Is there a standard process for finding linear relationships with matching?

Question

Most of my reference on matching has been Rosenbaum's book "Observational Studies". From what I understand, the usual process of matching involves administering a treatment that is represented by a binary value of either 0 or 1, depending on whether the patient got the treatment or not. In the case that I am looking at, the treatment is represented as a continuous value with no particular meaning given to "0" treatment, making it difficult/non-applicable to assign a control.

In my hypothesis there is a linear relationship between the amount of treatment and the outcome and I want to test for that. During these tests I need to apply all the corrections for overt biases included in the Matching procedure. What is the best method for testing for a linear relationship?

What I have been doing, perhaps naively, is sampling at 2 different levels of treatment, $x_1$ and $x_2$, and with their mean outcomes, $u_1$ and $u_2$, I've been estimating the relationship as the line given by the two-point equation passing through $(x_1, u_1)$ and $(x_2, u_2)$. This turns out to be too inaccurate. Would it be better for me to sample at several levels of treatment $x_1, x_2, x_3, \dots,x_n$ and do a linear regression on that data or is there another standard way of doing it?

Thanks

Answer 1

In the simple case of a single treatment and a control group, the idea of matching is for the two groups to look identical in every pre-treatment respect. The theory behind matching, which Rosenbaum lays out, is that any post-treatment differences in the groups can be ascribed to the treatment status.

To apply that logic to your case, you would need every treatment level to look similar on pre-treatment characteristics. Alternatively, you could pick a reference group and perform matching from each other group to that one.

For example, suppose that you have a treatment level of 0 and you want this to be your reference group. You could match from treatment group 1 to group 0 and get an estimated impact of going from group 0 to group 1. Then, match from group 2 to group 0 and get the impact of going from group 0 to group 2, and so on. All of your impacts become measured relative to the base group.

As is likely obvious, this only works if you have a relatively small number of groups. You could discretize your treatment into a few groups (maybe based upon the quartiles or quantiles of treatment level?) and perform this analysis. Then, ask whether the estimated impacts increase in a relatively linear fashion. If so, you might try a standard linear regression model and see how the results compare. If the pattern isn't linear, you might be able to apply a function to treatment (take the log for impacts that grow slower than linearly or instead take the log of the outcome for impacts that grow faster than linearly) and apply the linear model.