In measuring food intake, three common methods are used: 24-hour recalls, food frequency questionnaires and food records. Food records or 24-hour recalls are often thought to be the most reliable, but they are difficult and expensive to obtain. The question of interest to us is to use the food records or 24-hour recalls to examine possible systematic biases in questionnaires as a measure of usual food intake. In Freedman, et al. (1991), this problem is addressed through a linear errors in variables analysis. Their model assumes that all measurements on a given individual have the same mean and variance. However, such assumptions may be violated in at least two circumstances, as in for example the Women's Health Trial Vanguard Study and in the Finnish Smokers' Study. First, some studies occur over a period of years, and diets may change over the course of the study. Second, measurements might be taken at different times of the year, and it is known that diets differ on the basis of seasonal factors. In this paper, we will suggest new models incorporating mean and variance offsets, i.e., changes in the population mean and variance for observations taken at different time points. The parameters in the model are estimated by simple methods, and the theory of unbiased estimating equations (M-estimates) is used to derive asymptotic covariance matrix estimates. The methods are illustrated with data from the Women's Health Trial Vanguard Study.