1. EHS 655 Lecture 4: Descriptive statistics, censored data

2. What we’ll cover today Descriptive analysis and visualization
Distribution
Central tendency
Dispersion
Censored data
Stata – basic commands

3. DESCRIPTIVE ANALYSIS
Before we can make inference from data we must thoroughly examine variables
Catch mistakes
Look for patterns
Find violations of statistical assumptions
Generate hypotheses
Avoid headaches later

4. Scope of dataset/analysis
Univariate
Measurements on one variable per subject
Bivariate
Measurements on two variables per subject
Multivariate
Measurements on many variables per subject
Today’s focus

5. UNIVARIATE ANALYSES Characteristics of single variable Typically
Distribution (frequency distribution)
Central tendency (mean, median, mode)
Dispersion (range, quartiles, absolute deviation, variance, standard deviation)

6. Distribution: categorical (table)
Stata: “tab varname”

7. Distribution: categorical (ordinal)
Stata: “graph bar (percent), over(varname)”

8. Distribution – quantitative (ratio) histogram
Stata: “histogram varname, freq” (add “normal” to superimpose normal curve)

9. Distribution: cumulative distribution
Stata: “cumul varname, gen (newvar) line newvar varname, sort”

10. Distribution: exceedance fraction

11. Central tendency: mean, median, mode
Use: identify “center” around which data are distributed
Mean: Best for symmetric, non-skewed distributions
Median: Best for skewed distribution or data with outliers
Mode: Dataset may be bimodal, or may lack mode

12. Examples of central tendency
Symmetrical, unimodal
Symmetrial, bimodal
Positively skewed, unimodal
Negatively skewed, unimodal

13. When to use mean, median, mode
Stata: “tabstat varname, stat (mean median)”
Note: Stata does not have an easy way to identify mode
Type of variable
Best measure of central tendency
Nominal
Mode
Ordinal
Median
Interval/ratio (not skewed)
Mean
Interval/ratio (skewed)

14. Dispersion
Measures which identify spread of data (i.e., how far measurements are from “center”)
Range
Quartiles
Standard deviation (SD)
Variance
Coefficient of variation
Stata: “sum varname”
Provides n, range, SD or
Stata: “sum varname, detail”
Provides n, range, quartiles, SD, variance
Stata: “tabstat varname, stat(mean sd median range iqr cv)

15. Dispersion: range Simplest measure of dispersion Range
Maximum - minimum
Range

16. Dispersion: quartiles
3 points that divide data set into 4 equal groups
1st quartile (Q1) marks lowest 25% of data = 25th percentile
2nd quartile (Q2) splits data set in half = 50th percentile
3rd quartile (Q3) marks highest 25% of data = 75th percentile
Upper – lower quartile is interquartile range (IQR)

17. Dispersion: boxplot Stata: “graph box varname1, over(varname2)”

18. Dispersion: standard deviation
Variation of data
Not dependent on n
Not affected by number of measurements
Expressed in same units as data
Commonly used in exposure analysis

19. Variance Square of standard deviation
Squaring eliminates negative values
Unit is square of measurement unit (!)
Values farther from mean contribute more to variance
Commonly used in exposure analysis

20. Coefficient of variation
Normalized measure of dispersion
Dimensionless, often expressed as %
Allows comparison of datasets with different units or means
Unlike σ, cannot be used to construct confidence intervals around mean
Mean close to 0 = Cv will approach infinity

21. BIVARIATE ANALYSES
Allow us to begin to explore relationships between variables
Scatter plot
Correlation
Cross-tabulation

22. Bivariate: scatter plot
Stata: “scatter varname1 varname2”

23. Bivariate: Pearson correlation
r (Pearson’s correlation coefficient) is amount of change in one value you expect from change in another value
Assumptions:
Both variables interval or ratio data
Both variables normally distributed
Absence of outliers
Linear relationship
Homeskedasticity
Stata: “pwcorr varname1 varname2, sig”

24. Bivariate: correlation

25. Bivariate: Spearman correlation
Spearman’s rank correlation coefficient (rs or ρ)
Pearson correlation coefficient between ranked variables
Raw scores Xi, Yi converted to ranks xi, yi
Nonparametric (no distributional assumptions)
Assumptions:
Ordinal, interval, or ratio data
Monotonic relationship
Stata: “spearman varname1 varname2, stats(rho p)”

26. Bivariate: spearman vs. Pearson correlation coefficient examples

27. Bivariate: Cross-tabulation
Stata: tab varname1 varname2

28. CENSORED DATA Uncensored/complete Left censored Interval censored
Value of each sample unit observed/known
Default assumption
Left censored
Data <max value
Interval censored
Data between min and maxi value
Right censored
Data >max value

29. Exercise
Come up with one example of exposure data where you might find each type of censoring
Right censoring
Left censoring
Interval censoring

30. Common approach #1 to dealing with censored data
Assign all censored data ½ LOD
Assumes data uniformly distributed below LOD

31. Common approach #2 to dealing with censored data
Hornung and Reed (1990)
More accurate than LOD/2 when data normally or lognormally distributed
Okay if <~50% data censored if low to moderate variability
Less accurate than LOD/2 for highly skewed data

32. On to Stata Basic data manipulation commands
Define label/name for a variable (“label variable”)
Create labels (“label define”)
Assign labels to variable (“label values”)
Rename variable (“rename”)
Generate a new variable (“generate”)
Replace an existing variable (“replace”)

33. On to Stata
Anyone have to use the “Break” button?