Isotonic regression and an alternative nonparametric trend test plus a parametric version
There are two nonparametric methods which test to see if a single response is ordered with respect to groups which have an assumed apriori ordering.
Isotonic regression finds the best fitting model which has an unknown number of changepoints separating linear fits. An alternative nonparametric approach, the Jonckheere-Terpstra test, compares the ordering in the groups to that which would be expected by chance.
Both these tests may be fitted in the statistical package, R, using code [http://www.stat.umn.edu/geyer/5601/examp/oneway.html here.] The specific code of interest is listed [:FAQ/isoR_eg: here.] The more recent ISOREG procedure in R will also perform an isotonic regression. In either case a p-value is produced to assess whether the groups form an ordering with respect to the observed response. The basic isotonic regression model fitted in R pools means of responses at particular time points until they form a strictly increasing set of means with respect to the categories. The least squares fit from these poolings is then compared to that using the grand (overall) mean (and hence no changepoints) to assess the improvement in fit.
Jonckheere's trend test is also fitted in the analyse>nonparametric tests procedure in SPSS. Further details of [:FAQ/JonckheereTrendTest: Jonckheere's Trend Test] with an example, are also given.
There is also a parametric version which fits linear regression lines and finds the minimum residual sums of squares to determine the location of a fixed number of changepoints. R code for one and two changepoint examples are given [:FAQ/Rcpt: here.]
Howell (2013, p.646-9) illustrates a related issue allowing the fitting of separate slopes and intercepts for baseline and intervention trials on each of four single cases using a standard regression analysis of phase (baseline or intervention), trial number and phase by trial interaction.
In addition a comprehensive suite of changepoint programs with R code are illustrated [http://cran.r-project.org/web/packages/changepoint/changepoint.pdf here] (or [attachment:cp.pdf here] if the link is not working). The programs obtain locations of changepoints using information functions which minimize lack of fit using likelihood functions which assume different distributions of the data and use a penalty function which favours models with smaller numbers of changepoints.
References
Howell, D.C. (2013). Statistical methods for psychology. 8th Edition. International Edition. Wadsworth:Belmont,CA.
SPSS Version 3 reference manual. To be confirmed.