# Delta method

## Sampling variance of Pearson r in a two-level design

Consider Pearson’s correlation coefficient, (r), calculated from two variables (X) and (Y) with population correlation (\rho). If one calculates (r) from a simple random sample of (N) observations, then its sampling variance will be approximately [ \text{Var}® \approx \frac{1}{N}\left(1 - \rho^2\right)^2. ] But what if the observations are drawn from a multi-stage sample? If one uses the raw correlation between the observations (ignoring the multi-level structure), then the (r) will actually be a weighted average of within-cluster and between-cluster correlations (see Snijders & Bosker, 2012).

## The multivariate delta method

The delta method is surely one of the most useful techniques in classical statistical theory. It’s perhaps a bit odd to put it this way, but I would say that the delta method is something like the precursor to the bootstrap, in terms of its utility and broad range of applications—both are “first-line” tools for solving statistical problems. There are many good references on the delta-method, ranging from the Wikipedia page to a short introduction in The American Statistician (Oehlert, 1992).

## 2SLS standard errors and the delta-method

I just covered instrumental variables in my course on causal inference, and so I have two-stage least squares (2SLS) estimation on the brain. In this post I’ll share something I realized in the course of prepping for class: that standard errors from 2SLS estimation are equivalent to delta method standard errors based on the Wald IV estimator. (I’m no econometrician, so this had never occurred to me before. Perhaps it will be interesting to other non-econometrician readers.