diffpwr.dep {diffcor}

Monte Carlo Simulation for the correlation difference between dependent correlations

Description

Computation of a Monte Carlo simulation to estimate the statistical power of the comparison between the correlations of a variable with two competing variables that are also correlated with each other.

Usage

diffpwr.dep(n,
            rho12,
            rho13,
            rho23,
            alpha = 0.05,
            n.samples = 1000,
            seed = 1234)

Arguments

n

Sample size to be tested in the Monte Carlo simulation.

rho12

Assumed population correlation between the criterion with which both competing variables are correlated and the first of the two competing variables.

rho13

Assumed population correlation between the criterion with which both competing variables are correlated and the second of the two competing variables.

rho23

Assumed population correlation between the two competing variables.

alpha

Type I error. Default is .05.

n.samples

Number of samples generated in the Monte Carlo simulation. The recommended minimum is 1,000 iterations, which is also the default.

seed

To make the results reproducible, it is recommended to set a random seed.

Details

Depending on the number of generated samples (n.samples), correlation coefficients simulated. For each simulated sample, it is checked whether the correlations r12 and r13 differ, given the correlation r23. The ratio of simulated z-tests of the correlation difference tests exceeding the critical z-value, given the intended alpha-level and sample size, equals the achieved statistical power(see Muthén & Muthén, 2002 <doi:10.1207/S15328007SEM0904_8>; Robert & Casella, 2010 <doi:10.1007/978-1-4419-1576-4>, for overviews of the Monte Carlo method).

It should be noted that the Pearson correlation coefficient is sensitive to linear association, but also to a host of statistical issues such as univariate and bivariate outliers, range restrictions, and heteroscedasticity (e.g., Duncan & Layard, 1973 <doi:10.1093/BIOMET/60.3.551>; Wilcox, 2013 <doi:10.1016/C2010-0-67044-1>). Thus, every power analysis requires that specific statistical prerequisites are fulfilled and can be invalid with regard to the actual data if the prerequisites do not hold, potentially biasing Type I error rates.

Value

As dataframe with the following parameters

rho12

Assumed population correlation between the criterion with which both competing variables are correlated and the first of the two competing variables.

cov12

Coverage. Indicates the ratio of simulated confidence intervals including the assumed effect size rho12.

bias12_M

Difference between the mean of the distribution of the simulated correlations and rho12, divided by rho12.

bias12_Md

Difference between the median of the distribution of the simulated correlations and rho12, divided by rho12.

rho13

Assumed population correlation between the criterion with which both competing variables are correlated and the second of the two competing variables.

cov13

Coverage. Indicates the ratio of simulated confidence intervals including the assumed effect size rho13.

bias13_M

Difference between the mean of the distribution of the simulated correlations and rho13, divided by rho13.

bias13_Md

Difference between the median of the distribution of the simulated correlations and rho13, divided by rho13.

rho23

Assumed population correlation between the two competing variables.

cov23

Coverage. Indicates the ratio of simulated confidence intervals including the assumed effect size rho23.

bias23_M

Difference between the mean of the distribution of the simulated correlations and rho23, divided by rho23.

bias23_Md

Difference between the median of the distribution of the simulated correlations and rho23, divided by rho23.

n

Sample size to be tested in the Monte Carlo simulation.

pwr

Statistical power as the ratio of simulated difference tests that yielded statistical significance.

Biases should be as close to zero as possible and coverage should be ideally between .91 and .98 (Muthén & Muthén, 2002 <doi:10.1207/S15328007SEM0904_8>).

Author(s)

Christian Blötner [e-mail]

References

Duncan, G. T., & Layard, M. W. (1973). A Monte-Carlo study of asymptotically robust tests for correlation coefficients. Biometrika, 60, 551–558. https://doi.org/10.1093/BIOMET/60.3.551

Muthén, L. K., & Muthén, B. O. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling: A Multidisciplinary Journal, 9(4), 599–620. https://doi.org/10.1207/S15328007SEM0904_8

Robert, C., & Casella, G. (2010). Introducing Monte Carlo methods with R. Springer. https://doi.org/10.1007/978-1-4419-1576-4

Wilcox, R. (2013). Introduction to robust estimation and hypothesis testing. Elsevier. https://doi.org/10.1016/C2010-0-67044-1

Examples

diffpwr.dep(n.samples = 1000,
                      n = 250,
                      rho12 = .30,
                      rho13 = .45,
                      rho23 = .50,
                      alpha = .05,
                      seed = 1234)