1. Exposing Students to Statistical Methods Based on Counts and Ranks
Beyond The Formula VIII
Monroe Community College
August 5, 2004
Brad Hartlaub
Kenyon College

2. Getting the ball rolling

3. Why Study Alternative Methods for Inference?
These procedures are applicable to data with far fewer assumptions than the usual classical procedures.
They are less sensitive to departures from the assumptions - with tests, the level is maintained even if the assumed model is incorrect. Procedures are less sensitive to extreme or unusual observations.

4. More Reasons Simplicity and intuitive nature of the procedures.
Applicable to specific types of data for which classical procedures are not appropriate. E.g., rank and count data.
Technology advances have increased accessibility for the general practitioner. (bootstrapping, jackknifing, simulation, etc.)

5. The Distribution-Free Property
Many nonparametric tests, intervals, statistics, confidence bands, and multiple comparisons procedures are distribution-free. Roughly speaking, this means that these methods do NOT depend on the specific distributional form of the population.

6. General Description of One-Sample Location Problem
Paired Replicates Data - We have an experiment such that each time it is conducted we obtain two observations. We refer to these as “pre-treatment” (X) and “post-treatment” (Y) observations. We are interested in assessing the effect of the treatment on our measured observations.
We are developing a nonparametric competitor to the paired t-test.

7. Model Let refer to the “typical effect” of the treatment.
Let Zi=Yi-Xi, for i=1,…, n and assume that:
The n Zi’s are mutually independent;
Each Zi has a continuous distribution with median .
Note that we do NOT need to assume that the Zi’s are normally distributed. Also, we do not need to assume that the Zi’s have the same distribution.

8. We want to make inferences for
Three ways to use the information in the Zi’s
Use all of the information (Paired t-test)
Use the information on Y beating X and the relative magnitudes of the Zi’s. (signed rank)
Use “minimal” information on whether or not Y beats X. (sign)

9. Sign Test We want to test Test Stat: Example versus Privgov.mtw
SignTest.doc

10. Large Sample Approximation
Since the distribution of B under the null hypothesis is Binomial(n, 1/2), the approximate level test is based on the standardized statistic

11. Wilcoxon Signed Rank Wilcoxon signed rank test statistic
where Ri is the rank of |Zi| among |Z1|, …, |Zn|.
Find the null distribution for T+ when n=4
Large sample approximation for the signed rank test

12. Another Example
An experiment was conducted to test a method for reducing faults on telephone lines. Fourteen matched pairs of areas were used. Do the data provide significant evidence to conclude that the faults have been reduced?

13. Fault Data Test Control 676 88 206 570 230 605 256 617 280 653 433
2913
337
924
466
286
497
1098
512
982
794
2346
428
321
452
615
519

14. Two-sample location problem
Wilcoxon rank sum test statistic
W=sum of the joint ranks for the Y-sample
Mann-Whitney statistic
U=the number of pairs where Y is larger than X
Obtaining the null distribution for W
Large sample approximation for the Wilcoxon rank sum test

15. Trauma and Metabolic Expenditure
Metabolic expenditures (kcal/kg/day) were collected for 8 patients admitted to a hospital for reasons other than trauma and for 7 patients admitted for multiple fractures (trauma). Are the expenditures significantly different for the two groups of patients?

16. Patient Data Nontrauma Trauma 20.1 38.5 22.9 25.8 18.8 22.0 20.9 23.0
20.0

17. Point and Interval Estimation
Form the mn differences, Yj - Xi; i=1,…, m; j=1,…n.
The point estimator for the location parameter is the median of these differences.
Another common parameter of interest in this two sample location problem is P(Y>X). How would you estimate this probability?

18. Conclusion
The more practice your students get with statistical inference, the more comfortable they will be in making appropriate inferences.
Including alternative procedures for inference clearly illustrates why it is so important to check all assumptions and conditions.