1. EHS 655 Lecture 11: Transformations, inferential statistics (t-test, ANOVA)

2. What we’ll cover today Transformations Inferential statistics
t-test
ANOVA
Review of midterm report requirements

3. TRANSFORMING VARIABLES
Many inferential statistical methods assume data are normally distributed
t-test
ANOVA
Linear regression
However, many exposures positive and right-skewed

4. One solution: log-transform data
Yi=ln(xi)
where
Yi is log-transformed data point
xi is original data point
ln is natural logarithmic function
Natural log (ln) transform of lognormally distributed variable has properties of normal distribution
i.e., bell-shaped and symmetric
Described by geometric mean (GM) and geometric standard deviation (GSD)

5. Log transformation Exposure distributions – original and transformed
Rappaport and Kupper, 2008

6. Log transformation

7. Evaluating lognormal distribution
Quantile-quantile plots
Untransformed Log-transformed
Stata: qnorm varname1

8. Interpreting log transformed estimates
Arithmetic mean of log transformed exposures
Arithmetic SD of log transformed exposures
Geometric mean
Antilog of mean of log-transformed exposures
Geometric standard deviation
Stata: can use two combinations for transformation
ln() or log() and exp() … OR … log10() and 10log10value

9. Caution about transformation
Back-transformed mean ≠ original variable mean
GM isn't easily interpreted
Proper to run statistical tests on transformed values
But often report means in unit of untransformed scale as well
“If it ain’t broke, don’t fix it.” Transformation bad if:
Distribution more or less symmetrical, few outliers
Variances reasonably homogeneous
Transformation may be useful
Markedly skewed data or heterogeneous variances

10. INFERENTIAL STATISTICS
Descriptive statistics applied to populations are called parameters
Inferential statistics apply to samples
We’ll focus on two inferential approaches today
t-test
ANOVA

11. t-test

12. t-test
Detect differences between means of (normally-distributed) samples
Significant t-statistic = means differ
Student’s (unpaired) t-test
Test hypothesis that means of two samples are equal; null is
Stata: ttest varname1, by(groupvar)
Paired sample t-test
Test whether two measurements on same individual are equal
Stata: ttest varname1 == varname2

13. Things we can do with a t-test
Single-sample t-test: identify differences in the mean of a group and a reference value
Unpaired t-test: identify differences in mean exposures between two groups
Paired-sample t-test: identify differences in exposure before and after an intervention in a group of subjects

14. Interpreting a single-sample t-test in Stata

15. Interpreting a t-test between groups in Stata

16. Interpreting a paired t-test in Stata

17. ANOVA (ANalysis Of Variance)
Technique for assessing how categorical independent variables affect continuous dependent variable
Like a t-test generalized to three or more means
Tells use whether means from k groups are same or not
Null hypothesis:

18. Things we can do with ANOVA
Identify differences in mean exposures between more than two groups
Evaluate relationship of within-worker variance within exposure group to between-worker variance
Within-worker > between worker = good exposure grouping
Within-worker < between worker = poor exposure grouping

19. ANOVA assumptions Continuous dependent variable
Independent variable is 2+ categorical groups
Data independent from each other
Errors normally distributed
Variances same for all groups
ANOVA fairly robust for these assumptions
But data should not be extremely far off

20. ANOVA illustrated

21. Generic ANOVA components

22. ANOVA – F-test
Compares variability in exposure accounted for by predictor variable vs error variability
Error variability (mean squared error) measures inherent randomness of observations
Large differences between groups = significant F test

23. F-statistic

24. F-statistic

25. Stata ANOVA output Stata: oneway responsevar groupvar
Bigger F = significant

26. Stata ANOVA output

27. Stata ANOVA output Stata: anova responsevar groupvar
Note different output: now get R2, adj R2, RMSE, etc. More in regression lecture

28. Stata ANOVA output Stata: oneway responsevar groupvar, tabulate
Tabulate gives results by group

29. Why use ANOVA instead of t-test?
Could do t-tests for all pairs of predictor variable categories
Not a good idea
As number of exposure groups grows, so does number of needed pair comparisons
Each comparison introduces risk of error
ANOVA puts all data into one number (F) and gives one P for null hypothesis

30. What if I want to know which groups are different
Multiple comparisons possible
After you run oneway command, use this second command
Stata: pwcompare groupvar, effects sort mcompare(tukey)

31. Multiple comparison ANOVA output

32. Measure of agreement between categorical and continuous variables
Stata: loneway responsevar groupvar
Intraclass correlation coefficient = measure of agreement, same scale as Cohen’s kappa

33. ANOVA in action
Enough with words already. Let’s see how ANOVA actually works
Stata ANOVA commands: oneway responsevar groupvar Option (to get more detailed output by group) oneway responsevar groupvar, tabulate means standard

34. Resources Choosing statistical tests
Stata annotated output from various tests

35. Review of midterm report

36. Example of noise exposure calculation requiring transformation
Can describe noise exposures (in dBA) across individuals arithmetically
In other words, to estimate a group mean for individuals in, say, the same trade, compute arithmetic mean
To estimate average noise exposures within individual (in dBA) is computing dose
Requires temporary transformation
LEQi= 10 log [1/N (10 (TWA1/10) +10 (TWA2/10) + …+ 10 (TWAn/10))]
Where N is total number of TWAs used to estimate average LEQ for person i
How to operationalize in Stata?
Note temporary transformation