survey sampling l sampling & non-sampling error l bias l simple sampling methods l sampling...
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Survey sampling
Sampling & non-sampling error Bias Simple sampling methods Sampling terminology Cluster sampling Design effect Stratified sampling Sampling weights
Why sample?
To make an inference about a population
Studying entire pop is impractical or impossible
Example of sampling
Estimate the proportion of adults, ages 18-65, in Port Elizabeth that have type 2 diabetes
Select a sample from which to estimate the proportion
Population: adults aged 18-65 living in Port Elizabeth
Inference: proportion with type 2 diabetes
Probability sampling
Each individual has known (non-zero) probability of selection
Precision of estimates can be quantified
Non-probability sampling
Cheaper, more convenient Quality of estimates cannot be
assessed May not be representative of
population
Sampling errorv.
Non-sampling error
Sampling error
Random variability in sample estimates that arises out of the randomness of the sample selection process
Precision can be quantified (estimation of standard errors, confidence intervals)
Non-sampling error
Estimation error that arises from sources other than random variation– non-response– undercoverage of survey– poorly-trained interviewers– non-truthful answers– non-probability sampling
This type of error is a bias
What is bias?
We want to estimate the mean weight of all women aged 15-44 living in Coopersville. Suppose there are 50,000 such women and the true mean weight is 61.7 kg.
We select a sample of 200 such women and interview them, asking each woman what her weight is.
The sample mean weight is 59.4 kg.
Is our estimate biased?
Bias
Suppose we could repeat the survey many, many times.
Then we compute the mean of all the sample means.
Say the mean of the means = 62.9
Bias = (mean of means) - (true mean)
= 62.9 - 61.7 = 1.2 kg
Unbiased estimation
If . . .
(mean of the means) = (true mean)
then the bias is zero, and we say that the estimator is unbiased.
The “mean of the means” is called the “expected value” of the estimator.
Simple sampling methods
Task: Select a sample of n individuals or items from a population of N individuals or items
Common methods– simple random sampling– systematic sampling
Simple sampling methods
Simple random sampling (SRS)– each item in population is equally likely
to be selected– each combination of n items is equally
likely to be selected Systematic sampling (typical method)– randomly select a starting point– select every kth item thereafter
Systematic sampling example
Stack of 213 hospital admission forms; select a sample of 15
213/15 = 14.2 Select every 14th form Starting point: random number between 1 and 14
(we choose 11) First form selected is 11th from top Second form selected is 25th from top (11 + 14 = 25) Third form selected is 39th from top (11 + 2x14 = 39) And so forth . . .
Systematic sampling, continued
What is the probability that the 146th form will be selected? The 195th?
Does this qualify as a simple random sample? Why or why not?
Is there any potential problem arising from the use of systematic sampling in this situation?
Example was typical quick method
In the preceding example, we selected every 14th form
Ideally, we would select every 14.2th form (see later example on 2-stage sample of nurses)
Example is a quick and easy method, commonly used in the field; it is a good approximation to the more rigorous procedure
Systematic sampling: + and -
Advantages of systematic sampling– typically simpler to implement than SRS– can provide a more uniform coverage
Potential disadvantage of systematic sampling– can produce a bias if there is a
systematic pattern in the sequence of items from which the sample is selected
Role of simple sampling methods
These simple sampling methods are necessary components of more complex sampling methods:– cluster sampling– stratified sampling
We’ll discuss these more complex methods next (following some definitions)
Definitions
Listing units (or enumeration units)– the lowest level sampled units (e.g.,
households or individuals) PSUs (primary sampling units)– the first units sampled (e.g., states or
regions) Sampling probability– for any unit eligible to be sampled, the
probability that the unit is selected in the sample
More definitions
EPSEM sampling– “equal probability of selection method”,
thus a method in which each listing unit has the same sampling probability
Sampling frame– the set of items from which sampling is
done--often a list of items.
More definitions
Undercoverage: the degree to which we fail to identify all eligible units in the population– incomplete lists– incomplete or incorrect eligibility
information
Still more definitions
Non-response: failure to interview sampled listing units (study subjects)– refusal– death– physician refusal– inability to locate subject– unavailability
Still more definitions
Precision: the amount of random error in an estimate– often measured by the width or half-
width of the confidence interval– standard error is another measure of
precision– estimates with smaller standard error or
narrower CI are said to be more precise
CLUSTER SAMPLINGsingle stage
Clusters
Subsets of the listing units in the population
Set of clusters must be mutually exclusive and collectively exhaustive– counties– townships– regions– institutions
ExampleSingle-stage cluster sampling
There are 361 nurses working at the 31 hospitals and clinics in Region 4
We wish to interview a sample of these nurses– select a simple random sample of 5
hospitals/clinics– interview all nurses employed at the 5
selected institutions
Assessing the example
Hospitals/clinics are the PSUs Nurses are the listing units Sampling probability for each nurse
is 5/31 Thus, this is an EPSEM sample Sampling frame is the list of 31
hospitals and clinics
CLUSTER SAMPLINGtwo stage
Cluster sampling -- two stage
Select a sample of clusters, as in the single-stage method
From each selected cluster, select a subsample of listing units
Cluster sampling -- two stage
It is always nice to do EPSEM sampling because such samples are self-weighting– don’t need sampling weights in analysis
A common EPSEM method for two-stage sampling is PPS (probability proportional to size)
PPS sampling
The key to the method is that the sampling probabilities of clusters in the first stage are proportional to the “sizes” of the clusters– size = number of listing units in cluster
At stage 2, select the same number of listing units from each selected cluster
Nurse example revisitedTwo-stage sampling
We want to interview a sample of 36 nurses
We can afford to visit 9 different hospitals/clinics
Thus, we need to interview 36/9 = 4 nurses at each institution
Nurse example revisitedTwo-stage sampling
Stage 1: select a sample of 9 hospitals/clinics– Selection prob. proportional to “size”
Stage 2: select a sample of 4 nurses from each selected institution
At each stage, use one of the simple sampling methods
Nurse example revisitedTwo-stage sampling
PSUs are the hospitals/clinics Listing units are the nurses Sampling frames– Stage 1: List of 31 hospitals/clinics– Stage 2: Lists of nurses at each
selected hospital/clinic
Selecting 2-stage nurse sample
Sampling interval, I = 361/9 = 40.1 Starting point, random number between 1
and 40; we choose R = 14 First sampling number = R = 14 2nd sampling number = 14 + 1x40.1 = 54.1 3rd sampling number = 14 + 2x40.1 = 94.2 We have selected institutions 2, 5, 9, . . .
Two-stage nurse sampleInstitutionNumber
No. ofNurses
CumulativeNurses
SamplingNumber
1 12 122 7 19 143 9 284 18 465 11 57 54.16 7 647 10 748 14 889 8 96 94.2..
.
...
31 9 361Total 361
Applying the sampling numbers
For each sampling number, choose the first unit with cumulative “size” equal to or greater than the sampling number
Example: sampling number 54.1– first unit with cumulative size 54.1 is
unit 5 (cum. no. of nurses = 57)
–so we select unit 5 for the sample
Optional challenge
What is the selection probability for institution 1?
12/40.1 = 0.299
What is the selection probability for a nurse in institution 1?
(12/40.1) x (4/12) = 0.998 = 36/361
What is the selection probability for a nurse in institution 2?
(7/40.1) x (4/7) = 0.998 = 36/361
All nurses have the same selection probability.
Why do cluster sampling instead Of a simple sampling method?
Advantages– reduced logistical costs (e.g., travel)– list of all 361 nurses may not be available
(reduces listing labor) Disadvantages– estimates are less precise– analysis is more complicated (requires
special software)
Design effect
Relative increase in variance of an estimate due to the sampling design– “variance” = (standard error)2
Formula– s1 = standard error under simple
random sampling– s2 = standard error under complex
sampling design (e.g., cluster sampling)– design effect = (s2/s1)2
Design effect for cluster sampling
For cluster sampling designs, the design effect is always >1
This means that estimates from a survey done with cluster sampling are less precise than corresponding estimates obtained from a survey having the same sample size done with simple random sampling
Cluster sizes Recommended “take” per cluster is
20-40 for multi-purpose surveys Time and resource limitations will
often dictate the maximum number of clusters you can include in the study
Including more clusters improves the precision of your estimates more than a corresponding increase in sample size within the clusters already in the sample
STRATIFIEDSAMPLING
Strata
Subsets of the listing units in the population
Set of strata must be mutually exclusive and collectively exhaustive
Strata are often based on demographic variables– age– sex– race
Stratified sampling
Sample from each stratum Often, sampling probabilities vary
across strata
Stratified sampling
Advantages– guarantees coverage across strata– can over-sample some strata in order to obtain
precise within-stratum estimates– typically, design effect < 1
Disadvantages– with unequal sampling probabilities, sampling
weights must be included in analysis• more complicated • requires special software
Example: sampling breast cancer cases for the Women’s CARE Study
Stratification variables– geographic site– race (2 races)– five-year age group
Over-sampled younger women Over-sampled black women
Example: Sampling households for a reproductive health survey in 11 refugee camps in Pakistan
Selected simple random sample of households from within each of the 11 camps
All households were selected with the same probability
Refugee camp sampling
Camp PopulationSample
SizeCompletedInterviews
Lakhte Banda 12,943 64 61Kotki 1 7,262 36 29Kotki 2 5781 29 21Kata Kanra 8,437 42 38Mohd Khoja 12,791 63 45Doaba 13,584 67 25Darsamand 17,797 88 53Kahi 11,061 55 32Naryab 5,543 28 19Thal 1 11,087 55 44Thal 2 17,130 85 60Dallan 10,990 55 45Total 134,406 667 472
The sampling operation
Must be carefully controlled– don’t leave to discretion in the field– use a carefully defined procedure
Document what you did– for reference during analysis– to defend your study
Sampling frames
A list containing all listing units is great if you can get it– ok if it includes some ineligibles
Problems associated with geographic location-based sampling–map-based sampling– EPI sampling
Sampling weights
Inverse of the net sampling probability
Interpretation: the sampling weight for an sampled individual is the number of individuals his/her data “represent”
Example--sampling weights
There are 150 employees in a firm– stratum 1: 50 employees aged 18-29– stratum 2: 100 employees aged 30-69
We sample 10 from each stratum Sampling probabilities are– stratum 1: 10/50 = 0.20– stratum 2: 10/100 = 0.10
Example: sampling weights (continued)
Sampling weights– stratum 1: 1/0.20 = 5– stratum 2: 1/0.10 = 10
Interpretation:– Each sampled employee in stratum 1
represents 5 employees– Each sampled employee in stratum 2
represents 10 employees
What about non-response?
1 employee in the stratum 1 sample and 3 employees in the stratum 2 sample refuse to participate in the survey
Net sampling probabilities– stratum 1: 9/50 = 0.18– stratum 2: 7/100 = 0.07
Revised sampling weights
Sampling weights revised for non-response– stratum 1: 1/0.18 = 5.56– stratum 2: 1/0.07 = 14.29
This computation is often done by multiplying the original sampling weights by adjustment factors to account for non-response rates
Post-stratification weighting
Define strata, which may or may not have been used as strata in the sampling design
Compute sampling probabilities = proportion of each stratum that was actually sampled
Compute sampling weights from these sampling probabilities
Allows post-hoc treatment of unequal representation of population segments in the sample
Discussion topics
What is the population of interest? Infinite populations Selecting random numbers Selecting simple random samples– from finite populations– from infinite populations
Analysis software for complex surveys