Multiple Correlation (Root(R²)) Interpretation according to Cohen (1992)

[Im Poster]

Hi,

this might constitute an obvious / dumb question.

I ran a path analysis (investigation of a multiple mediation).

The model explained R² ~ 28% of the variance in the primary outcome / DV.

Which appears to be  a reasonable result. However, I was asked

if it would be possible to quantify this in terms of an effect size measure.

So since R² is just the squarred mutliple correlation, I was wondering,

if I could just take the root of the determination coefficient (R²) and

then interpret the "effect size" according to Cohen (1992) - Power Primer.

I created an MRE with reprex. So here we have R² = 47.8%.

Which would be equal to, r = .691, thus r > .5 a strong effect?

``` r
# Example taken from here:
# https://lavaan.ugent.be/tutorial/mediation.html
library(lavaan)
#> This is lavaan 0.6-12
#> lavaan is FREE software! Please report any bugs.
set.seed(1234)
X <- rnorm(100)
M <- 0.5*X + rnorm(100)
Y <- 0.7*M + rnorm(100)
Data <- data.frame(X = X, Y = Y, M = M)
model <- ' # direct effect
             Y ~ c*X
           # mediator
             M ~ a*X
             Y ~ b*M
           # indirect effect (a*b)
             ab := a*b
           # total effect
             total := c + (a*b)
         '
fit <- sem(model, data = Data)
summary(fit,
        fit.measures=T,
        rsquare=T,
        standardized=T,
        estimates=T,
        ci=T)
#> lavaan 0.6-12 ended normally after 1 iterations
#>
#>   Estimator                                         ML
#>   Optimization method                           NLMINB
#>   Number of model parameters                         5
#>
#>   Number of observations                           100
#>
#> Model Test User Model:
#>                                                      
#>   Test statistic                                 0.000
#>   Degrees of freedom                                 0
#>
#> Model Test Baseline Model:
#>
#>   Test statistic                                84.319
#>   Degrees of freedom                                 3
#>   P-value                                        0.000
#>
#> User Model versus Baseline Model:
#>
#>   Comparative Fit Index (CFI)                    1.000
#>   Tucker-Lewis Index (TLI)                       1.000
#>
#> Loglikelihood and Information Criteria:
#>
#>   Loglikelihood user model (H0)               -281.061
#>   Loglikelihood unrestricted model (H1)       -281.061
#>                                                      
#>   Akaike (AIC)                                 572.122
#>   Bayesian (BIC)                               585.148
#>   Sample-size adjusted Bayesian (BIC)          569.357
#>
#> Root Mean Square Error of Approximation:
#>
#>   RMSEA                                          0.000
#>   90 Percent confidence interval - lower         0.000
#>   90 Percent confidence interval - upper         0.000
#>   P-value RMSEA <= 0.05                             NA
#>
#> Standardized Root Mean Square Residual:
#>
#>   SRMR                                           0.000
#>
#> Parameter Estimates:
#>
#>   Standard errors                             Standard
#>   Information                                 Expected
#>   Information saturated (h1) model          Structured
#>
#> Regressions:
#>                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
#>   Y ~                                                                  
#>     X          (c)    0.036    0.104    0.348    0.728   -0.168    0.241
#>   M ~                                                                  
#>     X          (a)    0.474    0.103    4.613    0.000    0.273    0.675
#>   Y ~                                                                  
#>     M          (b)    0.788    0.092    8.539    0.000    0.607    0.969
#>    Std.lv  Std.all
#>                  
#>     0.036    0.028
#>                  
#>     0.474    0.419
#>                  
#>     0.788    0.679
#>
#> Variances:
#>                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
#>    .Y                 0.898    0.127    7.071    0.000    0.649    1.147
#>    .M                 1.054    0.149    7.071    0.000    0.762    1.346
#>    Std.lv  Std.all
#>     0.898    0.522
#>     1.054    0.825
#>
#> R-Square:
#>                    Estimate
#>     Y                 0.478
#>     M                 0.175
#>
#> Defined Parameters:
#>                    Estimate  Std.Err  z-value  P(>|z|) ci.lower ci.upper
#>     ab                0.374    0.092    4.059    0.000    0.193    0.554
#>     total             0.410    0.125    3.287    0.001    0.166    0.654
#>    Std.lv  Std.all
#>     0.374    0.285
#>     0.410    0.312
```

^{Created on 2022-11-16 with [reprex v2.0.2](https://reprex.tidyverse.org)}

Also, I know that one can in case of a multiple regression transform R² to

an Cohen's f (as explained here). I am not quite sure if this holds for path models

or more general SEMs as well.

Is either of these two approaches valid?

[Terrence Jorgensen]

The model explained R² ~ 28% of the variance in the primary outcome / DV.

I was asked if it would be possible to quantify this in terms of an effect size measure.

R-squared is an effect size.  Cohen provided arbitrary guidelines about it being small/large, corresponding very closely to the squared values of his guidelines about small/large correlations.

I know that one can in case of a multiple regression transform R² to

an Cohen's f

Sure, why not.  But that value is not as easy to interpret as "variance explained".  Cohen just made f to parse out the effect size from sample size from the test statistic F used in ANOVA.  That is, F = f times the ratio of num/denom df.   It is useful for power calculations in OLS regression, but I'm not sure it is particularly meaningful for path models.  The interpretation of R-squared is just as straight-forward.

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen