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11.220 Quantitative Reasoning & Statistical Methods for Planners I
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Quantitative Reasoning and Statistical Methods
Ezra Glenn
March 31, 2009
Ezra Glenn
Quantitative Reasoning and Statistical Methods
An Example: Immunization Study
Before
After
Immunization
56
34
Ezra Glenn
No Immunization
144
166
Total
200
200
Quantitative Reasoning and Statistical Methods
An Example: Immunization Study
Before
After
Immunization
56
34
No Immunization
144
166
Total
200
200
“Before” Sample:
�
p̂b = 56 ÷ 200 = .28
seb =
√
.28×.78
√
200
= 0.0330
(1)
“After” Sample:
�
p̂a = 34 ÷ 200 = .17
sea =
√
.17×.83
√
200
= 0.0266
(2)
Pooled Error:
sed =
�
.03302 + .02662 = 0.0424
Ezra Glenn
Quantitative Reasoning and Statistical Methods
(3)
Difference Between Two Groups: Proportions
⎧
⎪
⎨p̂b = 0.28
p̂a = 0.17
⎪
⎩
sed = 0.0424
(4)
Then use t-statistic:
t=
p̂b − p̂a
.28 − .17
=
= 2.59
sed
.0424
which exceeds our t-critical for df=398.
∴ we can reject H0 (i.e., we can conclude that the difference is
significant)
Ezra Glenn
Quantitative Reasoning and Statistical Methods
(5)
Looking at Confidence Intervals
15
Confidence Intervals for Two Samples
0
5
Probability Density
10
Before
After
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Estimate
95% confidence intervals
Ezra Glenn
Quantitative Reasoning and Statistical Methods
Looking at the T-Distribution
8
T−Test for Difference Between Two Proportions
4
6
Observed Difference
2
●
0
Probablity Density
●
−0.2
−0.1
0.0
0.1
0.2
Difference in Proportions
95% confidence interval
Ezra Glenn
Quantitative Reasoning and Statistical Methods
Controlling for Race
Table: All Respondents
Before
After
Immunization No Immunization
56
144
34
166
From 28% down to 17%
Total
200
200
Table: With Health Insurance
Table: Without Health Insurance
Imm. No
Total
Imm.
Before 48
10
58
After
6
1
7
From 83% up to 85%
Imm. No
Total
Imm.
Before 8
134
142
After
28
165
193
From 5.6% up to 17%
You could also now test each of these for significance. . .
Ezra Glenn
Quantitative Reasoning and Statistical Methods
The χ2 Test: Rolling a Die
Table: Observed Counts (n = 60)
count
1
14
2
8
Ezra Glenn
3
8
4
8
5
8
6
14
Quantitative Reasoning and Statistical Methods
The χ2 Test: Rolling a Die
Table: Observed Counts (n = 60)
count
1
14
2
8
3
8
4
8
5
8
6
14
Table: Expected Counts (n = 60)
count
1
10
2
10
Ezra Glenn
3
10
4
10
5
10
6
10
Quantitative Reasoning and Statistical Methods
Computing χ2
Table: Observed-Expected Counts
observed
expected
1
14
10
2
8
10
3
8
10
4
8
10
5
8
10
6
14
10
total
60
60
(observed − expected)2
expected
42
22
22
22
22
42
+
+
+
+
+
=
10 10 10 10 10 10
2 × 1.6 + 4 × .4 =
χ2 = Σ
3.2 + 1.6 = 4.8
From table: χ2critical ≥ 11.07 (5 df, α = .05, two-tailed).
4.8 < 11.07, ∴ we cannot reject H0
Ezra Glenn
Quantitative Reasoning and Statistical Methods
Another χ2 Test
Table: DUSP Applications, Observed (2008)
non-minority applicants
minority applicants
total applicants
CDD
98
16
114
Ezra Glenn
IDG
70
6
76
EPP
40
5
45
HCED
43
23
66
Total
251
50
301
Quantitative Reasoning and Statistical Methods
Another χ2 Test
Table: DUSP Applications, Observed (2008)
non-minority applicants
minority applicants
total applicants
CDD
98
16
114
IDG
70
6
76
EPP
40
5
45
HCED
43
23
66
Total
251
50
301
Table: DUSP Applications, Expected (2008)
Non-minority
Minority
Total
CDD
95.06
18.94
114
IDG
63.38
12.62
76
Ezra Glenn
EPP
37.52
7.48
45
HCED
55.04
10.96
66
Total
251
50
301
Quantitative Reasoning and Statistical Methods
DUSP χ2 Calculation
Table: DUSP
(Observed−Expected)2
Expected
Non-minority
Minority
CDD
0.09
0.46
IDG
0.69
3.48
Cell Contributions
EPP
0.16
0.82
HCED
2.63
13.21
χ2 = 0.09 + 0.69 + 0.16 + 2.63 + 0.46 + 3.48 + 0.82 + 13.21 = 21.54
df = 3, significant at p < .001
Ezra Glenn
Quantitative Reasoning and Statistical Methods
The χ2 Distribution
0.15
0.10
0.05
chi−square critical, df=3
chi−square of 21.54
●
●
0.00
Probability Density
0.20
0.25
Chi−Squared Distribution, df=3
0
5
10
15
20
25
30
Chi−Square Value
95% confidence
Ezra Glenn
Quantitative Reasoning and Statistical Methods
DUSP Applications, Removing HCED
Table: DUSP Applications, Observed (2008)
non-minority applicants
minority applicants
total applicants
CDD
98
16
114
IDG
70
6
76
EPP
40
5
45
Total
208
27
235
Table: DUSP Applications, Expected (2008)
Non-minority
Minority
Total
CDD
100.90
13.10
114
Ezra Glenn
IDG
67.27
8.73
76
EPP
39.83
5.17
45
Total
208
27
235
Quantitative Reasoning and Statistical Methods
Calculating χ2
Table: DUSP Applications, Expected (2008)
Non-minority
Minority
Total
Table: DUSP
CDD
100.90
13.10
114
IDG
67.27
8.73
76
(Observed−Expected)2
Expected
Non-minority
Minority
CDD
0.08
0.64
EPP
39.83
5.17
45
Total
208
27
235
Cell Contributions
IDG
0.11
0.85
EPP
0.00
0.01
χ2 = 1.7
df = 2, not significant at p < .05
Ezra Glenn
Quantitative Reasoning and Statistical Methods