sampling and its variability
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TRANSCRIPT
Presentation by: Dr. Bhushan Kamble
Moderator: Dr. Poornima Tiwari
Professor,
Department of Community Medicine,
VMMC & SJH
Sampling and sampling variability
Outline of presentation• Definitions• Need for sampling• Types of sampling design
Probability sampling Non probability sampling
• Factors affecting choice of sampling design• Sample size
Factors affecting sample size Calculation of sample size for• Descriptive studies• Comparison studies
• Sampling variability• Sampling errors• References
Definitions
Population: The target group to which the findings (of a study) would ultimately apply is called population1
OrPopulation is the term statisticians use to describe a large set or
collection of items that have something in common2.
Sample: is that part of the target population which is actually enquired upon or investigated1.
OrSample is a subset of population, selected in such a way that it
is representative of the larger population2
1. Indrayan A., Satyanarayana L., Medical Biostatistics, third edition, 20092. Last JM. Dictionary of Epidemiology, 3rd edition, 2000.
Definitions (cont.)
Sampling: is the process of selecting a small number of elements from a larger defined target group of elements such that the information gathered from the small group will allow judgments to be made about the larger groups.
conclusions based on the sample results may be attributed only to the population sampled*. .
*Dawson B., Trapp RG, Basic and Clinical Biostatistics, second edition, 1994
Definitions cont..
Sampling unit: is the unit of selection
Unit of study or element: is the subject on which information is obtained.
Sampling frame: list of all sampling units in the target population is called a sampling frame.
Sample size: the number of units or subjects sampled for inclusion in the study is called sample size.
Sampling technique: Method of selecting sampling units from sampling frame
Population Vs. SamplePopulation Vs. SampleSample Population Sample
Parameter StatisticWe measure the sample using statistics in order to draw inferences about the population and its parameters.
Population of Interest
Target population
Sampling frame
Sample
Population you want to generalize results to
Population you have access to for your study
Study population
How can you get access to study population?
Study actually done on?
1.2.3…..
Need for sampling
1. Complete enumeration may not be possible.
2. Resources: Lower cost, Lesser demand on personnel.
3. Speed: Faster results due to lesser coverage.
4. Reliable information: Due to small size - better trained personnel, more accurate methods, better supervision.
To draw conclusions about population from sample, there are two major requirements for a sample.
Firstly, the sample size should be large. Secondly, the sample has to be selected appropriately so that
it is representative of the population. Sample should have all the characteristics of the population.
Disadvantages of sampling
1. Sampling entails an argument from the fraction to the whole. Validity depends on representativeness of the sample.
2. Fails to provide precise information in case of small segments containing few individuals.
3. Not necessary in studies where complete enumeration is needed.
4. May cause a feeling of discrimination among the subjects who are not included in the study.
Types of sampling
Probability sampling Non probability samplingProbability of selection of
each individual is known and pre determinedSimple random samplingSystematic random
samplingStratified random
samplingCluster random samplingMultistage random
sampling
Probability of selection of each individual is not knownQuota samplingPurposive/ Judgmental
samplingSnowball/ Network
samplingConvenience/ Grab
sampling (man in the street)
Simple random sampling
Equal probability of selection of units for inclusion in the studyRequires a list of all sampling units (sampling frame)Each individual is chosen randomly.Methods:
Lottery method (possible for finite population)Random number tablesSoftware that generate random numbers
Lottery method
Lottery method
Random number table
76 58 30 83 64
47 56 91 29 34
10 80 21 38 84
00 95 01 31 76
07 28 37 07 61
Simple random sampling (contd.)
Simple random methodWith replacementWithout replacement
AdvantageVery scientific methodEqual chance of all subjects for selection
Disadvantage Requires sampling frame
Example:Blood sampling – TLC, Hb estimation
Stratified random sampling
Preferred method when the population is heterogeneous with respect to characteristic under study.
Population is divided into groups or strata on the basis of certain characteristics.
A simple random sample is selected from each strata.Ensures representation of different strata/ groups in the study
population.Can be done by selecting individuals from different strata in
certain fixed predetermined proportions.Proportional stratified samplingDis-proportionate stratified sampling
Stratified random sampling(contd.)
For example, if we draw a simple random sample from a population, a sample of 100 may contain 10 to 15 from high socioeconomic group20 to 25 from middle socioeconomic
group70 to 75 from low socioeconomic group
To get adequately large representation for all the three socio economic structures, we can stratify on socioeconomic class and select simple random samples from each of the three strata.
POPULATION
LOW SOCIOECONOMIC
MIDDLE SOCIOECONOMIC
HIGH SOCIOECONOMIC
Stratified random sampling(contd.)Advantage:
All groups, however small are equally represented.When we want to highlight a specific subgroup within the
population. Ensures presence of the subgroup.Observe existing relationships between two or more
subgroups.Can representatively sample even the smallest and most
inaccessible subgroups in the population. To sample the rare extremes of the given population.
Higher statistical precision compared to simple random sampling. (d/t lesser variability). So less time and money.
Disadvantage:Requires a sampling frame for each stratum separately.Requires accurate information on proportions of each stratum
Systematic random sampling
Systematic sampling is a commonly employed technique, when complete and up to date list of sampling units is available.
A systematic random sample is obtained by Selecting the first unit on a random basis Then others are included on the basis of
sampling interval I = N/n.
For example, if there are 100 patients (N) in a hospital and to select a sample of 20 patients (n) by systematic random sampling procedure,
Step 1: write the names of 100 patients in alphabetical order or their roll numbers one below the other.
Step 2: sampling fraction: divide N by n to get the sampling fraction (k).In the example k=100/20 = 5.
Step 3: randomly select any number between 1 to k i.e. between 1 to 5. Suppose the number we select is 4.
Step 4: patient number 4 is selected in the sample. Step 5: Thereafter every 4+k th patient is selected
in the sample until we reach the last one.
Systematic random sampling(contd.)
Systematic random sampling(contd.)
Advantage: easy to draw, simplicity.assurance that the population will be evenly sampled.
Disadvantage: Requires sampling frame.
Eg. Random blinded rechecking of slides under RNTCP. Slides are drawn from the register by systematic random sampling.
Systematic random sampling(contd.)
Cluster samplingThe population is divided into subgroups (clusters) like
families. A simple random sample is taken of the subgroups and then all members of the cluster selected are surveyed.
Cluster sampling is used when the population is heterogeneous.
Clusters are formed by grouping units on the basis of their geographical locations.
Cluster sampling is a very useful method for the field epidemiological research and for health administrators.
Cluster sampling
Cluster 4
Cluster 5
Cluster 3
Cluster 2Cluster 1
Types: One stage – when all units in the selected cluster are selected.Two stage – only some units from a selected cluster are taken
using simple random or systematic random sampling.Advantages
Simple as complete list of sampling units within population not required
Low costCan estimate characteristics of both cluster and populationLess travel/resources required
DisadvantagesPotential problem is that cluster members are more likely to be
alike, than those in another cluster (homogenous).Each stage in cluster sampling introduces sampling error—
the more stages there are, the more error there tends to be Usually less expensive than SRS but not as accurate
Cluster sampling (contd.)
A special form of cluster sampling called the “30 X 7 cluster sampling”, has been recommended by the WHO for field studies in assessing vaccination coverage.
In this a list of all villages (clusters) for a given geographical area is made.
30 clusters are selected using Probability Proportional to Size (PPS).
From each of the selected clusters, 7 subjects are randomly chosen.
Thus a total sample of 30 x 7 = 210 subjects is chosen. The advantage of cluster sampling is that sampling frame is not
required
Cluster sampling (contd.)
Steps:List of all clusters (villages and sectors/wards) is made.Population of each cluster is written against them.Cumulative population is then written in serial order.Sampling interval is calculated = Total cumulative population/30
Choose a random number between 1 and the SI. This is the Random Start (RS). The first cluster to be sampled contains this cumulative populationCalculate the following series: RS; RS + SI; RS + 2SI; …. RS+(d-
1)*SI.The clusters selected are those for which the cumulative population
contains one of the serial numbers.
Probability proportional to size (PPS)
Multistage random samplingMultistage sampling refers to sampling plans where the sampling is
carried out in stagesusing smaller and smaller sampling units at each stage.
Not all Secondary Units Sampled normally used to overcome problems associated with a geographically dispersed population
Multistage random samplingIn this method, the whole population is divided in first stage
sampling units from which a random sample is selected.The selected first stage is then subdivided into second stage units
from which another sample is selected. Third and fourth stage sampling is done in the same manner if
necessary.Example:
NFHS data is collected by multistage sampling.Rural areas – 2 stage sampling – Villages from list by PPS,
Households from villageUrban areas – Wards (PPS) – CEB (PPS) – 30 households
from each CEB
CEBWARD HOUSHOLD
Non probability sampling
The probability of each case being selected from the total population is not known
Units of the sample are chosen on the basis of personal judgment or convenience
There are NO statistical techniques for measuring random sampling error in a non-probability sample. Therefore, generalizability is never statistically appropriate
• Involves non random methods in selection of sample
• All have not equal chance of being selected
• Selection depend upon situation
• Considerably less expensive
• Convenient
• Sample chosen in many ways
Non probability sampling
Types of Non probability sampling
Convenience/Grab/Availability
Judgment/Purposive sampling
Quota sampling
Snowball/Network
Convenience/Grab/Availability sampling Subjects selected because it is easy to access them.No Students in your class, people on Street, friends etcAdvantages:
In pilot studies, convenience sample is usually used to obtain basic data and trends.
In documenting that a particular quality of a substance or phenomenon occurs within a given sample.
Disadvantages:Not representative of the entire population – skewed results.Limitation in generalization and inference making about the entire
population – low external validity.
Snowball/Network sampling If the sample for the study is very rare or is limited to a very
small subgroup of the population.Works like a chain referral.Initial subject helps identify people with a similar trait.Advantages:
To reach rare and difficult to access populations.Cheap, cost – efficient.Lesser workforce, lesser planning.
Disadvantages:Little control over sampling technique.Representativeness is not guaranteed.Sampling bias d/t people referring known people who are
more likely to be similar.
Purposive or judgmental samplingThe specialty of an authority can select a more representative
sample. Knowledge of research question required.Subjects selected for a good reason tied to purposes of research.Advantages:
Hard-to-get populations that cannot be found through screening general population.
Usually used when a limited number of individuals possess the trait of interest.
Disadvantages:No way to evaluate the reliability of the expert or the
authority.Biased since no randomization was used in obtaining the
sample. So results cannot be generalised.
Quota sampling
• The population is divided into cells on the basis of relevant control characteristics.
• A quota of sample units is established for each cell.• A convenience sample is drawn for each cell until the quota is
met.• Pre-plan number of subjects in specified categories(e.g. 100
men, 100 women).• In uncontrolled quota sampling, the subjects chosen for those
categories are a convenience sample.• In controlled quota sampling, restrictions are imposed to limit
interviewer’s choice.
•To sample a subgroup that is of great interest to the study.•To observe relationships between subgroups.•Example – an interviewer may be told to sample 50 males and 50 females.Advantages: •Used when research budget limited•Introduces some elements of stratification
Disadvantages:•Variability and bias can not be controlled or measured •Time consuming
Factors affecting choice of sampling designs
Heterogeneity: need larger sample to study more diverse population
Desired precision: need larger sample to get smaller error Nature of analysis: complex multivariate statistics need
larger samples
Accuracy of sample depends upon sample size, not ratio of sample to population
Sample size
Factors affecting sample size
1. Study design: descriptive or comparison study
2. Sampling design: smaller if stratified, larger if cluster
3. Type and number of variables being studied.
4. Maximum tolerable probability of type I error.
5. Required power for a specified clinically important difference.
6. Specification of the magnitude of difference that would be considered significant.
7. The extent of variability among measurements( S.D.)
8. Whether underlying distribution is normal or skewed
9. Heterogeneity of population: need larger sample to study more diverse population
10. Desired precision: need larger sample to get smaller error
11. Nature of analysis: complex multivariate statistics need larger samples
12. Resources and time at hand
Calculation of sample size
SAMPLE SIZE FOR QUALITATIVE OUTCOME VARIABLE
n=4 / 2𝑃𝑄 𝐿 n= sample sizeP= estimated prevalenceQ= 1-PL= allowable errorA survey is to estimate prevalence of influenza virus infection in school kids. Suppose the available evidence suggests that approximately 20% (P=20) of the children will have antibodies to the virus. Assume the investigator wants to estimate the prevalence within 6% of the true value (6% is called allowable error; L)
The required sample size is :
n = (4 x 20 x 80) / (6 x 6) = 177.78Thus approximately 180 kids would be needed for the survey
Sample size for estimation of mean
n= z2a/2s2
l2
Where, n= sample sizes= standard deviation l= absolute precisionz= relative deviatea= alpha error
Za/2 = 1.96 for a= 0.05
n = 4 s2
l2
Example Suppose that it was required to estimate diastolic blood pressure in a
population to within ±2mmHg (using a 95% confidence interval) and the standard deviation of diastolic blood pressure was known to be 15mmHg.
S= 15 l= 2
n = 4 s2
l2
N= 4× (225/4)= 216.09
The next highest integer is taken, giving a requirement of 217 subjects
Sample size for estimation of proportion
n= z2a/2p(1-p)
l2
Where, n= sample size
p= anticipated value of proportion in population
l= absolute precision
z= relative deviate
a= alpha error
Za/2 = 1.96 for a= 0.05
n= 4 p(1-p) l2
Example Suppose it is thought that there are about
28% smokers in the population and it is required to estimate the percentage of smokers to within ±3% (in absolute terms), using a 95% confidence interval.
p= 0.28 l= 0.03 n= 4 p(1-p) n= 4 ×0.28(1-0.28) l2 (0.03)2 n= 860.5so that a survey of 861 persons is required,
Sample size for estimation of rate
n= 4 r2
l2
where: r = estimated rate in the population
l = absolute precision Suppose that a rate is expected to be around 25 per million (per
year) and it is required to estimate it with a 95% confidence interval to within ± 5 per million. The number of cases required to achieve this level of precision is
n= 4 (25)2
(5)2
n= 96.04
which means that 97 cases would have to be observed
Sample size for estimation of difference between two population means
n= z2a/2 (s12 + s2
2 )
l2
Where, n= sample size
s= standard deviation ( subscript 1,2 refer to two populations)
l= absolute precision
z= relative deviate
a= alpha error
Za/2 = 1.96 for a= 0.05
n= 4 (s12 + s2
2 )
l2
Sample size for estimation of difference between two population proportion
n= z2a/2[ p1(1-p1) + p2 (1-p2) ] l2
Where, n= sample sizep= anticipated value of proportion in population
( subscript 1,2 refer to two populations) l= absolute precisionz= relative deviatea= alpha error
Za/2 = 1.96 for a= 0.05
n= 4 [ p1(1-p1) + p2 (1-p2) ] l2
Sampling variability refers to the different values which a given function of the data takes when it is computed for two or more samples drawn from the same population.
Factors affecting sampling variability:1.Inherent variation in the population2. Sample size 3.Sampling distribution of the mean4.Sampling error and bias.
Sampling variability
Eg. Population of 7000 children and their birth weight. The mean and standard deviation for this distribution are 3.36 and 0.56 respectively.
N Sample 1 Sample 2 Sample 3 Sample 4 Sample 5
1 3.09 4.28 4.09 2.34 4.29
2 3.74 2.82 2.96 3.06 2.87
3 2.56 3.80 3.09 3.35 3.43
4 3.63 1.89 3.14 3.30 3.40
5 2.96 4.04 3.14 4.36 3.58
6 2.76 2.39 4.38 3.99 3.96
7 3.98 3.41 3.87 4.62 3.18
8 3.76 3.95 4.34 3.18 3.07
9 2.66 5.83 3.81 2.80 2.70
10 3.16 3.30 4.16 3.14 3.21
N 10 10 10 10 10
Mean 3.23 3.57 3.70 3.41 3.37
SD 0.51 1.10 0.56 0.71 0.48
Minimum 2.56 1.89 2.96 2.34 2.70
maximum 3.98 5.83 4.38 4.62 4.28
Irrespective of sample size , the sample means are expected to fluctuate evenly about the true population mean.
The variation in sample means exhibited in the table is an example of sampling variation due to chance.
If we take 50 observations ,mean is 3.46 kg. sampling error 3.46-3.36= 0.10
The means vary less(by chance) if the sample size is large; that is sampling error is smaller,the larger is the sample.
The distribution more closely clustered around a middle value as the sample size increases.
The mean do not systematically increase or decrease with increasing sampling and have more variability(larger SD) when the sample size is small.
The standard deviation of the means steadily decrease as sample size increases, more quickly when the sample size is small.
The sampling distribution of the mean
A sampling experiment(based on the distribution of birth weights): what happens to mean and variability of a sample mean when we keep doubling the sample size
N Mean of population values=3.36
Mean of sample means(kg)
SD of population values=0.56
SD of sample means(observed SE OF Mean;kg)
2 3.50 0.40
4 3.51 0.28
8 3.46 0.19
16 3.45 0.11
32 3.44 0.080
64 3.46 0.06
Sampling error • Types of sampling error: 1. sample error • 2. non sample error
SAMPLE ERROR: is incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population.
For example, if one measures the height of a thousand individuals from a country of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country.
Sample error (random error) • Error caused by the act of taking a sample• They cause sample results to be different from the results of
census• Size of error can be measured in probability samples• Expressed as “standard error”
• of mean, proportion…• We have no control over• Sample error depends upon:
• Size of the sample (larger size lesser error)• Distribution of character of interest in population
Non sample error Non response error: A non-response error occurs when
units selected as part of the sampling procedure do not respond in whole or in part
Response error: A response or data error is any systematic bias that occurs during data collection, analysis or interpretation • Respondent error (e.g., lying, forgetting, etc.)• Interviewer bias• Recording errors• Poorly designed questionnaires
References
1. Indrayan A., Satyanarayana L., Medical Biostatistics, third edition, 2009
2. Last JM. Dictionary of Epidemiology, 3rd edition, 2000.
3. Dawson B., Trapp RG, Basic and Clinical Biostatistics, second edition, 1994
4. Daly LE, Bourke GJ, Interpretation and uses of medical statistics, fifth edition, 2003
5. Detels R., Beaglehole R., Oxford Textbook of public health, fifth edition,2011.
Thank You