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In the paper “Advanced Quantitative Methods” the author identifies the dependent and independent variables, the level of measurement for each variable, the value of the appropriate measure of association, and the proportionate reduction of error (PRE) interpretation of the association measure…
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Seminar in Advanced Quantitative Methods I In each of questions through 3, a) identify the dependent and independent variables, b) the level of measurement for each variable, c) the value of the appropriate measure of association, and d) the proportionate reduction of error (PRE) interpretation of the association measure.
1. The following table considers the relationship between social class and health condition.
Health Status
Social Class
Lower class
Working class
Middle or Upper Class
Poor
9
17
11
Fair
11
58
37
Good or Excellent
14
133
150
A contingency table shows frequency distribution for bivariate data. There are two variables Health Status and Social Class in above contingency table. From above table, it appears that Health Status is dependent variable and Social Class is independent variable.
The level of measurement for variable Health Status is ordinal and for variable Social Class is nominal.
Table 1 shows the Chi-square Contingency Table Test for Independence using MegaStat’s (an Excel add-in).
The Health Status depend on Social Class, 2(4, N = 440) = 25.61, p < .001. In other words, Health Status is dependent on Social Class.
The appropriate measure of association for Health Status and Social Class is Cramers V coefficient.
Cramers V = = = 0.1706 (n is sample size, and k is smaller of row or column)
The value of Cramers V coefficient is 0.1706 that suggest there is weak relationship between Health Status and Social Class.
The proportionate reduction of error (PRE), λ
If we were asked to guess Heath Status, but we did not know the Social Status (independent variable) than the possible guess for Health Status is Good or Excellent because the mode for dependent variable Health Status is Good or Excellent. By doing this, we make 143 (= 440 - 297) mistakes out of the 440 choices, for an error rate of 143/440 or 32.5% (say old error).
Now suppose we know both Health Status and Social Class (dependent and independent variable). Now the guess will that Lower Class has Good or Excellent Health Status, Working Class has Good or Excellent Health Status, and Middle or Upper Class has Good or Excellent Health Status. Now we make 143 (= (34 – 14) + (208 – 133) + (198 – 150)) mistakes out of the 440 choices, for an error rate of 143/440 or 32.5% (say new error).
The difference between the old error and the new error, divided by the old error is the proportionate reduction of error, or PRE, λ . λ is not symmetric its value depends on which is the independent variable.
The proportionate reduction of error (PRE), λ suggests that the Health Status and Social Class are independent.
Table 1
Chi-square Contingency Table Test for Independence
Lower class
Working class
Middle or Upper Class
Total
Poor
Observed
9
17
11
37
Expected
2.86
17.49
16.65
37.00
(O - E)² / E
13.19
0.01
1.92
15.12
Fair
Observed
11
58
37
106
Expected
8.19
50.11
47.70
106.00
(O - E)² / E
0.96
1.24
2.40
4.61
Good or Excellent
Observed
14
133
150
297
Expected
22.95
140.40
133.65
297.00
(O - E)² / E
3.49
0.39
2.00
5.88
Total
Observed
34
208
198
440
Expected
34.00
208.00
198.00
440.00
(O - E)² / E
17.64
1.65
6.32
25.61
25.6075
chi-square
4
df
0.0000
p-value
0.2412
Phi coefficient
0.1706
Cramérs V
2. The use of cellular phones in automobiles has increased dramatically in the last few years. Of concern to traffic experts, as well as manufacturers of cellular phones, is the effect on accident rates. Is someone who is using a cellular phone more likely to be involved in a traffic accident? What is your conclusion from the following sample information? Use the .05 significance level.
Had Accident in the Last Year
Did Not Have an Accident in the Last Year
Cellular phone in use
35
200
Cellular phone not in use
50
400
Use of cellular phone is independent variable and traffic accident is dependent variable.
The level of measurement for both variables use of cellular phone and traffic accident is nominal.
Table 2 shows the Chi-square Contingency Table Test for Independence using MegaStat’s (an Excel add-in).
The traffic accident is independent of use of cellular phone, 2(1, N = 600) = 2.03, p = .154. In other words, someone who is using a cellular phone is not more likely to be involved in a traffic accident.
The appropriate measure of association for use of cellular phone and traffic accident is Phi (φ) coefficient.
The value of Phi (φ) coefficient is 0.0545 that suggest there is no association between use of cellular phone and traffic accident.
The proportionate reduction of error (PRE), λ
Here, column represents the dependent variable (traffic accident). Therefore, if no information is available for cellular phone use than error is
Misclassified in situation 1 (E1) = 85
If information is available for cellular phone use than error is
Misclassified in situation 2 (E2) = (235 – 200) + (450 – 400) = 85
Therefore, the proportionate reduction of error, or PRE, λ is
The proportionate reduction of error (PRE), λ suggests that use of cellular phone and traffic accident are independent.
Table 2
Chi-square Contingency Table Test for Independence
Had Accident in the Last Year
Did Not Have an Accident in the Last Year
Cellular phone in use
Observed
35
200
Expected
29.16
205.84
(O - E)² / E
1.17
0.17
Cellular phone not in use
Observed
50
400
Expected
55.84
394.16
(O - E)² / E
0.61
0.09
Total
Observed
85
600
Expected
85.00
600.00
(O - E)² / E
1.78
0.25
2.0322
chi-square
1
df
.1540
p-value
.0545
Phi coefficient
.0545
Cramérs V
3. A researcher believes that women are happier in marriage than men. To examine the relationship between sex and marital happiness, the following table was obtained for a sample of 300 adults.
Happiness Rating
Men
Women
Very Happy
103
75
Pretty Happy
51
60
Not Too Happy
5
6
Sex (men or women) is independent variable and Marital Happiness Rating is dependent variable.
The level of measurement for variable Marital Happiness Rating is ordinal and for variable Sex (men or women) in nominal.
Table 3 shows the Chi-square Contingency Table Test for Independence using MegaStat’s (an Excel add-in).
Marital happiness does not differ by sex, 2(2, N = 300) = 4.16, p = .125. In other words, the men and women have same marital happiness.
The appropriate measure of association for Sex and Marital Happiness Rating is Cramers V coefficient.
Cramers V = = = 0.1178 (n is sample size, and k is smaller of row or column)
The value of Cramers V coefficient is 0.1178 that suggest there is very weak (not useful) relationship between Sex and Marital Happiness Rating.
The proportionate reduction of error (PRE), λ
Here, row represents the dependent variable (Happiness Rating). Therefore, if no information is available about sex than error is
Misclassified in situation 1 (E1) = 300 – 178 = 122
If information is available about sex than error is
Misclassified in situation 2 (E2) = (159 – 103) + (141 – 75) = 122
Therefore, the proportionate reduction of error, or PRE, λ is
The proportionate reduction of error (PRE), λ suggests that sex and marital happiness are independent.
Table 3
Chi-square Contingency Table Test for Independence
Happiness Rating
Men
Women
Total
Very Happy
Observed
103
75
178
Expected
94.34
83.66
178.00
(O - E)² / E
0.79
0.90
1.69
Pretty Happy
Observed
51
60
111
Expected
58.83
52.17
111.00
(O - E)² / E
1.04
1.18
2.22
Not Too Happy
Observed
5
6
11
Expected
5.83
5.17
11.00
(O - E)² / E
0.12
0.13
0.25
Total
Observed
159
141
300
Expected
159.00
141.00
300.00
(O - E)² / E
1.96
2.20
4.16
4.1601
chi-square
2
df
.1249
p-value
0.1178
Phi coefficient
0.1178
Cramérs V
4. A group of citizens has filed a complaint with the Chief of the Houston Police Department alleging
that poor neighborhoods receive significantly less protection than more affluent neighborhoods. The
Chief asks you to examine the complaint using the percent of the population on welfare and the
number of hours of daily police patrol in a sample of communities in Houston.
Community Number
Percent on Welfare
Hours of Daily Police Patrol
1
40
20
2
37
15
3
32
20
4
29
20
5
25
15
6
24
20
7
17
15
8
15
20
9
2
50
10
4
40
a. What are the dependent and independent variables for this study?
The dependent and independent variables for the study are Hours of Daily Police Patrol and Percent of the Population on Welfare, respectively.
b. Use EXCEL or SPPS to produce a regression analysis.
Regression Statistics
Multiple R
0.7589
R Square
0.5760
Adjusted R Square
0.5230
Standard Error
8.1478
Observations
10
ANOVA
df
SS
MS
F
Significance F
Regression
1
721.4114
721.4114
10.8669
0.0109
Residual
8
531.0886
66.3861
Total
9
1252.5000
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
39.0700
5.3803
7.2617
0.0001
26.6631
51.4769
Percent on Welfare
-0.6920
0.2099
-3.2965
0.0109
-1.1761
-0.2079
c. Find, and interpret, the value of the slope in the context of this study.
The value of the slope is -0.692. The value of slope suggests that every percent increase in percent of the population on welfare drops about 0.7 hours of daily police patrolling.
d. Find, and interpret, the value of the y-intercept in the context of this study.
The value of the y-intercept is 39.07. The value of y-intercept suggests that population without welfare has about 39 hours of daily police patrolling.
e. What is the prediction (also called least squares) equation?
The prediction equation is
Hour of Daily Police Patrol = 39.07 – 0.692(Percent on Welfare)
f. What is the expected number of hours of daily police patrol for a community in which 30 percent is on welfare.
Hour of Daily Police Patrol = 39.07 – 0.692(30) = 18.31
The expected number of hours of daily police patrol for a community in which 30 percent is on welfare is about 18 hours.
5. 5. As a researcher, you want to investigate the hypothesis that the number of crimes is related to police expenditures per capita because states with higher crime rates are likely to increase their police force, thereby spending more on the number of officers on the street. The following table shows X is the number of crimes (in thousands) and Y is Police Protection Expenditures per Capita (Dollars), per 10,000 people in the eastern and Midwestern United States.
State
X = number of crimes
Y = Police Protection Expenditures per Capita
Maine
2.88
122.50
New Hampshire
2.28
141.80
Vermont
2.81
102.80
Massachusetts
3.26
218.70
Rhode Island
3.58
179.20
Connecticut
3.38
193.60
New York
3.27
292.40
New Jersey
3.40
236.60
Pennsylvania
3.11
171.20
Ohio
4.00
179.40
Indiana
3.76
124.50
Illinois
4.51
224.40
Michigan
4.32
172.30
Wisconsin
3.29
196.60
Minnesota
3.59
166.80
Iowa
3.22
135.80
Missouri
4.57
153.90
North Dakota
2.39
102.90
South Dakota
2.64
115.30
TOTAL
64.26
3,230.70
a) Find the least-squares regression equation that predicts police expenditures per capita from the
number of crimes, and hence predict the police expenditure per capita for a state with 3.65 crimes (in thousands).
The least-squares regression equation that predicts police expenditures per capita from the number of crimes is
Police Protection Expenditures per Capita = 73.06 + 28.67(Number of Crimes)
Where, the number of crimes is in thousands and Police Protection Expenditures per Capita (Dollars) is per 10,000 people.
Police Protection Expenditures per Capita = 73.06 + 28.67(3.65) = 177.72
The police expenditure per capita for a state with 3.65 crimes (in thousands) is S177.72 per 10,000 people.
b) Find, and interpret (in terms of police expenditures per capita and the number of crimes), the coefficient of determination.
The coefficient of determination is 0.1402. This value suggests that the number of crimes explains about 14% variation in police expenditure per capita for a state. The number of crimes does not explain the other 86% variation in police expenditure per capita for a state.
Regression Statistics
Multiple R
0.3745
R Square
0.1402
Adjusted R Square
0.0897
Standard Error
47.5803
Observations
19
ANOVA
df
SS
MS
F
Significance F
Regression
1
6277.7787
6277.7787
2.7730
0.1142
Residual
17
38485.9855
2263.8815
Total
18
44763.7642
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
73.0594
59.2506
1.2331
0.2343
-51.9485
198.0672
number of crimes
28.6737
17.2190
1.6652
0.1142
-7.6552
65.0026
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