A mixture experiment is an experiment in which the q-ingredients {x_i,i=1,2,...,q} are nonnegative and ubject to the simplex restriction £Ux_i=1 on the (q-1)-dimensional probability simplex S^{q-1}. It is usually assumed that the observations are uncorrelated, although in many applications the observations are correlated. We study the difference between the ordinary least square estimator and the Gauss Markov estimator under correlated observations. It is shown that for certain models and a special covariance structure for the mixture experiments, the unknown parameter vector for the ordinary least square estimators and the Gauss Markov estimators are the same. Moreover, we also show that the corresponding optimal designs may be obtained from previous D- and A-optimal designs for uncorrelated observations. The models studied here includ Scheff'e models, log contrast models, models containing homogeneous functions, and models containing inverse terms.

The creation, storage, manipulation, and transmission of images have become less costly and more efficient. Consequently, the numbers of images and their users are growing rapidly. This poses challenges to those who organize and provide access to them. One of these challenges is similarity matching. Most current content-based image retrieval (CBIR) systems which can extract only low-level visual features such as color, shape, and texture, use similarity measures based on geometric models of similarity. However, most human similarity judgment data violate the metric axioms of these models. Tversky's (1977) contrast model, which defines similarity as a feature contrast task and equates the degree of similarity of two stimuli to a linear combination of their common and distinctive features, explains human similarity judgments much better than the geometric models. This study tested the contrast model as a conceptual framework to investigate the nature of the relationships between features and similarity of images as perceived by human judges. Data were collected from 150 participants who performed two tasks: an image description and a similarity judgment task. Qualitative methods (content analysis) and quantitative (correlational) methods were used to seek answers to four research questions related to the relationships between common and distinctive features and similarity judgments of images as well as measures of their common and distinctive features. Structural equation modeling, correlation analysis, and regression analysis confirmed the relationships between perceived features and similarity of objects hypothesized by Tversky (1977). Tversky's (1977) contrast model based upon a combination of two methods for measuring common and distinctive features, and two methods for measuring similarity produced statistically significant structural coefficients between the independent latent variables (common and distinctive features) and the dependent latent variable (similarity). This model fit the data well for a sample of 30 (435 pairs of) images and 150 participants (χ2 =16.97, df=10, p = .07508, RMSEA= .040, SRMR= .0205, GFI= .990, AGFI= .965). The goodness of fit indices showed the model did not significantly deviate from the actual sample data. This study is the first to test the contrast model in the context of information representation and retrieval. Results of the study are hoped to provide the foundations for future research that will attempt to further test the contrast model and assist designers of image organization and retrieval systems by pointing toward alternative document representations and similarity measures that more closely match human similarity judgments.