Many investigators are currently developing models to predict human performance in detecting a signal embedded in complex backgrounds. A common figure of merit for model performance is d', an index of detectability that can be mathematically related to the proportion correct (Pc) when the responses of the model are Gaussian distributed and statistically independent. However, in many multiple-alternative forced-choice (MAFC) detection tasks, the target appears in one of M different locations within an image. If the image contains slow spatially varying luminance changes (low-pass noise), the pixel luminance values at the possible signal locations are correlated and therefore the model/human responses to the different locations might also be correlated. We investigate the effect of response correlations on model performance and compare different figures of merit for these conditions. Our results show that use of the standard d' index of detectability assuming statistical independence can lead to erroneous underestimates of Pc and misleading comparisons of models. We introduce a novel figure of merit d'(r) that takes into account response correlations and can be used to accurately estimate Pc. Furthermore, we show that d'(r) can be readily related to the standard index of detectability d' by d'(r) = d'/square root of (1 - r), where r is the correlation between the responses in any MAFC detection task. We illustrate the use of the theory by computing figures of merit for two linear models detecting a signal in one of four locations within medical image backgrounds.