ss.aipe.reliability function

Sample Size Planning for Accuracy in Parameter Estimation for Reliability Coefficients.

This function determines a necessary sample size so that the expected confidence interval width for the alpha coefficient or omega coefficient is sufficiently narrow (when assurance=NULL) or so that the obtained confidence interval is no larger than the value specified with some desired degree of certainty (i.e., a probability that the obtained width is less than the specified width; assurance=.85). This function calculates coefficient alpha based on McDonald's (1999) formula for coefficient alpha, also known as Guttman-Cronbach alpha. It also uses coefficient omega from McDonald (1999). When the 'Parallel' or 'True Score' model is used, coefficient alpha is calculated. When the 'Congeneric' model is used, coefficient omega is calculated.

ss.aipe.reliability(model = NULL, type = NULL, width = NULL, S = NULL, 
conf.level = 0.95, assurance = NULL, data = NULL, i = NULL, cor.est = NULL, 
lambda = NULL, psi.square = NULL, initial.iter = 500, 
final.iter = 5000, start.ss = NULL, verbose=FALSE)

Arguments

• model: the type of measurement model (e.g., "parallel items", "true-score equivalent", or "congeneric model") for a homogeneous single common factor test
• type: the type of method to base the formation of the confidence interval on, either the "Factor Analytic" (McDonald, 1999) or "Normal Theory" (van Zyl, Neudecker, & Nel, 2000)
• width: the desired full width of the confidence interval
• S: a symmetric covariance matrix
• conf.level: the desired confidence interval coverage, (i.e., 1- Type I error rate)
• assurance: parameter to ensure that the obtained confidence interval width is narrower than the desired width with a specificied degree of certainty
• data: the data set that the reliability coefficient is obtained from
• i: number of items
• cor.est: the estimated inter-item correlation
• lambda: the vector of population factor loadings
• psi.square: the vector of population error variances
• initial.iter: the number of initial iterations or generations/replications of the simulation study within the function
• final.iter: the number of final iterations or generations/replications of the simulation study
• start.ss: the initial sample size to start the simulation at
• verbose: shows extra information one the current sample size and current level of assurance; helpful if the function gets stuck in a long iterative process

Details

Use verbose=TRUE if the function is taking a very long time to provide an answer.

Returns

• Required.Sample.Size: the necessary sample size

• width: the specified full width of the confidence interval

• specified.assurance: the specified degree of certainty

• empirical.assurance: the empirical assurance based on the necessary sample size returned

• final.iter: the specified number of iterations in the simulation study

References

McDonald, R. P. (1999). Test theory: A unified approach. Mahwah, New Jersey: Lawrence Erlbaum Associates, Publishers.

van Zyl, J. M., Neudecker, H., & Nel, D. G. (2000). On the distribution of the maximum likelihood estimator of Cronbach's alpha. Psychometrika, 65 (3), 271--280.

Author(s)

Leann J. Terry (Indiana University; ljterry@Indiana.Edu ); Ken Kelley (University of Notre Dame; KKelley@ND.Edu )

Note

Not all of the items can be entered into the function to represent the population values. For example, either 'data' can be used, or S, or i, cor.est, and psi.square, or i, lambda, and psi.square. With a large number of iterations (final.iter) this function may take considerable time.

Warning

In some conditions, you may receive a warning, such as "In sem.default(ram = ram, S = S, N = N, param.names = pars, var.names =vars,; Could not compute QR decomposition of Hessian. Optimizationprobably did not converge." This indicates that the model likely did not converge. In certain conditions this may occur because the model is not being fit well due to small sample size, a low number of iterations, or a poorly behaved covariance matrix.

See Also

CFA.1; sem; ci.reliability;

Examples

## Not run:

ss.aipe.reliability (model='Parallel', type='Normal Theory', width=.1, i=6, 
                     cor.est=.3, psi.square=.2, conf.level=.95, assurance=NULL, initial.iter=500, 
                     final.iter=5000)

# Same as above but now 'assurance' is used. 
ss.aipe.reliability (model='Parallel', type='Normal Theory', width=.1, i=6, 
cor.est=.3, psi.square=.2, conf.level=.95, assurance=.85, initial.iter=500, 
final.iter=5000)

# Similar to the above but now the "True Score" model is used. Note how the psi.square changes 
# from a scalar to a vector of length i (number of items). 
# Also note, however, that cor.est is a single value (due to the true-score model specified)
ss.aipe.reliability (model='True Score', type='Normal Theory', width=.1, i=5, 
                     cor.est=.3, psi.square=c(.2, .3, .3, .2, .3), conf.level=.95, 
                     assurance=.85, initial.iter=500, final.iter=5000)
                     
ss.aipe.reliability (model='True Score', type='Normal Theory', width=.1, i=5, 
                     cor.est=.3, psi.square=c(.2, .3, .3, .2, .3), conf.level=.95, 
                     assurance=.85, initial.iter=500, final.iter=5000)                 

# Now, a congeneric model is used with the factor analytic appraoch. This is likely the 
# most realistic scenario (and maps onto the ideas of Coefficient Omega). 
ss.aipe.reliability (model='Congeneric', type='Factor Analytic', width=.1, i=5, 
lambda=c(.4, .4, .3, .3, .5), psi.square=c(.2, .4, .3, .3, .2), conf.level=.95, 
assurance=.85, initial.iter=1000, final.iter=5000)

# Now, the presumed population matrix among the items is used. 
Pop.Mat<-rbind(c(1.0000000, 0.3813850, 0.4216370, 0.3651484, 0.4472136), 
c(0.3813850, 1.0000000, 0.4020151, 0.3481553, 0.4264014), c(0.4216370, 
0.4020151, 1.0000000, 0.3849002, 0.4714045), c(0.3651484, 0.3481553, 
0.3849002, 1.0000000, 0.4082483), c(0.4472136, 0.4264014, 0.4714045, 
0.4082483, 1.0000000))

ss.aipe.reliability (model='True Score', type='Normal Theory', width=.15, 
S=Pop.Mat, conf.level=.95, assurance=.85, initial.iter=1000, final.iter=5000) 
## End(Not run)