

Abstract

In order to study the effects of two or more factors on a response variable, factorial designs are usually used. By following these designs, all possible combinations of the levels of the factors are investigated. The factorial designs are ideal designs for studying the interaction effect between factors. By interaction effect, we mean that a factor behaves differently in the presence of other factors such that its trend of influence changes when the levels of other factors change. This has already been discussed in Chap. 1. In this chapter, we will learn more about factorial design and analysis of experimental data obtained by following such designs.

Problems

13.1

Cauliflowers were grown in a greenhouse under treatments consisting of four types of soils and three types of fertilizers. In order to examine the effects of soil and fertilizer, a factorial experiment was carried out. The results of experiments in terms of the yield (kg) of cauliflowers are shown in Table 13.31.

Table 13.31 Data for Problem 13.1

Analyze the data using $\alpha =0.05$.

13.2

In order to investigate the effects of type of resin, distance between blade and anvil, and weight fraction of nonvolatile content in a lacquer on the thickness of lacquer film on a substrate, the experiment shown in Table 13.32 was carried out.

Table 13.32 Data for Problem 13.2

Analyze the data using $\alpha =0.05$.

13.3

A $2^2$ factorial experiment was conducted to study the effects of temperature (25 and $65\,^{\circ }$C) and catalyst concentration (0.5 and 1.5$\%$) on transesterification of vegetable oil to methanol. The results are shown in Table 13.33.

Table 13.33 Data for Problem 13.3

• (a) Display the geometrical view of the aforementioned design.

• (b) Analyze the main and interaction effects of temperature and concentration on conversion.

• (c) Construct ANOVA for conversion. Which effects are statistically significant at 0.05 level of significance?

• (d) Develop an appropriate regression model for conversion.

13.4

In order to study the effects of pH (5.8 and 7.4) and buffer concentration of catholyte (25 and 150 mM) on the power density of a fuel cell, the $2^2$ factorial design of experiment was carried out and results are shown in Table 13.34.

Table 13.34 Data for Problem 13.4

• (a) Display the geometrical view of the aforementioned design.

• (b) Analyze the main and interaction effects of pH and concentration on power density.

• (c) Construct ANOVA for power density. Which effects are statistically significant at 0.05 level of significance?

• (d) Develop an appropriate regression model for power density.

13.5

An experiment was carried out to investigate the effects of ratio of PP and LLDRE $(X_1)$ and concentration of red mud particles $(X_2)$ on the tensile strength (Y) of red mud-filled PP/LLDPE-blended composites. The results of experiments are shown in Table 13.35.

Table 13.35 Data for Problem 13.5

• (a) Display the geometrical view of the aforementioned design.

• (b) Analyze the main and interaction effects of ratio of PP and LLDRE and concentration of red mud particles on the tensile strength.

• (c) Construct ANOVA for tensile strength. Which effects are statistically significant at 0.05 level of significance?

• (d) Develop an appropriate regression model.

13.6

An article entitled “Study of NADH stability using ultraviolet-visible spectrophotometric analysis and factorial design” published by L. Rovar et al. in Analytical Biochemistry, 260, 50–55, 1998, reported on the effects of pH, buffer, and temperature on percentage of degraded $1.0\times 10^{-4}$ M NADH. In the reported study, two levels of pH (6.8 and 7.8), two buffer solutions (phosphate and pipes), and two levels of temperature (25 and $30\,^{\circ }$C) were taken. The results are given in Table 13.36.

Table 13.36 Data for Problem 13.6

• (a) Display the geometrical view of the aforementioned design.

• (b) Analyze the main and interaction effects of pH, buffer, and temperature on percentage of degraded $1.0 \times 10^{-4}$ M NADH.

• (c) Construct ANOVA for percentage of degraded $1.0 \times 10^{-4}$ M NADH. Which effects are statistically significant at 0.05 level of significance?

13.7

An article entitled “Designed experiments to stabilize blood glucose levels”, published by R.E. Chapman and V. Roof in Quality Engineering, 12, 83–87, 1999, reported on the effects of amount of juice intake before exercise (4 oz or 8 oz), amount of exercise (10 min or 20 min), and delay between time of juice intake and beginning of exercise (0 min or 20 min) on the blood glucose level of patients. The data are shown in Table 13.37.

Table 13.37 Data for Problem 13.7

• (e) Display the geometrical view of the aforementioned design for blood glucose level.

• (f) Analyze the main and interaction effects of amount of juice intake, amount of exercise, and delay between juice intake and exercise in determining blood glucose level.

• (g) Construct ANOVA for blood glucose level. Which effects are statistically significant at 0.05 level of significance?

• (h) Develop an appropriate regression model for blood glucose level.

13.8

An article entitled “Rotary ultrasonic machining of ceramic matrix composites: feasibility study and designed experiments”, published by Z.C. Li et al. in International Journal of Machine Tools & Manufacture, 45, 1402–1411, 2005, described the use of a full factorial design to study the effects of rotary ultrasonic machining on the cutting force, material removal rate, and hole quality (in terms of chipping dimensions). The process variables and their levels were taken and are shown in Table 13.38.

Table 13.38 Data for Problem 13.8

The results of experiments are stated in Table 13.39.

Table 13.39 Data for Problem 13.8

• (a) Display the geometrical views of the design for cutting force, material removal rate, chipping thickness, and chipping size.

• (b) Analyze the main and interaction effects of spindle speed, feed rate, and ultrasonic power in determining cutting force, material removal rate, chipping thickness, and chipping size.

• (c) Construct ANOVA for cutting force, material removal rate, chipping thickness, and chipping size. Which effects are statistically significant at 0.05 level of significance?

• (d) Develop appropriate regression models for cutting force, material removal rate, chipping thickness, and chipping size.

13.9

Consider the experiment described in Problem 13.6. Analyze the data using Yates’ algorithm. Check if the results obtained are same with those obtained in Problem 13.6.

13.10

Consider the experiment described in Problem 13.7. Analyze the data using Yates’ algorithm. Check if the results obtained are same with those obtained in Problem 13.7.

13.11

Consider the experiment described in Problem 13.1. Analyze the experiment assuming that each replicate represents a block and draw conclusions.

13.12

Consider the experiment described in Problem 13.4. Analyze the experiment assuming that each replicate represents a block and draw conclusions.

13.13

Consider the data from the first replicate of Problem 13.6. Suppose that these experiments could not all be run using the same batch of material. Construct a design with two blocks of four observations each with SFP confounded. Analyze the data.

Table 13.40 Data for Problem 13.15

13.14

Consider the data from the first replicate of Problem 13.4. Suppose that these experiments could not all be run using the same batch of material. Suggest a reasonable confounding scheme and analyze the data.

13.15

An article entitled “Screening of factors influencing Cu(II) extraction by soybean oil-based organic solvents using fractional factorial design” published by S.H. Chang et al. in Journal of Environment Management 92, 2580–2585, 2011, described a fractional factorial experiment to study the effect of Cu (II) extraction process on extraction efficiency $(\eta )$. The process variables and their levels are given in Table 13.40.

The results of experiments are stated in Table 13.41.

Analyze the main and interaction effects of the five process factors, and calculate percent contribution of them. Which effects are statistically significant at 0.05 level of significance? Construct ANOVA for extraction efficiency.

Table 13.41 Results of Problem 13.15