View
214
Download
0
Tags:
Embed Size (px)
Citation preview
PSY 340Statistics for the
Social Sciences
Chi-Squared Test of Independence
Statistics for the Social SciencesPsychology 340
Spring 2010
PSY 340Statistics for the
Social Sciences
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog).
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog).
The question: is there a relationship between age and watch preference?
Young (under 30)
Old (over 30)
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog).
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog).
The question: is there a relationship between age and watch preference?
Young (under 30)
Old (over 30)
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
Statistical analysis follows design
PSY 340Statistics for the
Social Sciences
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference?
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference?
PREFERENCE
Digital Analog Undecided Row totals
Under 30
Age
Over 30
Column totals
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
Chi-Squared Test for Independence
Step 1: State the hypotheses– H0: Preference is
independent of age (“no relationship”)
– HA: Preference is related to age (“there is a relationship”)
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference?
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference?
Observed scores
PREFERENCE
Digital Analog Undecided Row totals
Under 30
90
40
10
Age
Over 30
10
40
10
Column totals
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
Step 2: Compute your degrees of freedom & get critical valuedf = (#Columns - 1) * (#Rows - 1) = (3-1) * (2-1) = 2
Level of significance df 0.05 0.025 0.01 1 3.84 5.02 6.64 2 5.99 7.38 9.21 3 7.81 9.35 11.34 : : : :
• For this example, with df = 2, and α = 0.05
• The critical chi-squared value is 5.99
– Go to Chi-square statistic table and find the critical value
PREFERENCE
Digital Analog Undecided Row totals
Under 30
90
40
10
Age
Over 30
10
40
10
Column totals
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences Chi-Squared Test for Independence
PREFERENCE
Digital Analog Undecided Row totals
Under 30
90
40
10
Age
Over 30
10
40
10
Column totals
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies
Observed scores
PSY 340Statistics for the
Social Sciences Chi-Squared Test for Independence
PREFERENCE
Digital Analog Undecided Row totals
Under 30
90
40
10
140
Age
Over 30
10
40
10
60
Column totals
100
80
20
200
Spot check: make sure the row totals and column totals add up to the same thing
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies
Observed scores
PSY 340Statistics for the
Social Sciences
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies
PREFERENCE
Digital Analog Undecided Row totals
Under 30
90
40
10
140
Age
Over 30
10
40
10
60
Column totals
100
80
20
200
Under 30
Over 30
Digital Analog Undecided
fe =100(140)
200=70
70
fe =80(140)
200=56
56
fe =20(140)
200=14
14
fe =100(60)
200=30
30
fe =80(60)200
=24
24
fe =20(60)200
=6
6
Observed scores
fe =fcolumnfrow
n
Expected scores
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
Step 3: Collect the data. Obtain row and column totals (sometimes called the marginals) and calculate the expected frequencies
PREFERENCE
Digital Analog Undecided Row totals
Under 30
90
40
10
140
Age
Over 30
10
40
10
60
Column totals
100
80
20
200
Under 30
Over 30
Digital Analog Undecided
fe =100(140)
200=70
70
fe =80(140)
200=56
56
fe =20(140)
200=14
14
fe =100(60)
200=30
30
fe =80(60)200
=24
24
fe =20(60)200
=6
6
Observed scores
fe =fcolumnfrow
n
Expected scores
“expected frequencies” - if the null hypothesis is correct, then these are the frequencies that
you would expect
“expected frequencies” - if the null hypothesis is correct, then these are the frequencies that
you would expect
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
χ 2 = ∑fo − fe( )
2
fe
Cell fo fe (fo-fe) (fo-fe)2 (fo-fe)
2/fe <30 digital
90 70
<30 analog
40 56
<30 undecided
10 14
>30 digital
10 30
>30 analog
40 24
>30 undecided
10 6
PREFERENCE
Digital Analog Undecided Row totals
Under 30
70 90
56 40
14 10
140
Age
Over 30
30 10
24 40
6 10
60
Column totals
100
80
20
200
• Find the residuals (fo - fe) for each cell
Step 4: compute the χ2
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
Step 4: compute the χ2
Cell fo fe (fo-fe) (fo-fe)2 (fo-fe)
2/fe <30 digital
90 70 20
<30 analog
40 56 -16
<30 undecided
10 14 -4
>30 digital
10 30 -20
>30 analog
40 24 16
>30 undecided
10 6 4
PREFERENCE
Digital Analog Undecided Row totals
Under 30
70 90
56 40
14 10
140
Age
Over 30
30 10
24 40
6 10
60
Column totals
100
80
20
200
• Find the residuals (fo - fe) for each cell
χ 2 = ∑fo − fe( )
2
fe
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
• Square these differences
Cell fo fe (fo-fe) (fo-fe)2 (fo-fe)
2/fe <30 digital
90 70 20 400
<30 analog
40 56 -16 256
<30 undecided
10 14 -4 16
>30 digital
10 30 -20 400
>30 analog
40 24 16 256
>30 undecided
10 6 4 16
PREFERENCE
Digital Analog Undecided Row totals
Under 30
70 90
56 40
14 10
140
Age
Over 30
30 10
24 40
6 10
60
Column totals
100
80
20
200
• Find the residuals (fo - fe) for each cell
χ 2 = ∑fo − fe( )
2
fe
Step 4: compute the χ2
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
• Square these differences
Cell fo fe (fo-fe) (fo-fe)2 (fo-fe)
2/fe <30 digital
90 70 20 400 5.71
<30 analog
40 56 -16 256 4.57
<30 undecided
10 14 -4 16 1.14
>30 digital
10 30 -20 400 13.33
>30 analog
40 24 16 256 10.67
>30 undecided
10 6 4 16 2.67
PREFERENCE
Digital Analog Undecided Row totals
Under 30
70 90
56 40
14 10
140
Age
Over 30
30 10
24 40
6 10
60
Column totals
100
80
20
200
• Find the residuals (fo - fe) for each cell
• Divide the squared differences by fe
χ 2 = ∑fo − fe( )
2
fe
Step 4: compute the χ2
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
• Square these differences
Cell fo fe (fo-fe) (fo-fe)2 (fo-fe)
2/fe <30 digital
90 70 20 400 5.71
<30 analog
40 56 -16 256 4.57
<30 undecided
10 14 -4 16 1.14
>30 digital
10 30 -20 400 13.33
>30 analog
40 24 16 256 10.67
>30 undecided
10 6 4 16 2.67
PREFERENCE
Digital Analog Undecided Row totals
Under 30
70 90
56 40
14 10
140
Age
Over 30
30 10
24 40
6 10
60
Column totals
100
80
20
200
• Find the residuals (fo - fe) for each cell
• Divide the squared differences by fe
• Sum the results
=38.09 = χ 2
χ 2 = ∑fo − fe( )
2
fe
Step 4: compute the χ2
Chi-Squared Test for Independence
PSY 340Statistics for the
Social Sciences
Chi-Squared, the final stepStep 5: Compare this computed statistic (38.09)
against the critical value (5.99) and make a decision about your hypotheses
χ2(2,n=200)=38.09,p<0.05
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference?
A manufacturer of watches takes a sample of 200 people. Each person isclassified by age and watch type preference (digital vs. analog). The question: is there a relationship between age and watch preference?
here we reject the H0 and conclude that there is a relationship between age and watch preference
PSY 340Statistics for the
Social Sciences In SPSSIn SPSS• Each person gets a row• Each person has data about the
variables – age & watch preference
PSY 340Statistics for the
Social Sciences In SPSSIn SPSS• Analyze => Descriptives =>
Crosstabs• Select the two variables (usually
they are nominal or ordinal) you want to examine and click the arrow to move one into the “rows” and one into the “columns” box.
• Click on “statistics” button, and check the “Chi-square” box.
• Click “continue.”• Click “OK.”
PSY 340Statistics for the
Social Sciences SPSS OutputLook at the “Chi-square tests”
box.The top row of this box gives
results for “Pearson’s Chi-Square”
• “Value” is the value of the χ2 statistic,
• “df” is the degrees of freedom for the test
• “Asymp. Sig. (2-sided)” is the probability (p-value) associated with the test.
• The chi-squared distribution, like the F-distribution, is “squared” so 1-tailed test is not possible.