The Scientific Study of Politics (POL 51)
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Transcript The Scientific Study of Politics (POL 51)
The Scientific Study of
Politics (POL 51)
Professor B. Jones
University of California, Davis
Today
Sampling Plans
Survey Research
More fun with simulations
samplesize<-10000
population<-rnorm(samplesize, 5, 2)
truth<-mean(population)
sdtruth<-sd(population)
truth
Sdtruth
Here’s what I know in
> truth
[1] 5.002265
> sdtruth
[1] 2.003601
the “population”:
What do my samples look like?
ten<-sample(population, 10, replace=F)
m1<-mean(ten); m1
sd1<-sd(ten)
hist(ten)
fifty<-sample(population, 50, replace=F)
m2<-mean(fifty); m2
sd2<-sd(fifty)
hist(fifty)
hundred<-sample(population, 100,
replace=F)
m3<-mean(hundred); m3
sd3<-sd(hundred)
hist(hundred)
.
.
.
Sampling Sizes
In general, we’ve seen larger sample
sizes yield more accurate conclusions.
Though the differences between very
large and just “merely” large samples
may in fact be negligible.
Requires us to turn to the concept of
repeated sampling and sample
variability.
Polls and Repeated Sampling
As individual researchers, you usually
have one “shot” at it.
Statistical theory (classical) relies on the
concept of long-run probability
Repeated trials
…law of large numbers
…central limit theorem
Maybe concepts you have heard of
before? …or not.
Side-trip to the 2008
Presidential Election
Pollster.com allows us to think about
“repeated” sampling.
This cite basis its analysis on all
available polls
Why might this be a good thing?
There is sampling variability in individual
samples.
Let’s look at polls leading up to the Nov.
4th Election
What are the “dots”
The blue dots are Obama percentage
(estimates)
The red dots are McCain
Why are they different?
Variability in samples…sampling frames,
methodologies differ.
Combine them, and you get a better
picture.
Look at solid red and blue states.
Polls
Note how the polls seem to be
“clustering” as the election gets closer.
Why?
Undecideds deciding?
More certainty?
Let’s look at close states.
Polls, Projections and the EC
EC Projections
Tied to Polls
Variability
340.2-197.8 fivethirtyeight.com
311-142-85
311-174-53
353-185
278-132-128
260-160-118
pollster.com
zogby.com
electoral-vote.com
realclearpolitics.com
rasmussenreports.com
Understanding variability
We kind of see “repeated sampling”
The basic idea:
The “truth” will be revealed if you just
sample enough
But any one sample may be off in one
direction or another.
Back to sampling
Let’s simulate repeated sampling in R
More Simulation
The Population
N=1,000,000
Mean of the Population is 0.4992135
R Code:
#"The Population"
X<-runif(1000000,.01,.99)
meanX <- mean(X); meanX
Let’s Sample n=500, 1000, 5000.
First Sample: Mean=.4692207
Second Sample: Mean=.5004778
Third Sample: Mean=.5027007
#Some Samples: First, sample 1, n=500, evaluate:
set.seed(52151)
nsamp <- 1
res <- numeric(nsamp)
for (i in 1:nsamp) res[i] <- mean(sample(X, 500, replace = FALSE))
mean(res)
#Some Samples: Second, sample 2, n=1000, evaluate:
set.seed(110789008)
nsamp <- 1
res <- numeric(nsamp)
for (i in 1:nsamp) res[i] <- mean(sample(X, 1000, replace = FALSE))
mean(res)
#Some Samples: Third, sample 3, n=5000, evaluate:
set.seed(16978)
nsamp <- 1
res <- numeric(nsamp)
for (i in 1:nsamp) res[i] <- mean(sample(X, 5000, replace = FALSE))
mean(res)
Repeated Sampling
Suppose we were to take 10 samples of
size 500?
[1,] 0.4922826
[2,] 0.5114829
[3,] 0.5006157
[4,] 0.5180107
[5,] 0.5083638
[6,] 0.5054319
[7,] 0.4992882
[8,] 0.4612303
[9,] 0.4897318
[10,] 0.5016498
Mean: 0.4988088
S.D.: 0.01568156
Lessons?
Sampling variability is a real issue.
Range in estimates went from .46 to .52
Way under and way over estimate the
mean in certain trials.
However, on average, “we’re close.”
More simulations.
Repeated Sampling
Experiment 1: 1000 samples, n=500
Mean: 0.4994611
S.D.: 0.01209907
set.seed(7869324)
nsamp <- 1000
res <- numeric(nsamp)
for (i in 1:nsamp) res[i] <- mean(sample(X,
500, replace = FALSE))
mean(res); sd(res)
hist(res, br=10, xlim=range(.5))
abline(v =meanX)
N=500, 1000 Samples
Repeated Sampling
Experiment 2: 1000 samples, n=1000
Mean: 0.4988333
S.D: 0.008994245
set.seed(7454)
nsamp <- 1000
res <- numeric(nsamp)
for (i in 1:nsamp) res[i] <- mean(sample(X,
1000, replace = FALSE))
mean(res); sd(res)
hist(res, br=10, xlim=range(.5) )
abline(v =meanX)
N=1000, 1000 Samples
Repeated Sampling
Experiment 3: 1000 samples, n=5000
Mean: 0.499128
S.D.: 0.004016436
set.seed(13433)
nsamp <- 1000
res <- numeric(nsamp)
for (i in 1:nsamp) res[i] <- mean(sample(X,
5000, replace = FALSE))
mean(res); sd(res)
hist(res, br=10, xlim=range(.5))
abline(v =meanX)
N=5000, 1000 Samples
What’s going on?
Sampling Variability
If we “fix” the number of samples, what
happened?
As n increases, variability decreases.
“On average, our sample estimate is
“close” to the true value…
AND, the variation across samples is
decreasing.
Theory
Population Parameter
θ is the unknown parm.
What does this equality tell us?
How does it relate to samples?
^
E ( )
Sample Proportions
In our examples, we wanted to estimate
a proportion.
We knew it’s true value (we usually do
not!)
We therefore must sample.
The same concept as before applies:
^
E ( P) P
Probability
“Over repeated samples, the expected
value of the proportion will equal the true
population proportion.”
This is a good thing.
Sample estimates can do a good job of
approximating the population value.
This permits generalizability.
Good sampling technique will produce
“unbiased estimates.”
Repeated Sampling Redux
Suppose we were to take 10 samples of
size 500?
[1,] 0.4922826
[2,] 0.5114829
[3,] 0.5006157
[4,] 0.5180107
[5,] 0.5083638
[6,] 0.5054319
[7,] 0.4992882
[8,] 0.4612303
[9,] 0.4897318
[10,] 0.5016498
Mean: 0.4988088
S.D.: 0.01568156
Mean of the Population is 0.4992135
E(P)=.4988; Population “P”=.4992
E(P)≈P
Note, any single sample might
be “off”; however, the idea is that
there is no systematic tendency to be
off one direction or the other.
Sampling Distribution
What we’ve just gone through are
simulations of SAMPLING
DISTRIBUTONS
Defined: the distribution of a statistic that
you obtain from repeated samples of
size n from some population.
The Concept of Variance
How far might you be off in a particular
sample?
Why, by the way, might you like to know
this?
You usually only have ONE sample!!
Is there a way we can determine this
degree of variability?
Standard Error of a Proportion
Variance: “Average “squared” deviations
Standard Error: square root of the
variance.
2
P
P
P (1 P )
N
P (1 P )
N
Standard Error in Action
Suppose the true population parameter
is P.
P=.50
In repeated samples, you would expect the
average sample statistic to approach .50
Recall prior simulation
What is the “sampling error”?
Using formula from previous slide:
[.5(1-.5)/100]1/2 =.05
Interpretation?
If the true population proportion is .50
and we took repeated (random) samples
of size 100, the expected value of P
would be .50 but the standard deviation
would be .05.
.05 is our standard error of the sampling
distribution. This is what ought to
happen in repeated sampling.
More to it…that comes later.
Put it to the test.
> #"The Population"
> X<-runif(1000000,.01,.99)
> meanX <- mean(X); meanX
[1] 0.500889
> sdX<-sd(X); sdX
[1] 0.2832314
>
> #Sample 100, 1000 times
>
> set.seed(7324)
> nsamp <- 1000
> res <- numeric(nsamp)
> for (i in 1:nsamp) res[i] <- mean(sample(X, 100,
replace = FALSE))
> mean(res); sd(res)
[1] 0.5007463
[1] 0.02781522
Result
What conclusions would I draw from my
simulation?
“Best guess” of P is .50.
The average deviation across samples is
about .03.
My guess + my error allows me to
compute a CONFIDENCE INTERVAL
Estimate +/- Error=C.I.
Confidence Interval
What I’ve really done in my simulation is
computed a “68 percent confidence interval.”
.50 plus or minus .03
68 percent of all samples give a value for P
between (about) .47 and .53
Classical interpretation: In repeated samples of
size 100, the expected value of P will lie in the
range .47 to .53, 68 percent of the time.
Why “68 percent”?
68-95-99.7 Rule and the Normal Distribution
One Sample
You have one
sample.
What makes the C.I.
big versus small?
The Standard Error
As n goes up, s.e.
goes down.
Therefore, C.I. must
get smaller.
P (1 P )
N
P (1 P )
P
N
2
P
Illustration
Relationship Between Sample Size and
Sampling Error
0.10
0.08
0.06
0.04
0.02
0.00
25
10
0
20
0
30
0
40
0
50
0
60
0
70
0
80
0
90
0
10
00
20
00
30
00
40
00
Standard Error
0.12
Sample Size
Implications?
If we want to cut our s.e. in half, we must
quadruple the sample size.
N exponentially related to s.e.
S.E. for N=100 is .05
S.E. for N=400 is .025
.05/.025=2
S.E. for N=1000 is .0158
S.E. for N=4000 is .0079
.0158/.0079=2
There are trade-offs between precision and
design.