#### 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.