smdMES: Compute Effect Sizes for Multiple End-point Studies

Description

It computes the standardized mean differences and their asymptotic sampling covariance matrix for two multiple end-point studies with p effect sizes.

Usage

smdMES(m1, m2, V1, V2, n1, n2,
       homogeneity=c("covariance", "correlation", "none"), 
       bias.adjust=TRUE, list.output=TRUE, lavaan.output=FALSE)

Arguments

A vector of p sample means of the first group.

A vector of p sample means of the second group.

A p by p sample covariance matrix of the first group.

A p by p sample covariance matrix of the second group.

The sample size of the first group.

The sample size of the second group.

homogeneity

If it is covariance (the default), homogeneity of covariance matrices is assumed. The common standard deviations are used as the standardizers in calculating the effect sizes. If it is correlation, homogeneity of correlation is not assumed. The standard deviations of the first group are used as the standardizer in calculating the effect sizes. If it is none, no homogeneity assumption is made. The standard deviations of the first group are used as the standardizer in calculating the effect sizes.

bias.adjust

If it is TRUE (the default), the effect sizes are adjusted for small bias by multiplying \(1-3/(4*(n1+n2)-9)\).

list.output

If it is TRUE (the default), the effect sizes and their sampling covariance matrix are outputed as a list. If it is FALSE, they will be stacked into a vector.

lavaan.output

If it is FALSE (the default), the effect sizes and its sampling covariance matrix are reported. If it is TRUE, it outputs the fitted lavaan-class object.

Details

Gleser and Olkin (2009) introduce formulas to calculate the standardized mean differences and their sampling covariance matrix for multiple end-point studies under the assumption of homogeneity of the covariance matrix. This function uses a structural equation modeling (SEM) approach introduced in Chapter 3 of Cheung (2015) to calculate the same estimates. The SEM approach is more flexible in two ways: (1) it allows homogeneity of covariance or correlation matrices or not; and (2) it allows users to test this assumption by checking the fitted lavaan-class object.

References

Cheung, M. W.-L. (2015). Meta-analysis: A structural equation modeling approach. Chichester, West Sussex: John Wiley & Sons, Inc.

Cheung, M. W.-L. (2018). Computing multivariate effect sizes and their sampling covariance matrices with structural equation modeling: Theory, examples, and computer simulations. Frontiers in Psychology, 9(1387). https://doi.org/10.3389/fpsyg.2018.01387

Gleser, L. J., & Olkin, I. (2009). Stochastically dependent effect sizes. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis. (2nd ed., pp. 357-376). New York: Russell Sage Foundation.

Examples

Run this code

# NOT RUN {
## Sample means for the two constructs in Group 1  
m1 <- c(2.5, 4.5)

## Sample means for the two constructs in Group 2     
m2 <- c(3, 5)

## Sample covariance matrix in Group 1    
V1 <- matrix(c(3,2,2,3), ncol=2)

## Sample covariance matrix in Group 2
V2 <- matrix(c(3.5,2.1,2.1,3.5), ncol=2)

## Sample size in Group 1
n1 <- 20

## Sample size in Group 2    
n2 <- 25

## SMD with the assumption of homogeneity of covariance matrix    
smdMES(m1, m2, V1, V2, n1, n2, homogeneity="cov", bias.adjust=TRUE,
       lavaan.output=FALSE)

## SMD with the assumption of homogeneity of correlation matrix 
smdMES(m1, m2, V1, V2, n1, n2, homogeneity="cor", bias.adjust=TRUE,
       lavaan.output=FALSE)

## SMD without any assumption of homogeneity
smdMES(m1, m2, V1, V2, n1, n2, homogeneity="none", bias.adjust=TRUE,
       lavaan.output=FALSE)

## Output the fitted lavaan model
## It provides a likelihood ratio test to test the null hypothesis of
## homogeneity of variances.     
fit <- smdMES(m1, m2, V1, V2, n1, n2, homogeneity="cov", bias.adjust=TRUE,
              lavaan.output=TRUE)

lavaan::summary(fit)

lavaan::parameterestimates(fit)
# }