# Calculate correlations and correlation tables with complex survey data

#### Jacob Long

#### 2023-07-06

Source:`vignettes/svycor.Rmd`

`svycor.Rmd`

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 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.

```
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.

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

```
api00 api99
api00 1.00 0.98
api99 0.98 1.00
```

You can specify with the `digits =`

argument how many
digits past the decimal point should be printed.

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

```
api00 api99
api00 1.0000 0.9759
api99 0.9759 1.0000
```

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`

.

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

```
api00 api99
api00 1 0.9759*
api99 0.9759* 1
```

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.

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

```
api00 api99
api00 1.0000000 0.9759047
api99 0.9759047 1.0000000
```

`c$p.values`

```
api00 api99
api00 0 0
api99 0 0
```

`c$std.err`

```
api00 api99
api00 0.000000000 0.003407475
api99 0.003407475 0.000000000
```

## Technical details

The heavy lifting behind the scenes is done by `svyvar`

,
which from its output you may not realize also calculates
covariance.

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

```
variance SE
api00 15191 1255.7
api99 16518 1318.4
```

But if you save the `svyvar`

object, you can see that
there’s more than meets the eye.

```
api00 api99
api00 15190.59 15458.83
api99 15458.83 16518.24
attr(,"var")
api00 api00 api99 api99
api00 1576883 1580654 1580654 1561998
api00 1580654 1630856 1630856 1657352
api99 1580654 1630856 1630856 1657352
api99 1561998 1657352 1657352 1738266
attr(,"statistic")
[1] "variance"
```

Once we know that, it’s just a matter of using R’s
`cov2cor`

function and cleaning up the output.

```
cor <- cov2cor(var)
cor
```

```
api00 api99
api00 1.0000000 0.9759047
api99 0.9759047 1.0000000
attr(,"var")
api00 api00 api99 api99
api00 1576883 1580654 1580654 1561998
api00 1580654 1630856 1630856 1657352
api99 1580654 1630856 1630856 1657352
api99 1561998 1657352 1657352 1738266
attr(,"statistic")
[1] "variance"
```

Now to get rid of that covariance matrix…

```
api00 api99
api00 1.0000000 0.9759047
api99 0.9759047 1.0000000
```

`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.

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

```
api99 api00
api99 1.0000000 0.9759047
api00 0.9759047 1.0000000
```