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Power Analysis for Indirect Effects in a Mediation Model (Z, Joint, and Monte Carlo Tests)

Source: R/regression.mediation.R
power.z.mediation.Rd

Calculates power or sample size (only one can be NULL at a time) to test indirect effects in a mediation model (Z-Test, Joint Test, and Monte Carlo Interval Test). One can consider explanatory power of the covariates in the mediator and outcome model via specifying R-squared values accordingly. power.z.mediation() and power.z.med() are the same functions.

Formulas are validated using Monte Carlo simulations.

Usage

power.z.mediation(
  beta.a,
  beta.b,
  beta.cp = 0,
  sd.predictor = 1,
  sd.mediator = 1,
  sd.outcome = 1,
  r.squared.mediator = beta.a^2 * sd.predictor^2/sd.mediator^2,
  r.squared.outcome = (beta.b^2 * sd.mediator^2 + beta.cp^2 *
    sd.predictor^2)/sd.outcome^2,
  n = NULL,
  power = NULL,
  alpha = 0.05,
  alternative = c("two.sided", "one.sided"),
  method = c("sobel", "aroian", "goodman", "joint", "monte.carlo"),
  n.simulation = 1000,
  n.draws = 1000,
  ceiling = TRUE,
  verbose = 1,
  utf = FALSE
)

Arguments

beta.a

regression coefficient for X -> M path. One can use standardized regression coefficient, but should keep sd.predictor = 1 and sd.mediator = 1 or leave them out as they are default specifications.

beta.b

regression coefficient for M -> Y path. One can use standardized regression coefficient, but should keep sd.mediator = 1 and sd.outcome = 1 or leave them out as they are default specifications.

beta.cp

regression coefficient for X -> Y path (the direct path). One can use standardized regression coefficient, but should keep sd.predictor = 1 and sd.outcome = 1 or leave them out as they are default specifications.

sd.predictor

standard deviation of the predictor (X). For a binary predictor, sd.predictor = sqrt(p * (1 - p)) where p is the proportion of subjects in one of the groups.

sd.mediator

standard deviation of the mediator (M).

sd.outcome

standard deviation of the outcome (Y).

r.squared.mediator

R-squared value for the mediator model (M ~ X). The default is r.squared.mediator = beta.a ^ 2 * sd.predictor ^ 2 / sd.mediator ^ 2 assuming that X is the only predictor. Thus, an r.squared.mediator below this value will throw a warning. To consider other covariates in the mediator model provide a value greater than the default.

r.squared.outcome

R-squared value for the outcome model (Y ~ M + X). The default is r.squared.outcome = (beta.b ^ 2 * sd.mediator ^ 2 + beta.cp ^ 2 * sd.predictor ^ 2) / sd.outcome ^ 2 assuming that M and X are the only predictors. Thus, an r.squared.outcome below this value will throw a warning. To consider other covariates in the outcome model provide a value greater than the default.

n

integer; sample size.

power

statistical power, defined as the probability of correctly rejecting a false null hypothesis, denoted as \(1 - \beta\).

alpha

type 1 error rate, defined as the probability of incorrectly rejecting a true null hypothesis, denoted as \(\alpha\).

alternative

character; the direction or type of the hypothesis test: "two.sided" or "one.sided".

method

character; "sobel", "aroian", "goodman", "joint" or "monte.carlo". "joint" and "monte.carlo" methods cannot be used for sample size calculation.

n.simulation

integer; number of replications (applies when method = "monte.carlo").

n.draws

integer; number of draws from the distribution of the path coefficients for each replication (applies when method = "monte.carlo").

ceiling

logical; whether sample size should be rounded up. TRUE by default.

verbose

1 by default (returns test, hypotheses, and results), if 2 a more detailed output is given (plus key parameters and definitions), if 0 no output is printed on the console.

utf

logical; whether the output should show Unicode characters (if encoding allows for it). FALSE by default.

Value

parms

list of parameters used in calculation.

test

type of the statistical test ("Z-Test", "Joint Test", or "Monte Carlo Interval Test").

mean

mean of the alternative distribution.

sd

standard deviation of the alternative distribution.

null.mean

mean of the null distribution.

null.sd

standard deviation of the null distribution.

z.alpha

critical value(s).

power

statistical power \((1-\beta)\).

n

sample size.

Details

  • NOTE: The function pwrss.z.mediation() (or its alias pwrss.z.med()) are no longer supported. However, they will remain available as wrappers for the power.z.mediation function.

References

Aroian, L. A. (1947). The probability function of the product of two normally distributed variables. Annals of Mathematical Statistics, 18(2), 265-271.

Goodman, L. A. (1960). On the exact variance of products. Journal of the American Statistical Association, 55(292), 708-713.

MacKinnon, D. P., & Dwyer, J. H. (1993). Estimating mediated effects in prevention studies. Evaluation Review, 17(2), 144-158.

MacKinnon, D. P., Warsi, G., & Dwyer, J. H. (1995). A simulation study of mediated effect measures. Multivariate Behavioral Research, 30(1), 41-62.

Preacher, K. J., & Hayes, A. F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models. Behavior Research Methods, Instruments, & Computers, 36, 717-731.

Preacher, K. J., & Hayes, A. F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40, 879-891.

Sobel, M. E. (1982). Asymptotic intervals for indirect effects in structural equations models. In S. Leinhart (Ed.), Sociological methodology 1982 (pp. 290-312). Jossey-Bass.

Examples

# with standardized coefficients

## statistical power
power.z.mediation(beta.a = 0.25,
            beta.b = 0.25,
            beta.cp = 0.10,
            n = 200)
#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Indirect Effect in a Mediation Model
#> 
#>   Method : Sobel
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : beta[a*b]  = 0
#>   H1 (Alternative) : beta[a*b] != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 200
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.279
#>   Statistical Power    = 0.721  <<
#> 

## minimum required sample size
power.z.mediation(beta.a = 0.25,
            beta.b = 0.25,
            beta.cp = 0.10,
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Indirect Effect in a Mediation Model
#> 
#>   Method : Sobel
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : beta[a*b]  = 0
#>   H1 (Alternative) : beta[a*b] != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 242  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

## adjust for covariates in the outcome model
power.z.mediation(beta.a = 0.25,
            beta.b = 0.25,
            beta.cp = 0.10,
            r.squared.outcome = 0.50,
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Indirect Effect in a Mediation Model
#> 
#>   Method : Sobel
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : beta[a*b]  = 0
#>   H1 (Alternative) : beta[a*b] != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 185  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.199
#>   Statistical Power    = 0.801
#> 

# with binary predictor X such as treatment/control variable
# in this case standardized coefficients for path a and cp would be Cohen's d values

## statistical power
p <- 0.50 # proportion of subjects in one group
power.z.mediation(beta.a = 0.40,
            beta.b = 0.25,
            beta.cp = 0.10,
            sd.predictor = sqrt(p*(1-p)),
            n = 200)
#> +--------------------------------------------------+
#> |                POWER CALCULATION                 |
#> +--------------------------------------------------+
#> 
#> Indirect Effect in a Mediation Model
#> 
#>   Method : Sobel
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : beta[a*b]  = 0
#>   H1 (Alternative) : beta[a*b] != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 200
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.387
#>   Statistical Power    = 0.613  <<
#> 

## minimum required sample size
power.z.mediation(beta.a = 0.40,
            beta.b = 0.25,
            beta.cp = 0.10,
            sd.predictor = sqrt(p*(1-p)),
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Indirect Effect in a Mediation Model
#> 
#>   Method : Sobel
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : beta[a*b]  = 0
#>   H1 (Alternative) : beta[a*b] != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 311  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#> 

## adjust for covariates in the outcome model
power.z.mediation(beta.a = 0.40,
            beta.b = 0.25, beta.cp = 0.10,
            r.squared.outcome = 0.50,
            sd.predictor = sqrt(p*(1-p)),
            power = 0.80)
#> +--------------------------------------------------+
#> |             SAMPLE SIZE CALCULATION              |
#> +--------------------------------------------------+
#> 
#> Indirect Effect in a Mediation Model
#> 
#>   Method : Sobel
#> 
#> ----------------------------------------------------
#> Hypotheses
#> ----------------------------------------------------
#>   H0 (Null)        : beta[a*b]  = 0
#>   H1 (Alternative) : beta[a*b] != 0
#> 
#> ----------------------------------------------------
#> Results
#> ----------------------------------------------------
#>   Sample Size          = 254  <<
#>   Type 1 Error (alpha) = 0.050
#>   Type 2 Error (beta)  = 0.200
#>   Statistical Power    = 0.800
#>