Delta 2 and chi(2)I2 Fit Indices for Structural Equation Models: A Brief Note of Clarification

Marsh and Balla (1986) and Marsh, Balla, and McDonald (1988) proposed an index of fit called chi(2)I2, but McDonald and Marsh (1990) subsequently demonstrated that the index is biased and recommended that it not be used. Bollen (1989) independently proposed Delta 2 which is the same as chi(2)I2 (hereafter referred to as chi(2)I2-Delta 2), indicating that it adjusts for sample size and degrees of freedom (df). Gerbing and Anderson (1992), apparently based on the assumption that the chi(2)I2-Delta 2 index is unbiased and appropriately corrects for df (penalizes a lack of parsimony), recommended its use, and the index is routinely presented by major computer programs (e. g., EQS and LISREL 8). However, a more critical evaluation of the chi(2)I2-Delta 2 index reveals that: (a) it is systematically biased (i.e., its value varies systematically with N) although the size of the bias may be small; (b) the adjustment for df is inappropriate in that it penalizes model parsimony instead of model complexity; and (c) the inappropriate penalty for model parsimony is larger for small N. Because of these undesirable properties, the chi(2)I2-Delta 2 index is not recommended for routine use.