Correct a Correlation for Attenuation Due to Measurement Error

Applies the Spearman (1904) correction for attenuation: the observed correlation between two fallible measures understates the correlation between the constructs they measure, and dividing by the square root of the product of the two reliabilities recovers it, $$r_c \;=\; \frac{r_{XY}}{\sqrt{\rho_{XX'}\,\rho_{YY'}}}.$$ Within classical test theory (Lord & Novick, 1968), the disattenuated correlation estimates the correlation between the true scores, that is, how strongly the two constructs would correlate if each were measured without error. When N is supplied, a confidence interval for the corrected correlation is formed by disattenuating the endpoints of the Fisher's Z interval for the observed correlation, the standard practice when the reliabilities are treated as known.

Usage

correction_for_attenuation(
  r,
  reliability_x,
  reliability_y,
  N = NULL,
  conf_level = 0.95
)

Arguments

r: The observed correlation between the two measures, in $[-1, 1]$.
reliability_x: Reliability of the first measure, in $(0, 1]$. Any reliability estimate appropriate to the use may be supplied (for example, coefficient alpha or omega from reliability).
reliability_y: Reliability of the second measure, in $(0, 1]$. For a correlation between a measure and an error-free criterion, supply 1 for that side (correcting for criterion unreliability only yields what the validity generalization literature calls the operational validity).
N: Optional sample size on which r is based. When supplied, a conf_level confidence interval for the corrected correlation is reported by correcting the endpoints of the Fisher's Z interval for r.
conf_level: Confidence level for the interval when N is supplied. Defaults to 0.95.

Value

A data.frame (class dmar_tbl) in term / value layout with the observed correlation (correlation_observed), the corrected correlation (correlation_corrected), the corrected interval (lower_limit, upper_limit; present only when N is supplied), and the reliability_x, reliability_y, and N inputs.

Details

The correction treats the two reliabilities as known constants, which is the conventional assumption; uncertainty in the reliabilities themselves would widen the interval further. Because the observed correlation can exceed what the supplied reliabilities allow (sampling error, or reliabilities that understate the truth), the corrected value can exceed 1 in magnitude; when that happens the value is reported as computed, with a warning, rather than silently truncated, since a corrected correlation beyond 1 is itself diagnostic information about the inputs.

Prefer the factor model when you have the items. The Spearman formula is the summary-statistics route: it is exactly right when all you have are the observed correlation and reliability estimates. When the item-level data are available, the better practice is to estimate the construct-level correlation directly as the factor correlation in a two-factor model (each scale loading on its own factor, factors free to correlate): the latent correlation is then estimated jointly with the measurement model rather than assembled from plug-in reliabilities, and it comes with a standard error that propagates the sampling variability of all the moving parts. The example below shows both routes on the same data, the formula route using reliability_omega, the model route using lavaan; with congeneric items the two agree closely, and when they disagree the factor model is the one to trust.

References

Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores. Addison-Wesley.

Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72–101.

Author

Ken Kelley kkelley@nd.edu

Examples

# An observed correlation of .30 between measures with reliabilities .80
# and .70 corresponds to a construct-level correlation of about .40.
correction_for_attenuation(r = 0.30, reliability_x = 0.80, reliability_y = 0.70)
#>  term                  value
#>  correlation_observed  0.3  
#>  correlation_corrected 0.401
#>  reliability_x         0.8  
#>  reliability_y         0.7  

# With the sample size, the corrected interval comes along.
correction_for_attenuation(r = 0.30, reliability_x = 0.80, reliability_y = 0.70,
                    N = 120)
#>  term                  value
#>  correlation_observed  0.3  
#>  correlation_corrected 0.401
#>  lower_limit           0.171
#>  upper_limit           0.608
#>  reliability_x         0.8  
#>  reliability_y         0.7  
#>  N                     120  
#> 
#> Confidence level: 95%

# Correct one side only (error-free criterion).
correction_for_attenuation(r = 0.30, reliability_x = 0.80, reliability_y = 1)
#>  term                  value
#>  correlation_observed  0.3  
#>  correlation_corrected 0.335
#>  reliability_x         0.8  
#>  reliability_y         1    

# The two routes to the construct-level correlation, on the same data
# (requires lavaan). Two congeneric scales of three items each whose
# latent variables correlate .50:
set.seed(113)
n <- 400
fx <- rnorm(n); fy <- 0.5 * fx + sqrt(1 - 0.25) * rnorm(n)
lam <- c(.8, .7, .6)
items <- data.frame(
  x1 = lam[1] * fx + rnorm(n, 0, sqrt(1 - lam[1]^2)),
  x2 = lam[2] * fx + rnorm(n, 0, sqrt(1 - lam[2]^2)),
  x3 = lam[3] * fx + rnorm(n, 0, sqrt(1 - lam[3]^2)),
  y1 = lam[1] * fy + rnorm(n, 0, sqrt(1 - lam[1]^2)),
  y2 = lam[2] * fy + rnorm(n, 0, sqrt(1 - lam[2]^2)),
  y3 = lam[3] * fy + rnorm(n, 0, sqrt(1 - lam[3]^2)))

# Route 1, summary statistics: omega reliabilities into the formula.
x_score <- rowMeans(items[, 1:3]); y_score <- rowMeans(items[, 4:6])
om_x <- reliability_omega(data = items[, 1:3])$value[1]
#> Robust omega is reported without a confidence interval by default because its interval is bootstrap based. Request it with ci_method = "percentile" (or "bca"); B = 10000 replications is the default when you do.
om_y <- reliability_omega(data = items[, 4:6])$value[1]
#> Robust omega is reported without a confidence interval by default because its interval is bootstrap based. Request it with ci_method = "percentile" (or "bca"); B = 10000 replications is the default when you do.
correction_for_attenuation(r = cor(x_score, y_score),
                           reliability_x = om_x, reliability_y = om_y,
                           N = n)
#>  term                  value
#>  correlation_observed  0.307
#>  correlation_corrected 0.413
#>  lower_limit           0.29 
#>  upper_limit           0.529
#>  reliability_x         0.76 
#>  reliability_y         0.725
#>  N                     400  
#> 
#> Confidence level: 95%

# Route 2, the factor model: the latent correlation estimated directly.
fit <- lavaan::cfa("X =~ x1 + x2 + x3\nY =~ y1 + y2 + y3",
                   data = items, std.lv = TRUE)
lavaan::parameterEstimates(fit)[
  lavaan::parameterEstimates(fit)$op == "~~" &
  lavaan::parameterEstimates(fit)$lhs == "X" &
  lavaan::parameterEstimates(fit)$rhs == "Y", ]
#>    lhs op rhs  est    se     z pvalue ci.lower ci.upper
#> 15   X ~~   Y 0.43 0.056 7.661      0     0.32    0.539