(Solution) IE 400 Homework 4 A. Chronopoulou Homework 4: ANOVA Diagnostics Due: 09/24 (beginning of class) 1. One proposes that deviations of the observations

1.A consultant discussing ANOVA applications in a seminar stated: “Sometimes I find that treatment

 

effects in an experiment do not show up through difference in the treatment means. Hence, it is

 

important to compare the residual plots for the treatments.” A member of the audience asked: “I

 

don’t think I understood your point regarding differences in treatment means being explored using

 

residual plots.” Discuss

 

2.One proposes that deviations of the observations of Yij,around the estimated overall mean yhat.. be plotted to assist in evaluating the appropriateness of the ANOVA cell means model.Would these

 

deviations be helpful in studying the independence of the error terms? The constancy of the variance of the error terms? The normality of the error terms? Discuss.IE 400

 

Homework 4

 

A. Chronopoulou

 

Homework 4: ANOVA Diagnostics

 

Due: 09/24 (beginning of class)

 

1. One proposes that deviations of the observations of

 

Y

 

ij

 

around the estimated overall mean

 

¯

 

Y

 

··

 

be

 

plotted to assist in evaluating the appropriateness of the ANOVA cell means model. Would these

 

deviations be helpful in studying the independence of the error terms? The constancy of the vari-

 

ance of the error terms? The normality of the error terms? Discuss.

 

2. A consultant discussing ANOVA applications in a seminar stated: “Sometimes I ?nd that treatment

 

e?ects in an experiment do not show up through di?erence in the treatment means. Hence, it is

 

important to compare the residual plots for the treatments.” A member of the audience asked: “I

 

don’t think I understood your point regarding di?erences in treatment means being explored using

 

residual plots.” Discuss.

 

3. Refer to the automobile data set (homework 2, problem 3).

 

(a) Obtain the residuals and prepare a residual plot against the ?tted values by factor level. What

 

departures from ANOVA model can be studied from these plots. What are your ?ndings?

 

(b) Prepare a normal probability plot of the residuals. Does the normality assumption appear to

 

be reasonable here? Explain.

 

(c) Assuming that the error terms are approximately normal, examine by means of the Hartley

 

test whether or not the treatment error variances are equal. Use

 

?

 

= 0

 

.

 

01. State the alterna-

 

tives, decision rule and conclusion.

 

4. Refer to the SENIC dataset.

 

(a) Referring to the analysis you did for variable 4 vs. variable 9 (problem 4, hw 2), obtain the

 

residuals and prepare a residual plot against the ?tted values by region. Are any serious

 

departures from ANOVA model suggested by your plots?

 

(b) Obtain a normal probability plot of the residuals. Is the normality assumption reasonable

 

here?

 

(c) Examine by means of the Hartley test whether or not the geographic region error variances

 

are equal; use

 

?

 

= 0

 

.

 

05. State the alternatives, decision rule, and conclusion.

 

(d) A test of whether or not mean length of stay (variable 2) is the same in the four geographic

 

regions (variable 9) is desired, but concern exists about the normality and equal variances

 

assumptions of the ANOVA model.

 

i. Obtain the residuals and plot them against the ?tted values to study whether or not the

 

error variances are equal for the four geographic regions. What are your ?ndings?

 

ii. For each geographic region, calculate

 

¯

 

Y

 

i

 

·

 

and

 

s

 

i

 

. Examine the three relations discussed in

 

‘Lecture 5’ slides and determine the transformation that is the most appropriate one here.

 

What do you conclude?

 

iii.

 

[For 4 credit students.]

 

Use the Box-Cox procedure to ?nd an appropriate power

 

transformation of

 

Y

 

. Evaluate the SSE for the values of

 

?

 

ranging from -1 to 1 with step

 

0.2; include also

 

±

 

1. Does

 

?

 

=

 

-

 

1, a reciprocal transformation, appear to be reasonable

 

based on the Cox-Box procedure?

 

iv. Use the reciprocal transformation

 

Y

 

= 1

 

/Y

 

to obtain transformed response data.

 

1

 

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