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# Sampling and its variability

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• Convenience = as the interviewer wishes.
• Accuracy of sample depends upon sample size, not ratio of sample to population
• n= 1.96 * 2 s2n= sample size
l2
• ### Sampling and its variability

1. 1. Presentation by: Dr. Bhushan Kamble Moderator: Dr. Poornima Tiwari Professor, Department of Community Medicine, VMMC & SJH Sampling and sampling variability
2. 2. 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
3. 3. Definitions Population: The target group to which the findings (of a study) would ultimately apply is called population1 Or Population 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. Or Sample 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, 2009 2. Last JM. Dictionary of Epidemiology, 3rd edition, 2000.
4. 4. 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
5. 5. 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
6. 6. Population Vs. Sample Sample Population Sample Parameter Statistic We measure the sample using statistics in order to draw inferences about the population and its parameters. Population of Interest
7. 7. 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…..
8. 8. 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.
9. 9. 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.
10. 10. Types of sampling Probability sampling Non probability sampling Probability of selection of each individual is known and pre determined Simple random sampling Systematic random sampling Stratified random sampling Cluster random sampling Multistage random sampling Probability of selection of each individual is not known Quota sampling Purposive/ Judgmental sampling Snowball/ Network sampling Convenience/ Grab sampling (man in the street)
11. 11. Simple random sampling Equal probability of selection of units for inclusion in the study Requires a list of all sampling units (sampling frame) Each individual is chosen randomly. Methods: Lottery method (possible for finite population) Random number tables Software that generate random numbers
12. 12. Lottery method Lottery method
13. 13. 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
14. 14. Simple random sampling (contd.) Simple random method With replacement Without replacement Advantage Very scientific method Equal chance of all subjects for selection Disadvantage Requires sampling frame Example: Blood sampling – TLC, Hb estimation
15. 15. 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 sampling Dis-proportionate stratified sampling
16. 16. 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 group 20 to 25 from middle socioeconomic group 70 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.
17. 17. POPULATION LOW SOCIOECONOMIC MIDDLE SOCIOECONOMIC HIGH SOCIOECONOMIC
18. 18. 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
19. 19. 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.
20. 20. 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.)
21. 21. Systematic random sampling(contd.)
22. 22. 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.)
23. 23. Cluster sampling The 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.
24. 24. Cluster sampling Cluster 4 Cluster 5 Cluster 3 Cluster 2Cluster 1
25. 25. 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 cost Can estimate characteristics of both cluster and population Less travel/resources required Disadvantages Potential 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.)
26. 26. 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.)
27. 27. 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 population Calculate 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)
28. 28. Multistage random sampling Multistage sampling refers to sampling plans where the sampling is carried out in stages using smaller and smaller sampling units at each stage. Not all Secondary Units Sampled normally used to overcome problems associated with a geographically dispersed population
29. 29. Multistage random sampling In 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 village Urban areas – Wards (PPS) – CEB (PPS) – 30 households from each CEB
30. 30. CEB WAR D HOUSHOLD
31. 31. 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
32. 32. • Involves non random methods in selection of sample • All have not equal chance of being selected • Selection depend upon situation • Considerably less expensive • Convenient Non probability sampling
33. 33. Types of Non probability sampling Convenience/Grab/Availability Judgment/Purposive sampling Quota sampling Snowball/Network
34. 34. Convenience/Grab/Availability sampling Subjects selected because it is easy to access them. No Students in your class, people on Street, friends etc Advantages: 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.
35. 35. 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.
36. 36. Purposive or judgmental sampling The 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.
37. 37. 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.
38. 38. •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
39. 39. 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
40. 40. Sample size
41. 41. 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
42. 42. Calculation of sample size
43. 43. SAMPLE SIZE FOR QUALITATIVE OUTCOME VARIABLE n=4𝑃𝑃/𝑃2  n= sample size P= estimated prevalence Q= 1-P L= allowable error A 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) = 177.78 Thus approximately 180 kids would be needed for the survey
44. 44. Sample size for estimation of mean n= z2a/2s2 l2 Where, n= sample size s= standard deviation l= absolute precision z= relative deviate a= alpha error Za/2 = 1.96 for a= 0.05 n = 4 s2 l2
45. 45. 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
46. 46. 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
47. 47. 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= 860.5 so that a survey of 861 persons is required,
48. 48. 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
49. 49. Sample size for estimation of difference between two population means n= z2a/2 (s1 2 + 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 (s1 2 + s2 2 )
50. 50. Sample size for estimation of difference between two population proportion n= z2a/2[ p1(1-p1) + p2 (1-p2) ] l2 Where, n= sample size p= anticipated value of proportion in population ( 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 [ p1(1-p1) + p2 (1-p2) ] l2
51. 51. 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 population 2. Sample size 3.Sampling distribution of the mean 4.Sampling error and bias. Sampling variability
52. 52. 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
53. 53. 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.
54. 54. 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.
55. 55. The sampling distribution of the mean
56. 56. 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
57. 57. 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.
58. 58. 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
59. 59. 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
60. 60. 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., OxfordTextbook of public health, fifth edition,2011.
61. 61. ThankYou