Designs with One Source of Variation

Abstract

The design of an experiment (including determination of the number of observations to be collected), the model, and the analysis of the experiment are all dependent upon one another. This chapter introduces the analysis of an experiment based on a completely randomized design. A one-way analysis of variance model together with its assumptions is described, and estimation of the model parameters using least squares is discussed in detail. The estimation of the error variance is described, together with the calculation of a confidence bound. A hypothesis test, based on the F distribution, for testing equality of treatment effects is developed, and the analysis of variance table (ANOVA) is introduced. The chapter provides a detailed discussion of the calculation of sample sizes in a one-way analysis of variance model setting using the power of a test. The concepts introduced in this chapter are illustrated through a real experiment and with the use of SAS and R software.

Exercises

• 1. Suppose that you are planning to run an experiment with one treatment factor having four levels and no blocking factors. Suppose that the calculation of the required number of observations has given $r_1=r_2=r_3=r_4=5$. Assign at random 20 experimental units to the $v=4$ levels of the treatments, so that each treatment is assigned 5 units.

• 2. Suppose that you are planning to run an experiment with one treatment factor having three levels and no blocking factors. It has been determined that $r_1=3,\, r_2=r_3=5$. Assign at random 13 experimental units to the $v=3$ treatments, so that the first treatment is assigned 3 units and the other two treatments are each assigned 5 units.

• 3. Suppose that you are planning to run an experiment with three treatment factors, where the first factor has two levels and the other two factors have three levels each. Write out the coded form of the 18 treatment combinations. Assign 36 experimental units at random to the treatment combinations so that each treatment combination is assigned two units.

• 4. For the one-way analysis of variance model (3.3.1), p. 33, the solution to the normal equations used by the SAS software is $\hat{\tau }_i=\overline{y}_{i.}-\overline{y}_{v.} (i=1,\ldots , v$) and $\hat{\mu }=\overline{y}_{v.}{}$.

  • (a) Is $\tau _i$ estimable? Explain.

  • (b) Calculate the expected value of the least squares estimator for $\tau _1-\tau _2$ corresponding to the above solution. Is $\tau _1-\tau _2$ estimable? Explain.

• 5. Consider a completely randomized design with observations on three treatments (coded $1, 2, 3$). For the one-way analysis of variance model (3.3.1), p. 33, determine which of the following are estimable. For those that are estimable, state the least squares estimator.

  • (a) $\tau _1+\tau _2-2\tau _3$.

  • (b) $\mu +\tau _3$.

  • (c) $\tau _1-\tau _2-\tau _3$.

  • (d) $\mu +(\tau _1+\tau _2+\tau _3)/3$.

• 6. (requires calculus) Show that the normal equations for estimating $\mu ,\,\tau _1,\ldots ,\,\tau _v$ are those given in Eq. (3.4.3) on p. 35.

• 7. (requires calculus) Show that the least squares estimator of $\mu +\tau { is}\overline{Y}_{..}{}$ for the linear model $Y_{it}=\mu +\tau +\epsilon _{it}^0 (t=1,\ldots , r_i;\, i=1, 2,\ldots , v$), where the $\epsilon _{it}^0$’s are independent random variables with mean zero and variance $\sigma ^2$. (This is the reduced model for the one-way analysis of variance test, Sect. 3.5.1, p. 41.)

• 8. For the model in the previous exercise, find an unbiased estimator for $\sigma ^2$. (Hint: first calculate $E[{{ ssE}}_0]$ in (3.5.10), p. 42.)

• 9. (requires calculus) Find the least squares estimates of $\mu _1,\,\mu _2$, ..., $\mu _v$ for the linear model $Y_{it}=\mu _i+\epsilon _{it} (t=1,\ldots , r_i;\, i=1, 2,\ldots , v$), where the $\epsilon _{it}{}$’s are independent random variables with mean zero and variance $\sigma ^2$. Compare these estimates with the least squares estimates of $\mu +\tau _i (i=1, 2,\ldots , v$) in model (3.3.1), p. 33.

• 10. For the model in the previous exercise, find an unbiased estimator for $\sigma ^2$. Compare the estimator with that in (3.4.7), p. 39.

• 11. Verify, for the one-way analysis of variance model (3.3.1), p. 33, that each treatment sample variance $S^2_i$ is an unbiased estimator of the error variance $\sigma ^2$, so that

  $$ E({{ SSE}}) = \sum _i(r_i-1)E(S^2_i) = (n-v)\sigma ^2. $$

  Table 3.13 Times (in seconds) for the balloon experiment

• 12. Balloon experiment

  Prior to 1985, the experimenter (Meily Lin) had observed that some colors of birthday balloons seem to be harder to inflate than others. She ran this experiment to determine whether balloons of different colors are similar in terms of the time taken for inflation to a diameter of 7 inches. Four colors were selected from a single manufacturer. An assistant blew up the balloons and the experimenter recorded the times (to the nearest 1/10 second) with a stop watch. The data, in the order collected, are given in Table 3.13, where the codes 1, 2, 3, 4 denote the colors pink, yellow, orange, blue, respectively.

  • (a) Plot inflation time versus color and comment on the results.

  • (b) Estimate the mean inflation time for each balloon color, and add these estimates to the plot from part (a).

  • (c) Construct an analysis of variance table and test the hypothesis that color has no effect on inflation time.

  • (d) Plot the data for each color in the order that it was collected. Are you concerned that the assumptions on the model are not satisfied? If so, why? If not, why not?

  • (e) Is the analysis conducted in part (c) satisfactory?

• 13. Heart–lung pump experiment, continued

  The heart–lung pump experiment was described in Example 3.4.1, p. 37, and the data were shown in Table 3.2, p. 38.

  • (a) Calculate an analysis of variance table and test the null hypothesis that the different number of revolutions per minute have the same effects on the fluid flow rate.

  • (b) Are you happy with your conclusion? Why or why not?

  • (c) Calculate a 90% upper confidence limit for the error variance $\sigma ^2$.

• 14. Meat cooking experiment (L. Alvarez, M. Burke, R. Chow, S. Lopez, and C. Shirk, 1998)

  An experiment was run to investigate the amount of weight lost (in grams) by ground beef hamburgers after grilling or frying, and how much the weight loss is affected by the percentage fat in the beef before cooking. The experiment involved two factors: cooking method (factor $A$, with two levels frying and grilling, coded 1, 2), and fat content (factor $B$, with three levels 10, 15, and 20%, coded 1, 2, 3). Thus there were six treatment combinations 11, 12, 13, 21, 22, 23, relabeled as treatment levels 1, 2, ..., 6, respectively. Hamburger patties weighing 110 g each were prepared from meat with the required fat content. There were 30 “cooking time slots” which were randomly assigned to the treatments in such a way that each treatment was observed five times ($r=5$). The patty weights after cooking are shown in Table 3.14.

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

  • (b) Write down a suitable model for this experiment.

  • (c) Calculate the least squares estimate of the mean response for each treatment. Show these estimates on the plot obtained in part (a).

  • (d) Test the null hypothesis that the treatments have the same effect on patty post-cooking weight.

  • (e) Estimate the contrast $\tau _1 - (\tau _2 + \tau _3)/2$ which compares the effect on the post-cooked weight of the average of the two higher fat contents versus the leanest meat for the fried hamburger patties.

  • (f) Calculate the variance associated with the contrast in part (e). How does the value of the variance compare with the variance $\sigma ^2$ of the random error variables?

  Table 3.14 Post-cooking weight data (in grams) for the meat cooking experiment

  Table 3.15 Data for the trout experiment

• 15. Trout experiment (Gutsell 1951, Biometrics)

  The data in Table 3.15 show the measurements of hemoglobin (grams per 100 ml) in the blood of brown trout. The trout were placed at random in four different troughs. The fish food added to the troughs contained, respectively, 0, 5, 10, and 15 g of sulfamerazine per 100 pounds of fish (coded $1, 2, 3, 4$). The measurements were made on ten randomly selected fish from each trough after 35 days.

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

  • (b) Write down a suitable model for this experiment, assuming trough effects are negligible.

  • (c) Calculate the least squares estimate of the mean response for each treatment. Show these estimates on the plot obtained in (a). Can you draw any conclusions from these estimates?

  • (d) Test the hypothesis that sulfamerazine has no effect on the hemoglobin content of trout blood.

  • (e) Calculate a 95% upper confidence limit for $\sigma ^2$.

• 16. Trout experiment, continued

  Suppose the trout experiment of Exercise 3.15 is to be repeated with the same $v=4$ treatments, and suppose that the same hypothesis, that the treatments have no effect on hemoglobin content, is to be tested.

  • (a) For calculating the number of observations needed on each treatment, what would you use as a guess for $\sigma ^2$?

  • (b) Calculate the sample sizes needed for an analysis of variance test with $\alpha = 0.05$ to have power 0.95 if: (i) $\Delta =1.5;\,({ ii}) \Delta =1.0;\,({ iii}) \Delta =2.0$.

• 17. Meat cooking experiment, continued

  Suppose the meat cooking experiment of Exercise 3.14 is to be repeated with the same $v=6$ treatments, and suppose the same hypothesis, that the treatments have the same effect on burger patty weight loss, is to be tested.

  • (a) Calculate an unbiased estimate of $\sigma ^2$ and a 90% upper confidence limit for it.

  • (b) Calculate the sample sizes needed for an analysis of variance test with $\alpha = 0.05$ to have power 0.90 if: (i) $\Delta =5.0;\,({ ii}) \Delta =10.0$.

• 18. The diameter of a ball bearing is to be measured using three different calipers. How many observations should be taken on each caliper type if the null hypothesis $H_0$:{effects of the calipers are the same} is to be tested against the alternative hypothesis that the three calipers give different average measurements. It is required to detect a difference of $0.01$ mm in the effects of the caliper types with probability $0.98$ and a Type I error probability of $\alpha = 0.05$. It is thought that $\sigma$ is about $0.03$ mm.

• 19. An experiment is to be run to determine whether or not time differences in performing a simple manual task are caused by different types of lighting. Five levels of lighting are selected ranging from dim colored light to bright white light. The one-way analysis of variance model (3.3.1), is thought to be a suitable model, and $H_0:\{\tau _1=\tau _2=\tau _3=\tau _4=\tau _5\}{}$ is to be tested against the alternative hypothesis $H_A$:{the $\tau _i$’s are not all equal} at significance level 0.05. How many observations should be taken at each light level given that the experimenter wishes to reject $H_0$ with probability $0.90$ if the difference in the effects of any two light levels produces a $4.5$-second time difference in the task? It is thought that $\sigma$ is at most $3.0$ seconds.