The Friedman Aligned Rank Test in Actual Practice

Friedman's Aligned Rank Test serves as an alternative to the regular Friedman Test. The aligned rank test is a better choice when the data have very few treatments and the blocks have different magnitudes of effect and scale.

The regular Friedman test is less sensitive for smaller number of treatments. This is due to the fact that in Friedman test, the observations are ranked within a block. Thus the comparison is made among the responses within each block only. The comparison of responses between blocks will not be meaningful as the blocks might have different magnitudes of effect and scale among themselves. The blocks may be such that some may give consistently high, and others consistently low, responses.

This Friedman Test shortcoming is addressed in the Friedman Aligned Rank Test by subtracting from the observations in each block some estimate of a measure of location of the block, such as the average of the observations in the block or the median of these observations. This estimate must be a symmetric function of the observations of the block. If the blocks show substantial difference in scale too, then the blocks could be aligned by standardizing the observations within the blocks. In all these alignment methods the blocks are made comparable. Once such data alignment is completed, further analysis is carried out by the regular Friedman test method.

Example - Newborn Behavior Levels Study

This data set is derived from Lehmann (1975). The behavior levels of 35 newborns were recorded under four different soothing conditions. Hence these data contain 35 blocks and 4 treatments.

The Relevant Study Data
(1 indicates quiet, 6 indicates extreme agitation)

Analysis utilizing a Friedman Test and StatXact 8's Friedman Aligned Rank Test

Conclusions

Note the large difference in p-values between the two methods:

- 0.7288 by standard Friedman Test

- 0.4958 by Friedman Aligned Rank Test using Exact Monte Carlo method

This is a typical pattern where Friedman Aligned Rank Test proves to be more sensitive in detecting differences than the Friedman Test.

Reference

Lehmann EL (1975). Nonparametrics: Statistical Methods Based on Ranks. Holden-Day, San Francisco.

One of the new capabilities in LogXact 8 is the procedure Profile Likelihood Confidence Intervals for parameters of binary logistic regression. Users can also choose to or not to apply Firth correction - see Penalized Maximum Likelihood Method (PMLE) for specifics regarding Firth's method and application in practice.

This new procedure obtains the Profile Likelihood based confidence intervals for the parameters using the algorithm of Venzon and Moolgavkar (1988).

References

- Venzon, DJ and Moolgavkar(1988). A method for computing Profile Likelihood Based Confidence Intervals. Applied Statist. 37, No 1.

- Georg Heinze and Michael Schemper (2002). A solution to the problem of separation in logistic regression. Statistics in Med. 2002; 21:2409-2419.

- George Heinze and Meihard Ploner (2004). Technical Report 2 : A sas macro, s-plus library and R package to perform logistic regression without convergence problems.

Questions?