Professor B. Jones University of California, Davis The Scientific Study of Politics (POL 51)

Today Sampling Plans Survey Research

Populations Key Concepts Population Defined by the research “All U.S. citizens age 18 or older.” All democratic countries Counties in the United States Characteristics of a Population Bounded and definable If you can’t define the population, you probably don’t have a well formed research question!

Populations vs. Samples Populations are often unattainable TOO BIG (U.S. population) Very Costly to Obtain May not be necessary The beauty of statistical theory Samples Simply Defined: a subset of the population chosen in some manner How you choose is the important question!

Moving Parts of a Sample Units of Analysis J is the population i is a member of J Then i is a “sample element” Sampling Frames The actual source of the data Literary Digest Poll (1936) “Dewey Defeats Truman” (1948) Exit Polls

More Moving Parts Sampling Unit Could be same as sample element (Unit of Analysis) But it could be collections of elements (cluster, stratified sampling) Sampling Plan Random? Nonrandom?

Kinds of Samples Simple Random Sample Major Characteristic: Every sample element has an equi- probable chance of selection. If done properly, maximizes the likelihood of a representative sample. What if your assumptions of randomness goes badly? Nonrandom samples (often) produce nonrepresentative surveys.

Why Randomness is Goodness Nonprobability Sampling Probability of “getting into” the sample is unknown All bets are off; inference most likely impossible Highly unreliable! Simple Random Sampling Every sample element has the same probability of being selected: Pr(selection)=1/N In practice, not always easy to guarantee or achieve An Example of a Bad Assumption

Some Data

More Data

Getting Probability Samples Wrong

Draft Lottery Simple random sampling did not exist. Avg. Lottery Number Jan.-June: 206 Avg. Lottery Number July-Dec.: 161 Avg. Deaths Jan.-June: 111 Avg. Deaths July-Dec.: 159 Differences highly significant. Its absence had profound consequences. Randomness should have ensured an equal chance of draft, invariant to birth date. It didn’t. By analogy, suppose college admissions were based on this kind of lottery… o.php?vid= o.php?vid=52539

How to Achieve Randomness Random number generation Modern computers are really good at this. Assign sample elements a number Generate a random numbers table Use a decision rule upon which to select sample. The Key: sampled units are randomly drawn. Why Important? Randomness helps ensure REPRESENTATIVENESS! Absent this, all bets are off: Convenience Polls Push Polls Person-on-the-Street Interviews

Populations and Samples A population is any well-defined set of units of analysis. The population is determined largely by the research question; the population should be consistent through all parts of a research project. A sample is a subset of a population. Samples are drawn through a systematic procedure called a sampling method. Sample statistics measure characteristics of the sample to estimate the value of population parameters that describe the characteristics of a population.

Populations and Samples

A population would be the first choice for analysis. Resources and feasibility usually preclude analysis of population data. Most research uses samples.

Probability Samples The goal in sampling is to create a sample that is identical to the population in all characteristics except size. Any difference between a population and a sample is defined as bias. Bias leads to inaccurate conclusions about the population.

Probability Samples Probability samples: Each element in the population has a known probability of inclusion in the sample. Probability samples are a better choice than nonprobability samples, when possible, because they are more likely to be representative and unbiased.

Probability Samples Simple random sample: Each element and combination of elements in a population have an equal chance of selection. Selection can be driven by a lottery, a random number generator, or any other method that guarantees an equal chance of selection.

Probability Samples Systematic sample: Generated by selecting elements from a list of the population at a predetermined interval. Start point for selection must be chosen at random or the list must be randomized; otherwise, the sample will not be as representative.

Probability Samples Stratified sample: Drawn from a population that has been subdivided into two or more strata based on a single characteristic. Elements are selected from each strata in proportion to the strata’s representation in the entire population.

Probability Samples Disproportionate stratified sample: Elements are drawn disproportionately from the strata. Used to over-represent a group that, due to its small size in the population, would not likely make up a large enough percentage of the sample to allow for quality inferences.

Probability Samples Cluster samples: Group elements for an initial sampling frame (50 states). Samples drawn from increasingly narrow groups (counties, then cities, then blocks) until the final sample of elements is drawn from the smallest group (individuals living in each household).

Nonprobability Samples Nonprobability samples: Each element in the population has an unknown probability of inclusion in the sample. These sampling techniques, while less representative, are used to collect data when probability samples are not feasible.

Nonprobability Samples Purposive samples: Used to study a diverse and limited number of observations. Case studies.

Nonprobability Samples Convenience samples: Include elements that are easy or convenient for the investigator; for example, college students in samples collected on college campuses.

Nonprobability Samples Quota sample: Elements are chosen for inclusion in a nonprobabilistic manner (usually in a purposive or convenient manner) in proportion to their representation in the population.

Nonprobability Samples Snowball sample: Relies on elements in the target population to identify other elements in the population for inclusion in the sample. Particularly useful when studying hard-to-locate or identify populations.

A Population and Some “Samples” A “Population” Striations represent “attitudes” Some “Samples”

Sampling come to life in…R!!! Suppose we have a population of 100,000 And in that population, we have 4 groups Group 1: 13,000 (13 percent) Group 2: 12,000 (12 percent) Group 3: 4,000 ( 4 percent) Group 4: 70,000 (70 percent) Racial/Ethnic Characteristics in the US: US Census White (69.13 percent) Black (12.06 percent) Hispanic (12.55 percent) Asian (3.6 percent) Some R Code

R #Creating a population of 100,000 consisting of 4 groups set.seed( ) population<- rep(1:4,c(13000, 12000, 4000, 70000)) #Tabulating the population (ctab requires package catspec) ctab(table(population)) #Tabulating the population (ctab requires package catspec) (btw, not sure why percents are not whole numbers) ctab(table(population)) Count Total % population

Sampling What do we expect from random sampling? That each sample reproduces the population proportions. Let’s consider SIMPLE RANDOM SAMPLES. Also, let’s consider small samples (size 100) …which is a.001 percent sample.

R: 3 samples of n=100 #Three Simple Random Samples without Replacement; n=100 which is a.001 percent sample #The set.seed command ensures I can exactly replicate the simulations set.seed(15233) srs1<-sample(population, size=100, replace=FALSE) ctab(table(srs1)) set.seed( ) srs2<-sample(population, size=100, replace=FALSE) ctab(table(srs2)) set.seed(5255) srs3<-sample(population, size=100, replace=FALSE) ctab(table(srs3))

R: Sample Results > set.seed(15233) > srs1<-sample(population, size=100, replace=FALSE) > ctab(table(srs1)) Count Total % srs > set.seed( ) > srs2<-sample(population, size=100, replace=FALSE) > ctab(table(srs2)) Count Total % srs > set.seed(5255) > srs3<-sample(population, size=100, replace=FALSE) > ctab(table(srs3)) Count Total % srs

Implications? Small samples? Variability in proportion of groups. Why does this occur? Let’s understand stratification. What does it do? You’re sampling within strata. Suppose we know the population proportions?

R: Identifying Strata and then Sampling from them. #Stratified Sampling #Creating the Groupings strata1<- rep(1,c(13000)) strata2<- rep(1,c(12000)) strata3<- rep(1,c(4000)) strata4<- rep(1,c(70000)) #Sampling by strata #Selection observations proportional to known population values: Proportionate Sampling set.seed( ) srs4<-sample(strata1, size=13, replace=FALSE) ctab(table(srs4)) set.seed( ) srs5<-sample(strata2, size=12, replace=FALSE) ctab(table(srs5)) set.seed(33325) srs6<-sample(strata3, size=4, replace=FALSE) ctab(table(srs6)) set.seed( ) srs7<-sample(strata4, size=70, replace=FALSE) ctab(table(srs7))

R: Results? Proportional Sampling w/small samples. > srs4<-sample(strata1, size=13, replace=FALSE) > ctab(table(srs4)) Count Total % srs > > set.seed( ) > srs5<-sample(strata2, size=12, replace=FALSE) > ctab(table(srs5)) Count Total % srs > > set.seed(33325) > srs6<-sample(strata3, size=4, replace=FALSE) > ctab(table(srs6)) Count Total % srs > > set.seed( ) > srs7<-sample(strata4, size=70, replace=FALSE) > ctab(table(srs7)) Count Total % srs

Proportionate Sampling What do we see? If we know the proportions of the relevant stratification variable(s)… Then sample from the groups. SMALL SAMPLES can reproduce certain characteristics of the sample. But of course, it is probabilistic.

Disproportionate Sampling Why? “Oversampling” may be of interest when research centers on small pockets in the population. Race is often an issue in this context.

R: Disproportionate Sampling > #Sampling by strata > #Selection observations disproportional to known population values: disproportionate Sampling > #"Oversampling by Race" > set.seed( ) > srs8<-sample(strata1, size=24, replace=FALSE) > ctab(table(srs8)) Count Total % srs > > set.seed( ) > srs9<-sample(strata2, size=22, replace=FALSE) > ctab(table(srs9)) Count Total % srs > > set.seed(103325) > srs10<-sample(strata3, size=14, replace=FALSE) > ctab(table(srs10)) Count Total % srs > > set.seed(11534) > srs11<-sample(strata4, size=70, replace=FALSE) > ctab(table(srs7)) Count Total % srs >

Disproportionate Samples What did I ask R to do? I “oversampled” for some groups. Again, understand why we, as researchers, might want to do this.

Side-trip: Sample Sizes Who is happy with a.001 percent SRS? On the other hand… What do we get from a stratified sample? Suppose we increase n in a SRS? It’s R time!

R: SRS with a 1 percent sample > #Sample Size=1000 > > set.seed( ) > srs1<-sample(population, size=1000, replace=FALSE) > ctab(table(srs1)) Count Total % srs > > set.seed( ) > srs2<-sample(population, size=1000, replace=FALSE) > ctab(table(srs2)) Count Total % srs > > set.seed(52909) > srs3<-sample(population, size=1000, replace=FALSE) > ctab(table(srs3)) Count Total % srs >

Implications? Sample Size MATTERS What do we see? Note, again, what stratification “buys” us. The issues with stratification? Another R example (code posted on website)

R We have again 4 sample elements > set.seed(52352) > urn<-sample(c(1,2,3,4),size=1000, replace=TRUE) > > ctab(table(urn)) Count Total % urn  My Population

R version of a person-on-the-street interview > #Convenience Sample: What shows up > > con<-matrixurn[1:10]; con [1] > > ctab(table(con)) Count Total % con

R and Samples, redux What do we find? Very unreliable sample: we oversample some groups, undersample others. Useless data more than likely. What do you imagine happens when we increase the sample sizes?

R and SRS with samples of size N /*Sample: Sizes 10, 50, 75, 100, 200, 250, 900, 1000*/ set.seed(562) s1<-sample(urn, 10, replace=FALSE) ctab(table(s1)) set.seed(58862) s1a<-sample(urn, 50, replace=FALSE) ctab(table(s1a)) set.seed(562657) s1b<-sample(urn, 75, replace=FALSE) ctab(table(s1b)) set.seed(58862) s2<-sample(urn, 100, replace=FALSE) ctab(table(s2)) set.seed(58862) s3<-sample(urn, 200, replace=FALSE) ctab(table(s3)) set.seed(10562) s4<-sample(urn, 250, replace=FALSE) ctab(table(s4)) set.seed(22562) s5<-sample(urn, 900, replace=FALSE) ctab(table(s5)) set.seed(56882) s6<-sample(urn, 1000, replace=FALSE) ctab(table(s6))

> /*Sample: Sizes 10, 50, 75, 100, 200, 250, 900, 1000*/ Error: unexpected '/' in "/" > > set.seed(562) > s1<-sample(urn, 10, replace=FALSE) > ctab(table(s1)) Count Total % s > > set.seed(58862) > s1a<-sample(urn, 50, replace=FALSE) > ctab(table(s1a)) Count Total % s1a > Sampling and Sample Size

> > set.seed(562657) > s1b<-sample(urn, 75, replace=FALSE) > ctab(table(s1b)) Count Total % s1b > > set.seed(58862) > s2<-sample(urn, 100, replace=FALSE) > ctab(table(s2)) Count Total % s > Sample Sizes

> set.seed(58862) > s3<-sample(urn, 200, replace=FALSE) > ctab(table(s3)) Count Total % s > > set.seed(10562) > s4<-sample(urn, 250, replace=FALSE) > ctab(table(s4)) Count Total % s > Sample Size

> set.seed(22562) > s5<-sample(urn, 900, replace=FALSE) > ctab(table(s5)) Count Total % s > > set.seed(56882) > s6<-sample(urn, 1000, replace=FALSE) > ctab(table(s6)) Count Total % s >

R: What did we learn? Sample size seems to have some impact here. But there are trade-offs.

Important Moving Parts Randomness (covered!) Sampling Frame Random sampling from a bad sampling frame produces bad samples. Sample Size What is your intuition about sample sizes? Must they always be large? Not necessarily so…although…

Bad Sampling Person-on-the-Street Interviews What do these imply? Small samples and inherently nonrandom Likely poor inference. Other examples? Not all non-random samples are necessarily bad Purposive Samples