# Correlation

## Explanation

In a conceptual model, the concepts are normally placed in a rectangular. We have two concepts, X and Y.
The double headed arrow indicates that you do not assume a causal relation but a simple correlation. Thus: if X is high, Y is high; when Y is low, X is low.
It is not always necessary to label the paths but for this tutorial it will turn out to be handy. Normally, when there is no sign (or label) it is assumed that the path has a positive valence. It is, however, good practice to include the valence (+/-) of the paths in your conceptual models.

Please note, I use the label concept and not variable. A variable is something that is part of your dataset, measured by, for example, a survey item. A concept is a theoretical construct, and this ‘concept as intended’ may have relations - according to your theory - with other concepts. This concept is measured by one or more variables, the ‘concept as measured’.
It is generally a good idea to use concepts that are (very) close to your actual measurements. Thus, although you may use the concept social cohesion in your conceptual model, this concept is overly broad and there is a fierce debate on how it should be defined theoretically and how we should measure it empirically. If you have measured social cohesion with for example a single item on generalized trust (“Generally speaking would you say that most people can be trusted or that you can’t be too careful in dealing with people?”) why not use the concept generalized trust in your conceptual model?

## Abstract hypothesis

Hypo: X is related to Y.
or:
Hypo: X and Y are related
better:
Hypo: X and Y are positively related

## Real life example

X is occupational success.
Y is health

Hypo 1a: People’s health and occupational success are positively related to each other.

You may have a more precise idea on why occupational success and health are related. Perhaps because of a third (unmeasured) concept such as social origin (e.g. income and parental education). That is, the relation is at least in part spurious (see menu on the left). Alternatively, you may assume two bidirectional causal mechanisms. In the latter case your hypothesis could read:

Hypo 1b: A higher occupational status will lead to a better health and a better health will lead to a higher occupational status.

Where hypo 1a is fairly easy to test, hypo1b is not. Hypo 1b implicitly assumes that health at time T will cause in part your health at time T + 1 and that occupational success at time T will cause in part your occupational success at time T + 1 (i.e. stability paths). It also assumes some correlation between health and occupational success at time T. Finally, Hypo 1b assumes that within-individual changes in occupational success (health) are related to within-individual changes in health (occupational success). You could test this Hypo 1b with, for example, a Random Intercept Cross-Lagged Panel Model (RI-CLPM).

## Structural equations

• Y=X

or, following the syntax of the R package Lavaan

• Y~~X

The double ~~ indicates a (error)variance.

## Formal test of hypotheses

rm(list = ls())  #empty environment
require(haven)
nells <- read_dta("../static/NELLS panel nl v1_2.dta")  #change directory name to your working directory

Operationalize concepts.

# We will use the data of wave 2.
nellsw2 <- nells[nells$w2cpanel == 1, ] # As an indicator of occupational success we will use income in wave 2. table(nellsw2$w2fa61, useNA = "always")
attributes(nellsw2$w2fa61) # recode (I will start newly created variables with cm from conceptual models) nellsw2$cm_income <- nellsw2$w2fa61 nellsw2$cm_income[nellsw2$cm_income == 1] <- 100 nellsw2$cm_income[nellsw2$cm_income == 2] <- 225 nellsw2$cm_income[nellsw2$cm_income == 3] <- 400 nellsw2$cm_income[nellsw2$cm_income == 4] <- 750 nellsw2$cm_income[nellsw2$cm_income == 5] <- 1250 nellsw2$cm_income[nellsw2$cm_income == 6] <- 1750 nellsw2$cm_income[nellsw2$cm_income == 7] <- 2250 nellsw2$cm_income[nellsw2$cm_income == 8] <- 2750 nellsw2$cm_income[nellsw2$cm_income == 9] <- 3250 nellsw2$cm_income[nellsw2$cm_income == 10] <- 3750 nellsw2$cm_income[nellsw2$cm_income == 11] <- 4250 nellsw2$cm_income[nellsw2$cm_income == 12] <- 4750 nellsw2$cm_income[nellsw2$cm_income == 13] <- 5250 nellsw2$cm_income[nellsw2$cm_income == 14] <- 5750 nellsw2$cm_income[nellsw2$cm_income == 15] <- 6500 nellsw2$cm_income[nellsw2$cm_income == 16] <- 7500 nellsw2$cm_income[nellsw2$cm_income == 17] <- NA # let us scale the variable a bit and translate into income per 1000euro nellsw2$cm_income <- nellsw2$cm_income/1000 # from household income to personal income attributes(nellsw2$w2fa62)
table(nellsw2$w2fa62, useNA = "always") nellsw2$cm_income_per <- nellsw2$w2fa62 nellsw2$cm_income_per[nellsw2$cm_income_per == 1] <- 0 nellsw2$cm_income_per[nellsw2$cm_income_per == 2] <- 10 nellsw2$cm_income_per[nellsw2$cm_income_per == 3] <- 20 nellsw2$cm_income_per[nellsw2$cm_income_per == 4] <- 30 nellsw2$cm_income_per[nellsw2$cm_income_per == 5] <- 40 nellsw2$cm_income_per[nellsw2$cm_income_per == 6] <- 50 nellsw2$cm_income_per[nellsw2$cm_income_per == 7] <- 60 nellsw2$cm_income_per[nellsw2$cm_income_per == 8] <- 70 nellsw2$cm_income_per[nellsw2$cm_income_per == 9] <- 80 nellsw2$cm_income_per[nellsw2$cm_income_per == 10] <- 90 nellsw2$cm_income_per[nellsw2$cm_income_per == 11] <- 100 nellsw2$cm_income_per[nellsw2$cm_income_per == 12] <- NA nellsw2$cm_income_ind <- nellsw2$cm_income * nellsw2$cm_income_per/100

# as an indicator of health we will use subjective well being from 5 (excellent) to 1 (bad) thus we
# have to reverse code original variable

attributes(nellsw2$w2scf1) table(nellsw2$w2scf1, useNA = "always")
nellsw2$cm_health <- 6 - nellsw2$w2scf1
##
##    1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17 <NA>
##   55   78  103  204  338  326  282  272  276  205  133   62   48   22   22   29  374    0
## $label ## [1] " wat is het netto inkomen per maand van u en uw partner samen?/van u?/ " ## ##$format.stata
## [1] "%8.0g"
##
## $labels ## Minder dan ¤150 per maand ¤150 - ¤299 per maand ¤300 - ¤499 per maand ## 1 2 3 ## ¤500 - ¤999 per maand ¤1.000 - ¤1.499 per maand ¤1.500 - ¤1.999 per maand ## 4 5 6 ## ¤2.000 - ¤2.499 per maand ¤2.500 - ¤2.999 per maand ¤3.000 - ¤3.499 per maand ## 7 8 9 ## ¤3.500 - ¤3.999 per maand ¤4.000 - ¤4.499 per maand ¤4.500 - ¤4.999 per maand ## 10 11 12 ## ¤5.000 - ¤5.499 per maand ¤5.500 - ¤5.999 per maand ¤6.000 - ¤6.999 per maand ## 13 14 15 ## ¤7.000 of meer per maand weet niet, wil niet zeggen ## 16 17 ## ##$class
## [1] "haven_labelled" "vctrs_vctr"     "double"
##
## $label ## [1] " hoe groot is uw bijdrage in dit inkomen ongeveer? kunt u een percentage noemen " ## ##$format.stata
## [1] "%8.0g"
##
## $labels ## vrijwel geen bijdrage ongeveer 10% ongeveer 20% ongeveer 30% ## 1 2 3 4 ## ongeveer 40% ongeveer 50% ongeveer 60% ongeveer 70% ## 5 6 7 8 ## ongeveer 80% ongeveer 90% ongeveer 100% weet niet ## 9 10 11 12 ## ##$class
## [1] "haven_labelled" "vctrs_vctr"     "double"
##
##
##    1    2    3    4    5    6    7    8    9   10   11   12 <NA>
##  253   48   89  259  233  242  183  229  114   63  887  229    0
## $label ## [1] " wat vindt u, over het algemeen genomen, van uw gezondheid? " ## ##$format.stata
## [1] "%8.0g"
##
## $labels ## uitstekend zeer goed goed matig slecht ## 1 2 3 4 5 ## ##$class
## [1] "haven_labelled" "vctrs_vctr"     "double"
##
##
##    1    2    3    4    5 <NA>
##  438  853 1211  247   48   32

And test the correlation. Naturally, there are many ways to test for a correlation in R but in this tutorial I will try to do everything at least also in the package Lavaan.

But first plot the association:

# I randomly select 200 respondents otherwise the plot will be too crowded
selection <- sample(1:length(nellsw2$cm_income_ind), 200, replace = FALSE) # because we are interested in a correlation, I plot the standardized variables plot(scale(nellsw2$cm_income_ind[selection]), scale(nellsw2$cm_health[selection]), xlab = "income", ylab = "health", main = "Association between income and health", sub = "(standardized variables)") abline(lm(scale(nellsw2$cm_health) ~ scale(nellsw2$cm_income_ind), data = nellsw2), col = "red") I don’t see a very clear association. Perhaps at most a weak correlation. And now,…the correlation cor.test(nellsw2$cm_income_ind, nellsw2$cm_health) ## ## Pearson's product-moment correlation ## ## data: nellsw2$cm_income_ind and nellsw2\$cm_health
## t = 3.886, df = 2353, p-value = 0.0001047
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.03959172 0.11986098
## sample estimates:
##       cor
## 0.0798558

And with Lavaan.

require(lavaan)

model <- "
cm_income_ind ~~ cm_health
"

fit <- cfa(model, data = nellsw2)  #I use cfa instead of lavaan. The only advantage is that I don't have to tell lavaan that I also need the variances of the variables.
summary(fit, standardized = TRUE)
# parameterEstimates(fit)
## lavaan 0.6-7 ended normally after 10 iterations
##
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of free parameters                          3
##
##                                                   Used       Total
##   Number of observations                          2355        2829
##
## Model Test User Model:
##
##   Test statistic                                 0.000
##   Degrees of freedom                                 0
##
## Parameter Estimates:
##
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
##
## Covariances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##   cm_income_ind ~~
##     cm_health         0.074    0.019    3.863    0.000    0.074    0.080
##
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
##     cm_income_ind     1.023    0.030   34.315    0.000    1.023    1.000
##     cm_health         0.839    0.024   34.315    0.000    0.839    1.000

The covariance between the standardized variables is of course the correlation and this is the same value as the one obtained with cor.test().