1. Random sampling Carlo Azzarri IFPRI Datathon APSU, Dhaka
9 November 2017
Random sampling
Carlo Azzarri
IFPRI

2. Random Sampling
Random Sampling (a.k.a. Scientific Sampling) is a selection procedure that gives each element of the population a non-zero, known positive probability of being included in the sample
Mathematical theory is available to predict the probability distribution of the Sampling Error (the error caused by observing a sample instead of the whole population) and Confidence Intervals
Other sampling procedures (purposive sampling, quota sampling, etc.) cannot do that

3. Random Sampling techniques
Single stage, equal probability sampling
Simple Random Sampling (SRS)
Systematic sampling with equal probability
Multi-stages sampling
Stratified sampling
In real life those techniques are usually combined in various ways – most sampling designs are complex
Technically different but in practice virtually the same (SRS & Systematic)
Exercises are all SRS.

4. Single stage, equal probability sampling/1
Techniques in Random Sampling
Single stage, equal probability sampling/1
Random selection of n “units” from a population of N units, so that each unit has an equal probability of selection
N (population ) → n (sample)
Probability of selection (sampling fraction) = f = n/N
It is the most basic form of probability sampling, the theoretical basis for more complicated techniques

5. Single stage, equal probability sampling/2
Techniques in Random Sampling
Single stage, equal probability sampling/2
Advantage
self-weighting (simplifies the calculation of estimates and variances)
Disadvantages
A list of all households in the country is generally not available to select the sample from (in other words, we don’t have a good sample frame)
Difficult management
May entail high transportation costs

6. Standard Deviation vs Standard Error
Population
= variance of the population
= standard deviation around the
mean
Sample
= variance of the sample
= standard error
Difference: The standard deviation is a descriptive statistic. It expresses the degree to which individuals in the population differ from the mean of the population. The standard error is an estimate of how close to the population mean your sample mean is likely to be.
Standard errors decrease with sample size. Standard deviations are left unchanged.

7. Bigger samples have smaller standard errors around the mean
Sample Standard Error
n = n = 750
Bigger samples have smaller standard errors around the mean

8. Confidence intervals
Estimates obtained from random samples can be accompanied by measures of the uncertainty associated with the estimate called confidence intervals.
It is not sufficient to simply report the sample proportion obtained by a candidate in the sample survey, we also need to give an indication of how accurate the estimate is.
Confidence interval for sample estimate
Interval of values (lower and upper limits) for an estimate which contain the true population value with a stated level of confidence (certainty)
Convenient way for the data user to interpret the sampling variability in the survey results

9. Confidence intervals for proportions
In a sample of 1,000 electors, 280 of them (28 percent) say they will vote Green.
Standard error is 1.42 percent.

10. Confidence intervals for averages
where:
tα = 1.28 for confidence level α = 80%
tα = 1.64 for confidence level α = 90%
tα = 1.96 for confidence level α = 95%
tα = 2.58 for confidence level α = 99%
T=normal deviate

11. Confidence intervals
In a sample of 1,000 households, 280 households (28 percent) say they will vote Green. Standard error is 1.42 percent.
Standard error
99 percent confidence interval: 28 ± 1.42 • 2.58
95 percent confidence interval:
28 ± 1.42 • 1.96

12. Sample size and population size
size needed for a given precision
Population size

13. Sample size and population size
Standard error e when estimating a prevalence P in a sample of size n taken from a population of size N

14. Sample size (cont’d)
As sample size increases, the distribution of the sample estimates becomes more concentrated around the expected value (central limit theorem)
Level of precision increases
Confidence intervals become more narrow
Standard error inversely proportional to square root of sample size
Quadruple sample size to double precision
When the population is very large or the sampling fraction is small (less than 5%), the size of the population does not affect the sample size
In the case of simple random sampling without replacement, a population correction factor (1-f) is applied to the variance when the sampling fraction (f) is higher than 5%

15. Sampling error and sample size
Standard
error
To halve sampling error...
...sample size must be quadrupled
Sample size

16. Sampling vs. non-sampling errors
Sample size

17. Sampling vs. non-sampling errors
Sample size

18. Sampling vs. non-sampling errors
Total error
Non-sampling error
Sampling error
Sample size

19. Techniques in Random Sampling
Two-stage sampling
Units of analysis are divided into groups called Primary Sampling Units (PSUs)
A sample of PSUs is selected first
Then a sample of units is chosen in each of the selected PSUs
More common than SRS
Reduces transportation costs to plausible level
Used with stratification for complex sample
This technique can be generalized (multi-stage sampling)

20. Two-stage sampling
The country

21. Two-stage sampling
The country can be divided…

22. Two-stage sampling
…into small Primary Sampling Units (PSUs) (and strata)

23. Two-stage sampling
The country is divided into small Primary Sampling Units (PSUs)
In the first stage, PSUs are selected

24. Two-stage sampling
The country is divided into small Primary Sampling Units (PSUs) (and strata)
In the first stage, PSUs are selected
In the second stage, households are chosen within the selected PSUs

25. Techniques in Random Sampling
Two-stage sampling
In planning the survey, selected PSUs should be allocated
Among teams
During the survey period

26. Techniques in Random Sampling
Stratified sampling
The population is divided into mutually exclusive subgroups called strata.
Then a random sample is selected from each stratum.
Common examples : Urban / Rural, Divisions, Male / Female
Over-sampling certain populations (such as urban populations in mainly rural country) May want to say something about urban dwellers in a country experiencing migration but SRS would still give too few respondents.

27. Techniques in Random Sampling
Stratified sampling
Parts of the country may need to be excluded from the sample for security or other reasons (excluded strata)

28. Simple linear regression model: population and sample!
Population regression function (PRF) linear in x. Graph Sample regression function (PRF) linear in x.
one-unit increase in x changes the expected value of y by the amount β1