How To Calculate Approximate Expectation Of Function Of A Binomial Random Variable?

For a bivariate data sample, is calculating the conditional expectation of variable Y given X directly from the samples (i.e. without any regression function), a principled way to show the qualitative relationship between the two?

• Suppose I have N samples of data consisting of tuples (X,Y) where X and Y are discrete quantities. Is plotting the conditional expectation E(Y|X) calculated directly from the data (i.e. without any regression fit) a principled way to show the relationship between X and Y? By relationship I don't mean a precise functional relationship, but rather a qualitative description, for example, "Y increases with X upto some value and then decreases", and so on.  If so, how do I calculate the error in these conditional expectation values, considering the number of data points may not be the same for all X?

• Answer:

  You have posed the question so that it essentially contradicts itself. First, you say you want a qualitiatitve description of the relationship between the two variables. Fine.  You also want to look at the relationship without any formal regression. Also fine; a good thing to do. You can plot the scatterplot and add a loess line. But you then say you want a description such as "Y increases with X upto some value and then decreases", I have absolutely nor problem with that. But... if you want a precise, quantified error, you need a precise, quantified statement.