Title: Nonlinear biomarker interactions in conversion from Mild Cognitive Impairment to Alzheimer’s disease

The multi-faceted nature of Alzheimer’s disease means that multiple biomarkers (e.g., amyloid- β , tau, brain atrophy) may improve the tracking of disease progression and the prediction of health outcomes. However, the mechanisms relating these biomarkers to the disease are inter-related to a degree. Hence, optimally modelling the relationships between different Alzheimer’s biomarker is warranted to reduce statistical uncertainty in predictions, to understand disease mechanisms, and to provide tools for clinical decision-making at the individual patient level. Here, we used Gaussian Processes to model nonlinear interactions when combining biomarkers to predict Alzheimer's disease conversion in 48 mild cognitive impairment participants who progressed to Alzheimer’s disease and 158 stable (over three years) people with mild cognitive impairment. Measures included: demographics, APOE4 genotype, CSF (amyloid- β 42, total tau, phosphorylated tau), neuroimaging markers of amyloid- β deposition ([18 F ]florbetapir) or neurodegeneration (hippocampal volume, brain-age). We examined: (i) the independent value each biomarker has in predicting conversion; and (ii) whether modelling nonlinear interactions between biomarkers improved prediction performance. Despite the correlations between different biomarkers, our results show each measured added complementary information when predicting conversion to Alzheimer's disease. A linear model predicting MCI group (stable versus progressive) explained over half the variance in group (R 2 = 0.51, P < 0.001), with the top independently-contributing biomarker being hippocampal volume (R 2 = 0.13). Next, we compared the sensitivity of different models to progressive MCI: independent biomarker models, additive models (with no interaction terms), nonlinear interaction models. We observed a significant improvement ( P < 0.001) for various two-way interaction models, with the best performing model including an interaction between amyloid- β -PET and P-tau, while accounting for hippocampal volume (sensitivity = 0.77). Our results showed that closely related biomarkers still contribute uniquely to the prediction of conversion, supporting the continued use of comprehensive biological assessments. However, our nonlinear interaction models suggest that for some patients, additional biomarkers may add little value (i.e., when hippocampal volume is high), but that for others (i.e., with low hippocampal volume) further invasive and expensive testing is warranted. Our results suggest that the interaction between hippocampal atrophy and amyloid-deposition influences progression to Alzheimer's disease over and above their independent contributions, to a moderate extent. Our Gaussian Processes framework enables visual examination of these nonlinear interactions, allowing projection of individual patients into biomarker ‘space’, providing a way to make personalise healthcare decisions or stratify subsets of patients for recruitment into trials of neuroprotective interventions. Processes based models; D) Partitioning of total variance explained into independent and shared variance explained for each biomarker; Gaussian Processes that allow just for main effects of biomarkers to be modelled are constructed by adding all the univariate squared exponential kernels corresponding to each biomarker. Full-order interactions between all biomarkers considered are captured through a single multivariate squared exponential kernel; Full set of statistics (sensitivity, specificity, accuracy, AUC) are computed for each replicate. Paired t-tests are conducted between accuracy scores stemming from main effects models and a model containing a multivariate kernel to assess if the increase in generalisation to unseen data is statistically significant. AUC Under Receiver Operator Characteristic Curve. T-tests (paired) based on comparison between accuracy and log likelihood scores on 100 bootstrapped sets are provided as a mean to assess the relative increase in generalisation attributable to the introduction of either two-way interactions or three-way interactions between biomarkers in comparison to main effects only variants.