Confounded Two-Level Factorial Experiments

Abstract

Factorial experiments that involve several treatment factors tend to be large, and many such experiments can only support one observation per treatment combination. When such experiments need to be arranged in blocks, some of the treatment contrasts are confounded (muddled with) some of the block contrasts. For factors at two levels, methods of designing such experiments so that information is available on as many important treatment contrasts as possible are developed and illustrated in this chapter 13. Partial confounding in multi-replicate factorial experiments in blocks is explored. The chapter also compares traditional incomplete block designs with the multiple use of single-replicate confounded designs. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software

Exercises

• 1. Construct a single-replicate $2^3$ design confounding ${ A\!B}$ with blocks. In other words, list the treatment combinations block by block.

• 2. Construct a single replicate $2^5$ design confounding ${{ A\!B\!C}}$ and ${{ C\!D\!E}}$. Determine the other effect that is confounded.

• 3. Projectile experiment

  N.L. Johnson and F.C. Leone, their 1977 book Statistics and Experimental Design in Engineering and the Physical Sciences, described a single-replicate $2^4$ experiment concerning the performance of a new rifle under test. Under study were the effects on projectile velocity of the factors charge weight (A), projectile weight (B), propellant web (C), and weapon (D), where two rifles were used. The design included two blocks each of size eight, corresponding to the two days on which data were collected, confounding ${{ A\!B\!C\!D}}$. The coded velocity data are given in Table 13.23.  

  Table 13.23 Projectile experiment data, confounding ${{ A\!B\!C\!D}}$

  • (a) Fit a model including block effects, treatment main effects, and 2-factor interactions. Use residual plots to check the standard model assumptions.

  • (b) Conduct the analysis of variance, and discuss the results.

  • (c) Construct simultaneous confidence intervals for any interesting treatment contrasts using an appropriate method of multiple comparisons.

  • (d) Reanalyze the data using the Voss–Wang method, including all estimable treatment effects in the analysis.

• 4. Field experiment, continued

  • (a) For the field experiment of Example 13.3.1, p. 437, verify that the sum of squares and the contrast estimate for ${{ B\!D}}$ are as shown in Table 13.5, p. 438.

  • (b) Draw the ${{ B\!D}}$ interaction plot. Does this plot also suggest that B and D should be at their low levels?

  • (c) Suppose that the experimenters had expected all of the 3-factor interactions to be negligible and had omitted the corresponding terms from the model (instead of those involving ${{ A\!D}}$). Reanalyze the experiment accordingly. What would have been concluded?

  • (d) Apply the Voss–Wang method to analyze the data of the field experiment. Relevant information is given in Table 13.5, p. 438.

• 5. Suggest a confounding scheme for a $2^6$ experiment in 8 blocks of 8, assuming that all 2-factor interactions are to be estimated, as are the 3-factor interactions involving both A and F. List all effects confounded. List the treatment combinations in the design block by block.

• 6. Suggest a confounding scheme for a $2^8$ experiment in 16 blocks of 16, assuming that all 2-factor and 3-factor interactions are to be estimated. List all effects confounded. List the treatment combinations in Block I and in two other blocks.

• 7. Mangold experiment, continued

  • (a) For the mangold experiment of Sect. 13.5, verify that the sum of squares and the contrast estimate for ${{ C\!D}}$ are as shown in Table 13.12, p. 449.

  • (b) Draw the ${{ CD}}$ interaction plot. Does this plot agree with the factor levels suggested in Sect. 13.5 for increasing the yield?

  • (c) Check that the assumption of normality of the error variables is satisfied. Also check that the variances of the errors appear to be equal for each level of the four factors.

  • (d) Draw a half-normal probability plot of all of the contrast estimates (including the higher-order interactions). Does it appear that the experimenters made the correct assumptions of negligible higher-order interactions?

• 8. Decontamination experiment—Beta particles

  An experiment was described by M. K. Barnett and F. C. Mead, Jr. the journal Applied Statistics in 1956 to explore the effect of four factors on the efficiency of a decontamination process for the removal of radioactive isotopes from liquid waste. The measurements taken after the decontamination process were the counts per minute per milliliter of alpha and beta particles. Data for the alpha particles and further description of the experiment were given in Sect. 13.8.1, p. 452. We consider here part of the data for the beta particles, shown in Table 13.24. The four treatment factors were:

  $$\begin{aligned}&A: \text {0.4}~\text {g and 2.5}~\text {g per liter of aluminum sulphate (coded 0}, \text {1)};\\&B: \text {0.4}~\text {g and 2.5}~\text {g per liter of barium chloride (coded 0}, \text {1)};\\&C: \text {0.08}~\text {g and 0.4}~\text {g per liter of carbon (coded 0}, \text {1)};\\&D: \text {Final pH of liquid waste (6 and 10}, \text {coded 0}, \text {1)}. \end{aligned}$$

  The experimenters selected a design in $b=2$ blocks of $k=8$ that confounded the four-factor interaction contrast ${{ A\!B\!C\!D}}$. The fitted model included all main-effects and all 2-factor and 3-factor interactions. The block$\times$treatment interaction was assumed to be negligible.

  Table 13.24 Randomized design for a $2^4$ experiment in 2 blocks of size 8, confounding ${{ A\!B\!C\!D}}$. Data for the decontamination beta-particle experiment are shown in parentheses.

  • (a) Use a half-normal probability plot to identify the important contrasts.

  • (b) Use the method of Voss–Wang to check your selection in part (a).

  • (c) Draw an interaction plot of any interaction that appears to be nonnegligible by either analysis.

  • (d) Looking at the results of your analysis, which settings of the factors would you recommend for reducing the beta particle counts?

  • (e) Suppose that the experimenters had believed before the experiment that the three-factor interactions were all negligible. What would the analysis of variance table have looked like? Would your recommendations have been any different?

• 9. Penicillin experiment

  An experiment is described in Example 9.2 of the book Design and Analysis of Industrial Experiments edited by O. L. Davies that investigates the effects of various factors on the yield of penicillin in surface culture experiments. The five factors of interest were added to the nutrient medium, which was inoculated with a spore suspension of P. Chrysogenum. The spores rise to the surface, causing the growth of mycelium accompanied by the formation of penicillin. The factors and their levels were corn steep liquor (factor A, 2% and 3% strength), lactose (factor B, 2% and 3% strength), precursor (factor C, 0% and 0.05%), sodium nitrate (factor D, 0% and 0.3%), and glucose (factor E, 0% and 0.5%). Only 16 of the 32 treatment combinations could be carried out at one time, and the experimenters decided to observe 16 treatment combinations in one week and the remaining 16 in the following week. Large week-to-week variations were known to exist, and therefore the experiment was designed as a block design with two blocks, confounding the 5-factor interaction ${{ A\!B\!C\!D\!E}}$. The observed yields of penicillin are shown in Table 13.25. Prior to the experiment, it was believed that all that all 3- and 4-factor interactions would be negligible, and also that the $C\!E$ interaction would be important.

  Table 13.25 Data for the penicillin experiment

  • (a) Analyze the data, assuming that all 3- and 4-factor interactions are negligible. Do not forget to check the assumptions on the model.

  • (b) The experimenters decided to use logarithms of the data. Does your assumption check in part (a) confirm that this should be done? If so, reanalyze the data and state your conclusions.

  • (c) Using the logarithms of the data, draw a half-normal probability plot of the contrast estimates without using any knowledge that the higher-order interactions are likely to be negligible. Do your conclusions remain the same? Which analysis do you prefer? Why?

• 10. Peas experiment

  The following experiment was run at Biggelswade, in England, and reported by F. Yates in his 1935 paper Complex Experiments. The three treatment factors were the standard fertilizers, nitrogen, phosphate, and potassium (factors N, P, and K) each at two levels. The experimental area was divided into $b=6$ blocks of 1 / 70 of an acre. Each block was large enough for four plots on which a certain variety of pea was sown, and the fertilizer combinations shown in Table 13.26 were added. The design consists of three identical single-replicate designs each of which confounds the 3-factor interaction ${{ N\!P\!K}}$. Each block has been separately randomized.

  • (a) Estimate the treatment contrasts for all main effects and interactions.

  • (b) Calculate the analysis of variance table for this experiment and test all relevant hypotheses. State the overall significance level.

  • (c) Draw interaction plots for any important interactions. Give a set of 95% confidence intervals for the main-effect contrasts, if appropriate.

  • (d) State your overall recommendations about the fertilizers in this experiment. Would you recommend a followup experiment? If so, what would you investigate?

• 11. Field experiment, continued

  The field experiment was described in Example 13.3.1, p. 437. There were four treatment factors (A, B, C, and D) at two levels each, and the $v=16$ treatment combinations were observed twice. Each of the $r=2$ sets of treatment combinations were divided into blocks of size 8. The first two blocks, which confounded the ${{ A\!B\!C\!D}}$ interaction, were shown in Table 13.4, p. 437. The complete design, which is shown in Table 13.27, consisted of two such single-replicate designs.

  Table 13.26 Data for the peas experiment

  Table 13.27 Data for the field experiment, by block and treatment combination (TC)

  • (a) Calculate the analysis of variance table for this experiment. Now that $r=2$, there is an estimate for error variability. Test any hypotheses of interest. Are the results similar to those obtained from the first two blocks only?

  • (b) Draw any interaction plots of interest. If the yield is to be increased, what recommendations would you make about the levels of the factors?

• 12. Construct a four-replicate $2^3$ design in eight blocks of size four, partially confounding each interaction effect. Compare the variance of each interaction contrast with that of each main effect, using divisor $v/2=4$ for each contrast.

• 13. Catalytic reaction experiment

  J.R. Bainbridge, his 1951 article in the journal Industrial and Engineering Chemistry, described a factorial experiment conducted at a small plant carrying out a catalytic gaseous synthesis reaction to remove the product as a liquid solution. A 2-replicate $2^3$ experiment was conducted to study the effects of converter reaction temperature (factor A), throughput rate through the converter (factor B), and the concentration of the active ingredient in the makeup gas (factor C) on each of several response variables, including the strength of the product solution ($y_{hijk}$). The design was composed of four blocks of size four, with the ${{ A\!B\!C}}$ interaction completely confounded. The design and data are provided in Table 13.28, including the run order. (The observations in Table 13.28 are “uncoded,” each value being 80 plus one-tenth the coded value given by Bainbridge.)

  Table 13.28 Data for the catalytic reaction experiment

  • (a) Based on the run order, discuss how the design was probably randomized.

  • (b) Fit an appropriate model, and use residual plots to check the standard model assumptions.

  • (c) Conduct the analysis of variance, and discuss the results.

  • (d) Using a simultaneous confidence level of 95% for all six factorial effects, construct confidence intervals for those effects found to be significant in the analysis of variance.

  Table 13.29 Confounding schemes for $2^p$ experiments in $b=2^{s}$ blocks of size $k=2^{p-s}$. For each design, s independent generators are underlined, and s corresponding equations are given. To obtain Block I of a design, list all k combinations of the first $a_i$’s shown, then use the equations modulo 2 to complete each treatment combination

• 14. Catalytic reaction experiment, continued

  In the experiment described in Exercise 13, the covariate “makeup gas purity” was measured. The covariate values were 17, 12, 10, 10, 13, 14, 10, 16, 12, 13, 13, 11, 16, 11, 12, and 11, corresponding to runs 1–16, respectively. Repeat Exercise 13, but for an analysis of covariance.

• 15. Design of a follow-up experiment

  An experiment was run in 2007 by Joanne Sklodowski, Josh Svenson, Adam Dallas, Tim Degenero and Paul Cotellesso to examine the compressive strength of various mortar mixes. They examined the effects of four factors: Amount of water (Factor A, 0.75lb and 0.85lb), Sand type (Factor B, play sand and medium grain sand), Temperature of water (Factor C, 58 and 96 $^{\circ }$C), Cure time (Factor D, 4 and 6 days). The type and age of cement and the mixing time were held fixed throughout the experiment.

  The ingredients were mixed and poured into a cylindrical mold. After the allotted curing time, the cylinder was crushed on a compression machine. The maximum pressure exerted before the cylinder failed was recorded in pounds per square inch (psi).

  • (a) The analysis of the experiment showed large effects of $B\!C\!D$, B, C, $A\!B$. Design a follow-up experiment to examine the interactions $A\!B$, $B\!C$, $B\!D$, $C\!D$ and $B\!C\!D$, as well as all main effects. You can only take $r=1$ observation on each treatment combination and you need to run the experiment in 4 blocks of 4. Write out two of the four blocks of your design and state how to find the other two.

  • (b) Write down a model for this experiment. Write out the “Degrees of Freedom” columns of the analysis of variance table.

• 16. Design of an experiment

  • (a) Design an experiment with five factors A, B, C, D, E, each having two levels, in 4 blocks of 8 and $r=1$ observation per treatment combination. Make sure that you can estimate all main effects, all two-factor interactions, as well as all three-factor interactions that involve both D and E.

  • (b) Write out the 8 treatment combinations in Block I, and indicate how to find the treatment combinations in the other blocks. Illustrate this with two treatment combinations in the second block.

  • (c) Write out the degrees of freedom column for the analysis of variance table corresponding to the model that you would fit.