• In most cases a complete enumeration is not possible, but if done properly, properties of a population can be inferred from a randomly selected sample

• Overall planning of the survey
• Define the population of interest
• Constructing the sampling frame
• Choice of sampling design
• Selection of sample
• Data collection
• Data editing and imputation
• Estimation
• Dissemination

• The fact that not all units go into the inference introduces an uncertainty about the estimates which we call sampling error
• The sampling error is a consequence of the fact that we would get (slightly) different results if we could repeatedly draw new samples
• Given a sampling design the sampling error will generally be lower if the units in the population are alike (low variance) or the sample size is bigger
• Note: For the same sample size the sampling error will be smaller if a suitable design is chosen

# synthetic population
set.seed(236542)
N <- 1000
yi <- rnorm(N, mean=5000, s=1000)

# attributes at population level
mean (yi)

## [1] 4965.94

hist (yi)

# take a sample of size n
n <- 10
s <- sample(1:length(yi), n, replace = FALSE)

# mean of sample
mean(yi[s])

## [1] 4729.023

mean(yi)

## [1] 4965.94

# repeat sampling K times
K <- 1000
mu_hat <- numeric(K)

for (i in 1:K){
  s <- sample(1:length(yi), n, replace = FALSE)
  mu_hat[i] <- mean(yi[s])
}

mean(mu_hat)  

## [1] 4964.755

# sampling distribution
hist(mu_hat)  

• Playing with population variance and sample size…

• Representativity is not dependent upon the sample size
• For a sample to be representative, all units must have the possibility to be in the sample (inclusion probability > 0)
• The intuitive meaning of a representative sample normally is that has a certain size, but also that the marginal properties of the sample resemble those of the population
  • Certain procedures exist that makes a balanced sample in exactly this meaning
  • A balanced sample is also said to be well spread and actual measures for this have been constructed (see eg. Grafström and Tillé)

• SRS is one of the fundamental building block
• Assume a population of size \(N\) and that a sampling frame (a list of units) is available
• Randomly select \(n\) out of \(N\) units without replacement
• The implied inclusion probability is \(\pi = n/N\)
• Note: This is equivalent to considering the \(C(N,n)\) possible \(n\)-subsets and randomly select one of these as the sample!

• Assume a population with \(N\) units and that the units have some quantitative attribute \(y\)
• We seek to estimate the total sum of this attribute over the entire population: \[ \tau_y = \sum _{j=1}^N y_j\]
• For this purpose we draw a sample of size \(n\) by simple random sampling (SRS)
• The inclusion probability \(\pi\) for each unit is \(n/N\)

• We can now estimate \(\tau_y\) by applying the Horvitz-Thomson estimator
• Every single observation from the sample is weighted by the inverse of the inclusion probability: \[\hat{\tau}_y = \sum_{j=1}^n \frac{N}{n}y_j\]

• The variance of \(\hat{\tau}_y\) is given by \[ V(\hat{\tau}_y) = N^2 (1-\frac{n}{N}) \frac{\hat{s}^2}{n} \] where \(\hat{s}^2\) is an estimate for the variance of \(y\) in the population \(V(y) = \sigma^2\)
• The factor \((1-n/N)\) is the finite population correction — this quantity can be ignored when the sampling fraction \(n/N\) is very small
• We normally estimate \(s\) from the sample by the standard deviation \[ s = \sqrt{\frac{1}{n-1}\sum_{j=1}^{n}(y_i - \bar{y})^2} \]

• Normally we are more interested in the square root of the variance of the estimate — this quantity is called the standard error
• The standard error is conveniently measured on the same scale as the estimate
• We can also calculate the coefficient of variation (or relative standard error) \[ CV(\hat{\tau}) = \frac{s_\hat{\tau}}{\hat{\tau}} \]

• Note: If instead we wanted to estimate the mean \(\mu\) of \(y\) in the population then we simply divide by the population size \(N\) which yields \[ \hat{\mu}_y = \frac{1}{N} \hat{\tau}_y = \sum_{i=1}^n \frac{1}{n}y_i \] This is simply the sample mean
• Consequently the variance for the estimated mean can be found as \[ V(\hat{\mu}_y) = V(\frac{1}{N}\hat{\tau}_y) = \frac{1}{N^2}V(\hat{\tau}_y) = (1-\frac{n}{N}) \frac{\hat{s}^2}{n} \]
• Note that \(CV(\hat{\tau}) = CV (\hat{\mu}_y)\)

• Example from Cochran (pp. 27-28)

• We start by entering the data, which can be done in any number of ways

yi <- c(rep(42,23), rep(41,4), 36, 32, 29, rep(27,2), 23, 19, rep(16,2), 
        rep(15,2), 14, 11, 10, 9, 7, rep(6,3), rep(5,2), 4, 3)

• We also need to declare \(N\) and \(n\): The former is hard coded while the latter is derived from the \(y_i\)’s

N <- 676
(n <- length(yi))

## [1] 50

hist(yi, breaks=seq(0,42,2))

• An estimate for the total number of votes is calculated as \(\hat{Y} = N\frac{\sum y_i}{n}\):

(Yhat <- N * sum(yi) / n)

## [1] 19887.92

• The 80 pct. confidence limits can be calculated by hand as follows

Yhat + c(-1,1) * qnorm(0.9) * N * sd(yi) * sqrt(1-n/N) / sqrt(n) 

## [1] 18103.84 21672.00

• This corresponds to the way the calculations are presented in Cochran…

• An easier approach is to use the survey package written by Thomas Lumley (@tslumley)

library(survey)

• In order to use survey data has to be in a data frame

sample.data <- data.frame(i=1:n, y=yi, N=N)
head(sample.data, n=5)

##   i  y   N
## 1 1 42 676
## 2 2 42 676
## 3 3 42 676
## 4 4 42 676
## 5 5 42 676

• We now create a survey object using the function svydesign
• The object is created by describing the survey design and point to the data
• In our case the function requires a minimal number of argument since the design is very simple

srs.design <- svydesign(id=~1,
                        fpc=~N,
                        data=sample.data)

• There is a summary method for the class of survey objects

summary(srs.design)

## Independent Sampling design
## svydesign(id = ~1, fpc = ~N, data = sample.data)
## Probabilities:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.07396 0.07396 0.07396 0.07396 0.07396 0.07396 
## Population size (PSUs): 676 
## Data variables:
## [1] "i" "y" "N"

• This is very useful with more complicated designs