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In unrelated individuals, their phenotypes at two gene loci may appear correlated. The two major reasons for such a deviation from independence are linkage disequilibrium (allelic association) and an interaction between the phenotypes. In this paper, it is shown how the usual chi-square test for association of phenotypes can be partitioned into two components, one being due to allelic association and the other being due to other causes of phenotypic association. The test is demonstrated with an example.

Background Odor identification (OI) dysfunction is an early marker of Alzheimer’s disease (AD), but it remains unclear how olfactory-related regions change from stages of subjective cognitive decline (SCD) and mild cognitive impairment (MCI) to AD dementia. Methods Two hundred and sixty-nine subjects were recruited in the present study. The olfactory-related regions were defined as the regions of interest, and the grey matter volume (GMV), low-frequency fluctuation (ALFF), regional homogeneity (ReHo) and functional connectivity (FC) were compared to explore the changing pattern of structural and functional abnormalities across AD, MCI, SCD and normal controls. Results From the SCD, MCI to AD groups, the reduced GMV, increased ALFF, increased ReHo and reduced FC of olfactory-related regions became increasingly severe, and only the degree of reduced GMV of hippocampus and caudate nucleus clearly distinguished the 3 groups. In particular, SCD subjects exhibited reduced GMV (hippocampus, etc.), increased ReHo (caudate nucleus) and reduced FC (hippocampus- hippocampus and hippocampus-parahippocampus ) in olfactory-related regions compared with normal controls. Additionally, reduced GMV of the bilateral hippocampus and increased ReHo of the right caudate nucleus were associated with OI dysfunction and global cognitive impairment, and they exhibited partially mediated effects on the relationships between OI and global cognition across all subjects. Conclusion Structural and functional abnormalities of olfactory-related regions present early with SCD and deepen with disease severity in the AD spectrum. The hippocampus and caudate nucleus may be the hub joining OI and cognitive function in the AD spectrum.