TITLE: RELIABILITY ESTIMATION FOR (APPROXIMATELY) CONTINUOUS COMPONENTS
DATA: FILE = ;
VARIABLE: NAMES = U1-UR; ! R = # ITEMS IN SET U. (SEE MAIN TEXT.)
MODEL: F BY U1*(B_1)
U2-UR (B_2-B_R);
U1-UR (THETA_1-THETA_R);
F@1;
MODEL CONSTRAINT:
NEW(SC_REL); ! DEFINING PLACE-HOLDER FOR SCALE RELIABILITY ?U .
SC_REL =(B_1+B_2+…+B_R)**2/ ! WITH NEXT LINE, = ?U IN MAIN TEXT.
((B_1+B_2+…+B_R)**2+THETA_1+THETA_2+…+THETA_R); ! WRITE FULLY OUT
! FOR A GIVEN NUMBER R IN AN EMPIRICAL SETTING. SEE EQ. (17).
OUTPUT: CINTERVAL; ! USE ESTIMATE AND S.E. OF SC_REL WITH NEXT R-FUNCTION
Note. With deviations from normality (not resulting from ceiling/floor effects, clustering effects, or highly discrete components), add the line ANALYSIS: ESTIMATOR = MLR; in order to invoke the robust ML method of estimation. U1 through UR = components in item set U under consideration, B_1 through B_R = component loadings on common true score, THETA_1 through THETA_R = component error variances (see Equation (17)), SC_REL = reliability of the scale score U (overall sum score). (See next for obtaining a CI of reliability.) For binary scored items, see Raykov et al. (2010) for point and interval estimation of scale reliability and revision effect upon it. With component error correlations, extend correspondingly the last line of the MODEL CONSTRAINT section (see main text).
To obtain a CI for scale reliability, ?U, call the following R-function using as its arguments the scale reliability estimate and its standard error obtained with the last Mplus command file. (The R-function next is computationally identical to the R-function ‘ci.cid’ in Appendix 1, and named differently here only to emphasize its applicability also for interval estimation of reliability.)
ci.rel = function(r, se){
l = log(r/(1-r))
sel = se/(r*(1-r))
ci_l_lo = l-1.96*sel
ci_l_up = l+1.96*sel
ci_lo = 1/(1+exp(-ci_l_lo))
ci_up = 1/(1+exp(-ci_l_up))
ci = c(ci_lo, ci_up)
ci
}
Note. In the lines for the logit-transformed reliability coefficient ‘r’, use corresponding standard normal cut-offs if CIs at other confidence levels are desired. (See Raykov & Marcoulides, 2011a, for further details and illustrations.)