STAT461留学生代做、代做R编程语言、代写Longnecker、代做R

日期：2019-03-11 10:28

STAT461 Lab #7

Fruit Yield Example (From Ott and Longnecker, 2010)

An experiment was conducted to determine the effects of four different pesticides (Factor B) on the

yield of fruit from three different varieties (Factor A) of a citrus tree. Eight trees from each variety

were randomly selected from an orchard. The four pesticides were then randomly assigned to two

trees of each variety and applications were made according to recommended levels. Yields of fruit

(in bushels per tree) were obtained after the test period.

Pesticide, B

Variety, A 1 2 3 4

1 49 50 43 53

39 55 38 48

2 55 67 53 85

41 58 42 73

3 66 85 69 85

68 92 62 99

Use the following commands to read the data into R and define the factors and response; the

sep=’\t’ option indicates that columns are separated by tabs.

data = read.table(’citrus.dat’,header=T,sep=’\t’)

variety = as.factor(data[,1])

pesticide = as.factor(data[,2])

yield = data[,3]

Next, use fit=lm(yield~variety+pesticide+variety*pesticide) to fit the two-way ANOVA

model with both factors and their interaction: Yijk = μ·· + αi + βj + (αβ)ij + εijk

1. Briefly explain what each of the terms Yijk, μ··, αi

, βj , and (αβ)ij represents.

2. In terms of εijk, state the assumptions for the two-way ANOVA model.

3. How are the residuals defined for this model? Use e=residuals(fit) to save them. Also,

check the assumptions in 2 with a QQ plot and a plot of residuals versus fits. Are they

reasonably satisfied?

4. Use interaction.plot(pesticide,variety,yield)to plot the yield means for each combination

of variety and pesticide. This plot will put pesticide on the horizontal axis and use

separate lines for the varieties. Comment on the evidence for

(a) interaction between variety and pesticide

(b) a variety effect

(c) a pesticide effect

5. Use anova(fit) to obtain the ANOVA table and relevant F-statistics. Is there significant

evidence for interaction between variety and pesticide? Use α = .05.

6. Recall the estimate for the effect of the ith variety is (Y i·· ? Y ···), which averages over all

pesticides. Comment on the appropriateness of this, given the results of the interaction test.

7. Conduct the test for a variety effect. State the null and alternative hypotheses in terms of

the model parameters, and also state these words. What is your conclusion with α = .05?

8. Obtain 95.0% simultaneous confidence intervals for the mean yield of each variety with a

Bonferroni adjustment for multiplicity. Hint: tapply(yield,variety,mean).

9. Use Tukey’s procedure to conduct pairwise comparisons of mean yields for the varieties.

Recall Tukey’s critical value is

where a is the number of levels of Factor A (variety), and ab(n ? 1) is the degrees of freedom

for error.

10. If the interaction term is not significant, researchers sometimes remove it and re-fit the model.

(a) If the interaction term is removed, what happens to the sum of squares for the factors

(main effects)? What happens to the sum of squares for error? Hint: recall the

relationship

SST O = SSA + SSB + SSAB + SSE

(b) If the interaction term is removed, what happens to the degrees of freedom for the main

effects and for the error? How does this change the critical values for the tests?

(c) Given the changes observed above, does removing the interaction term necessarily make

the main effect tests more powerful? Explain.

(d) Use anova(lm(yield~variety+pesticide)) to fit the model without the interaction

term, and compare them with the results in 5. Comment on this, given the evidence for

interaction in this situation.