Exploring the Hierarchical Influence of Cognitive Functions for Alzheimer Disease: The Framingham Heart Study

Background Although some neuropsychological (NP) tests are considered more central for the diagnosis of Alzheimer disease (AD), there is a lack of understanding about the interaction between different cognitive tests. Objective This study aimed to demonstrate a global view of hierarchical probabilistic dependencies between NP tests and the likelihood of cognitive impairment to assist physicians in recognizing AD precursors. Methods Our study included 2091 participants from the Framingham Heart Study. These participants had undergone a variety of NP tests, including Wechsler Memory Scale, Wechsler Adult Intelligence Scale, and Boston Naming Test. Heterogeneous cognitive Bayesian networks were developed to understand the relationship between NP tests and the cognitive status. The performance of probabilistic inference was evaluated by the 10-fold cross validation. Results A total of 4512 NP tests were used to build the Bayesian network for the dementia diagnosis. The network demonstrated conditional dependency between different cognitive functions that precede the development of dementia. The prediction model reached an accuracy of 82.24%, with sensitivity of 63.98% and specificity of 92.74%. This probabilistic diagnostic system can also be applied to participants that exhibit more heterogeneous profiles or with missing responses for some NP tests. Conclusions We developed a probabilistic dependency network for AD diagnosis from 11 NP tests. Our study revealed important psychological functional segregations and precursor evidence of AD development and heterogeneity.


Supplemental Materials
The probability of the training data which has T participants and 11 NP tests is shown in Equation 1. To make parameters learning feasible, we made assumption that each participant is independent and identically distributed (iid) from the joint distribution defined by the BNs. With this assumption, the probability of participant t is shown in Equation 2.
( ) i Pa X represents the parent of node i X in the BN. Then we let w represent these parameters. Based on [5], the principle says that we should choose arg max w w P * = , because the logarithm function is strictly increasing monotonically, which is equivalent to choosing arg max log w w P * = . So the goal is to maximize Equation 3.
In other words, assuming that the parent of X i is π, the maximum likelihood estimate of the probability of X i = x is shown in , where the counts are relative to the training data.
Each estimated probability is proportional to the corresponding frequency in the training data. If the value x is never observed for some combination π, then its conditional probability is estimated to be 0. Below is an example. In the BN of total population, SIM has no parents, so here is the probability of SIM score equals to1 BNT30 has parent SIM. So when the SIM score equals to 1, 0.1742 is the probability of BNT30 score equals to 1.
All parameters in conditional probability tables are calculated in this manner.

AD Inference
The objective of averaging likelihood weighting simulation is to calculate the posterior probabilities of cognitive status, given some observed scores from NP tests. It generates a set of randomly selected participants (n = 1000) based on the structure and parameter of BN, and then approximate probabilities of cognitive status by the frequencies of appearances in the simulated participants. When sampling the simulated participants, the scores of some NP tests will be known while others not due to missing Supplemental Materials data or time limit of assessment that didn't allow administration of some tests. The NP tests with scores are referred to as evidence. Given the evidence, we need to query the remaining nodes in BN to determine the cognitive status, the sampling process is as follows.
We set a temporary variable w = 1 which holds the calculated weight of the simulated participant. A temporary variable, x is set to empty. If the pending participant has 3 NP tests (LMd = 2, VRd = 3, PASi = 3) and we want to infer whether this person has AD, this available information could be represented as x = {LMd = 2, VRd = 3, PASi = 3, Cognitive Status=AD}. Each NP test node in the BN is examined. If the score from a NP test is available, then we make the following calculation: w = w × P(current NP test | parents of current NP test). If the current NP test score is unknown, then it is sampled to determine its value. It does not contribute to the weight calculation. Whether the available NP test is considered as evidence, or whether the missing NP test is discovered through sampling, its state is added to x. After the entire network is examined for this participant, we will be left with x and w, representing the status of the participant and the likelihood weight, respectively. This is added to W using x as the simulated participant, w as the corresponding weight. If x already exists in W, then w is added to weight associated to x in W.