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Prof. Green
Stat 102
From TTests to Regression to Multiple Regression
The Ttest provides an instructive way to get acquainted with the idea of “degrees
of freedom.”
In contrast to the Ztest, where numbers like 1.65 and 1.96 have magical
qualities regardless of the number of observations in question, the Ttest requires one to
look up critical values in a table.
(Note that if the degrees of freedom are greater than
120, the Ztest and Ttest become equivalent, in which case you can once again rejoice at
the simple elegance of 1.65 and 1.96.)
The Ttest also provides a handy way to compare two samples drawn from normal
populations with an unknown variance.
However, anything a Ttest can do, regression can do, too.
The advantage of
regression is that it can do lots of other cool stuff.
In this lecture, I demonstrate the
equivalence of regression and Ttests in two important cases, unmatched and matched
experimental comparisons.
The (simulated) data presented below were inspired by recent experiments
designed to see how voters respond to political campaigns in 2002.
These hypothetical
data are meant to approximate the nevermadepublic but muchdiscussed experiments
that inspired the Republican “72 Hour Campaign.” (For results from alwaysmadepublic
experiments, see Green, Donald P., and Alan S. Gerber. 2004.
Get Out The Vote! How
to Increase Voter Turnout
. Washington, D.C.: Brookings Institution Press.)
Here’s the experiment.
Prior to the 2002 campaign, ten voting precincts of 100
voters apiece were grouped into pairs.
Each pair consisted of two precincts that had cast
the same proportion of votes for Republican candidates in 1998.
For each pair, a coin is
tossed, one member of the pair was placed in the treatment group.
The treatment group
received attention from the “ground campaign” – canvassers and local phone banks.
The
question is whether this treatment increased the number of voters casting Republican
ballots.
The data may be found in the following
dataset
.
The control group is denoted
treatment=0; the treatment group is treatment=1.
Notice that the dataset contains a precinct pair identification number.
The pairs
happen to vary widely in terms of past GOP voting.
Some leaned Republican in the past;
others had low GOP voting rates.
Let’s begin by treating the data as though they had been randomized without
pairing.
After all, flipping a coin for each pair still produces random assignment of 5
observations to treatment and control groups, so this is still a randomized experiment.
By
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View Full Documentignoring pairing now and taking it into account later, we can appreciate the importance of
this aspect of the research design.
To analyze whether the treatment and control groups differ, we might begin by
looking at the 5number summary for the two groups:
Descriptive Statistics: GOP Vote in 2002 by Treatment
Variable
Treatmen
N
Mean
Median
TrMean
StDev
GOP Vote
0
5
39.80
41.00
39.80
18.36
1
5
42.00
42.00
42.00
18.91
Variable
Treatmen
SE Mean
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 Fall '05
 JonathanReuningSchererDonaldGreen
 Degrees Of Freedom

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