Sampling Theory

Flashcards on Sampling Theory

Census

The oldest form of data collection, aiming for organization of tax payments or political representation, now supplemented with a wide variety of relevant information.

Survey

A well-targeted data collection method with a limited but precise scope, used in various fields like health, quality of life, unemployment, and agriculture.

Psychometric Concepts

Aspects such as reliability and validity that capture the quality of a survey.

Standardized Measurements

Measurement instruments that collect data in a standardized fashion, ensuring good psychometric properties like validity and reliability.

Probability Sampling

A family of probabilistic methods by which a subset of units from the sample frame is selected.

Survey Population

The collection of units (individuals) about which the researcher wants to make quantitative statements.

Sample Frame

The set of units (individuals) that has a non – zero probability of being selected.

Sample

The subset of units that have been selected.

Estimand

The true population quantity.

Estimator

A (stochastic) function of the sample data, with the aim to “come close” to the estimand.

Estimate

A particular realization of the estimator, for the particular sample taken.

List Quality

Considerations for a sample frame that requires knowing how a list has been composed and how updating takes place.

Simple Random Sampling

A sampling standard method, studied to compare other methods with.

Systematic Sampling

A sampling method chosen to increase precision and/or to ensure sampling with certainty for a subgroup of units.

Stratification

A sampling method performed to increase precision of population-level estimates or to allow for estimation at sub-population level.

Selection Probability

The probability of an individual to be selected, which should be known or estimable, but does not have to be constant.

Population Average

Y = (1/N) * Σ(I=1 to N) YI

Population Total

Y = Σ(I=1 to N) YI

Population Variance

σY^2 = (1/N) * Σ(I=1 to N) (YI - Y)^2 or SY^2 = (1/(N-1)) * Σ(I=1 to N) (YI - Y)^2

Population Covariance

σXY = (1/N) * Σ(I=1 to N) (XI - X)(YI - Y) or SXY = (1/(N-1)) * Σ(I=1 to N) (XI - X)(YI - Y)

Population Correlation

ρXY = σXY / (σX * σY) = SXY / (SX * SY)

Sampling Mechanisms

Assigning a probability PS (s = 1, …, S) to each sample.

Sample Fraction

f = n/N, relevant only in finite populations.

Expectation

The average of all possible estimates.

Bias

Y – E(^y)

Variance

The averaged squared deviation of a random variable around its mean σ ^y 2 = E( ^y−E ( ^y )) 2 = ∑s=1 S Ps(^ys−∑s=1 S Ps ^ys)2

Standard Error

In the specific case of an estimator, the standard deviation is termed the standard error.

MSE (Mean Squared Error)

MSE(^y) = σ ^y 2 + [bias(^y)]²

Sample Fraction f

f = n/N

Variance of the Mean

σy^2 = (1/n) * σY^2 (with replacement) or σy^2 = (1/n) * (1-f) * SY^2 (without replacement)

Variance of the Total

σ^y^2 = (N^2/n) * σY^2 (with replacement) or σ^y^2 = (N^2/n) * (1-f) * SY^2 (without replacement)

Population Proportion (P)

Denoted by P or . The proportion of units belonging to the subgroup, at the population level P = (1/N) * Σ(I=1 to N) ZI

Population Variance for the Proportion

σZ^2 = (N/(N-1)) * PQ ≃ PQ

Estimating the Size of a Subgroup

Estimated from a sample of size n by n^g = N × ^p, with variance σ^ng^2 = N^2 * (1/n) * (1-f) * (N/(N-1)) * PQ

Delta Method

A method used to assess the random error in a function of random variables, which can use covariances of other variables.

Relative Variance

RVar(X) = Var(X) / X^2 and RCov(X, Y) = Cov(X, Y) / (X * Y)

SRS

Simple Random Sampling

Auxiliary variable

Variables used for correcting mechanisms such as stratification.

Stratification

Partitioning the population in subgroups according to the levels of an auxiliary variable, so that the survey variable is more homogeneous within such a subgroup, or stratum, than in the population as a whole.

Post-stratification

Stratified analysis of a sample that was taken in an un-stratified way.

Multi-stage sampling

Hierarchy of units that is selected. Starting with a primary sampling units (PSU), then with the secondary sampling units (SSU) are subselected, then within which are tertiary sampling units (TSU) are subselected etc.

Weights

Arises naturally in a variety of context such as stratification, clustering and probabilities of section. Represented as ∑wi yi/ ∑wi and N ∑wi yi/ ∑wi

Design Effect

The ratio of two variances: The variance of an estimator taking design aspects into account and The variance of the SRS estimator

PROC SURVEYSELECT

A software that Allows SRS, URS, SYS, SEQ and PPS which can be combined with STRATIFICATION.

Missing Completely at Random (MCAR)

A missing framework represented by f(Ri |ψ).

Missing at Random (MAR)

A missing framework represented by f(Ri |Yi 0 , ψ)

Multiple Imputation

A process involving fillign in missing values M times, analyzing the M complete data sets by using standard procedures and combining the results from the M analyses into a single inference