Δ2 and χ² I2 fit indices for structural equation models: A brief note of clarification

Abstract

Marsh and Balla (1986) and Marsh, Balla, and McDonald (1988) proposed an index of fit called χ²I2, but McDonald and Marsh (1990) subsequently demonstrated that the index is biased and recommended that it not be used. Bollen (1989) independently proposed Δ2 which is the same as χ²I2 (hereafter referred to as χ²I2‐Δ2), indicating that it adjusts for sample size and degrees of freedom (df). Gerbing and Anderson (1992), apparently based on the assumption that the χ²I2‐Δ2 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 χ²I2‐Δ2 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 χ²I2‐Δ2 index is not recommended for routine use.