Some Useful Formulas From the Statistics of Survey Sampling, I

Equally-Likely Outcomes & Counting

• If K outcomes O_{1}, …, O_{K} are equally likely, then the probability of any one of them is 1/K.

• Consider taking a sample of n objects from a population of N objects.

– Sampling with replacement, there are N^{n} possible samples of size n; the probability of any one of them is 1/N^{n}.

– Sampling without replacement, there are \binom{N}{n} = \frac{N!}{n!(N−n)!} possible samples of size n [where N! = N · (N − 1) · (N − 2) ⋯ 3 · 2 · 1], so the probability of any one of them is 1 \begin{matrix}/ \\ ( & {N \atop n} & ) \end{matrix}.

Discrete Random Variables Let X and Y be random variables with sample spaces {x_{1}, …, x_{K}} and {y_{1}, …, y_{K}} and distributions

P[X = x_{i}, Y = y_{j}] = p_{ij} , P[X = x_{i}] = p_{i·} = ∑_{j=1}^{K} p_{ij} , P[Y = y_{j}] = p_{·j} = ∑_{i=1}^{K} p_{ij}

Then, for example

E[X] = ∑_{i=1}^{K} x_{i}p_{i·} , Var(X) = ∑_{i=1}^{K} (x_{i} − E[X])^{2}p_{i·} , Cov(X, Y) = ∑_{i=1}^{K} (x_{i} − E[X])(y_{i} − E[Y])p_{ij}

P[X = x_{i}|Y = y_{j}] = p_{ij}/p_{·j} , E[X|Y = y_{j}] = ∑_{i=1}^{K} x_{i}P[X = x_{i}|Y = y_{j}] , E[aX + bY + c] = aE[X] + bE[Y] + c

Random Sampling From a Finite Population

Consider a population of size N and a sample of size n. Let y_{i} be the (fixed) values of some variable of interest in the population (such as a person’s age, or whether they would vote for Obama). Let

Z_{i} = \{ \begin{matrix}1, if i is in the sample \\ 0, else \end{matrix}

be the random sample inclusion indicators, and let Y_{i} be the random observations in the sample. Then the sample average can be written

\overline{Y} = \frac{1}{n} ∑_{i=1}^{n} Y_{i} = \frac{1}{n} ∑_{i=1}^{N} Z_{i}y_{i}

The Z_{i}’s are Bernoulli random variables with

E[Z_{i}] = \frac{n}{N} , Var(Z_{i}) = \frac{n}{N} (1 − \frac{n}{N}) , Cov(Z_{i}, Z_{j}) = −\frac{1}{N − 1} \frac{n}{N} (1 − \frac{n}{N})

Confidence Intervals and Sample Size

(a) A CLT-based 100(1 − α)% confidence interval for the population mean is (\overline{Y}− z_{α/2}SE , \overline{Y}+ z_{α/2}SE).

(b) For sampling with replacement from an infinite population, SE = SD/ \sqrt{n}.

(c) For sampling without replacement from a finite population, the SE has to be multiplied by the finite population correction (FPC).

(d) For a given margin of error (ME, half the width of the CI) and confidence level 1 − α, we can find the sample

size by solving

z_{α/2}SE < ME

for n. The same approach works for both SRS with replacement (using the SE in (b)) and SRS without replacement (using the SE in (c)).

Some Useful Formulas From the Statistics of Survey Sampling, II

Stratified Sampling Consider H strata with population counts N = ∑ _{h=1}^{H} N_{h} and sample counts n = ∑ _{h=1}^{H} n_{h}. Let f_{h} = n_{h}/N_{h}; W_{h} = N_{h}/N; and \overline{y}_{h} = \frac{1}{n_{h}} ∑ _{i=1}^{n_{h}} y_{ih} in each stratum, and let s_{h}^{2} = \frac{1}{n_{h}−1} ∑ _{i}(y_{ih} − \overline{y}_{h})^{2} be the sample variance in each stratum. Then

\overline{y}_{st} = ∑_{h=1}^{H} W_{h}\overline{y}_{h} , Var(\overline{y}_{st}) ≈ ∑_{h=1}^{H} W_{h}^{2}(1 − f_{h})\frac{s_{h}^{2}}{n_{h}} , DEFF = \frac{Var(\overline{y}_{st})}{Var(\overline{y}_{srs})} = \frac{∑ _{h=1}^{H} W_{h}^{2}(1 − f_{h})\frac{s_{h}^{2}}{n_{h}}}{(1 − f)\frac{s^{2}}{n}}

Cluster Sampling Consider a population of N clusters. We take an SRS S of n clusters, and all units within each sampled cluster (onestage clustering). Assume clusters all have same size M. Let \overline{y}_{i} = \frac{1}{M} ∑ _{j=1}^{M} y_{ij} in each cluster. Then

\overline{y}_{cl} = \frac{1}{n} ∑_{i∈S} \overline{y}_{i} , Var(\overline{y}_{cl}) ≈ (1 − \frac{n}{N}) \frac{1}{n}s_{\overline{y}_{i}}^{2} = (1 − \frac{n}{N}) \frac{1}{n} [\frac{1}{n − 1} ∑_{i∈S} (\overline{y}_{i} − \overline{y}_{cl})^{2}]

and

DEFF = \frac{Var(\overline{y}_{cl})}{Var(\overline{y}_{srs})} = \frac{Ms_{\overline{y}_{i}}^{2}}{s_{y_{ij}}^{2}} ≈ 1 + (M − 1)ρ

where s_{\overline{y}_{i}}^{2} is the sample varance of the cluster means, s_{y_{ij}}^{2} is the sample variance of the individual observations, and ρ is the intraclass (intracluster) correlation, or ICC. Post-Stratification Weights and Means As part of survey data collection it is a good idea to get general demographic information (e.g. in our surveys: sex, age, class, major, hometown, etc.). After data collection we compare the proportions in each of these categories in our sample with the same proportions in the population. If they agree, great. If not, calculate

w_{i} = (N_{h}/N)/(n_{h}/n) for each i in post-stratum h , and \overline{y}_{w} = \frac{∑ _{i} w_{i}y_{i}}{∑ _{i} w_{i}}

Post-Stratification Variance Calculations \underline{Taylor series}:

Var_{TS}(\overline{y}_{w}) ≈ \frac{1}{(∑ _{i} w_{i})_{2}} [Var(∑_{i} w_{i}y_{i}) − 2\overline{y}_{w}Cov(∑_{i} w_{i}y_{i}, ∑_{i} w_{i}) + (\overline{y}_{w})^{2}Var(∑_{i} w_{i})]

where \overline{y}_{w} is as above, \overline{w} = \frac{1}{n} ∑ _{i} w_{i}, \overline{wy}= \frac{1}{n} ∑ _{i} w_{i}y_{i},

Var(∑ _{i=1}^{n} w_{i}) ≈ n · \frac{1}{n−1} ∑ _{i=1}^{n}(w_{i} − \overline{w})^{2}, Var(∑ _{i=1}^{n} y_{i}w_{i}) ≈ n · \frac{1}{n−1} ∑ _{i=1}^{n}(w_{i}y_{i} − \overline{wy})^{2},
Cov(∑ _{i=1}^{n} y_{i}w_{i}, ∑ _{i=1}^{n} w_{i}) ≈ n · \frac{1}{n−1} ∑ _{i=1}^{n}(w_{i}y_{i} − \overline{wy})(w_{i} − \overline{w})

\underline{Jackknife}:

• Replicate n times (by removing one obs. each time and recalculating weights):

\overline{y}_{w}^{(r)} = \frac{∑ _{i=1}^{n} w_{i}^{(r)}y_{i}^{(r)}}{∑ _{i=1}^{n} w_{i}^{(r)}}

• Calculate

\overline{y}_{JK} = \frac{1}{n} ∑_{r=1}^{n} \overline{y}_{w}^{(r)} , Var_{JK}(\overline{y}_{w}) ≈ \frac{n − 1}{n} ∑_{r=1}^{n} (\overline{y}_{w}^{(r)} − \overline{y}_{jk})^{2}