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Types of Surveys Cross-sectional

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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

3 Cross Sectional Surveys
Sampling Design Terminology

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

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

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 kth unit for some chosen number k

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

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

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%

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

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

12 SYC Design strata sampling units
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

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

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

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

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

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

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

19 Longitudinal Surveys Sampling Design

20 Schematic Representation

21 Schematic Representation

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.

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

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 16 13 waves of the survey from 1991 to 2004

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)

26 BHPS: Schema for Sampling

27 Survey Weights

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

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

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

31 Unweighted Histogram

32 Weighted Histogram

33 Weighted versus Unweighted

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

35 Issues and Simple Examples from Graphical Methods
Survey Data Analysis Issues and Simple Examples from Graphical Methods

36 Basic Problem in Survey Data Analysis

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

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 0.539 assumption to obtain a valid estimate of correlation: pairs of observations are iid

39 Scatterplot of Evaluations
Rank correlation is 0.539

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

41 Scatterplot by Pay Category

42 Correlations within Level
Correlations within each pay level Level 2: –0.293 Level 3: –0.010 Level 4: 0.317 Level 5: 0.496 Only Level 4 is significantly different from 0

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

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

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

46 BMI: Women

47 BMI: Men

48 BMI: Comparisons

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

50 Ontario Health Survey

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


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


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

56 Computing Packages STATA and R

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

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

59 R: survey package define the sampling design: svydesign output
wk1de<- svydesign(id=~ea,strata=~newstrata,weight=~indiv_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)

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

61 Available Survey Commands
STATA Descriptive Yes Regression Yes (More) Resampling Longitudinal No PMLE Calibration

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

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)

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 ith row and jth column:

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?

66 STATA Commands

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

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

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

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

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

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

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

74 Linear and Logistic Regression
Survey Data Analysis Linear and Logistic Regression

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

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

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

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

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 

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

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

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

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

84 Partial STATA Output No interaction terms Interaction terms present

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.

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

87 STATA Output

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

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

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

91 STATA Commands

92 STATA Output part I

93 STAT Output part II

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 [pweight = axrwght], family(poisson) link(identity) corr(exchangeable)

95 GEE Results

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

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

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

99 STATA Output

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