Exposing Students to Statistical Methods Based on Counts and Ranks

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Presentation transcript:

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

Getting the ball rolling

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.

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.)

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.

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.

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.

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)

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

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

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

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?

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

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

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?

Patient Data Nontrauma Trauma 20.1 38.5 22.9 25.8 18.8 22.0 20.9 23.0 20.1 38.5 22.9 25.8 18.8 22.0 20.9 23.0 20.9 37.6 22.7 30.0 21.4 24.5 20.0

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?

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.