Analysis of Covariance

Abstract

Experiments which involve variables (covariates) that affect the response but that are not of direct interest nor can be controlled during the design of the experiment can be analyzed by the technique of analysis of covariance. This technique adjusts the treatment parameter estimates for the estimated values of the covariates. This chapter describes standard analysis of covariance models. Treatment parameter estimates are obtained via least squares, and analysis of covariance tests and confidence interval methods for the comparison of treatment effects are also developed. The concepts introduced in this chapter are illustrated through examples and use of SAS and R software.

Exercises

• 1. Consider the hypothetical data of Example 9.3.1, in which two treatments are to be compared.

  • (a) Fit the analysis of covariance model (9.2.1) or (9.2.2) to the data of Table 9.1, p. 289.

  • (b) Plot the residuals against the covariate, the predicted values, and normal scores. Use the plots to evaluate the model assumptions.

  • (c) Test for inequality of slopes, using a level of significance $\alpha = 0.05$.

  • (d) Test for equality of the treatment effects, using a significance level of $\alpha = 0.05$. Discuss the results.

  • (e) Construct a 95% confidence interval for the difference in treatment effects. Discuss the results.

• 2. (optional) Assume that the analysis of covariance model (9.2.2) holds, so that $Y_{it} = \mu + \tau _i + \beta (x_{it}-\overline{x}_{..}) + \epsilon _{it}$.

  • (a) Compute $E[Y_{it}]$.

  • (b) Verify that ${\text{ sp }}^*_{xY} = \sum _i\sum _t (x_{it}-\overline{x}_{i.})Y_{it}$, given that ${\text{ sp }}^*_{xY} = \sum _i\sum _t (x_{it} - \overline{x}_{i.}) (Y_{it}-\overline{Y}_{i.})$.

  • (c) Show that $E[\hat{\beta }]=\beta$, where $\hat{\beta }= {\text{ sp }}^*_{xY}/{\text{ ss }}^*_{xx}$ and ${\text{ ss }}^*_{xx} = \sum _i\sum _t (x_{it}-\overline{x}_{i.})^2$.

  • (d) Verify that $\text{ Var }(\hat{\beta }) = \sigma ^2/{\text{ ss }}^*_{xx}$ and Cov$(\overline{Y}_{i.},\hat{\beta })=0$.

  • (e) Verify that $E[\hat{\mu }+ \hat{\tau }_i] = \mu + \tau _i$, where $\hat{\mu }+ \hat{\tau }_i = \overline{Y}_{i.} - \hat{\beta }(\overline{x}_{i.} - \overline{x}_{..})$.

  • (f) Using the results of (c) and (e), argue that $\mu + \tau _i$ and $\beta$ and all linear combinations of these are estimable.

• 3. Zinc plating experiment

  The following experiment was used by C. R. Hicks (1965), Industrial Quality Control, to illustrate the possible bias caused by ignoring an important covariate. The experimental units consisted of 12 steel brackets. Four steel brackets were sent to each of three vendors to be zinc plated. The response variable was the thickness of the zinc plating, in hundred-thousandths of an inch. The thickness of each bracket before plating was measured as a covariate. The data are reproduced in Table 9.8.

  Table 9.8 Bracket thickness $x_{it}$ and plating thickness $y_{it}$ in $10^{-5}$ inches for three vendors (Hicks 1965)

  • (a) Plot $y_{it}$ versus $x_{it}$, using the vendor index i as the plotting symbol. Discuss the relationship between plating thickness and bracket thickness before plating. Based on the plot, discuss appropriateness of the analysis of covariance model. Based on the plot, discuss whether there appears to be a vendor effect.

  • (b) Fit the analysis of covariance model (9.2.1) or (9.2.2) to the data.

  • (c) Plot the residuals against the covariate, predicted values, and normal scores. Use the plots to evaluate model assumptions.

  • (d) Test for equality of slopes, using a level of significance $\alpha = 0.05$.

  • (e) Test for equality of the vendor effects, using a significance level $\alpha = 0.05$.

  • (f) Fit the analysis of variance model to the data, ignoring the covariate.

  • (g) Using analysis of variance, ignoring the covariate, test for equality of the vendor effects using a significance level $\alpha = 0.05$.

  • (h) Compare and discuss the results of parts (e) and (g). For which model is msE smaller? Which model gives the greater evidence that vendor effects are not equal? What explanation can you offer for this?

• 4. Paper towel absorbancy experiment

  S. Bortnick, M. Hoffman, K.K. Lewis and C. Williams conducted a pilot experiment in 1996 to compare the effects of two treatment factors, brand and printing, on the absorbancy of paper towels. Three brands of paper towels were compared (factor A at 3 levels). For each brand, both white and printed towels were evaluated (factor B, 1=white, 2=printed). For each observation, water was dripped from above a towel, which was horizontally suspended between two pairs of books on a flat surface, until the water began leaking through to the surface below. The time to collect each observation was measured in seconds. Absorbancy was measured as the number of water drops absorbed per square inch of towel. The rate at which the water droplets fell to the towel was measured (in drops per second) as a covariate. The data are reproduced in Table 9.9.

  Table 9.9 Data for the paper tower absorbancy experiment

  • (a) Plot absorbancy versus rate, using the treatment level as the plotting symbol. Based on the plot, discuss appropriateness of the analysis of covariance model, and discuss whether there appear to be treatment effects.

  • (b) Fit the one-way analysis of covariance model to the data.

  • (c) Plot the residuals against the covariate, run order, predicted values, and normal scores. Use the plots to evaluate model assumptions.

  • (d) Test for equality of slopes, using a level of significance $\alpha = 0.05$.

  • (e) Test for equality of treatment effects, using a significance level $\alpha = 0.05$.

  • (f) Conduct a two-way analysis of covariance. Test the main effects and interactions for significance.

• 5. Catalyst experiment, continued

  The catalyst experiment was described in Exercise 5 of Chap. 5. The data were given in Table 5.18, p. 134. There were twelve treatment combinations consisting of four levels of reagent, which we may recode as $A=1,\, B=2,\, C=3,\, D=4$, and three levels of catalyst, which we may recode as $X=1,\, Y=2,\, Z=3$, giving the treatment combinations $11,\, 12,\, 13,\, 21, \ldots , 43$.

  The order of observation of the treatment combinations is also given in Table 5.18.

  • (a) Fit a two-way complete model to the data and plot the residuals against the time order. If you are happy about the independence of the error variables, then check the other assumptions on the model and analyze the data. Otherwise, go to part (b).

  • (b) Recode the treatment combinations as $1,\, 2,\ldots , 12$. Fit an analysis of covariance model (9.2.1) or (9.2.2) to the data, where the covariate $x_{it}$ denotes the time in the run order at which the tth observation on the ith treatment combination was made. Check all of the assumptions on your model, and if they appear to be satisfied, analyze the data.

  • (c) Plot the adjusted means of the twelve treatment combinations in such a way that you can investigate the interaction between the reagents and catalysts. Test the hypothesis that the interaction is negligible.

  • (d) Check the model for lack of fit; that is, investigate the treatment $\times$ time interaction. State your conclusions.