Designs with Two Blocking Factors

Abstract

This chapter examines row-column designs which involve two blocking factors that do not interact. Latin square designs and Youden designs are examined as simple examples of row-column designs. The model and an overview of analysis of variance, confidence intervals, and multiple comparisons for general row-column design is provided. The analysis simplifications that occur in Latin square and Youden designs are then described. The chapter provides a brief description of model assumption checking as well as an extension of the model to cover factorial experiments in row-column designs. The concepts introduced in this chapter are illustrated through examples and through the use of SAS and R software.

Exercises

• 1. Randomization

  • (a) Randomize the plan in Table 12.1 (p. 400) so that it can be used for an experiment with seven subjects, each being assigned a sequence of four out of a possible seven antihistamines over four time periods.

  • (b) Discuss whether or not one would need to include a carryover effect in the model, or whether this could be avoided through the design of the experiment.

• 2. Latin squares

  • (a) Show that there is only one standard 3$\times$3 Latin square. (Hint: Given the letters in the first row and the first column, show that there is only one way to complete the Latin square.)

  • (b) Show that there are exactly four standard 4$\times$4 Latin squares.

• 3. Sample sizes

  Consider an experiment to compare 4 degrees of twist in a cotton-spinning experiment with respect to the number of breaks per 100 pounds. A replicated Latin square design is to be used, with time periods and machines being the row and column blocking factors.

  • (a) Determine the number s of Latin squares and the number r of observations per degree of twist to include in the experiment if each interval in a simultaneous set of 95% confidence intervals for all pairwise comparisons is to have a minimum significant difference (half-width) of 5 breaks per 100 pounds. The error standard deviation is thought to be at most 6 breaks per 100 pounds. Investigate both the Tukey and the Bonferroni methods.

  • (b) Discuss how the resulting design would be randomized.

• 4. Youden design randomization

  • (a) Find a Youden square (plan of treatment labels in rows and columns) for 5 treatments in 5 rows and 4 columns.

  • (b) Randomize the design found in part (a), assigning the rows to 5 different drying temperatures, the columns to 4 different paint nozzles, and the treatment labels to 5 different paint formulations.

• 5. Row–column design randomization

  Consider an experiment to compare 5 protocols with respect to a resting metabolism rate measurement. A row–column design is to be used, blocking on subjects and time periods. Since subjects prefer to stay in a study for a short length of time, only 3 time periods will be used, with each subject assigned a different protocol in each of the 3 time periods. For 10 subjects, the following experimental plan with 10 rows and 3 columns could be used:

  	Column				Column
  Row	I	II	III	Row	I	II	III
  I	1	2	3	VI	1	2	4
  II	2	3	4	VII	2	3	5
  III	3	4	5	VIII	3	4	1
  IV	4	5	1	IX	4	5	2
  V	5	1	2	X	5	1	3

  Table 12.16 Latin square design showing treatments and data for the video game experiment

  • (a) Does this experimental plan have treatments evenly distributed across rows and columns? Explain what you mean by “evenly distributed.”

  • (b) Determine the number of replicates s of this experimental plan, and the corresponding number of observations r per protocol to include in the experiment if each interval in a set of simultaneous 95% confidence intervals for all pairwise comparisons is to have a minimum significant difference (half-width) of 150 units. The error standard deviation is thought to be at most 250 units. Investigate both the Tukey and the Bonferroni methods.

  • (c) Discuss how the resulting design would be randomized.

• 6. Video game experiment

  Professor Robert Wardrop, of the University of Wisconsin, conducted an experiment in 1991 to evaluate in which of five sound modes he best played a certain video game. The first three sound modes corresponded to three different types of background music, as well as game sounds expected to enhance play. The fourth mode had game sounds but no background music. The fifth mode had no music or game sounds. Denote these sound modes by the treatment factor levels 1–5, respectively.

  The experimenter observed that the game required no warm up, that boredom and fatigue would be a factor after 4–6 games, and that his performance varied considerably on a day-to-day basis. Hence, he used a Latin square design, with the two blocking factors being “day” and “time order of the game.” The response measured was the game score, with higher scores being better. The design and resulting data are given in Table 12.16.

  • (a) Write down a possible model for these data and check the model assumptions. If the assumptions appear to be approximately satisfied, then answer parts (b)–(f).

  • (b) Plot the adjusted data and discuss the plot.

  • (c) Complete an analysis of variance table.

  • (d) Evaluate whether blocking was effective.

  • (e) Construct simultaneous 95% confidence intervals for all pairwise comparisons, as well as the “music versus no music” contrast

    $$\frac{1}{3}(\tau _1 + \tau _2 + \tau _3)-\frac{1}{2}(\tau _4 + \tau _5)$$

    and the “game sound versus no game sound” contrast

    $$\frac{1}{4}(\tau _1 + \tau _2 + \tau _3 + \tau _4) - \tau _5\,.$$

  • (f) What are your conclusions from this experiment? Which sound mode(s) should Professor Wardrop use?

  Table 12.17 Latin square design and data for the air freshener experiment

• 7. Video game experiment, continued

  Suppose that in the video game experiment of Exercise 6, Professor Wardop had run out of time and that only the first four days of data had been collected. The design would then have been a Youden design. Repeat parts (c), (e), and (f) of Exercise 6. Do your conclusions remain the same? Is this what you expected? Why or why not?

• 8. Air freshener experiment

  A. Cunningham and N. O’ Connor (1968, European Journal of Marketing 2, 147–151) conducted two-replicate Latin square design to compare the effects of four price-and-display treatments on the sales of a brand of air fresheners. Treatments 1–3 corresponded to high, middle, and low prices, respectively, and each had an extra display. Treatment 4 corresponded to the middle price and no extra display. The response variable was the unit sales for a one-week period. The experiment involved two blocking factors defined by stores ($c=8$ levels) and one-week periods ($b=4$ levels). The design and data are given in Table 12.17.

  • (a) Factors such as product location and shelf stocking could affect sales. Discuss how these factors might be controlled in such an experiment.

  • (b) Check the model assumptions.

  • (c) Plot the adjusted data and comment on the results.

  • (d) Complete an analysis of variance table.

  • (e) Evaluate whether blocking was effective.

  • (f) Test for equality of treatment effects using a 5% significance level.

  • (g) Construct simultaneous 95% confidence intervals for all pairwise comparisons of the treatments. What would you recommend for the sales of air fresheners if the results of this experiment are still valid today?

  Table 12.18 2-replicate Latin square design and data in parentheses for the quantity perception experiment—data are ‘true number’ minus ‘guessed number’

• 9. Air freshener experiment, continued

  Suppose that the air freshener experiment of Exercise 8 had to be stopped after only 3 weeks. The resulting design would then be a replicated Youden design. Repeat parts (d)–(g) of Exercise 8. Do your conclusions remain the same? Is this what you expected? Why or why not?

• 10. Quantity perception experiment

  An was run in 1996 by M. Gbenado, A. Veress, L. Heimenz, J. Monroe, and S. Yu to investigate the effect of color on the perception of quantity. Subjects were recruited at random from the student population. A number of small candies of a specific color were tipped onto a flat tray. A subject was allowed to view the tray for 3 sec and then asked to make a guess as to the number of candies on the tray. The response was the difference between the actual number of candies on the tray and the number guessed by the subject.

  The treatment factors of interest were “actual number of candies on the tray” and “color.” The selected levels were 17, 29, and 41 for the treatment factor “number”, and yellow, orange, brown for the factor “color.” The experimenters decided that each subject should view all nine treatment combinations, and they based their design on $9\times 9$ Latin squares.

  We consider only part of the original study, constituting a 2-replicate Latin square, as shown in Table 12.18. Subjects represent the row blocks, and time order represents the column blocks. The treatment combinations have been coded as follows:

  $$ \begin{array}{ccc} 1 = \text{(17, } \text{ yellow) } &{} \qquad 2 = \text{(17, } \text{ orange) } &{} \qquad 3 = \text{(17, } \text{ brown) }\\ 4 = \text{(29, } \text{ yellow) } &{} \qquad 5 = \text{(29, } \text{ orange) } &{} \qquad 6 = \text{(29, } \text{ brown) }\\ 7 = \text{(41, } \text{ yellow) } &{}\qquad 8 = \text{(41, } \text{ orange) } &{} \qquad 9 = \text{(41, } \text{ brown) } \end{array}$$

  Table 12.19 Latin square design and systolic blood pressure readings (mm Hg) for the caffeine experiment

  • (a) The experiment was conducted in a busy hallway at The Ohio State University. Subjects were recruited from the population of noncolorblind students walking past the table. Recruited subjects were not allowed to view the experiment in progress with previous subjects, but they were paid for their participation in candies. Discuss whether or not the subjects in this study are likely to be representative of some larger population of subjects. Are the conclusions of the study likely to be relevant to noncolorblind people in general?

  • (b) Fit a model that includes the effects of the two blocking factors, “subject” and “time-order”, the treatment effect, and the treatment$\times$time-order interaction. By looking at a computer calculated analysis of variance table, verify that the interaction (adjusted for subjects) can be measured with the full set of $(v-1)^2 =64$ degrees of freedom.

  • (c) For the model that includes treatment$\times$time-order interaction, check whether the residuals are approximately normally distributed and whether they have approximately the same variance for each treatment. Do you prefer to use the original response variable “guessed number” or the transformed response “square root of guessed number” or “(true number − guessed number)/(true number)” or some other transformation?

  • (d) Present an analysis of variance table and test any hypotheses that you think are of interest. State your conclusions.

  • (e) Rewrite your model in terms of main effects and interactions of the two treatment factors. Redo your analysis of variance table. What can you conclude from the experiment?

  • (f) If you were to plan a followup experiment, what would you wish to study? Write up a checklist for such an experiment.

  Table 12.20 Driver-distance (yards) pairs for the golf driver experiment

• 11. Caffeine experiment

  An experiment was run by Lisa Carpinello in 2001 to study the effects of caffeine on a subject’s blood pressure readings. Two treatment conditions were studied; subject abstaining from caffeine for two hours in advance of the blood pressure readings (level 0), and subject consuming 12 ounces of coffee with one tablespoon of non-dairy creamer one hour before the readings (level 1).

  The investigator measured the subject’s blood pressure at 9am and 2pm each day for eight days, with the subject sitting quietly for five minutes before the readings were taken. The ordered pairs of treatments were randomly assigned to the days in such a way that on four of the days, the treatment order was level 0 at 9am, level 1 at 2pm, and, on the other four days, the treatment order was reversed (i.e. 1 at 9am, 0 at 2pm). The design can be represented as a randomized 4-replicate Latin square design with rows representing days and columns time of day. The design and data are shown in Table 12.19.

  • (a) Plot the data and comment on the results.

  • (b) Write down a model for this experiment and check the assumptions on your model.

  • (c) Complete an analysis of variance table and evaluate whether blocking was effective.

  • (d) Test for equality of treatment effects using a 5% significance level.

  • (e) Construct a 95% confidence interval for comparing the treatments, and interpret the results.

• 12. Golf driver experiment

  An experiment was conducted by Dale Meyer in 2001 to compare four different golf clubs (drivers) in terms of the distance golf balls travel when hit. Two expensive drivers (coded 1 and 2) and two cheaper ones (coded 3 and 4) were used. Four golfers each hit four shots in each of four rounds, each golfer using each of the four drivers once per round, and balancing for the order in which each driver was used. A different Latin square was used as the design in each round. The final design is shown in Table 12.20 with driver as the treatment factor, and rows and columns defined by golfer and order. Notice that, not only do the combinations of golfer/order/driver constitute a Latin square in each round, but so do combinations of round/order/driver for each golfer, and so do combinations of round/golfer/driver for each order. So the design is extremely well balanced and this allows the golfer$\times$driver interaction to measured without adjusting for order or round.

  The experimenter was interested in comparing the driver effects and the golfer$\times$driver interaction. No other interaction effects were anticipated. If the golfer$\times$driver interaction were to be significant, then the experimenter also wanted to compare the driver effects separately for each golfer. For each observation, a golf ball was hit in the golf simulator of a golf store, and the simulated distances of the shots (in yards) were recorded and are shown in Table 12.20.

  • (a) Suggest a model for this experiment. By looking at computer calculated interaction degrees of freedom and the adjusted and unadjusted sums of squares, verify that the golfer$\times$driver interaction can be measured based on the full set of $(v-1)^2=9$ degrees of freedom, and without adjusting for order or round. Test whether the golfer$\times$driver interaction is significantly different from zero using $\alpha =0.01$.

  • (b) If there is no significant golfer$\times$driver interaction, compare the driver effects (averaged over golfer, order and round) using simultaneous 99% confidence intervals. Which method of multiple comparisons did you use and why? What can you conclude?

  • (c) Compare the pairwise effects of the drivers for each golfer, using individual 99% confidence intervals. Which method of multiple comparisons did you use and why? State your overall confidence level and interpret your results.