Transcript Fundamentals of Sampling Methods

Fundamentals of
Sampling Method
Week 4
Research Methods & Data
Analysis
Dr. Mario Mazzocchi
Research Methods & Data Analysis
1
Tutorials
• Thursday 30th October
9-11 AG GL 20 (M. Mazzocchi)
• Tuesday 4th November
11-1pm (H.Neeliah)
• You may attend:
– One (the most convenient for you)
– Both (it may be very useful)
– None (not really advised…)
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Lecture outline
•
•
•
•
•
Key notions of statistics
Simple random sampling
Sampling error
Sampling size
Other sampling methods
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Distributions
• A set of values of a set
of data together with
their
Count
– Absolute frequencies
– Relative frequencies
(probabilities)
60
40
20
0
200.00
300.00
400.00
500.00
600.00
700.00
Amount spent
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Relative and cumulate
frequencies
fi=ni/N
Fi  f1  f 2 
i
 fi   f h
h 1
100%
8%
75%
Perce nt
Percent
6%
4%
50%
25%
2%
0%
200.00
300.00
400.00
500.00
600.00
700.00
200.00
300.00
Amount spe nt
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400.00
500.00
600.00
700.00
Amount s pent
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Distributions of random
variables
• The distribution of possible values
together with their probabilities
(probability density function, p.d.f.)
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The normal (Gaussian)
distribution
• …is the distribution representing perfect
randomness around a mean value
• In statistics, the normal distribution play a
key role in the theory of errors
• The central limit theorem implies that
“averaging” almost always give origin to a
normal distribution (error on the average is
random), provided that the number of
observation is large (>40)
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The normal distribution
p
95% of
values
0,025
0,025
m-1.96s
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m
m+1.96s
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The student-t distribution
• When the parameter in the population has a normal
distribution (with unknown variance), within the
sample the parameter assumes a t distribution
• The t-distribution is similar to the normal
distribution, apart from having higher tailprobabilities
• The bigger is the sample, the more similar the tdistribution is to the normal distribution
• For samples with more than 30-40 units, the
difference between the two distributions is negligible
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The t-distribution
x-ta/2sx
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x
x+ta/2sx
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ta/2 and za/2 – tabled values
Level of confidence
99%
95%
90%
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t according to sample size
a
a /2
10
20 30
0.01 0.005 3.17 2.85 2.75
0.05 0.025 2.23 2.09 2.04
0.10 0.050 1.81 1.72 1.70
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z
40
2.70
2.02
1.68
2.58
1.96
1.64
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Population parameters
(in a population of N elements)
1 N
• Mean
m   xi
N i 1
N
1
• Variance 2
s   ( xi  m )2
N i 1
• Standard deviation
N
1
2
s  s2 
(
x

m
)

i
N i 1
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Sampling
• A sample is a subgroup of the
population selected for the study
• Sample statistics allow to make
inference about the population
parameters, through estimation and
hypothesis testing
• The sample space is a complete set of
all possible results of the sampling
procedure
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Simple random sampling
• Each element of the population has a known and
equal probability of selection
• Every element is selected independently from other
elements
• The probability of selecting a given sample of n
elements is computable (known)
• The Central Limit Theorem guarantees that for
simple random samples with sample size (n)
sufficiently large (>40), the sample mean in a S.R.S.
follows the normal distribution
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Sample statistics
• Sample mean
1 n
x   xi
n i 1
• Sample variance
n
1
2
2
s 
( xi  x )

n  1 i 1 unbiasedness
• Sample standard deviation
n
1
2
2
s s 
( xi  x )

n  1 i 1
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Standard deviation and
standard error
• The standard deviation measures
the variability of a given variable (e.g.
X) within the population or sample
• The standard error refers to the
accuracy (variability) of the sample
statistics (e.g. mean), i.e. the error due
to the fact that the statistic is computed
on a sample rather than on the
population (sampling error)
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Basic SRS sample statistics
(unknown pop. variance)
n
Mean case
x
x
i
i 1
Proportion case (p)
n
n
s
2
(
x

x
)
 i
i 1
n 1
sx 
s2
n
Sample
standard
deviation of X
Standard error of the
mean/proportion
s
n
p(1  p)
n 1
sp 
p(1  p)
n 1
ACCURACY of
sample estimates
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Finite population
correction factor
• For finite population (…i.e. all in social
research), large samples (more than 10% of
N) tend to overestimate the standard error of
the sample mean (proportion)
• In order to account for that, the following
correction is necessary
sx 
n
s2
1
N
n
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sp 
p(1  p)
n
1
n 1
N
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Level of confidence a
and z parameter
The level of confidence a refers to the
probability that the true population mean falls
in the identified confidence interval
For the normal distribution, given a
value of a, the corresponding za/2
values is tabulated
a/2
x  za / 2 sx
a/2
x
x  za / 2 sx
a=0.05
za/2 =1.96
Confidence interval for x at a level of confidence a
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The t-distribution
x-ta/2sx
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x
x+ta/2sx
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Confidence intervals
• Calculate the sample mean
• Decide a level of confidence (usually
95% or 99%)
• Choose whether using the Student-t
distribution or the Normal distribution
• Compute the sample standard error
• Define the lower and upper bound of
the confidence interval
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Exercise
• Suppose that you have interviewed 20
students out of 200 in the agricultural
building, asking them how much they
paid for lunch yesterday
• You get an average of £ 3.67
• The standard deviation is 1.25
• Compute the 95% confidence interval
• Compute the 99% confidence interval
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Determining sample size
Factors influencing sample size (n):
• Size of the population (N)
• Variability of the population (s)
• Desired level of accuracy (q)
• Level of confidence (a)
• Budget constraint
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Simple random sampling:
determining sample size
• Relative sampling error (r.s.e)
r
ta / 2 sx
nX
n
1
N
• Determining sampling size for a given
r.s.e. (approximate formula)
 ta / 2 sx 
n0  

 rX 
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The sampling design
process
1. Define the target population, its elements and
the sampling units
2. Determine the sampling frame (list)
3. Select a sampling technique
• Sampling with/without replacement
• Probability/Nonprobability sampling
4. Determine the sample size
• Precision versus costs
• The marginal value in terms of precision of
additional sampling units is decreasing
5. Execute the sampling process
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The sampling techniques
• Probabilistic samples
– Simple random sampling
– Systematic sampling
– Stratified sampling
– Cluster sampling
– Other sampling techniques
• Nonprobabilistic samples
– Convenience sampling
– Judgmental sampling
– Quota sampling
– Snowball sampling
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Representativeness
• A sample can be considered as
“representative” when it is expected to
exhibit the average properties of the
population
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Selection bias
• Improper selection of sample units (ignoring
a relevant “control variable” that generate
bias), so that the values observed in the
sample are biased and the sample is not
representative.
Example:
A survey is conducted for measuring goat milk
consumption, but the interviewers just select
people in urban areas, that on average drink
less goat milk.
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Simple random sampling
• Each element of the population has a known and
equal probability of selection
• Every element is selected independently from other
elements
• The probability of selecting a given sample of n
elements is computable (known)
–Statistical inference is possible
–It is easily understood
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–Representative samples are large
and expensive
–Standard errors are larger than in
other probabilistic sampling
techniques
–Sometimes it is difficult to execute
a really random sampling
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Systematic sampling
• A list of N elements in the population is compiled,
ordered according to a specified variable
– Unrelated to the target variable (similar to SRS)
– Related to the target variable (increased
representativeness)
• A sampling size n is chosen
• A systematic step of k=N/n is set
• A random number s between 1 and N is extracted and
represents the first element to be included
• Then the other elements selected are s+k, s+2k, s+3k…
–Cheaper and easier than SRS
–More representative if order is related
to the interest variable (monotone)
–Sampling frame not always necessary
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–Less representative (biased) if the
order is cyclical
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Stratified sampling
• Population is partitioned in strata through control
variables (stratification variables), closely related
with the target variable, so that there is homogeneity
within each stratum and heterogeneity between strata
• A simple random sampling frame is applied in each
strata of the population
– Proportionate sampling: size of the sample from each stratum is
proportional to the relative size of the stratum in the total
population
– Disproportionate sampling: size is also proportional to the
standard deviation of the target variable in each stratum
–Gains in precision
–Stratification variables may not be
easily identifiable
–Include all relevant subpopolation even
if small
–Stratification can be expensive
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Cluster sampling
•
•
1.
The population is partitioned into clusters
Elements within the cluster should be as
heterogeneous as possible with respect to the
variable of interests (e.g. area sampling)
A random sample of clusters is extracted through
SRS (with probability proportional to the cluster
size)
–
–
2a. All the elements of the cluster are selected (onestage)
2b. A probabilistic sample is extracted from the cluster
(two-stage cluster sampling)
–Reduced costs
–Higher feasibility
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–Less precision
–Inference can be difficult
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Non probabilistic samples
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Convenience sampling
• Only “convenient” elements enter the
sample
–Cheapest method
–Quickest method
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–Selection bias
–Non representativeness
–Inference is not possible
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Judgmental sampling
• Selection based on the judgment of the
researcher
–Low cost
–Quick
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–Non representativeness
–Inference is not possible
–Subjective
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Quota sampling
1.
Define control categories (quotas) for the
population elements, such as sex, age…
2. Apply a “restricted judgmental sampling”,
so that quotas in the sample are the same of
those in the population
–Cheapest method
–Quickest method
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–There is no guarantee that the
sample is representative (relevance
of control characteristic chosen)
–Many sources of selection bias
–No assessment of sampling error
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Snowball sampling
• A first small sample is selected randomly
• Respondents are asked to identify others who
belong to the population of interests
• The referrals will have demographic and
psychographic characteristics similar to the
referrers
–Lower costs
–Low variability
–Useful for “rare” populations
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–Inference is not possible
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