Using autoregressive and random walk models to detect trends and shifts in unequally spaced tumour biomarker data

Continuous time autoregressive (CAR(1)) and random walk models of time series data are provided for detecting non-random shifts and trends of tumour markers in breast cancer patients following resection for cure. The continuous time random walk model with observation error is extended to the case of multiple patient time series. These models can be used to monitor large numbers of patients with time series with few sampling events that are serially correlated and unequally spaced. Further, the methodologies can be used to recommend appropriate testing intervals. A Kalman filter recursive algorithm is used to calculate the likelihood functions arising from the CAR(1) and random walk models and to calculate recursive residuals, which are monitored by Shewhart-cusum schemes.