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Fundamentals of Sampling Method. Week 4 Research Methods & Data Analysis. Tutorials. Thursday 30 th October 9-11 AG GL 20 (M. Mazzocchi) Tuesday 4 th 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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fundamentals of sampling method

Fundamentals of Sampling Method

Week 4

Research Methods & Data Analysis

Research Methods & Data Analysis

tutorials
Tutorials
  • Thursday 30th October9-11 AG GL 20 (M. Mazzocchi)
  • Tuesday 4th November11-1pm (H.Neeliah)
  • You may attend:
    • One (the most convenient for you)
    • Both (it may be very useful)
    • None (not really advised…)

Research Methods & Data Analysis

lecture outline
Lecture outline
  • Key notions of statistics
  • Simple random sampling
  • Sampling error
  • Sampling size
  • Other sampling methods

Research Methods & Data Analysis

distributions
Distributions
  • A set of values of a set of data together with their
    • Absolute frequencies
    • Relative frequencies (probabilities)

Research Methods & Data Analysis

relative and cumulate frequencies
Relative and cumulate frequencies

fi=ni/N

Research Methods & Data Analysis

distributions of random variables
Distributions of random variables
  • The distribution of possible values together with their probabilities (probability density function, p.d.f.)

Research Methods & Data Analysis

the normal gaussian distribution
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)

Research Methods & Data Analysis

the normal distribution
The normal distribution

p

95% of values

0,025

0,025

m-1.96s

m

m+1.96s

Research Methods & Data Analysis

the student t distribution
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 tail-probabilities
  • The bigger is the sample, the more similar the t-distribution is to the normal distribution
  • For samples with more than 30-40 units, the difference between the two distributions is negligible

Research Methods & Data Analysis

the t distribution
The t-distribution

x-ta/2sx

x

x+ta/2sx

Research Methods & Data Analysis

t a 2 and z a 2 tabled values
ta/2 and za/2 – tabled values

Research Methods & Data Analysis

population parameters in a population of n elements
Population parameters(in a population of N elements)
  • Mean
  • Variance
  • Standard deviation

Research Methods & Data Analysis

sampling
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

Research Methods & Data Analysis

simple random sampling
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

Research Methods & Data Analysis

sample statistics
Sample statistics
  • Sample mean
  • Sample variance
  • Sample standard deviation

unbiasedness

Research Methods & Data Analysis

standard deviation and standard error
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)

Research Methods & Data Analysis

basic srs sample statistics unknown pop variance
Basic SRS sample statistics (unknown pop. variance)

Mean case

Proportion case (p)

Sample standard deviation of X

Standard error of the mean/proportion

ACCURACY of sample estimates

Research Methods & Data Analysis

finite population correction factor
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

Research Methods & Data Analysis

level of confidence a and z parameter
Level of confidence aand 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/2values is tabulated

a=0.05

za/2 =1.96

a/2

a/2

x

Confidence interval for x at a level of confidence a

Research Methods & Data Analysis

the t distribution1
The t-distribution

x-ta/2sx

x

x+ta/2sx

Research Methods & Data Analysis

confidence intervals
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

Research Methods & Data Analysis

exercise
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

Research Methods & Data Analysis

determining sample size
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

Research Methods & Data Analysis

simple random sampling determining sample size
Simple random sampling: determining sample size
  • Relative sampling error (r.s.e)
  • Determining sampling size for a given r.s.e. (approximate formula)

Research Methods & Data Analysis

the sampling design process
The sampling design process
  • Define the target population, its elements and the sampling units
  • Determine the sampling frame (list)
  • Select a sampling technique
    • Sampling with/without replacement
    • Probability/Nonprobability sampling
  • Determine the sample size
    • Precision versus costs
    • The marginal value in terms of precision of additional sampling units is decreasing
  • Execute the sampling process

Research Methods & Data Analysis

the sampling techniques
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

Research Methods & Data Analysis

representativeness
Representativeness
  • A sample can be considered as “representative” when it is expected to exhibit the average properties of the population

Research Methods & Data Analysis

selection bias
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.

Research Methods & Data Analysis

simple random sampling1
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
  • 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

Research Methods & Data Analysis

systematic sampling
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
  • Less representative (biased) if the order is cyclical

Research Methods & Data Analysis

stratified sampling
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
  • Include all relevant subpopolation even if small
  • Stratification variables may not be easily identifiable
  • Stratification can be expensive

Research Methods & Data Analysis

cluster sampling
Cluster sampling
  • 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 (one-stage)
    • 2b. A probabilistic sample is extracted from the cluster (two-stage cluster sampling)
  • Reduced costs
  • Higher feasibility
  • Less precision
  • Inference can be difficult

Research Methods & Data Analysis

non probabilistic samples

Non probabilistic samples

Research Methods & Data Analysis

convenience sampling
Convenience sampling
  • Only “convenient” elements enter the sample
  • Cheapest method
  • Quickest method
  • Selection bias
  • Non representativeness
  • Inference is not possible

Research Methods & Data Analysis

judgmental sampling
Judgmental sampling
  • Selection based on the judgment of the researcher
  • Low cost
  • Quick
  • Non representativeness
  • Inference is not possible
  • Subjective

Research Methods & Data Analysis

quota sampling
Quota sampling
  • Define control categories (quotas) for the population elements, such as sex, age…
  • Apply a “restricted judgmental sampling”, so that quotas in the sample are the same of those in the population
  • Cheapest method
  • Quickest method
  • There is no guarantee that the sample is representative (relevance of control characteristic chosen)
  • Many sources of selection bias
  • No assessment of sampling error

Research Methods & Data Analysis

snowball sampling
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
  • Inference is not possible

Research Methods & Data Analysis