A Linear Spatial Correlation Model, with Applications to Positron Emission Tomography

Abstract A simple spatial-correlation model is presented for repeated measures data. Correlation between observations on the same subject is assumed to decay as a linear function of the squared distance separating the regions in three-dimensional space where the observations are made. This quadratic decay (QD) model has many attractive theoretical properties. The covariance structure of the "normalized" data, formed by subtracting the subject average, is the same as that generated by random linear trends plus centered white noise, which leads to a simple method for simulating the original data. It also implies that the first three principal components of the normalized observations are the spatial coordinates of the regions. Thus principal components analysis can be used as an exploratory tool to verify the QD model, provided the white noise component is small. Certain simple predictions can be made, such as the fact that the variances of the normalized observations increase linearly with the squared distance from the centroid of the regions, even though the original observations are assumed to have equal variance. This implies that it is harder to detect abnormal regional measurements in the outlying regions using normalized observations. Assuming multivariate normality, generalized least squares can be used to find maximum likelihood estimates of the QD model; calculations can be reduced using an expression for the inverse of the covariance matrix that only involves the inverse of a 3 × 3 matrix. It is shown, however, that although normalization simplifies the model by removing the random subject effect, it reduces information about the spatial correlation effect, making it harder to detect. This may have implications for other spatial-correlation models that are fitted to residuals from a sample mean. The QD model is applied to 30 cerebral regional glucose metabolism measurements from positron emission tomography (PET) images on a group of 20 normal subjects.