Multinomial Logistic Regression: Model fit and predictive properties. Breast tumour type predicted from FNA

Fine needle aspiration (FNA) is a non-invasive technique for retrieving a small piece of tissue from a breast tumour. Tissue smears are studied under the microscope by a pathologist, and classified according to cytological criteria such as nuclear pleomorphism (none/moderate/profound), bipolar naked nuclei (absent/present), and myoephithelial cells (absent/present). The pathologist also predicts the tumour type. The true tumour type can be found from histological diagnosis after surgery. Using data from 133 patients, we used multinomial logistic modelling for prediction of tumour class, with 9 cytological criteria as candidate independent variables. The dependent variable was histological diagnosis, in 3 groups from benign through non-invasive to invasive (NBPD/PBD, AIDH/DCIS, and IDC) (Bofin et al, 2004). We assessed the predictive properties by crossvalidation, deleting one observation at a time, fitting a separate model by selecting variables (in our case stepwise) and estimating parameters for each deleted data set, resulting in deleted fitted probabilities for each observation. We propose a visual examination of probabilities versus observed outcomes, which for a trinomial model can be viewed in a 3-1=2 dimensional space (plane) if appropriately tilted. The predicted diagnosis may be the one with highest probability, or by a classification rule reflecting different misclassification “costs”. We compared logistic regression classifications and original cytological diagnosis with the final histological diagnoses, using classification tables, and by computing sensitivity and specificity with different cut-offs between diagnosis groups. This was a valuable procedure for assessing the fits of different logistic models, and for comparing these to the original predictions by the pathologists.