1. Exploratory Analysis of Survey Data Lisa Cannon Luke Peterson

2. Presentation Outline Density Estimation  Nonparametric kernel density estimates  Properties of kernel density estimators  Other methods Graphical Displays  NHANES data

3. Three features that distinguish survey data: 1. Individuals in the sample represent differing numbers of individuals in the population - sampling weights used to estimate this. 2. Some data imputed due to item nonresponse. 3. Sample sizes can be quite large.

4. The Need for Nonparametric Methods We often study point estimation that assumes iid random variables. Stratification may result in violation of identically distributed random variables Clustering may result in violation of independence Methods we discuss use asymptotic properties that allow nonparametric methods for estimating shape of a distribution

5. Kernel Density Estimates Bellhouse and Stafford (1999) looked at kernel density estimation for  The whole data set  Binned data (groups the data after it is smoothed)  Smoothing binned data (smooths the data after it is grouped) Asymptotic integrated MSE for model-based and design-based derived.

6. Why Binning? To simplify estimation of large samples The shape of the data can be distorted by binning Smoothing helps to recover lost structure

7. Design-Based and Model-Based Different ways to handle the asymptotics Model-based: N finite population units are a sample of identically distributed units from infinite super-population Design Based: A nested sequence of N finite populations, where the distribution function of these populations converges as Weights do not affect bias, but the estimation of variance is inflated by the value for the design effect

8. Buskirk and Lohr (2005) Also addressed kernel density estimation Considers use of whole data (no binning) Also considered a combination of design- based and model-based approaches Explore conditions for consistency and asymptotic normality Defined confidence bands for the density

9. Applications Ontario Health Survey US National Crime Victimization Survey (NCVS) US National Health and Nutrition Examination Survey (NHANES)

10. Other Methods Bellhouse, Stafford (2001)– Polynomial regression methods Bellhouse, Chipman, Stafford (2004)– Additive models for survey data via penalized least squares method Korn et al. (1997) – Smoothing the empirical cumulative distribution function Graubard, Korn (2002)– Variance estimation Many others

11. Plotting Survey Data Common difficulties with plotting survey data:  Dealing with sampling weights  Plotting a large number of observations can be difficult to interpret See Korn and Graubard (1998).

12. National Health and Nutrition Survey (NHANES) Has been conducted on a periodic basis since 1971. Completes about 7,000 individual interviews annually. Analyzes risk factor for selected diseases and conditions. Sample implemented is a stratified multistage design. Data available at http://www.cdc.gov/nhaneshttp://www.cdc.gov/nhanes

13. Glycohemoglobin Level (Ghb) A blood test that measures the amount of glucose bound to hemoglobin. Normally, about 4% to 6%. People with diabetes have more glycohemoglobin than normal. The test indicates how well diabetes has been controlled in the 2 to 3 months before the test. Source: http://my.webmd.comhttp://my.webmd.com

14. Histograms Histograms provide a nice summary of the distribution of large data sets. Suppose that we would like to assess the distribution of glycohemoglobin levels. Sampling weights must be considered before plotting a histogram.

15. SAS Code: Account for Weights proc univariate data=explore.glyco noprint; var glyco; freq weight; histogram / nrows=2 cfill=red midpoints=3 to 15 by 0.5 cgrid=grayDD; run; The variable weight indicates the number of population units the sample unit represents.

16. Histograms – Effect of Sampling Weights

17. Boxplots Boxplots indicate location of important summary statistics along with distribution. See Figures 7.8 and 7.10 in Lohr. The boxplot procedure in SAS will not accept any arguments to account for weights. The survey library in R will.

18. Graphs for Regression – Bubble Plots Scatterplots are inadequate for survey data as they fail to account for sampling weights. Bubble plots incorporate the weights by making the area of each circle proportional to the number of population observations at those coordinates (See Lohr, Chapter 11). The ordinary least squares regression line is then replaced by a weighted least squares line. See Figure 11.5 in Lohr

20. Bubble Plot for NHANES Data

21. Dealing with Large Samples Bubble plots are hard to interpret for large data sets due to overlapping bubbles. Potential solutions:  Create a “sampled scatterplot” in which we sample from the original data where probability of selection is proportional to sample weights.  “Jitter” the data by adding some random noise to the values before plotting. These and others discussed in Korn and Graubard (1998).

22. SAS Code: Plotting a representative subsample proc surveyselect data=explore.glyco out=plotdata method=pps sampsize=300 seed=3452; size weight; run; symbol1 v=circle i=r c=black ci=green w=2; proc gplot data=plotdata; plot glyco*age; run;

23. Subsample: Glycohemoglobin vs. Age

24. Plotting Recommendations For univariate displays, adjust for the sampling weights. For scatterplots, sampling weights can be accounted for by using bubble plots. If the sample is large, a subsampling procedure that incorporates the weights might be more appropriate.

25. References Bellhouse,D.R. and Starfford, J.E. (1999). Density Estimation from complex surveys. Statistica Sinica. Bellhouse, D. R. and Stafford, J.E. (2001). Local polynomial regression in complex surveys. Survey Methodology. Bellhouse, D.R. and Stafford, J.E. (2004). Additive models for survey data via penalized least squares. Technical Report. Buskirk, T.D. and Lohr, S.L. (2005). Asymptotic properties of kernel density estimation with complex survey data. Journal of Statistical Planning and Inference. Graubard, B.I. and Korn E.L. (2002). Inference for superpopulation parameters using sample surveys. Statistical Science. Korn, E.L., Midthune, D., and Graubard, B.I. (1997). Estimating interpoloated percentiles from grouped data with large samples. J. Official Statist. Korn, E.L. and Graubard, B.I. (1998). Scatterplots with survey data. The American Statistician.