Introduction: This study aims to determine the stages of Alzheimer's disease (AD) using different machine learning algorithms, and compares the performance of these models. Methods: Demographic, genetic, and neurocognitive inventory data from the National Alzheimer's Coordinating Center (NACC) database as well as brain volume/thickness data from magnetic resonance imaging (MRI) scans were used. Deep Neural Networks, Ordinal Logistic Regression, Random Forest, Gaussian Naive Bayes, XGBoost, and LightGBM models were used to identify four different ordinal stages of AD. Results: Although the performance measures of the developed models were similar, the highest classification rate of AD stages was achieved by the Random Forest model (accuracy: 0.86; F1 score: 0.86; AUC: 0.95). The outputs of the model with the best performance were explained by the SHapley Addictive exPlanations (SHAP) method. Conclusions: This indicates that non-invasive markers and machine learning models can be used effectively in early diagnosis and decision support systems to predict stages of AD.