I have some data to which I am fitting piecewise linear models. I want to select different subsets of the data, fit a model to each of them, and then compare which subsets are best able to be described by a single line or by a multiple part line. Each subset that I am testing shares points with other subsets and are of different sizes. All smaller subsets are completely contained by larger subsets.
Someone has suggested that I calculate and compare the reduced chi-squared of the best-fit models for each subset to do so. Their point is that since the normal chi-squared statistic is modified to account for the number of samples within each subset the reduced chi-squared's can be directly compared between subsets to answer this question.
I'm not clear whether this is statistically justified. It seems to me intuitively that since the models are being fit to different data it doesn't make sense to compare this statistic in this way, but I'm not sure and would be grateful for any insight.