Fit indices are highly frequently used for assessing the goodness of fit of latent variable models. Most prominent fit indices, such as the root-mean-square error of approximation (RMSEA) or the comparative fit index (CFI), are based on a noncentrality parameter estimate derived from the model fit statistic. While a noncentrality parameter estimate is well suited for quantifying the amount of systematic error, the complex weighting function involved in its calculation makes indices derived from it challenging to interpret. Moreover, noncentrality-parameter-based fit indices yield systematically different values, depending on the indicators’ level of measurement. For instance, RMSEA and CFI yield more favorable fit indices for models with categorical as compared to metric variables under otherwise identical conditions. In the present article, approaches for obtaining an approximation discrepancy estimate that is independent from any specific weighting function are considered. From these unweighted approximation error estimates, fit indices analogous to RMSEA and CFI are calculated and their finite sample properties are investigated using simulation studies. The results illustrate that the new fit indices consistently estimate their true value which, in contrast to other fit indices, is the same value for metric and categorical variables. Advantages with respect to interpretability are discussed and cutoff criteria for the new indices are considered.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11336-023-09909-6.