STAT3600
Statistical Analysis
Assignment 4 (submit Q4, Q5, Q7)
Deadline: 2 May, 2025
Note: (1) Numeric values should be presented in 4 decimal places. (2) Show the intermediate steps for Q4 – Q10.
1. A rehabilitation center researcher was interested in examining the relationship between physical prior to surgery of persons undergoing corrective knee surgery and time required in physical therapy until successful rehabilitation. Patient records in the rehabilitation center were examined, and 24 male subjects ranging in age from 18 to 30 years who had undergone similar corrective knee surgery during the past year were selected for the study. The number of days (Y) required for successful completion of physical therapy and the prior physical fitness status (A: 1 = below average, 2 = average, 3 = above average) of each patient are recorded. The rehabilitation researcher wishes to use age of patient (X) as a concomitant variable. The data file is ‘rehabit.csv’.
(a) Read the data into ‘mydata’. Run and explain the following codes.
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mydata$A<-factor(mydata$A,levels=1:3,labels=c('below','average','above'))
mydata$A<-relevel(mydata$A,ref='below')
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(b) Prepare a scatter plot of the data. Does it appear that there are effects of physical fitness status on the mean number of days required for therapy? Discuss.
(c) State the regression model equivalent to covariance model according to the setting in (a). Also stat the reduced regression model for testing for treatment effects.
(d) Fit the full regression model and report the least squares estimates of the regression coefficients.
(e) Fit the reduced regression models and test for treatment effects; use α = 0.01. State the hypotheses, decision rule and conclusion.
(f) Estimate the mean number of days required for therapy for patients of average physical fitness and age 24 years; use a 99% confidence interval.
(g) Make all pairwise comparisons between the treatment effects; use a 95% confidence level. Comment on the comparisons.
2. A sociologist selected a random sample of 45 adjunct professors who teach in the evening division of a large metropolitan university for a study of special problems associated with teaching in the evening division. The data collected include the amount of payment received by the faculty member for teaching a course during the past semester. The sociologist classified the faculty members by subject matter of course (factor A, 1 = humanities, 2 = social sciences, 3 = engineering, 4 = management) and highest degree earned (factor B, 1 = bachelor, 2 = master, 3 = doctorate). The earnings per course (in thousand dollar’s, Y) are stored in ‘adjunct.txt’.
(a) Consider an ANOVA model for this case. State the equivalent regression model; use ‘humanities’ and ‘doctorate’ as the reference levels.
(b) Read the data into ‘mydata’. Run and apply to factor B in a similar manner.
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mydata$A<-factor(mydata$A,levels=1:4,labels=c('humanities','social','engineering','management'))
mydata$A<-relevel(mydata$A,ref='humanities')
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(c) Report the least squares estimates for the regression coefficient in the regression model in (a).
(d) Find the treatment means obtained, yij. by the model. Plot the treatment means. Does it appear that any factor effects are present? Explain.
(e) What is the reduced model for testing for interaction effects? Fit the reduced model and thus, test whether or not interaction effects are present by fitting the full and reduced models; use a = 0.01. State the alternatives, decision rule and conclusion. What is the p-value of the test?
(f) State the reduced regression models for testing for subject matter and highest degree main effects, respectively, and conduct each of the tests. Use α = .01 each time and state the alternatives, decision rule, and conclusion. What is the p-value of each test?
(g) Based on the full model, make all pairwise comparisons between the subject matter means. Use a 95 percent (individual) confidence coefficient.
(h) Based on the full model, make all pairwise comparisons between the highest degree matter means. Use a 95 percent (individual) confidence coefficient.
3. A consumer organization studied the effect of age and gender of automobile owner on size of cash offer (Y: in hundred dollars) for a used car by utilizing 12 persons in each of three age groups (A: 1 = young, 2 = middle, 3 = elderly) who acted as the owner of a used car. Six male (B = 1) and six female (B = 2) volunteers were used in each age group. An analyst wishes to use each dealer’s sales volume (X: in hundred thousand dollars) as a concomitant variable. The data are stored in ‘cash.txt’. Assume that covariance model is applicable.
(a) State the regression model equivalent to covariance model. Fit this full model.
(b) State the reduced regression models for testing for interaction and factor A and factor B main effects, respectively. Fit these reduced regression models.
(c) Test for interaction effects; use α = .05. State the alternatives, decision rule, and conclusion. What is the p-value of the test?
(d) Test for factor A main effects; use α= .05. State the alternatives, decision rule, and conclusion. What is the p-value of the test?
(e) Test for factor B main effects; use α= .05. State the alternatives, decision rule, and conclusion. What is the P-value of the test?
(f) For each factor, make all pairwise comparisons between the factor level main effects. Use a 90% confidence level for each comparison.
4. Five observations of a response variable Y and a treatment A are given as follows.
(a) Use level 3 of A as the reference level. Using the zero constraint (ii) in 7.2.1, state the regression model equivalent to an ANOVA model. State the data matrix, X, of the regression model.
(b) Calculate XTX, X TY and YTY.
(c) It is given that
Calculate the LSE of the regression coefficients.
(d) Compile the ANOVA table. Test the effects of A at the 5% level of significance. State the hypotheses, decision rule and conclusion.
(e) Conduct a T test for each of the coefficients at the 5% level of significance. Interpret the results.
(f) Express mean Y for each level of A in terms of the parameters in (a).
(g) Express the pairwise comparisons of mean Y for the 3 levels of A in terms of the parameters in (a). Thus, make all pairwise comparisons between the treatment effects; use a 90% confidence level by the Bonferroni’s method. Thus, comment of the comparisons of the treatment effects.
5. Six observations of Y are given for four treatments defined by two factors A and B.
(a) Use levels 2 of A and B, respectively, as the reference levels. Using the zero constraint (ii) in
7.2.1, state the regression model equivalent to a 2-way ANOVA model with interaction. State the data matrix, X, of the regression model.
(b) Calculate XTX, X TY and YTY.
(c) It is given that
Calculate the LSE of the regression coefficients.
(d) Test whether there is a regression of Y on the main effects ofA, B and their interaction at the 5% level of significance. Compile the ANOVA table, state the hypotheses, decision rule and conclusion.
(e) Express mean Y for each of the four treatments in terms of the parameters in (a).
(f) Express the pairwise comparisons among the means for Y of the four treatments in terms of the parameters in (a). Thus, make all pairwise comparisons between among the treatments; use a 90% confidence level by the Bonferroni’s method. Thus, comment of the comparisons of the treatment effects.
6. Refer to Q5.
(a) State the reduced model to test for the interaction effect. State the data matrix, Xr, of the regression model.
(b) It is given that
Calculate SSE for the reduced model.
(c) Test the interaction effect at the 5% level of significance by an F test. state the hypotheses, decision rule and conclusion. State the hypotheses, decision rule and conclusion.
(d) Consider a model without interaction. Test the main effects ofA and B, respectively, by an F test at the 5% level of significance. State the hypotheses, decision rule and conclusion.
7. A randomized block design is considered. There are 4 levels of factor A and two blocks. The observations of Y are given as follows.
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A
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Block
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1
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2
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3
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1
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1
0
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10
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-5
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2
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3
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13
12
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-4
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(a) Use level 3 of A as the reference level. Using the zero constraint (ii) in 7.2.1. State the regression model equivalent to the block design. State the data matrix X.
(b) Calculate the LSE for the regression coefficients. It is given that
(c) Compile the ANOVA table.
(d) Test at the 5% level of significance the effect of factor A. State the hypotheses, decision rule and conclusion.
(e) Use the Bonferroni’s method at 95% confidence level, conduct a pairwise comparison ofmean Y among the three levels ofA.
8. Five observations of a response variable Y, a treatment A and a covariate X are given as follows.
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A
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1
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2
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2
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3
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3
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X
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-1
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0
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1
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-1
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2
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Y
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7
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8
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6
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4
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-3
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It is given that
(a) Use level 2 of A as the reference level. Using the zero constraint (ii) in 7.2.1. State the regression model equivalent to an ANCOVA model. State the data matrix X.
(b) Calculate the LSE for the regression coefficients and SSE in (a).
(c) State the reduced model for testing the effects ofA and calculate SSE for the reduced model.
(d) Test the effects ofA; use α = 0.05. State the hypotheses, decision rule and conclusion.
(e) Use the Bonferroni’s method at 90% confidence level, conduct a pairwise comparison ofmean Y among the three levels ofA.
9. Refer to Q8. Construct a 95% confidence interval for the average of the means of Y for the values given as follows.
10. Seven observations ofa response variable Y, two treatments A and B, and a covariate X are given as follows.
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A
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1
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1
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1
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2
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1
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2
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2
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2
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B
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1
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1
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2
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2
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2
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1
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2
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1
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X
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-1
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-3
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3
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-2
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-5
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1
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5
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4
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Y
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8
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10
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3
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3
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9
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2
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-4
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-2
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(a) State the regression model equivalent to ANCOVA model with interaction between A and B.
Calculate least squares estimates of the regression coefficients and SSE. It is given that
Calculate the standard errors of the regression coefficients.
(b) Test for interaction effects; use α = .05. State the alternatives, decision rule, and conclusion. What is the p-value of the test?
(c) Construct the Bonferroni 95% confidence intervals for the pairwise comparisons among all four treatments defined by A and B. Comment on the differences.