---
title: "Calculate correlations and correlation tables with complex survey data"
author: "Jacob Long"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Calculate correlations and correlation tables with complex survey data}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r echo=FALSE}
required <- c("survey", "weights")
knitr::opts_chunk$set(message=F, warning=F, comment = "")
if (!all(sapply(required, requireNamespace, quietly = TRUE)))
  knitr::opts_chunk$set(eval = FALSE)
library(jtools)
```

The `survey` package is one of R's best tools for those working in the social sciences. 
For many, it saves you from needing to use commercial software for research that uses
survey data. However, it lacks one function that many academic researchers often 
need to report in publications: correlations. The `svycor` function in `jtools` helps to fill that gap. 

A note, however, is necessary. The initial motivation to add this feature comes from a [response to a question](https://stackoverflow.com/questions/34418822/pearson-correlation-coefficient-in-rs-survey-package#41031088) about
calculating correlations with the `survey` package written by Thomas Lumley, the `survey`
package author. All that is good about this function should be attributed to Dr. Lumley;
all that is wrong with it should be attributed to me (Jacob).

With that said, let's look at an example. First, we need to get a `survey.design` 
object. This one is built into the `survey` package.

```{r}
library(survey)
data(api)
dstrat <- svydesign(id = ~1,strata = ~stype, weights = ~pw, data = apistrat, fpc=~fpc)
```

## Basic use

The necessary arguments are no different than when using `svyvar`. Specify, using 
an equation, which variables (and from which design) to include. It doesn't matter
which side of the equation the variables are on.

```{r}
svycor(~api00 + api99, design = dstrat)
```

You can specify with the `digits =` argument how many digits past the decimal point
should be printed.

```{r}
svycor(~api00 + api99, design = dstrat, digits = 4)
```

Any other arguments that you would normally pass to `svyvar` will be used as well,
though in some cases it may not affect the output.

## Statistical significance tests

One thing that `survey` won't do for you is give you *p* values for the null hypothesis
that $r = 0$. While at first blush finding the *p* value might seem like a simple 
procedure, complex surveys will almost always violate the important distributional
assumptions that go along with simple hypothesis tests of the correlation coefficient.
There is not a clear consensus on the appropriate way to conduct hypothesis tests in
this context, due in part to the fact that most analyses of complex
surveys occurs in the context of multiple regression rather than simple bivariate cases.

If `sig.stats = TRUE`, then `svycor` will use the `wtd.cor` function from the `weights`
package to conduct hypothesis tests. The *p* values are derived from a bootstrap 
procedure in which the weights define sampling probability. The `bootn =` argument
is given to `wtd.cor` to define the number of simulations to run. This can significantly
increase the running time for large samples and/or large numbers of simulations. The 
`mean1` argument tells `wtd.cor` whether it should treat your sample size as the number of
observations in the survey design (the number of rows in the data frame) or the 
sum of the weights. Usually, the former is desired, so the default value of `mean1` is 
`TRUE`.

```{r}
svycor(~api00 + api99, design = dstrat, digits = 4, sig.stats = TRUE, bootn = 2000, mean1 = TRUE)
```

When using `sig.stats = TRUE`, the correlation parameter estimates come from the 
bootstrap procedure rather than the simpler method based on the survey-weighted
covariance matrix when `sig.stats = FALSE`.

By saving the output of the function, you can extract non-rounded coefficients, *p* 
values, and standard errors.

```{r}
c <- svycor(~api00 + api99, design = dstrat, digits = 4, sig.stats = TRUE, bootn = 2000, mean1 = TRUE)

c$cors

c$p.values

c$std.err
```


## Technical details

The heavy lifting behind the scenes is done by `svyvar`, which from its output you
may not realize also calculates covariance.

```{r}
svyvar(~api00 + api99, design = dstrat)
```

But if you save the `svyvar` object, you can see that there's more than meets the eye.

```{r}
var <- svyvar(~api00 + api99, design = dstrat)
var <- as.matrix(var)
var
```
Once we know that, it's just a matter of using R's `cov2cor` function and cleaning 
up the output.

```{r}
cor <- cov2cor(var)
cor
```

Now to get rid of that covariance matrix...

```{r}
cor <- cor[1:nrow(cor), 1:nrow(cor)]
cor
```

`svycor` has its own print method, so you won't see so many digits past the decimal
point. You can extract the un-rounded matrix, however.

```{r}
out <- svycor(~api99 + api00, design = dstrat)
out$cors
```
