What is the difference between segmented regression analysis and cate-nelson analysis? I observed a possible inflection point for my dataset and was introduced to segmented regression analysis and cate-nelson analysis. The segmented regression analysis seems to fit better for my dataset. I would like to know what is the theoretical difference between these two types of analysis? How do I decide or justify which analytical tool to use?
 A: Cate-Nelson analysis and segmented regression are really quite different, though they are sometimes used in similar situations.  They are both used in cases of bivariate data where the variables have a relationship that wouldn't be described well with a simple linear regression or nonlinear regression.
They are also often used to determine a "critical value".  In segmented regresion, this is the x value where the two segments meet.  In Cate-Nelson, this is the x value which separates the two groups of values.  I find it useful if software can report a confidence interval about this critical value.
Segmented regression is useful when the data follow the shape of two linear --- or curvilinear --- segments.
In the following example of a linear-plateau model, Life expectancy increases linearly, up to a point (about 2500 $/yr).  Above this x value, the data are fit with a linear segment with 0 slope.
http://rcompanion.org/handbook/images/image042.png
Note that this data could also be fit with a quadratic-plateau model, which has a more more natural transition between the two segments (critical x about 3500 $/yr).
http://rcompanion.org/handbook/images/image041.png
Cate-Nelson analysis is very different.  It is used to simply divide the bivariate data so that most of the data fall into a group with a high x and high y value or a low x and low y value.
In the following example, operations to the left of the vertical line (smaller than about 4 hectares) tend to have low rates of adoption, while those to the right of the vertical line tend to have high rates of adoption. 87% of observations fall within quadrants II and IV of the plot.  
http://rcompanion.org/rcompanion/images/h_02_03.png
Cate-Nelson analysis is useful in cases where the data could be seen to fall into quadrants like this, but don't follow a pattern such that it would be easy to fit a linear or nonlinear model to them.  For example, in the preceding plot, for x values below about 5, the y values don't really increase as x increases.  Traditionally, Cate-Nelson was also very useful because it didn't require much computing power:  The critical x and y values could be estimated manually and the points falling into quadrants could be counted.  It's also possible to set the y critical value.  For example, if the y-axis is crop yield, it could be set to 90% of the maximum to find the x value that separates > 90% yield from < 90% yield.
