Sampling: The Basics

5.1 Sampling: The Basics

This Chapter can be difficult to follow, but within this Chapter are some key concepts. Don’t get lost in the details or formulas. Focus on the big picture. There are many books on sampling and this Chapter will not make you an expert within the field of survey sampling and data collection, but it will get you started.

Focus on the following key concepts that will be covered:

Garbage In Garbage Out (G.I.G.O.)
Sampling Error
Non-Sampling Error
- Selection Bias
- Measurement Error
Unbiased
Central Limit Theorem

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Often a subset of data is collected from a larger group in order to learn about the larger group. This is the basis of sampling. We wish to take a subset of data, sample, and use this sample to learn about what is called a target population. To learn about the target population, certain population quantities of interest are estimated from the sample.

Terminology and Notation

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Target population
- Collection of objects that we want to study and learn about.
- Example might be all residents in Bangkok.
- Population size is denoted by upper case N.
- The set of unit labels representing the population units is denoted $u = (1,2, ...,N )$
- The vector of the associated y-values of the population variable of interest is denoted $y = (y1,y2,...,yN)′$
- Often within the sampling context the y-values are considered fixed but unknown constants, not random variables. The ”random” part is introduced within the sampling.
- In some context the y-values, y = (y₁,y₂,…,y_N)′ are considered one realization/observation from Y = (Y ₁,Y ₂,…,Y _N)′ random variables.
Sample
- A set of units from a larger or equal size collection of units.
- Example: 100 residents in Bangkok selected by any manner from all residents in Bangkok.
- Example: 100 residents in Bangkok selected by any manner from 1000 residents in Bangkok.
- Sample size is denoted by lower case n.
- The set of units selected, units in the sample is denoted $s = (i1,...,in).$
- The vector of observed y-values associated with the sample, s, is denoted $′ ys = (yi1,...,yin).$
- For simplicity often unit labels are reordered from 1 to n and the vector of observed y-values associated with the sample, s, is denoted $ys = (y1,...,yn)′.$
  This notation is often used without explanation.
- The data in survey sampling is comprised of both s and y_s, denoted as d, where d = (s,y_s)
Sampling frame
- A sampling frame is the set of unit labels from which the units to be sampled can come from. Often the sample frame does not include all units in the target population due to various possible reasons.
Sampling design
- The way in which a sample is taken. In other words, the sampling methodology used to collect the data.
Sampling Error
- The difference between the estimate from a sample versus the value one would obtain from looking at all units within the sampling frame.
Non-Sampling Error
- Selection Bias exists if units within the target population have zero probability of being observed. For example, a mobile phone survey to estimate the average income of people in Bangkok. This type of survey would omit all people without mobile phones.
- Measurement Bias exists when the instrument measuring the item of interest is consistently different from the truth in a certain direction. In other words measurement bias is a result of the instrument measuring the item of interest yielding measurement with error, i.e. measurement error. A simple but illustrative example is a scale that consistently reports the weight of individuals 1 kilogram higher than the truth.

Typical Population Quantities of Interest

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Population Mean :

N -1-∑ μ = N yi i=1

Population Total :

N ∑ τ = yi = N ⋅ μ i=1

Finite Population Variance :

∑N σ2 = --1--- (y - μ )2 N - 1 i i=1

Population Standard Deviation :

√--2 σ = σ

Population Proportion :

∑N π = -1- zi, N i=1

where

{ 1, if units i satisfies the specified condition of interest. zi = 0,Otherwise.

There are a various ways to take a sample and collect data. Some ways are better than others. Often a sample is taken by those with little knowledge on sampling leading to various data analysis issues. Garbage in garbage out (G.I.G.O.) is very important to think about before sampling. If a sample is taken properly, often simple statistics can yield great insight. On the other hand, if the sample is not taken properly, simple statistics and advanced statistical techniques may not yield insight, and can actually yield misleading information. Unfortunately it is easy to understand the cost of sampling but not the importance of sampling. Sampling can be very costly and as a result sampling is often driven by financial concerns, as mentioned the cost of sampling is the easiest to understand.

Many samples are collected by convenience sampling. A convenience sample is collected by convenience. Convenience samples are not very costly, but the information obtained can be very unreliable. An extreme example of a convenience sample is collecting data on ones friends to learn about the target population, say people living in Bangkok. A more realistic example of a convenience sample, is collecting data from people at various malls in Bangkok on weekends to learn about people in Bangkok. Many statistical techniques you have learned and will learn do not apply to a convenience sample. It is easy to understand the number of observations one can obtain through convenience sampling versus more complicated sampling techniques. As with many things, in sampling quality is more important than quantity, again think G.I.G.O. Better to collect fewer observations through proper sampling techniques than convenience sampling.

Most statistical techniques do apply to samples taken by simple random sampling with replacement (SRSWR) and simple random sampling without replacement (SRSWOR) from a sufficiently large population. In SRSWR it is possible to observe the same unit (e.g. person) more than once in the sample. In SRSWOR, once a unit is selected to be in the sample it is removed from the list of units for subsequent selections. Thus in SRSWOR, it is not possible to sample the same unit more than once. SRSWR and SRSWOR are two sampling designs that involve probability sampling. A probability sample is a sample in which the units in the target population have a specified probability of being selected. In addition the probability of the sample, s, being observed is independent of the y-values in the population, y, that is, P(s|y) = P(s). For a SRSWR and SRSWOR the units are sampled with equal probability, they are types of equal probability sampling. Unequal probability sampling is when the units are selected with different probabilities. There are various probability sampling designs:

Simple random sampling with replacement
Simple random sampling without replacement
Stratified sampling
Cluster sampling
Systematic sampling
etc.

This chapter will not go into depth of the various sampling designs mentioned above, but will focus more on the fundamental concepts within sampling. The major benefit of taking a probability sample is that it is most reliable for extrapolating results to the target population of interest. In addition, of lessor benefit, often there exists an unbiased estimator, , for the population quantity of interest, θ. Most commonly, the population quantity of interest, θ, is the population mean, μ or the population total, τ. An unbiased estimator is an estimator that has an expected value equal to the parameter of interest. A (sampling) design unbiased estimator is such that
$∑ E [ˆθ] = P (s)ˆθ = θ, s∈S$ (5.1)

where denotes the collection of all possible samples. Note: In this Chapter and when referring to sampling, expectation will be considered taken over all possible samples. The equation ?? can be viewed in a similar manner to equation ??, E[X] = ∑ p_ix_i = μ_x, where is analogous to x_i and P(s), the probability of observing a specific sample which yields , is analogous to p_i, the probability of observing x_i. Equation ?? is the expectation of a random variable, X, and equation ?? is the expectation of an estimator, .

For a biased estimator, the expectation of does not equal θ, i.e. E[ ]≠θ. The bias of is defined as the difference between the expectation of ˆ
θ and the population quantity of interest θ,

Bias(ˆθ) = E (ˆθ) - θ.

One factor in determining which sampling design and which estimator to use to estimate θ, is the mean square error, MSE, of the sampling design and the estimator. The MSE of equals the expected squared error,

MSE( )	= E(- θ)²
	= E²
	= E
	= E² + E² + 0
	= Var( ) + ²

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The MSE for an unbiased estimator equals the variance since the bias of the estimator equals zero.

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