Fox Module 16 analysis of variance: explanation of Duncan’s prestige data


Fox Module 16 analysis of variance: explanation of Duncan’s prestige...

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Fox Module 16 analysis of variance:explanation of Duncan’s prestige data

(The attached PDF file has betterformatting.)

This file explains one-way analysis ofvariance on pages 147-148 of the Fox textbook. Final exam problems compute R2and F-statistics from the TSS, RSS, and RegSS.

Chapter 8 of the Fox textbook usesDuncan’s prestige data to illustrate a one-way ANOVA analysis. The data shows45 occupations, with four attributes:


Type of occupation:professional or managerial, white collar, and blue collar.

Average income

Education

Measure of prestige


Income, education, and prestige arescaled from 0 to 100.

The table below shows the data byoccupation. The Excel workbook attached to this posting shows all the figuresfor the one-way analysis of variance. The computations are straight-forward;the final exam problems test these computations on a small data set.

Occupation

Type

Income

Education

Prestige

accountant

prof

62

86

82

pilot

prof

72

76

83

architect

prof

75

92

90

author

prof

55

90

76

chemist

prof

64

86

90

minister

prof

21

84

87

professor

prof

64

93

93

dentist

prof

80

100

90

reporter

wc

67

87

52

engineer

prof

72

86

88

undertaker

prof

42

74

57

lawyer

prof

76

98

89

physician

prof

76

97

97

welfare.worker

prof

41

84

59

teacher

prof

48

91

73

conductor

wc

76

34

38

contractor

prof

53

45

76

factory.owner

prof

60

56

81

store.manager

prof

42

44

45

banker

prof

78

82

92

bookkeeper

wc

29

72

39

mail.carrier

wc

48

55

34

insurance.agent

wc

55

71

41

store.clerk

wc

29

50

16

carpenter

bc

21

23

33

electrician

bc

47

39

53

RR.engineer

bc

81

28

67

machinist

bc

36

32

57

auto.repairman

bc

22

22

26

plumber

bc

44

25

29

gas.stn.attendantbc

bc

15

29

10

coal.miner

bc

7

7

15

streetcar.motormanbc

bc

42

26

19

taxi.driver

bc

9

19

10

truck.driver

bc

21

15

13

machine.operator

bc

21

20

24

barber

bc

16

26

20

bartender

bc

16

28

7

shoe.shiner

bc

9

17

3

cook

bc

14

22

16

soda.clerk

bc

12

30

6

watchman

bc

17

25

11

janitor

bc

7

20

8

policeman

bc

34

47

41

waiter

bc

8

32

10

The one-way ANOVA analysis testswhether the type of occupation affects prestige. Professional occupations havehigher prestige than white collar or blue collar; we test if the differencesare statistically significant.

Jacob: Prestige depends on education andincome, not type of occupation. The highest blue collar prestige level (67) isfor RR engineer, which also has the highest blue collar income (81). Thehighest professional prestige level is for physicians (97), who have the thirdhighest education. In Duncan’s study, dentists and lawyers have highereducation, but this is probably measurement error: medical school along withinternship and residency is longer for doctors than for dentists or lawyers.

Rachel: You are correct; the full ANOVAanalysis considers also income and education. One-way ANOVA considers a simplerquestion: does prestige differ by type of occupation? Our goal is to explainthe statistical technique. Education and income affect prestige, and thissimple analysis is not complete.

The explanatory variable is type ofoccupation; the response variable is prestige. Regression analysis assumes theresponse variable has a normal distribution. But prestige is a value from 0 to100; it does not have a normal distribution. We transform prestige to logit(prestige/ 100). The transformed response variable is closer to a normal distribution.

Jacob: How do we test if the responsevariable has a normal distribution?

Rachel: We use QQ plots. The QQ plot forprestige is thin-tailed; the QQ plot for logit(prestige/100) fits betterto a normal distribution.

The textbook shows the computationsfor both prestige and logit(prestige/100).

The overall mean prestige is 47.68889.The mean prestige by type of occupation is


Professional: 80.44444

White collar: 36.66667

Blue collar: 22.76190


The total sum of squares (TSS) is thesquare of the (prestige minus the average prestige). For accountants, this is(82 – 47.68889)2 = 1,177.25

The residual sum of squares (RSS) isthe square of the (prestige minus the average prestige for that type of occupation).For accountants, this is (82 – 80.44444)2 = 2.42.

The regression sum of squares (RegSS)is the square of the (average prestige for the occupation minus the overallaverage prestige). For accountants, this is (80.44444 – 47.68889)2 =1,072.93.

Occupation

Type

Inc

Edu

TSS

Prestige

Mn(prs)

RSS

RegSS

accountant

prof

62

6

1,177.25

82

80.4444

2.42

1,072.92

pilot

prof

72

76

1,246.87

83

80.4444

6.53

1,072.93

architect

prof

75

92

1,790.23

90

80.4444

91.31

1,072.93

author

prof

55

90

801.52

76

80.4444

19.75

1,072.93

chemist

prof

64

86

1,790.23

90

80.4444

91.31

1,072.93

minister

prof

21

84

1,545.36

87

80.4444

42.98

1,072.93

professor

prof

64

93

2,053.10

93

80.4444

157.64

1,072.93

dentist

prof

80

100

1,790.23

90

80.4444

91.31

1,072.93

reporter

wc

67

87

18.59

52

36.6667

235.11

121.49

engineer

prof

72

86

1,624.99

88

80.4444

57.09

1,072.93

undertaker

prof

42

74

86.70

57

80.4444

549.64

1,072.93

lawyer

prof

76

98

1,706.61

89

80.4444

73.20

1,072.93

physician

prof

76

97

2,431.59

97

80.4444

274.09

1,072.93

welfare.worker

prof

41

84

127.94

59

80.4444

459.86

1,072.93

teacher

prof

48

91

640.65

73

80.4444

55.42

1,072.93

conductor

wc

76

34

93.87

38

36.6667

1.78

121.49

contractor

prof

53

45

801.52

76

80.4444

19.75

1,072.93

factory.owner

prof

60

56

1,109.63

81

80.4444

0.31

1,072.93

store.manager

prof

42

44

7.23

45

80.4444

1,256.31

1,072.93

banker

prof

78

82

1,963.47

92

80.4444

133.53

1,072.93

bookkeeper

wc

29

72

75.50

39

36.6667

5.44

121.49

mail.carrier

wc

48

55

187.39

34

36.6667

7.11

121.49

insurance.agent

wc

55

71

44.74

41

36.6667

18.78

121.49

store.clerk

wc

29

50

1,004.19

16

36.6667

427.11

121.49

carpenter

bc

21

23

215.76

33

22.7619

104.82

621.35

electrician

bc

47

39

28.21

53

22.7619

914.34

621.35

RR.engineer

bc

81

28

372.92

67

22.7619

1,957.01

621.35

machinist

bc

36

32

86.70

57

22.7619

1,172.25

621.35

auto.repairman

bc

22

22

470.41

26

22.7619

10.49

621.35

plumber

bc

44

25

349.27

29

22.7619

38.91

621.35

gas.stn.attendantbc

bc

15

29

1,420.45

10

22.7619

162.87

621.35

coal.miner

bc

7

7

1,068.56

15

22.7619

60.25

621.35

streetcar.motormanbc

bc

42

26

823.05

19

22.7619

14.15

621.35

taxi.driver

bc

9

19

1,420.45

10

22.7619

162.87

621.35

truck.driver

bc

21

15

1,203.32

13

22.7619

95.29

621.35

machine.operator

bc

21

20

561.16

24

22.7619

1.53

621.35

barber

bc

16

26

766.67

20

22.7619

7.63

621.35

bartender

bc

16

28

1,655.59

7

22.7619

248.44

621.35

shoe.shiner

bc

9

17

1,997.10

3

22.7619

390.53

621.35

cook

bc

14

22

1,004.19

16

22.7619

45.72

621.35

soda.clerk

bc

12

30

1,737.96

6

22.7619

280.96

621.35

watchman

bc

17

25

1,346.07

11

22.7619

138.34

621.35

janitor

bc

7

20

1,575.21

8

22.7619

217.91

621.35

policeman

bc

34

47

44.74

41

22.7619

332.63

621.35

waiter

bc

8

32

1,420.45

10

22.7619

162.87

621.35

Total / average

43,687.64

47.6889

10,597.59

33,090.05

The total / average row shows that TSS(43,687.64) = RSS (10,597.59) + RegSS (33,090.05).

The prestige scores do not have anormal distribution. For a better ANOVA analysis, Fox uses the logit of the prestigescores divided by 100. Let Pr = prestige / 100, so logit (Pr) = ln(Pr)/ (1 – ln(Pr) ). We dot show the analysis of variance table forunadjusted prestige levels, though you can compute them easily from the lastrow of the table.

Logit of (Prestige / 100)

We form the same table using logit(prestige / 100). The attached Excel workbook has the same figures.

Occupation

Type

I

E

TSS

Pres

logit(Pr)

Mn(pr)

RegSS

RSS

accountant

prof

62.00

6.00

2.66960

82

1.5163

1.632114

3.06130

0.01340

pilot

prof

72.00

76.00

2.90079

83

1.5856

1.632114

3.06130

0.00216

architect

prof

75.00

92.00

5.35815

90

2.1972

1.632114

3.06130

0.31935

author

prof

55.00

90.00

1.61347

76

1.1527

1.632114

3.06130

0.22986

chemist

prof

64.00

86.00

5.35815

90

2.1972

1.632114

3.06130

0.31935

minister

prof

21.00

84.00

4.07435

87

1.9010

1.632114

3.06130

0.07228

professor

prof

64.00

93.00

7.31287

93

2.5867

1.632114

3.06130

0.91121

dentist

prof

80.00

100.00

5.35815

90

2.1972

1.632114

3.06130

0.31935

reporter

wc

67.00

87.00

0.03904

52

0.0800

-0.590384

0.22358

0.44947

engineer

prof

72.00

86.00

4.45199

88

1.9924

1.632114

3.06130

0.12983

undertaker

prof

42.00

74.00

0.15952

57

0.2819

1.632114

3.06130

1.82321

lawyer

prof

76.00

98.00

4.87652

89

2.0907

1.632114

3.06130

0.21034

physician

prof

76.00

97.00

12.91426

97

3.4761

1.632114

3.06130

3.40028

welfare.worker

prof

41.00

84.00

0.23185

59

0.3640

1.632114

3.06130

1.60820

teacher

prof

48.00

91.00

1.23691

73

0.9946

1.632114

3.06130

0.40640

conductor

wc

76.00

34.00

0.13839

38

-0.4895

-0.590384

0.22358

0.01017

contractor

prof

53.00

45.00

1.61347

76

1.1527

1.632114

3.06130

0.22986

factory.owner

prof

60.00

56.00

2.45722

81

1.4500

1.632114

3.06130

0.03316

store.manager

prof

42.00

44.00

0.00691

45

-0.2007

1.632114

3.06130

3.35910

banker

prof

78.00

82.00

6.55304

92

2.4423

1.632114

3.06130

0.65648

bookkeeper

wc

29.00

72.00

0.10875

39

-0.4473

-0.590384

0.22358

0.02047

mail.carrier

wc

48.00

55.00

0.29784

34

-0.6633

-0.590384

0.22358

0.00532

insurance.agent

wc

55.00

71.00

0.06072

41

-0.3640

-0.590384

0.22358

0.05127

store.clerk

wc

29.00

50.00

2.37371

16

-1.6582

-0.590384

0.22358

1.14029

carpenter

bc

21.00

23.00

0.34886

33

-0.7082

-1.482151

1.86216

0.59902

electrician

bc

47.00

39.00

0.05650

53

0.1201

-1.482151

1.86216

2.56735

RR.engineer

bc

81.00

28.00

0.68183

67

0.7082

-1.482151

1.86216

4.79757

machinist

bc

36

32

0.15952

57

0.2819

-1.482151

1.86216

3.11171

auto.repairman

bc

22

22

0.86197

26

-1.0460

-1.482151

1.86216

0.19026

plumber

bc

44

25

0.60504

29

-0.8954

-1.482151

1.86216

0.34430

gas.stn.attendantbc

bc

15

29

4.32508

10

-2.1972

-1.482151

1.86216

0.51133

coal.miner

bc

7

7

2.61488

15

-1.7346

-1.482151

1.86216

0.06373

streetcar.motormanbc

bc

42

26

1.77547

19

-1.4500

-1.482151

1.86216

0.00103

taxi.driver

bc

9

19

4.32508

10

-2.1972

-1.482151

1.86216

0.51133

truck.driver

bc

21

15

3.18057

13

-1.9010

-1.482151

1.86216

0.17540

machine.operator

bc

21

20

1.07151

24

-1.1527

-1.482151

1.86216

0.10855

barber

bc

16

26

1.60973

20

-1.3863

-1.482151

1.86216

0.00919

bartender

bc

16

28

6.09668

7

-2.5867

-1.482151

1.86216

1.22000

shoe.shiner

bc

9

17

11.27990

3

-3.4761

-1.482151

1.86216

3.97583

cook

bc

14

22

2.37371

16

-1.6582

-1.482151

1.86216

0.03100

soda.clerk

bc

12

30

6.93792

6

-2.7515

-1.482151

1.86216

1.61134

watchman

bc

17

25

3.89351

11

-2.0907

-1.482151

1.86216

0.37038

janitor

bc

7

20

5.40471

8

-2.4423

-1.482151

1.86216

0.92198

policeman

bc

34

47

0.06072

41

-0.3640

-1.482151

1.86216

1.25034

waiter

bc

8

32

4.32508

10

-2.1972

-1.482151

1.86216

0.51133

Total / avg

134.15390

47.689

-0.1175

95.55014

38.60376

These tables show how Fox calculatedthe figures on page 148. Fox doesn’t show all the work; the tables here showall the computations.


The logit of Prestige/ 100 for accountants is ln(0.82 / (1 – 0.82) ) = 1.51635.

The average logit forall occupations is –0.11754.

The average logit bytype of occupation is 1.6321 for professional, –0.5791 for white collar, and–14821 for blue collar.

The regression sum ofsquares (RegSS) for accountants is (1.6321 – –0.11754)2 = 3.06124.

The residual sum ofsquares (RSS) for accountants is (1.6321 – 1.5163)2 = 0.01341.


The total row in the table forms theanalysis of variance. Fox shows the following table:

Source

Sum of Squares

Degrees of Freedom

Mean Square

F

p

Groups

95.550

2

47.775

51.98

<< 0.001

Residuals

38.604

42

0.919

Total

134.154

44


The mean square isthe sum of squares divided by the degrees of freedom.

95.550 / 2 = 47.775;38.604 / 42 = 0.9191.

The F-statistic isthe mean square for the groups divided by the mean square of the residuals.

47.775 / 0.9191 =51.98

The R2 isthe sum of squares for the groups (RegSS) divided by the total sum of squares(TSS).

95.550 / 134.154 =71.22%


Jacob: How do we get the degrees of freedom?

Rachel: 45 data points minus 1 parameter (themean) = 44 degrees of freedom for the total sum of squares.

Three occupation types (groups) minusone relation = 2 degrees of freedom for the groups. The relation is that anoccupation is either professional, white collar, or blue collar.

Degrees of freedom for TSS – degreesof freedom for RegSS = degrees of freedom for RSS.


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Edited 8 Years Ago by NEAS
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Going from the occupational table to the type of occupation table, why is the sum of the residual sum of squares used as the RegSS for the group mean square, and on the other side the RegSS of the occupational table is used as the RSS for the Residual Mean Square?



[NEAS: Thank you for noticing the typo: the RSS and RegSScolumn headings were reversed on one of the exhibits, though all the figureswere correct and the final F-statistic was correctly computed.

Theregression sum of squares is square of the group mean minus the overall mean.

The residualsum of squares is the square of the individual value minus the group mean.

The post hasbeen corrected and re-posted.]




Edited 8 Years Ago by NEAS
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