Statistics Review - 1
What is the difference between a variable and a
constant?
Why are we more interested in variables than
constants?
What are the four levels of measurement?
Statistics Review - 2
What is the difference between a measure of
central tendency and a measure of
dispersion?
What are the three measures of central tendency
and under what circumstances do we use
each one?
What must we have in order to have a “social
science model”?
Why do we typically use regression rather than
measures of association?
Distributions (Normal & T)
What is the purpose of a Z score?
What is the utility of Tchebysheff’s Theorem?
What use of the normal curve did we make in
significance testing?
What are the two principles of any test of
statistical significance?
Statistics Review - 3
Variable | Obs
Mean Std. Dev. Min Max
-------------+-------------------------------------------------------tax | 100
46.54 28.731
7
97
cons | 100
35.11 31.242
0
100
party | 100
.62
.487
0
1
stinc | 100
9.20 1.524
6.1
12.4
WHAT CONCLUSIONS DO YOU DRAW ABOUT THE
DISPERSION OF SCORES ON EACH OF THESE
VARIABLES?
Recoding Tax and Conservatism
In the following exercise “Tax” and
“Conservatism” are recoded as follows:
0 – 33 = 1
34-66 = 2
67-100 = 3
Note: this procedure “costs” us much
information (i.e., 34 is the same as 66)
Cross Tabulation of Tax and
Conservatism
Tax
1
1
12.3%
2
40.4%
Conservatism
2
3
76.2%
95.5%
23.8%
4.5%
3 47.3%
0.0%
0.0%
What does the above data tell us?
Measures of Association
Association between Tax and Conservatism
Pearson’s Correlation: -.69
Gamma:
-.94
Kendall’s tau-b:
-.67
NOTE: if percentages rather than 1-3 scale are
used Pearson’s Correlation is -.80. Not using
all the information reduces the association.
WHAT DOESN’T THE ABOVE ANALYSIS TELL
US THAT WE USUALLY WANT TO KNOW?
20
40
60
80
Graph of .97 Correlation of
Brown10 and Boxer10
20
40
60
brown10
Fitted values
boxer10
80
20
40
60
80
Graph of .74 Correlation of
Coll00 and Boxer10
10
20
30
coll00
Fitted values
40
boxer10
50
20
40
60
80
Graph of -.58 Correlation of
%White in 2005 and Boxer10
60
70
80
white05
Fitted values
90
boxer10
100
20
40
60
80
Graph of -.23 Correlation of
%Senior in 2005 and Boxer10
8
10
12
14
16
senior05
Fitted values
boxer10
18
Regression Review - 1
Regression Review – 2- Regression of Tax
on Cons, Party and Stinc in Stata
Source |
SS
df
MS
-------------+-----------------------------Model | 54886.5757
3 18295.5252
Residual | 26840.2643
96 279.586087
-------------+-----------------------------Total |
81726.84
99 825.523636
Number of obs =
F( 3,
96) =
Prob > F
=
R-squared
=
Adj R-squared =
Root MSE
=
100
65.44
0.0000
0.6716
0.6613
16.721
-----------------------------------------------------------------------------tax |
Coef.
Std. Err.
t
P>|t|
Beta
-------------+---------------------------------------------------------------cons | -.64472
.07560
-8.53
0.000
-.7010575
party | 11.20792
4.67533
2.40
0.018
.1902963
stinc | -.56008
1.28316
-0.44
0.663
-.0297112
_cons | 67.38277
15.11393
4.46
0.000
.
------------------------------------------------------------------------------
Interpret both the unstandardized (“Coef.”
column) and standardized (“Beta”
column). Karl Marx’s thoughts on this?
Regression Review - 3
We might think of the value of “y”
(percentage of times the senator
supports the poor/middle income groups
on tax legislation) we observe is
conditional on the value of “x” (e.g., the
senator’s conservatism).
Take the mean of y at each value of x
We essentially have a frequency
distribution for the values y can take on
for each value of x
E(Y | xi)
The one time we observe
x, it is likely to be close to
the mean of its
probability distribution
Why Multiple Regression?
Example from the 300Reader: Value of “b”:
(1) if you use the senator’s conservatism
to explain tax voting: -.737
(2) if you use the senator’s party to
explain tax voting: 35.293
(3) if you use the median family income
in the senator’s state to explain tax
voting: 2.867
CAN YOU INTERPRET EACH “b”?
Why Multiple Regression?
Source |
SS
df
MS
-------------+-----------------------------Model | 54886.5757
3 18295.5252
Residual | 26840.2643
96 279.586087
-------------+-----------------------------Total |
81726.84
99 825.523636
Number of obs =
F( 3,
96) =
Prob > F
=
R-squared
=
Adj R-squared =
Root MSE
=
100
65.44
0.0000
0.6716
0.6613
16.721
-----------------------------------------------------------------------------tax |
Coef.
Std. Err.
t
P>|t|
Beta
-------------+---------------------------------------------------------------cons | -.64472
.07560
-8.53
0.000
-.7010575
party | 11.20792
4.67533
2.40
0.018
.1902963
stinc | -.56008
1.28316
-0.44
0.663
-.0297112
_cons | 67.38277
15.11393
4.46
0.000
.
------------------------------------------------------------------------------
Interpret both the unstandardized (“Coef.”
column) and standardized (“Beta”
column). Karl Marx’s thoughts on this?
Multiple Regression Interpretation
Notice how much smaller the impact of
senator party identification is when
senator ideology is in the same equation.
Also, note that the sign (i.e., direction of
the relationship) for state median family
income changes from positive to
negative once all three independent
variables are in the same equation.
Multiple Regression – Prediction - 1
From the previous output we know the following:
“a” = 67.382, the impact of senator
conservatism = -.644, the impact of senator
party affiliation = 11.207 and the impact of the
median household income in the senator’s
state = -.560. Senator #1’s scores on the three
independent variables are as follows:
conservatism = 26, party affiliation = 1 and
state median household income = 7.4 (i.e.,
$7,400 in 1970).
Multiple Regression – Prediction - 2
To predict the score on “tax” for senator #1 the
computer works the following equation:
67.382 + (26)(-.644) + (1)(11.207)
+ [(7.4)(-.560)]
= 67.382 – 16.744 + 11.207 – 4.144 = 57.701
Multiple Regression – Prediction - 3
Senator #1 is “predicted” to support the poor
57.701% of the time . Since senator #1
“actually” supported the poor on 54% of their
tax votes, the prediction error (“e” or “residual”)
for senator #1 is: 54 - 57.701 = -3.701
The computer then squares this value (i.e.,
-3.701 x -3.701 = 13.69). The computer
performs this same operation for all 100
senators. The sum of the squared prediction
errors for all 100 senators is 26,840.
Multiple Regression – Prediction - 4
If any of the values of the coefficients (i.e.,
67.382, -.644, 11.207 or -.560) were
changed, the sum of the squared prediction
errors would have been greater than 26,840.
This is known as the “least squared errors
principle.”
Regression Model Performance - 1
Let’s see how well our regression model
performed. From the following we know
that the mean score on “tax” is 46.5 (i.e.,
the average senator supported the
poor/middle class 46.5% of the time).
Variable |
Obs
Mean Std. Dev.
-------------+---------------------------------------------tax |
100
46.54 28.73193
Regression Model Performance - 2
We also know that senator #1 supported
the poor/middle class 54% of the time.
If we subtract the average score from
senator #1s score, we obtain senator
#1s deviation from the mean. Thus,
54 – 46.54 = 7.46. If we squared this
deviation (i.e., 7.46 x 7.46) we obtain the
squared deviation from the mean for
senator #1 (7.46 x 7.46 = 55.65).
Regression Model Performance - 3
If we repeat this process for all remaining 99
senators and add this total, we obtain the total
variation in the dependent variable that we
could explain: 81,776. From the previous
discussion we know that the total squared
prediction errors equal 26,840. If take [1 –
(26,840/81,776 = 1 - .328 = 67.1) we find that
variation in senator conservatism, party
affiliation and state median household income
explained 67.1% of the variation in senatorial
voting on tax legislation.
Review of Nonlinear Models
What are nonlinear models?
Under what circumstances should we use
probit/logit instead of regression?
Multicollinearity
An independent variable may be statistically
insignificant because it is highly correlated with
one, or more, of the other independent
variables. For example, perhaps state median
family income is highly correlated with senator
conservatism (e.g., if wealthier states elected
more conservative senators). Multicollinearity
is a lack of information rather than a lack of
data.
Visualizing Multicollinearity - 1
Visualizing Multicollinearity - 2
Visualizing Multicollinearity - 3
Multicollinearity Check in Stata
1 - 1/vif yields the percentage of the
variation in one independent explained
by all the other independent variables.
Variable |
VIF
1/VIF
-------------+---------------------cons |
1.98
0.506218
party |
1.84
0.542894
stinc |
1.35
0.738325
What would Karl Marx think now?
Multicollinearity - Interpretation
Unfortunately for Karl Marx, only 26% of
the variation in state median family
income is explained by the variation in
senator conservatism and senator party
affiliation (1- .738 = .262). Since this is
low (i.e., well below the .70 threshold
mentioned in the readings), Marx can’t
legitimately claim high multicollinearity
undermined his hypothesis.
Bread and Peace Model - 1
The Bread and Peace Model explain presidential
voting on the basis of the percentage change
in real disposable income and U.S. casualties
in post-WWII wars.
a = 46.2 (y intercept)
b1 = 3.6 (average per capita real income
growth – annual lag operator .91)
b2 = -.052 (thousands of post-WWII casualties)
Bread and Peace Model - 2
65
Bread and Peace Voting in US Presidential Elections 1952-2008
60
1972
55
1956
1964
1984
1988
1996
50
2008
1992
1952
40
45
1976
1968
1980
2000
2004
1960
-2
-1
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Real income growth and military fatalities combined
Combination of real growth and fatalities weights each variable by its estimated coefficient.
Estimated fatalities effects: -0.7% 2008, -7.6% 1968, -9.9% 1952; negligible in 1964, 1976, 2004.
Source: www.douglas-hibbs.com
Government Benefits - 1
The following slide contains the percentage of
people who (a) benefit from various programs,
and (b) claim in response to a government
survey that they 'have not used a government
social program.’ Government social programs
are stigmatized as “welfare.” But many people
benefit from such programs without realizing it.
This results in a likely underprovision of such
benefits.
Government Benefits - 2
529 or Coverdell - 64.3
Home mortgage interest deduction - 60.0
Hope or Lifetime Learning Tax Credit- 59.6
Student Loans - 53.3
Child and Dependent Tax Credit - 51.7
Earned income tax credit - 47.1
Pell Grants – 43.1
Medicare – 39.8
Food Stamps – 25.4
Regression in Value Added Teacher
Evaluations – LA Times - 3/28/11
The general formula for the "linear mixed model" used in
her district is a string of symbols and letters more than
80 characters long: y = Xβ + Zv + ε where β is a p-by1 vector of fixed effects; X is an n-by-p matrix; v is a qby-1 vector of random effects; Z is an n-by-q matrix;
E(v) = 0, Var(v) = G; E(ε) = 0, Var(ε) = R; Cov(v,ε) = 0.
V = Var(y) = Var(y - Xβ) = Var(Zv + ε) = ZGZT + R. In
essence, value-added analysis involves looking at
each student's past test scores to predict future
scores. The difference between the prediction and
students' actual scores each year is the estimated
"value" that the teacher added — or subtracted.
California Election 2010 - 1
correlate boxer10 brown10 coll00 medinc08
(obs=58)
| boxer10 brown10
coll00 medinc08
-------------+-----------------------------------boxer10 |
1.0000
brown10 |
0.9788
1.0000
coll00 |
0.7422
0.6885
1.0000
medinc08 |
0.6022
0.5401
0.8321
1.0000