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# A Bit More on ? Survey Sample Size Design

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A Bit More on ? Survey Sample Size Design
Professor Ron Fricker Naval Postgraduate School Monterey, California
1/28/13

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Some Basic Considerations
?? Previously gave sample size expressions for means, totals, and proportions ?? Survey usually a combination of Likert and binary (yes/no) questions
ЁC? If so, use proportions calculation to estimate required sample size ЁC? Most conservative (i.e., gives the largest sample size)
?? If sample size expression involves standard deviations, also estimate conservatively
ЁC? I.e., use larger estimates vice smaller
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For Clustered Designs
?? Almost always using clustering because of or under budget restrictions
ЁC? Implicit sampling frame Ўъ area sampling Ўъface- to-face survey mode Ўъ expensive
?? Basic approach: Within budget constraints, maximize the number of clusters (PSU, SSU, etc.)
ЁC? Spread the sample size across as many clusters as possible ЁC? That is, youЎҜre trying to minimize the correlation as much as possible
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For Stratified Designs
?? Two broad reasons to use stratification:
ЁC? Requirement that results for subpopulations have a particular margin of error ?? Generally related to oversampling proportionally smaller or rare groups ЁC? Homogeneity on the measures of interest among subpopulations ?? Stratification will then result in more precise population estimates
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Sample Size Determination & Allocation
?? Compared to SRS, calculations more complicated as need to
ЁC? Choose overall sample size n, and ЁC? Allocate sample to strata, n
1
+ n
2
+Ўӯ + n
L
= n
?? Ў°BestЎұ allocation depends on
ЁC? Purpose of the stratification ЁC? Number of sampling units in each stratum ЁC? Variability of sampling units within each stratum ЁC? Cost of surveying each sampling unit from each stratum
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Sample Size Calculation for ? Estimating the Mean
?? Begin as with SRS, setting ?? Now, define a
i
as the proportion of the sample size to allocate to strata i, n
i
= n x a
i
?? Then,
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B = 2 Var ? y
st
( )
n = N
i 2ҰТ i 2 a i i=1 L
ЎЖ
N2B2 4+ N
i 2ҰТ i 2 i=1 L
ЎЖ

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Proportionate Allocation to Strata
?? Sample size within each strata is proportional to strata size in population ?? If N is population size and n is total sample size, then where
ЁC? N
i
is the population size of stratum i ЁC? n
i
is the sample size for stratum i
?? Then
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/ /
i i
n n N N =
( ) (
)
2 2 2 2 2 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 1
4 4 4
L L L i i i i i i i i i i i L L L i i i i i i i i i
N a N N N N n N B N N B N NB N
ҰТ ҰТ ҰТ ҰТ ҰТ ҰТ
= = = = = =
= = = + + +
ЎЖ ЎЖ ЎЖ ЎЖ ЎЖ ЎЖ

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Other Allocation Schemes
?? Rather than allocating sample proportional to strata size, can
ЁC? Allocate according to variability of the strata ?? Idea is to allocate more of the sample to strata that are more variable ?? Done right, can provide most precise population estimates ЁC? Allocate according to the cost of collection per strata ?? Idea is to allocate more of the sample to strata that cost less
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Design Effect
?? The design effect (aka deff) compares how a complex sampling design, in this case stratified sampling, compares to SRS ?? Design effect can be greater or less than 1
ЁC? But with reasonably homogeneous strata, almost always get decrease in variance ЁC? I.e. a deff < 1
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d2 = Var Y
st
( )
Var Y
SRS
( )
= 1 N2 N
i 2 1?n i
/ N
i
( )s
i 2 / n i i=1 L
ЎЖ
1?n
i
/ N
i
( )s2 / n

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Effective Sample Size
?? I like to think about design effects in terms of
effective sample size
ЁC? What size SRS would give the same precision as the complex sample design?
?? Consider a simple clustered example, with m clusters, and d2 = 3.13
ЁC? The effective sample size is ЁC? So we could have done a SRS of a sample of 64 and achieved the same precision ЁC? Would have meant going to 64/m times as many sites ЁC perhaps unaffordable
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200/3.13 64
eff
n = = 200
i
n =
ЎЖ 