Transcript Lecture notes for 11/21/00 - University of Maryland

Sociology 601 Class 25: November 24, 2009
• Homework 9
• Review
– dummy variable example from ASR (finish)
– regression results for dummy variables
• Quadratic effects
– example: earnings and age
– plotting
• F-tests comparing models
• Example from Sociology of Religion
1
Review: Regression with Dummy Variables
Create dummy variables for age:
why? age is an interval variable, what advantage is there to creating a series of dummies?
gen byte age25=0 if age<. /* new variable, age25, will be missing if age is missing */
replace age25=1 if age>=25 & age<=29
gen byte age30=0 if age<.
replace age30=1 if age>=30 & age<=34
gen byte age35=0 if age<.
replace age35=1 if age>=35 & age<=39
gen byte age40=0 if age<.
replace age40=1 if age>=40 & age<=44
gen byte age45=0 if age<.
replace age45=1 if age>=45 & age<=49
gen byte age50=0 if age<.
replace age50=1 if age>=50 & age<=55
* check age dummies (agecheck should =1 for all cases)
egen byte agecheck=rowtotal(age25-age50)
tab agecheck, missing
2
Stata Shortcut for Dummy Variables
gen byte agecat= floor(age/5)*5
tab agecat, gen(age)
* floor function deletes decimal places:
* e.g., at age 23: floor(23/5)*5 = floor(4.6)*5 = 4*5 = 20
* check age dummies (agecheck should =1 for all cases)
egen byte agecheck=rowtotal(age1-age6)
tab agecheck, missing
drop if age<25 | age>54
3
Regression with Age Dummy Variables
. regress conrinc age2-age6 if sex==1
Source |
SS
df
MS
Number of obs =
725
-------------+-----------------------------F( 5,
719) =
12.79
Model | 3.8044e+10
5 7.6089e+09
Prob > F
= 0.0000
Residual | 4.2773e+11
719
594895739
R-squared
= 0.0817
-------------+-----------------------------Adj R-squared = 0.0753
Total | 4.6577e+11
724
643334846
Root MSE
=
24390
-----------------------------------------------------------------------------conrinc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age2 |
8220.236
3143.413
2.62
0.009
2048.872
14391.6
age3 |
16495.6
3122.571
5.28
0.000
10365.16
22626.05
age4 |
17274.8
3112.55
5.55
0.000
11164.03
23385.57
age5 |
21532.53
3288.812
6.55
0.000
15075.7
27989.35
age6 |
20013.57
3406.607
5.87
0.000
13325.48
26701.66
_cons |
26954.2
2325.541
11.59
0.000
22388.54
31519.86
------------------------------------------------------------------------------
Same R-squared and overall F, but different b’s and t’s (although same relative order):
. regress conrinc age1-age5 if sex==1
Source |
SS
df
MS
Number of obs =
725
-------------+-----------------------------F( 5,
719) =
12.79
Model | 3.8044e+10
5 7.6089e+09
Prob > F
= 0.0000
Residual | 4.2773e+11
719
594895739
R-squared
= 0.0817
-------------+-----------------------------Adj R-squared = 0.0753
Total | 4.6577e+11
724
643334846
Root MSE
=
24390
-----------------------------------------------------------------------------conrinc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age1 | -20013.57
3406.607
-5.87
0.000
-26701.66
-13325.48
age2 | -11793.33
3266.455
-3.61
0.000
-18206.26
-5380.405
age3 | -3517.968
3246.403
-1.08
0.279
-9891.531
2855.595
age4 | -2738.771
3236.766
-0.85
0.398
-9093.413
3615.872
age5 |
1518.956
3406.607
0.45
0.656
-5169.13
8207.043
_cons |
46967.77
2489.343
18.87
0.000
42080.52
51855.02
------------------------------------------------------------------------------
4
Plot Earnings by Age
. tab age, sum(conrinc)
Age and Earnings, men 25-54
60,000
50,000
Annual Earnings
|
Summary of respondent income in
age of |
constant dollars
respondent |
Mean
Std. Dev.
Freq.
------------+-----------------------------------25 |
16277.936
10757.323
47
26 |
22712.5
12540.689
46
27 |
21188.725
11802.539
40
28 |
25593.444
18395.24
54
29 |
27021.244
17314.169
45
30 |
29687.902
16242.466
61
31 |
30723.709
21631.857
55
32 |
30218.871
19739.067
62
33 |
26096.263
15751.154
57
34 |
30685.51
20528
51
35 |
37709.106
26704.259
47
36 |
29178.255
21877.287
51
37 |
33702.843
20378.26
70
38 |
39046.871
30994.531
62
39 |
40338.326
29449.024
43
40 |
35442.909
23448.711
55
41 |
38218.979
31804.641
48
42 |
34377.678
26582.113
59
43 |
37867.069
25189.647
58
44 |
34885.268
23017.34
41
45 |
35212.378
20559.449
45
46 |
41641.308
28233.297
39
47 |
39708.14
29503.584
50
48 |
41391.807
26493.252
57
49 |
38324.964
23601.741
55
50 |
42443.892
29193.688
37
51 |
37255.357
25395.935
42
52 |
35165.655
20471.181
29
53 |
44005.892
30812.439
37
54 |
36918.065
26556.129
31
------------+-----------------------------------Total |
33571.775
24047.119
1474
40,000
30,000
20,000
10,000
0
25
30
35
40
45
50
Age
Source: General Social Survey, 1996.
5
Regression Test for Curvilinearity
•test whether x has a curvilinear relationship with y:
•testing for a quadratic relationship is the most common, but not the only
method of testing for curvilinearity.
•yi = β0 + β1xi + β2xi2 + ei
•test whether β2 ≠ 0
o
if β2 > 0, then U-shape curve (or part)
o
if β2 < 0, then inverted-U curve (or part)
o
if β2 !> 0 & β2 !< 0, then revert to linear equation by dropping x2
•β1 is rather irrelevant in this test
o
if p(β2 ≠ 0)>.05 and p(β1 ≠ 0)>.05, that does not mean there is no
linear relationship.
6
Curvilinear Regression Equation: β2
yi = β0 + β1xi + β2xi2 + ei
β2 (quadratic coefficient) determines how steeply the curve accelerates:
y = 2x2 ;
y = x2 ;
y = .5 x2
900
800
700
600
500
400
300
200
100
0
-20
-10
0
10
20
7
Curvilinear Regression Equation: β2< 0
yi = β0 + β1xi + β2xi2 + ei
β2 (quadratic coefficient) < 0 then curve is inverted-U
y = -2x2 ; y = -x2 ; y = -.5 x2
-20
-10
0
10
20
0
-100
-200
-300
-400
-500
-600
-700
-800
-900
8
Curvilinear Regression Equation:
Inflexion Point = Maximum | Minimum
yi = β0 + β1xi + β2xi2 + ei
inflexion point
= value of x when y is a maximum or minimum
= - β1 / 2β2
75000
y = -20x2 + 800x + 62000
inflexion= -800 / (-20 * 2) = 20
(i.e., below observed x values)
70000
65000
y = -100x2 + 8000x – 90000
inflexion = -8000 / (-100 *2) = 40
(i.e., within the x range)
60000
55000
50000
y = -20x2 + 2400x + 800
inflexion = -2400 / (-20 * 2) = 60
(i.e., above observed values)
45000
9
40000
25
30
35
40
45
50
55
Curvilinear Regression Equation:
Inflexion Point = Maximum | Minimum
yi = β0 + β1xi + β2xi2 + ei
for completeness, when β2 is positive:
inflexion point = value of x when y is a maximum or minimum
= - β1 / 2β
70000
y = 20x2 - 800x + 50000
inflexion= --800 / (20 * 2) = 20
(i.e., below observed x values)
65000
60000
y = 100x2 - 8000x + 205000
inflexion = -8000 / (-100 *2) = 40
(i.e., within the x range)
55000
50000
y = 20x2 - 2400x + 114000
inflexion = -2400 / (-20 * 2) = 60
(i.e., above observed values)
45000
10
40000
25
30
35
40
45
50
55
Example: Regression with Curvilinear Age
. gen int agesq=age*age
. summarize age agesq
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------age |
1860
38.84355
8.309941
25
54
agesq |
1860
1577.839
655.309
625
2916
. regress conrinc age agesq if sex==1
Source |
SS
df
MS
-------------+-----------------------------Model | 3.8016e+10
2 1.9008e+10
Residual | 4.2776e+11
722
592463841
-------------+-----------------------------Total | 4.6577e+11
724
643334846
Number of obs
F( 2,
722)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
725
32.08
0.0000
0.0816
0.0791
24341
-----------------------------------------------------------------------------conrinc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age |
4764.733
1134.778
4.20
0.000
2536.875
6992.591
agesq | -50.27083
14.30126
-3.52
0.000
-78.34785
-22.19381
_cons | -65221.92
21786.08
-2.99
0.003
-107993.6
-22450.29
------------------------------------------------------------------------------
tagesq = -3.52; p < .001, so: curvilinear;
bagesq = negative, so: inverted U;
inflexion point = -bage / (2 * bagesq)) = - 4764.7 / (2 * -50.27) = 47.4
so maximum earnings at age 47 and a half.
11
Cubic Polynomials
•
Occasionally (actually, rarely), it is worthwhile to investigate
whether a more complex polynomial would better describe
the curvilinear relationship.
•
Add a cubic term (x3) to the previous quadratic equation:
•
yi = β0 + β1xi + β2xi2 + β3xi3 + ei
•
Test β3 ≠ 0
o if you can’t show β3 ≠ 0, then revert to quadratic model
o if p(β3 ≠ 0) > .05, then don’t interpret β2 and β1
•
if β3 ≠ 0, then curve has at least two bends (although not
necessarily over the range of observed x’s)
12
Cubic Polynomials: Earnings and Age Example
. regress conrinc age agesq agecu if sex==1
Source |
SS
df
MS
-------------+-----------------------------Model | 3.8020e+10
3 1.2673e+10
Residual | 4.2775e+11
721
593278929
-------------+-----------------------------Total | 4.6577e+11
724
643334846
Number of obs
F( 3,
721)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
725
21.36
0.0000
0.0816
0.0778
24357
-----------------------------------------------------------------------------conrinc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age |
3971.837
8901.06
0.45
0.656
-13503.26
21446.93
agesq | -29.64795
230.0667
-0.13
0.897
-481.3286
422.0327
agecu | -.1739568
1.936886
-0.09
0.928
-3.976566
3.628653
_cons | -55354.68
112007
-0.49
0.621
-275253.4
164544.1
------------------------------------------------------------------------------
• Note: after age cubed in entered, none of the coefficients are statistically
significant (even though age and age squared were in the quadratic model).
• So, since βagecubed is not statistically significant, revert to the quadratic
model (DON’T conclude that age has no relationship with earnings!)
13
Cubic Polynomials: Actual Results
Annual Earnings by Occupation
Full-Time Year-Round Workers, 25-54
$50
Men (moving average)
Women (moving
average)
Men (4th power fit)
Median earnings (thousands)
$45
Women (cubic fit)
$40
$35
$30
$25
$20
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Percent Female in Occupation
14
Inferences: F-tests Comparing models
Comparing Regression Models, Agresti & Finlay, p 409:
F
R
c
2
 Rr
2
/ k  g
(1 Rc ) / [N  (k  1)]
df 1  k  g; df 2  N  (k  1)
2
Where:
Rc2 = R-square for complete model,
R r2 = R-square for reduced model,
k = number of explanatory variables in complete model,
g = number of explanatory variables in reduced model, and
N = number of cases.
15
Example: F-tests Comparing models
•
Complete model: men’s earnings on
•
•
•
•
•
•
Reduced model: men’s earnings on
•
•
•
age,
age square,
age cubed,
education, and
currently married dummy.
education and
currently married dummy.
F-test comparing model is whether age variables, as a
group, have a significant relationship with earnings after
controls for education and marital status
16
Example: F-tests Comparing models
•
Complete model: men’s earnings
. regress conrinc age agesq agecu educ married if sex==1
Source |
SS
df
MS
-------------+-----------------------------Model | 1.1116e+11
5 2.2233e+10
Residual | 3.5461e+11
719
493199914
-------------+-----------------------------Total | 4.6577e+11
724
643334846
Number of obs
F( 5,
719)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
725
45.08
0.0000
0.2387
0.2334
22208
-----------------------------------------------------------------------------conrinc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age |
5627.049
8127.377
0.69
0.489
-10329.18
21583.27
agesq | -75.30909
210.0421
-0.36
0.720
-487.6781
337.0599
agecu |
.1985975
1.768176
0.11
0.911
-3.272807
3.670003
educ |
3555.331
317.9738
11.18
0.000
2931.063
4179.599
married |
8664.627
1690.098
5.13
0.000
5346.51
11982.74
_cons | -127148.4
102508.3
-1.24
0.215
-328399.8
74103.01
------------------------------------------------------------------------------
•
•
Note: none of the three age coefficients are, by themselves,
statistically significant.
Rc2 = .2387; k = 5.
17
Example: F-tests Comparing models
•
Reduced model: men’s earnings
. regress conrinc educ married if sex==1
Source |
SS
df
MS
-------------+-----------------------------Model | 8.4666e+10
2 4.2333e+10
Residual | 3.8111e+11
722
527850916
-------------+-----------------------------Total | 4.6577e+11
724
643334846
Number of obs
F( 2,
722)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
725
80.20
0.0000
0.1818
0.1795
22975
-----------------------------------------------------------------------------conrinc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------educ |
3650.611
328.1065
11.13
0.000
3006.454
4294.767
married |
10721.42
1716.517
6.25
0.000
7351.457
14091.38
_cons |
-16381.3
4796.807
-3.42
0.001
-25798.65
-6963.944
------------------------------------------------------------------------------
•
Rr2 = .1818; g = 2.
18
Inferences: F-tests Comparing models
F
R
c
2
 Rr
2
/ k  g
(1 Rc ) / [N  (k  1)]
df 1  k  g; df 2  N  (k  1)
F=
2
( 0.2387 – 0.1818) / (5 – 2)
( 1 - .2387) / (725 – 6)
df1=5-2; df1=725-6
= 0.0569/3
0.7613/719
= 26.87, df=(3,719), p < .001
(Agresti & Finlay, table D, page 673)
19
Next: Regression with Interaction Effects
Examples with earnings:
• married x gender
• age x gender
• age x education
• marital status x gender
20