Why is the correlation coefficient parametric?

Question

I am a little confused on the definition of "parametric". The book that I'm reading writes that "the correlation coefficient attempts to estimate a particular parameter in the Normal model for two quantitative variables." I am a little confused as to what this "parameter in the Normal model" is and why, for example, Spearman's rho is not tied to a specific parameter. I've Googled around and still haven't found a good answer. The top answer here Why is Pearson parametric and Spearman non-parametric doesn't really answer why Pearson's coefficient is necessarily parametric.

Answer 1

The Pearson correlation coefficient itself is neither parametric nor nonparametric.

(Nor are means, variances, etc, nor for that matter are medians either parametric or nonparametric.)

Many basic books are quite misleading on the issue of what parametric and nonparametric mean and how they matter.

The term parametric at heart refers to a situation where you have a distribution that is defined up to a fixed, finite number of parameters. (Some parameters might not be free under this formulation; e.g. they might be set to zero or be constrained to equal a function of still other parameters depending on the situation and the parameterization at hand)

Nonparametric is what you have when the distribution may depend on a number of parameters that are not fixed and may potentially grow without bound (occasionally called 'infinite-parametric' though it's not a requirement that the number of parameters could ever be infinite). It may often refer to situations that make some assumptions about distributional form (e.g. symmetry), but not ones that yield an explicit parametric form of the population distribution from which the sample was drawn.

[The terminology was coined by Jacob Wolfowitz in the 1940s; his definitions were fairly similar to these.]

There's some variation in appropriate uses of the terms, but my description will apply to most conventional usages (including nonparametric regression methods, where the conditional distribution of the response may require a non-"fixed-and-finite" number of parameters to describe it - usually through its mean).

In a bivariate distribution, the Pearson correlation is a population parameter -- but there's no requirement that the remaining parameters be fixed or finite, nor even that the Pearson correlation define the dependence structure.

So in relation to the Pearson correlation whence 'parametric' at all?

It comes in when trying to construct confidence intervals and tests.

If you assume bivariate normality*, then the Pearson correlation is one of a fixed, finite number of parameters that together define that bivariate normal; moreover the properties of that Pearson correlation don't depend on the other parameters (outside some edge-cases). So, for example, when the population Pearson correlation is $0$ you can write down the distribution of the sample correlation (and thereby, construct a hypothesis test for the correlation) under that bivariate-normality assumption.

Specifically in the bivariate normal, the density can be written in this form (avoiding matrix notation, though to my mind that's a little more straightforward):

$f(x,y)={\frac {1}{2\pi \sigma _{X}\sigma _{Y}{\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {1}{2(1-\rho ^{2})}}\left[{\frac {(x-\mu _{X})^{2}}{\sigma _{X}^{2}}}+{\frac {(y-\mu _{Y})^{2}}{\sigma _{Y}^{2}}}-{\frac {2\rho (x-\mu _{X})(y-\mu _{Y})}{\sigma _{X}\sigma _{Y}}}\right]\right)$

The parameters in that formulation are $\mu_X,\mu_Y,\sigma_X,\sigma_Y$ and $\rho$. The population Pearson correlation is $\rho$. Together they specify completely the distribution of $(X,Y)$.

It's not the correlation itself that is parametric (though it is a population parameter); it's the additional assumptions that were put in place in order to derive a distribution of the test statistic under the null hypothesis that are "parametric".

It is not necessary to make this particular assumption to test a Pearson correlation. For example:

• You could make different parametric assumption, and in some cases identify a different distribution of the Pearson correlation under $H_0$, whether algebraically, via simulation, or by asymptotic (or other) approximation.

• You could make no parametric assumption whatever and instead construct a nonparametric test (e.g. permutation test or a bootstrap test).

*(or multivariate normality more generally, with $p \choose 2$ correlation parameters in all, along with $p$ means and $p$ variances)