Survey of Smoking Behavior. Samples and Elements. Survey of Smoking Behavior. Samples and Elements

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Ralph Nelson
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1 s and Elements Units are Same as Elementary Units Frame Elements Analyzed as a binomial variable 9 Persons from, Frame Elements N =, N =, n = 9 Analyzed as a binomial variable HIV+ HIV- % population smokes HIV- 9 9 s and Elements Units are Different from Elementary Units Frame Elements % are Smokers Frame Persons Analyzed as a ratio Shared Shared Shared Shared 9 s and Elements Units are Different from Elementary Units Frame Elements Analyzed as a ratio estimator t value Relation t value to z value Use t for Class Eample and later for Rapid Surveys with Clusters 9 t =.9 for sample 9 (i.e., 9 d.f.) z value = Size minus One

2 9% Confidence Intervals from all Possible s 9 Simple Random 9 Persons from Frame, Persons p surveys =.% Frame HH One Person HH Two Person HH Three Person HH 7 True value Smokers (%). Average SE =.% P = % N = N = N = Average HH size = Four Person HH Five Person HH confidence intervals did not bracket the true value * * * * * 7 9 Survey Number N = N = 7 HH from Frame HH One-Stage Cluster Survey Frame Elements N =, N =, n = 9 % population smokes Analyzed as a ratio estimator HH from Frame HH Elementary units are different from sampling units d HH but analyzing people Analysis is a ratio estimator, not a binomial variable Variable one (smokers) Variable two (persons) Same number persons as, Average persons per HH = Number HH =, Student's t Student's t for Confidence Interval Calculations 9% CL 9% CL 99% CL % CL.99 9% CL. 99% CL.7 Number Clusters z..9. Two Scenarios with HH from Frame HH Large and Small have the Same Percentage Smokers How do the findings in a cluster survey where the sample units and elements for analysis are different (i.e., estimating r) compare to a a population where the sample units and elements for analysis are the same (i.e., estimating p)? Smokers are randomly distributed in Smokers within are as variable as among Smokers are not randomly distributed in Smokers within are more or less variable than among 9

3 Frame with Random Distribution Persons % are Smokers Notions Variability among Clusters % Stated Size Low Variability Medium Variability (i.e., same as ) HH size = HH size = % % HH size = % HH size = % Random Distribution s Same as Smokers in HH Size not Associated with Smokers (%) HH from Frame HH Persons in HH % % smokers in HH Variable One Ratio estimator = Variable Two HH size = HH size = If there is no association between smoking (i.e., variable one) and HH size (i.e., variable two), then the analysis the ratio estimator behaves similar to the analysis a binomial variable in a people rather than Smokers (%). Random Distribution s (same variability as ) 9% Confidence Intervals from all Possible s Cluster with 9 Persons from 9 Frame, with Persons p surveys =.% 7 Average SE =.% Note: nearly the same as SE =.% for shown earlier confidence intervals did not bracket the true value * * * * * * 7 9 Survey Number True value P = % Relation t value to z value Use t for Class Eample Cluster Sampling and for Rapid Surveys t value t =. for sample (i.e., 9 d.f.) z value = Size minus One

4 Low Variability Smokers among Less Variability among Clusters (i.e., ) than HH Size not Associated with Smokers (%) % % smokers in HH Notions Variability among Clusters % Stated Size Medium Variability (i.e., same as ) HH size = High Variability HH size = Persons in HH % HH size = % % HH size = % 9 Low Variability (i.e., less than ) Smokers among Clusters (i.e. ) Smokers (%). 9% Confidence Intervals from all Possible s One-Stage Cluster with 9 Persons 9 from Frame, with Persons 7 Average SE =.% confidence intervals did not bracket the true value ** * * 7 9 Survey Number p surveys =.9% True value P = % 7 Smokers (%). 7 Medium Variability (i.e., same as ) Smokers among Clusters (i.e., ) 9% Confidence Intervals from all Possible s One-Stage Cluster with 9 Persons 9 from Frame, with Persons p surveys =.% Average SE =.% True value P = % confidence intervals did not bracket the true value * * * * * * 7 9 Survey Number Same as Slide Medium Variability Smokers among Same Variability among Clusters (i.e., ) as HH Size not Associated with Smokers (%) High Variability Smokers among More Variable among Clusters (i.e., ) than HH Size not Associated with Smokers (%) % % smokers in HH % % smokers in HH Persons in HH Same as Slide Persons in HH

5 High Variability (i.e., greater than ) Smokers among Clusters (i.e., ) Smokers (%) % Confidence Intervals from all Possible s One-Stage Cluster with 9 Persons from Frame, with Persons p surveys =.% True value P = % Average SE = 9.% confidence intervals did not bracket the true value * * * * * 7 9 Survey Number Variability Among and Within Clusters (i.e., ) with person Medium variability among clusters (i.e., ) is the same variability as a. Such variability is shown for the in the sample that have persons. p q =.. =. p q =.. =. p q =.. =. More heterogeneity (i.e., dissimilarity) within clusters p q =.. =. p q =.. =. p q = = Medium Variability HH size = % Medium variability among (i.e., between) clusters arises from medium variability within clusters Note: this is different than the variance a proportion, or p q n Variance (p q) Variance a Binomial Variable Smokers in a Group Five Persons p q Variability Among and Within Clusters (i.e., ) with person Low variability among clusters (i.e., ) is less variable than a. Such variability is shown for the in the sample that have persons. p q =.. =. p q =.. =. p q =.. =. Low Variability HH size = %. - - Number Smokers Much more heterogeneity (i.e., dissimilarity) within clusters Low variability among (i.e., between) clusters arises from high variability within clusters The variance the binomial variable smoking in a group five persons is greatest when - are smokers p q =.. =. p q =.. =. p q =.. =. Variability Among and Within Clusters (i.e., ) In the total population,, there are that are person In a typical survey smoking behavior involving, one-fifth (or ) will be -person High Variability HH size = % HH from Frame HH What happens to the ratio estimator if cluster (i.e., HH) size is associated with smoking? p q = = p q = = p q = = Ratio estimator = Variable One Variable Two No smokers High homogeneity (i.e., similarity) within clusters High variability among (i.e., between) clusters arises from etremely low variability within clusters Does the ratio estimator analyzed for a HH survey still behave the same as a proportion analyzed for a? HH = HH = Persons in large are more likely to smoke than persons in small p q = = p q = = p q = = 7

HH from Frame HH Complete distortion High variability smoking within Plus, smoking status varies dramatically with HH size Question to be answered Does this etreme situation bias the use a ratio

6 HH from Frame HH Complete distortion High variability smoking within Plus, smoking status varies dramatically with HH size Question to be answered Does this etreme situation bias the use a ratio estimator to determine the proportion who smoke? High Variability (i.e., greater than ) Smokers among Clusters (i.e., ) Smokers (%) % Confidence Intervals from all Possible s One-Stage Cluster with 9 Persons from Frame, with Persons p surveys =.% Slight bias True value P = % Average SE confidence intervals did not bracket the true value = 9.% * * * * * 7 9 Survey Number Notions Variability where Cluster (i.e., HH) Size is Associated with Smokers (%) % Stated Size Medium Variability (i.e., same as ) HH size = High Variability HH size = Summary Effects on SE and Mean What happens to the standard error and mean the ratio estimator as variability smoking changes among and within clusters (i.e., )? % % % % Type Survey Among -- Variability Within -- SE (%). Mean % Smokers. Bias No HH size = % HH size = % Cluster* Cluster* Cluster* Low Medium High High Medium Low No No No % % Cluster** High Low 9.. Slight 9 Through it all, the CI the ratio estimator still behaves like a 9% CI should behave * Smokers (%) not associated with HH size ** Smokers (%) associated with HH size High Variability Smokers among More Variable among Clusters (i.e., ) than HH Size Associated with Smokers (%) Reference % % smokers in HH Persons in HH Download from:

7 Random Addicts from a 9 Addicts Each Addict in the 9 has an Equal Appearing in the Nine Drug Addicts 9 = JOE HAL ROY JON SAM BOB NAT TED BEN 7 All Possible s from 9 SAMPLE SAMPLE Where is Joe? All Possible s from 9, Selected WOR, Disregarding Order ( samples) 9 The one sample that was, by chance, selected SAMPLE SAMPLE 7 = 7 possible samples selected WOR from a population 9, disregarding order 9 7 Proportion or Mean all Possible s Addicts from a 9 Addicts SAMPLE P or Y SAMPLE P or Y Equal into a addicts from a population 9 addicts, drawn WOR Frame P = (/9) P = (/9) P = (/9) The one sample that was, by chance, selected SAMPLE P or Y SAMPLE P or Y 7 being included in the sample is. P = (/9) P = (/9) 9 P = (/9) 9

8 Sampling from a, Households Multiple Strategies for Equal into a from a,, drawn WOR All Possible s Drawn WOR when Order is Disregarded All possible samples = N! n! (N-n)! (A),, =. from,,!! (,-)! =. possible samples (B),, =., Equal Method (EPSEM) Equal into a from a population,, drawn WOR Frame Multiple Strategies for Equal into a from a,, drawn WOR P = (/,), being included in the sample is. (C),, =.,, P. = (/,) EPSEM Simple Random from, UNITS Simple random sample probability Multiple Strategies for Equal into a from a,, drawn WOR, =. The selection probability for a sample (D),,,, =., =., EPSEM

9 Multiple Strategies for Equal into a from a,, drawn WOR (E),,,,,,,, EPSEM =., =., =., =., Two-stage Sampling with Equal Probabilities Study, Equal-sized Clusters (i.e., ) Study First Stage Second Stage CLUSTERS OF SIMILAR SIZE SIMPLE RANDOM SAMPLE EQUAL NUMBER =. =. =. =. All are EPSEM samples Cluster Frame Two-stage Sampling with Equal Probabilities Study, Unequal-sized Clusters Study First Stage Second Stage CLUSTERS OF VARYING SIZE SIMPLE RANDOM SAMPLE EQUAL FRACTION =. =. =. =. All are EPSEM samples 7 Selected Clusters Two-stage Sampling with Unequal Probabilities Study, Unequal-sized Clusters Study First Stage Second Stage CLUSTERS OF VARYING SIZE SIMPLE RANDOM SAMPLE EQUAL NUMBER =. 7 =.7 =. =. 9 The sample would need to be weighted in the analysis (i.e., not a self-weighted sample) Not EPSEM samples

10 Two-stage Sampling with Equal Probabilities Study, Unequal-sized Clusters Study First Stage Second Stage CLUSTERS OF VARYING SIZE PPS SAMPLE EQUAL NUMBER PPS This is a self-weighted sample since the selection probabilities compensate each other through the two stages selection.,,,, =. =. =. =. All are EPSEM samples List study population Divide into sampling intervals Start at beginning first sampling interval Start at end first sampling interval Random start in first sampling interval Sequential (or Systematic) Sampling (A), (B) 9, (C) (D) (E) Procedures for n= from N=, 7 9 9, 7 9 9, 7 9 9, PPS Sampling Method Step PPS Sampling List Good Better Best Tally number sampling units in each community Communities Cumulative List Net, take random sample list Identify location community in cumulative list Systematic PPS sample Natural order clusters with random start in first sampling interval Systematic PPS sample 7 7 Random order with random start in first sampling interval Random PPS sample from cumulative list Random PPS sample Random PPS sample Repeat until clusters are selected (WR) PPS Sampling Method Step Cumulative List then If random select number is Sampling Units and Elementary Units Frame Sampling Unit Typical situation with a Elementary Unit 7 Take cumulative list units in study population and draw random sample from list Continue until clusters are selected in the various communities 9 9 7

11 Sampling Units and Elementary Units Frame Sampling Unit Typical situation with a HH survey Elementary Unit From slides 7 and 7 Frame Two Stage Cluster Sampling First Stage PPS d 9 from Is this an EPSEM sample? Original ID number is changed to cluster number from to 9 Second Stage Frame Sampling Units and Elementary Units First Stage Second Stage Frame Frame Elements Dreaming Variable One Persons with casual se partners 9 Variable Two Regularly used condoms with casual partners 7 From slides 7 and 7 Frame Frame Two Stage Cluster Sampling First Stage PPS d 9 from Is this an EPSEM sample? Original ID number is changed to cluster number from to 9 7 Second Stage Ratio Estimator Condom user Ratio estimator household = With casual partner = = 7% Assume the sampling design is to select only one person in the household (HH). Is there one person who would represent the ratio estimator in the HH? 9 7 7

Hence, homogeneity would be increased within, resulting in a greater variance.

12 Sampling Representative sample Ratio estimators within the sampled represent the population Representative sample 7 Survey for HIV Representative sample Assume if one is infected that others will be infected. Hence, homogeneity would be increased within, resulting in a greater variance. Solution: randomly select only one person per HH Problem: ratio estimators within the sampled may not represent the population Representative sample 7 Survey for HIV Survey HIV prevalence Solve one problem and create another 7

Survey of Smoking Behavior. Survey of Smoking Behavior. Survey of Smoking Behavior

Survey of Smoking Behavior. Survey of Smoking Behavior. Survey of Smoking Behavior Sample HH from Frame HH One-Stage Cluster Survey Population Frame Sample Elements N =, N =, n = population smokes Sample HH from Frame HH Elementary units are different from sampling units Sampled HH but