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2 Types of Surveys Cross-sectional surveys a specific population at a given point in time will have one or more of the design components stratification clustering with multistage sampling unequal probabilities of selection Longitudinal surveys a specific population repeatedly over a period of time panel rotating samples

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3 Cross Sectional Surveys Sampling Design Terminology

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4 Methods of Sample Selection Basic methods simple random sampling systematic sampling unequal probability sampling stratified random sampling cluster sampling two-stage sampling

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5 Simple Random Sampling Why? basic building block of sampling sample from a homogeneous group of units How? physically make draws at random of the units under study computer selection methods: R, Stata

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6 Systematic Sampling Why? easy can be very efficient depending on the structure of the population How? get a random start in the population sample every k th unit for some chosen number k

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7 Additional Note Simplifying assumption: in terms of estimation a systematic sample is often treated as a simple random sample Key assumption: the order of the units is unrelated to the measurements taken on them

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8 Unequal Probability Sampling Why? may want to give greater or lesser weight to certain population units two-stage sampling with probability proportional to size at the first stage and equal sample sizes at the second stage provides a self-weighting design (all units have the same chance of inclusion in the sample) How? with replacement without replacement

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9 With or Without Replacement? in practice sampling is usually done without replacement the formula for the variance based on without replacement sampling is difficult to use the formula for with replacement sampling at the first stage is often used as an approximation Assumption: the population size is large and the sample size is small – sampling fraction is less than 10%

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10 Stratified Random Sampling Why? for administrative convenience to improve efficiency estimates may be required for each stratum How? independent simple random samples are chosen within each stratum

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11 Example: Survey of Youth in Custody first U.S. survey of youths confined to long-term, state-operated institutions complemented existing Children in Custody censuses. companion survey to the Surveys of State Prisons the data contain information on criminal histories, family situations, drug and alcohol use, and peer group activities survey carried out in 1989 using stratified systematic sampling

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12 SYC Design strata type (a) groups of smaller institutions type (b) individual larger institutions sampling units strata type (a) first stage – institution by probability proportional to size of the institution second stage – individual youths in custody strata type (b) individual youths in custody individuals chosen by systematic random sampling

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13 Cluster Sampling Why? convenience and cost the frame or list of population units may be defined only for the clusters and not the units How? take a simple random sample of clusters and measure all units in the cluster

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14 Two-Stage Sampling Why? cost and convenience lack of a complete frame How? take either a simple random sample or an unequal probability sample of primary units and then within a primary take a simple random sample of secondary units

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15 Synthesis to a Complex Design Stratified two-stage cluster sampling Strata geographical areas First stage units smaller areas within the larger areas Second stage units households Clusters all individuals in the household

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16 Why a Complex Design? better cover of the entire region of interest (stratification) efficient for interviewing: less travel, less costly Problem: estimation and analysis are more complex

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17 Ontario Health Survey carried out in 1990 health status of the population was measured data were collected relating to the risk factors associated with major causes of morbidity and mortality in Ontario survey of 61,239 persons was carried out in a stratified two-stage cluster sample by Statistics Canada

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18 OHS Sample Selection strata: public health units – divided into rural and urban strata first stage: enumeration areas defined by the 1986 Census of Canada and selected by pps second stage: dwellings selected by SRS cluster: all persons in the dwelling

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19 Longitudinal Surveys Sampling Design

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20 Schematic Representation

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21 Schematic Representation

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22 British Household Panel Survey Objectives of the survey to further understanding of social and economic change at the individual and household level in Britain to identify, model and forecast such changes, their causes and consequences in relation to a range of socio-economic variables.

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23 BHPS: Target Population and Frame Target population private households in Great Britain Survey frame small users Postcode Address File (PAF)

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24 BHPS: Panel Sample designed as an annual survey of each adult (16+) member of a nationally representative sample 5,000 households approximately 10,000 individual interviews approximately. the same individuals are re-interviewed in successive waves if individuals split off from original households, all adult members of their new households are also interviewed. children are interviewed once they reach the age of waves of the survey from 1991 to 2004

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25 BHPS: Sampling Design Uses implicit stratification embedded in two-stage sampling postcode sector ordered by region within a region postcode sector ordered by socio- economic group as determined from census data and then divided into four or five strata Sample selection systematic sampling of postcode sectors from ordered list systematic sampling of delivery points ( addresses or households)

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26 BHPS: Schema for Sampling

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27 Survey Weights

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28 Survey Weights: Definitions initial weight equal to the inverse of the inclusion probability of the unit final weight initial weight adjusted for nonresponse, poststratification and/or benchmarking interpreted as the number of units in the population that the sample unit represents

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29 Interpretation the survey weight for a particular sample unit is the number of units in the population that the unit represents

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30 Effect of the Weights Example: age distribution, Survey of Youth in Custody

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31 Unweighted Histogram

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32 Weighted Histogram

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33 Weighted versus Unweighted

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34 Observations the histograms are similar but significantly different the design probably utilized approximate proportional allocation the distribution of ages in the unweighted case tends to be shifted to the right when compared to the weighted case older ages are over-represented in the dataset

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35 Survey Data Analysis Issues and Simple Examples from Graphical Methods

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36 Basic Problem in Survey Data Analysis

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37 Issues iid (independent and identical distribution) assumption the assumption does not not hold in complex surveys because of correlations induced by the sampling design or because of the population structure blindly applying standard programs to the analysis can lead to incorrect results

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38 Example: Rank Correlation Coefficient Pay equity survey dispute: Canada Post and PSAC two job evaluations on the same set of people (and same set of information) carried out in 1987 and 1993 rank correlation between the two sets of job values obtained through the evaluations was assumption to obtain a valid estimate of correlation: pairs of observations are iid

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39 Scatterplot of Evaluations Rank correlation is 0.539

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40 A Stratified Design with Distinct Differences Between Strata the pay level increases with each pay category (four in number) the job value also generally increases with each pay category therefore the observations are not iid

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41 Scatterplot by Pay Category

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42 Correlations within Level Correlations within each pay level Level 2: –0.293 Level 3: –0.010 Level 4: Level 5: Only Level 4 is significantly different from 0

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43 Graphical Displays first rule of data analysis always try to plot the data to get some initial insights into the analysis common tools histograms bar graphs scatterplots

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44 Histograms unweighted height of the bar in the i th class is proportional to the number in the class weighted height of the bar in the i th class is proportional to the sum of the weights in the class

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45 Body Mass Index measured by weight in kilograms divided by square of height in meters 7.0 < BMI < 45.0 BMI < 20: health problems such as eating disorders BMI > 27: health problems such as hypertension and coronary heart disease

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46 BMI: Women

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47 BMI: Men

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48 BMI: Comparisons

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49 Bar Graphs Same principle as histograms unweighted size of the i th bar is proportional to the number in the class weighted size of the i th bar is proportional to the sum of the weights in the class

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50 Ontario Health Survey

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51 Scatterplots unweighted plot the outcomes of one variable versus another problem in complex surveys there are often several thousand respondents

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53 Solution bin the data on one variable and find a representative value at a given bin value the representative value for the other variable is the weighted sum of the values in the bin divided by the sum of the weights in the bin

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55 Bubble Plots size of the circle is related to the sum of the surveys weights in the estimate more data in the BMI range 17 to 29 approximately

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56 Computing Packages STATA and R

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57 Available Software for Complex Survey Analysis commercial Packages: STATA SAS SPSS Mplus noncommercial Package R

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58 STATA defining the sampling design: svyset –example svyset [pweight=indiv_wt], strata(newstrata) psu(ea) vce(linear) –output: pweight: indiv_wt VCE: linearized Strata 1: newstrata SU 1: ea FPC 1:

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59 R: survey package define the sampling design: svydesign –wk1de<- svydesign(id=~ea,strata=~newstrata,weight=~i ndiv_wt,nest=T,data=work1) output > summary(wk1de) Stratified 1 - level Cluster Sampling design With (1860) clusters. svydesign(id = ~ea, strata = ~newstrata, weight = ~indiv_wt, nest = T, data = work1)

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60 Syntax STATA: –svy: estimate –Example: least squares estimation –svyset [pweight=indiv_wt], strata(newstrata) psu(ea) –svy: regress dbmi bmi R: –svy***(*, design, data=,...) –Example: least squares estimation – wk2de<- svydesign(id=~ea,strata=~newstrata,weight=~indiv_wt, nest=T,data=work2) –svyglm(dbmi~bmi, data=work2,design=wk2de)

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61 Available Survey Commands RSTATA DescriptiveYes RegressionYesYes (More) ResamplingYes LongitudinalYesNo PMLEYes CalibrationYesNo

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62 Survey Data Analysis Contingency Tables and Issues of Estimation of Precision

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63 General Effect of Complex Surveys on Precision stratification decreases variability (more precise than SRS) clustering increases variability (less precise than SRS) overall, the multistage design has the effect of increasing variability (less precise than SRS)

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64 Illustration Using Contingency Tables two categorical variables that can be set out in I rows and J columns can get a survey estimate of the proportion of observations in the cell defined by the i th row and j th column:

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65 Example: Ontario Health Survey rows: five levels describing levels of happiness that people feel columns: four levels describing the amount of stress people feel Is there an association between stress and happiness?

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66 STATA Commands

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67 STATA Output table on stress and happiness estimated proportions in the table with test statistic

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68 Possible Test Statistics adapt the classical test statistic need the sampling distribution of the statistic Wald Test need an estimate of the variance-covariance matrix

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69 Estimation of Variance or Precision variance estimation with complex multistage cluster sample design: exact formula for variance estimation is often too complex; use of an approximate approach required NOTE: taking account of the design in variance estimation is as crucial as using the sampling weights for the estimation of a statistic

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70 Some Approximate Methods Taylor series methods Replication methods Balanced Repeated Replication (BRR) Jackknife Bootstrap

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71 Replication Methods you can estimate the variance of an estimated parameter by taking a large number of different subsamples from your original sample each subsample, called a replicate, is used to estimate the parameter the variability among the resulting estimates is used to estimate the variance of the full-sample estimate covariance between two different parameter estimates is obtained from the covariance in replicates the replication methods differ in the way the replicates are built

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72 Assumptions The resulting distribution of the test statistic is based on having a large sample size with the following properties the total number of first stage sampled clusters (or primary sampling units) is assumed large the primary sample size in each stratum is small but the number of strata is large the number of primary units in a stratum is large no survey weight is disproportionately large

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73 Possible Violations of Assumptions the complex survey (stratified two-sample sampling, for example) was done on a relatively small scale a large-scale survey was done but inferences are desired for small subpopulations stratification in which a few strata (or just one) have very small sampling fractions compared to the rest of the strata The sampling design was poor resulting in large variability in the sampling weights

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74 Survey Data Analysis Linear and Logistic Regression

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75 General Approach form a census statistic (model estimate or expression or estimating equation) for the census statistic obtain a survey estimate of the statistic the analysis is based on the survey estimate

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76 Regression Use of ordinary least squares can lead to badly biased estimates of the regression coefficients if the design is not ignorable underestimation of the standard errors of the regression coefficient if clustering (and to a lesser extent the weighting) is ignored

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77 Example: Ontario Health Survey Regress desired body mass index (DBMI) on body mass index (BMI)

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78 Simple Linear Regression Model typical regression model linear relationship plus random error errors are independent and identically distributed

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79 Census Statistic census estimate of the slope parameter Problem: the assumption of independent errors in the population does not hold Solution: the least squares estimate is a consistent estimate of the slope

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80 Survey Estimate the census estimate B is now the parameter of interest the survey estimate is given by estimate obtained from an estimating equation the estimate of variance cannot be taken from the analysis of variance table in the regression of y on x using either a weighted or unweighted analysis

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81 Variance Estimation Again, estimate of the variance of b is obtained from one of the following procedures Taylor linearization Jackknife BRR Bootstrap

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82 Issues in Analysis application of the large sample distributional results small survey regression analysis on small domains of interest multicollinearity survey data files often have many variables recorded that are related to one another

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83 Multicollinearity Example: Ontario Health Survey Two regression models: regress desired body mass index on actual body mass index, age, gender, marital status, smoking habits, drinking habits, and amount of physical activity all of the above variables plus interaction terms: marital status by smoking habits, marital status by drinking habits, physical activity by age

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84 Partial STATA Output No interaction terms Interaction terms present

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85 Comparison of Domain Means Domains and Strata both are nonoverlapping parts or segments of a population usually a frame exists for the strata so that sampling can be done within each stratum to reduce variation for domains the sample units cannot be separated in advance of sampling Inferences are required for domains.

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86 Regression Approach use the regression commands in STATA and declare the variables of interest to be categorical example: DBMI relative to BMI related to sex and happiness index STATA commands

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87 STATA Output

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88 Logistic Regression probability of success p i for the i th individual vector of covariates x i associated with i th individual dependent variable must be 0 or 1, independent variables x i can be categorical or continuous Does the probability of success p i depend on the covariates x i – and in what way?

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89 Census Parameter Obtained from the logistic link function and the census likelihood equation for the regression parameters Note: it is the log odds that is being modeled in terms of the covariate

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90 Example: Ontario Health Survey How does the chance of suffering from hypertension depend on: body mass index age gender smoking habits stress a well-being score that is determined from self- perceived factors such as the energy one has, control over emotions, state of morale, interest in life and so on

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91 STATA Commands

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92 STATA Output part I

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93 STAT Output part II

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94 GEE: Generalized Estimating Equations Dependent or response variable well-being measured on a 0 to 10 scale focus is on women only Independent or explanatory variables has responsibility for a child under age 12 (yes = 1, no = 2) marital status (married = 1, separated = 2, divorced = 3, never married = 5 [widowed removed from the dataset]) employment status (employed = 1, unemployed = 2, family care = 3) STATA syntax tsset pid year, yearly xi: xtgee wellbe i.mlstat i.job i.child i.sex [pweight = axrwght], family(poisson) link(identity) corr(exchangeable)

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95 GEE Results

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96 For each type of initial marital status Married Separated or divorced Never married

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97 Cox Proportional Hazards Model Dependent or outcome variable time to breakdown of first marriage Independent or explanatory variables gender race (white/non-white) Age in 1991 (restricted to 18 – 60) financial position: comfortable=1, doing alright=2, just about getting by=3, quite difficult=4, very difficult =5

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98 STATA Commands Command for survival data set up Command for Cox proportional hazards mode

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99 STATA Output

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