Slopes, standardized slopes, correlations, covariances

options(
  scipen = 999
)
library(tidyverse)

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## ✓ tibble  3.0.3     ✓ dplyr   1.0.0
## ✓ tidyr   1.1.0     ✓ stringr 1.4.0
## ✓ readr   1.3.1     ✓ forcats 0.5.0

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## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(MASS)

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:dplyr':
## 
##     select

#make some data
set.seed(1488)
d = MASS::mvrnorm(n = 10e3,
                  mu = c(0, 0),
                  Sigma = matrix(c(1, .5, .5, 1), nrow = 2),
                  empirical = T) %>% 
  as_tibble()

## Warning: The `x` argument of `as_tibble.matrix()` must have unique column names if `.name_repair` is omitted as of tibble 2.0.0.
## Using compatibility `.name_repair`.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_warnings()` to see where this warning was generated.

#stats
psych::describe(d)

#cors
cor(d)

##     V1  V2
## V1 1.0 0.5
## V2 0.5 1.0

#var and covar
var(d)

##     V1  V2
## V1 1.0 0.5
## V2 0.5 1.0

#slopes
lm(V1 ~ V2, data = d)

## 
## Call:
## lm(formula = V1 ~ V2, data = d)
## 
## Coefficients:
##            (Intercept)                      V2  
## 0.00000000000000001221  0.49999999999999983347

lm(V2 ~ V1, data = d)

## 
## Call:
## lm(formula = V2 ~ V1, data = d)
## 
## Coefficients:
##            (Intercept)                      V1  
## 0.00000000000000000222  0.49999999999999944489

#change the scale of V1 to SD = 2
d2 = d
d2$V1 = d2$V1 * 2

#stats
psych::describe(d2)

#cors
cor(d2)

##     V1  V2
## V1 1.0 0.5
## V2 0.5 1.0

#var and covar
var(d2)

##    V1 V2
## V1  4  1
## V2  1  1

#slopes
lm(V1 ~ V2, data = d2)

## 
## Call:
## lm(formula = V1 ~ V2, data = d2)
## 
## Coefficients:
##            (Intercept)                      V2  
## 0.00000000000000002442  0.99999999999999966693

lm(V2 ~ V1, data = d2)

## 
## Call:
## lm(formula = V2 ~ V1, data = d2)
## 
## Coefficients:
##            (Intercept)                      V1  
## 0.00000000000000000222  0.24999999999999972244

Notes:

• Correlations and slopes for standardized variables are the same (for single regression, and non-weighted data)
• Slopes are direction dependent (non-commutative). Slope of V1 on V2 is not necessarily the same as V2 on V1.
• Slopes for unstandardized variables are meaningful if their variables are in meaningful units. The slope is the increase in dependent variable per unit increase in the predictor.
• Covariances are in squared units, and do not have any intuitive meaning. They are symmetric (commutative), i.e., order does not matter.
• In general, you want to pay attention to slopes. If the variables lack sensible units, pay attention to the standardized slopes. Correlations and standardized slopes help compare to other research situations BUT beware that these are affected by the variance in your dataset (see range restriction). If you use an unrepresentative dataset in terms of variation in your variables, the standardized values will be misleading.