Mediation Analysis and Mixed-Effects Models for the Identification of Stage-specific Imaging Genetics Patterns in Alzheimer’s Disease

Abstract

Alzheimer’s disease (AD) is one of the most common and severe forms of Senile Dementia. Genome-wide association studies (GWAS) have identified dozens of AD susceptible loci. To better understand potential mechanism-of-action for AD, quantitative brain imaging features have been studied as mediators linking genetic variants to AD outcomes. In this study, Mediation analysis, Chow test and Mixed-effects Models are used to investigate the biological pathways by which genetic variants affect both brain structures/functions and disease diagnosis. We analyzed the imaging and genetics data collected from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) project, including a Polygenic Hazard Score (PHS) and 13 imaging quantitative traits (QTs) extracted from the AV45 PET scans quantifying the amyloid deposition in different brain regions of subjects from four separate diagnostic groups. Mediation analysis assessed the mediating effects of image QTs between PHS and diagnosis, whereas Chow test and Linear Mixed-Effects models were used to characterize intra-group differences in the associations between genetic scores and imaging QTs for different disease stages. Results show that promising stage-specific imaging QTs that mediate the genetic effect of the studied PHS on disease status have been identified, providing novel insights into the predictive power of the PHS and the mediating power of amyloid imaging QTs with respect to multiple stages over the AD progression.

I. Introduction

Alzheimer’s disease (AD) [1], the most common form of dementia characterized by an irreversible progressive memory loss followed by deterioration of cognitive function and memory recall, is currently afflicting over 6 million people in the United States with the number estimated to increase to 15 million by the year 2050. The disease’s pathology is complex and depends on a combination of genetic predisposition and anatomical deterioration [2]–[4]. In the last years, Genome-Wide Association Studies (GWAS) have been crucial in identifying multiple genetic variants that can be responsible for a higher risk of AD development, and dozens of susceptible AD loci have been identified in the human genome [4]–[8]. Polygenic Risk Scores (PRS) [9] are often used to compute the global risk of AD considering the effect of genetic variations such as Single Nucleotide Polymorphisms (SNPs).

Furthermore, biomedical imaging technologies such as PET scans, MRI or fMRI are commonly used to observe the phenotype differences in patients with diverse degrees of cognitive impairment, and they provide a valid support to perform the disease diagnosis and to observe the visible biological mechanisms related to AD in the brain [10]–[13]. For example, it has been observed that in AD patients’ brains, abnormal levels of a natural protein called Beta-Amyloid 42 are present and tend to clump together, forming pathologic plaques that disrupt neuronal function. These plaques can be easily observed and studied with quantitative imaging methods such as PET scans.

In this work, we used a combination of methods to investigate the biological pathways by which genetic variants affect both brain structures/functions and disease diagnosis. We analyzed the imaging and genetics data collected from the ADNI project [14]–[18], including a Polygenic Hazard Score (PHS) as an indication of genetic risk, and 13 imaging quantitative traits (QTs) extracted from the AV45 PET scans quantifying the amyloid deposition in different brain regions. The participants were from four diagnostic groups: Normal (NL), early mild cognitive impairment (EMCI), late mild cognitive impairment (LMCI), and Alzheimer’s dementia (AD). Mediation analysis [9] was used to identify imaging QTs that function as mediators between the PHS and the diagnosis, i.e. that take part in a causal relational mechanism between genetic risk and diagnosis. The Chow test and mixed-effects models were used to characterize the differences in the relationships between PHS and imaging QTs in the four diagnostic groups.

II. Datasets

Data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu) [18]. The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early AD. For up-to-date information, see www.adni-info.org.

In particular, two datasets were selected and merged: the first contained genetic information in the form of a Polygenic Hazard Score (PHS) and the second contained 13 different imaging QT phenotypes. 559 subjects were analyzed: 185 belonged to the Normal (NL) group, 196 to the early mild cognitive impairment (EMCI) group, 153 to the late mild cognitive impairment (LMCI) group, and 25 were diagnosed as AD patients. Figure 1 shows the data collection process from the ADNI database.

Fig. 1.

Data collection process.

A. The ADNI Polygenic Hazard Score

Genetic information was obtained in the form of a Polygenic Hazard Score, available in the ADNI dataset. This score quantifies the risk of developing AD considering the combination of the the effect sizes of 31 SNPs located on selected genes [19].

B. AV45 PET Scans

Thirteen quantitative traits coming from processed AV45 PET scans [20] of the ADNI2/GO cohorts were selected as imaging quantitative traits and candidate mediators. In particular, these QTs represent the quantity of amyloid protein deposition in several regions of the brain, therefore they can serve as an indication of the disease stage. Diagnosis data was also contained in this dataset.

III. Methods

A. Mediation Analysis

Mediation analysis is a technique used to find whether an established relationship between two variables (e.g. an independent and a dependent variable within a regression model) can be mediated by a third variable, referred to as the mediating variable or ‘mediator.’ In other words, a mediator is a variable that explains the underlying mechanism of the relation between the two other variables. A causal mediation analysis is usually performed in three steps, each relying on a distinct but related set of regression models. Considering an independent variable X, a dependent variable Y , and a mediating variable M, the following steps are performed:

• First, a (linear) regression model assesses the relationship between X and Y . In order to have a mediating relationship, the relation must be statistically significant.

  $$Y=b_0+b_{1}X+e$$ (1)
• Second, another (linear) regression model assesses the relationship between M and X. Similar to the first step, this relationship must also be statistically significant to establish a mediating relationship.

  $$M=b_0+b_{2}X+e$$ (2)
• Third, a final regression model determines if there is a significant relationship between independent variable X and dependent variable Y that also factors for mediator M. To have a significant mediating relationship, the relationship between the independent variable (X) and the dependent variable (Y ) should not be significant or the parameter b₄ has to be smaller than b₁ in eq. 1

  $$Y=b_0+b_{4}X+b_{3}M+e$$ (3)

Mediation analysis was performed using the mediate package in R.

B. Chow test

The Chow test is a statistical test used to verify whether or not different regression coefficients are equal. It was initially used in econometrics to determine if the relationship between an independent variable and a dependent variable changed after a major historical event (i.e., a war) or because of a categorical stratifying factor (i.e., race) [21]. More generally, this test can be used to determine whether the true coefficients of two analogous linear regression models built from stratified data sets are equal. If the coefficients differ a statistically significant amount (as determined by an F test), one can conclude the change is due to the stratifying factor or significant temporal event.

Within the context of this study, the Chow test will determine if there are statistically significant disease stage-related differences in associations between the PHS and each of 13 imaging-based QTs.

C. Mixed-effects Models

Linear Mixed-Effects Models (LMM) can be considered as an extension of classical linear regression models, where subjects can be grouped following certain criteria and within-groups differences are quantified [22]. A general LMM equation looks like the following:

$$y=X\beta +Zu+ϵ$$ (4)

The first part of the equation is equal to a standard linear model, where y is the dependent variable, X represents the independent variable(s) and β is a vector of coefficients, named fixed effects, that are common for the entire population. The vector u represents the so-called random effects, i.e. additional coefficients that are specific for each group, related to the observation through the matrix Z.

IV. Results

A. Mediation Analysis Results

Mediation analysis was used to assess the mediating effect of each image QT between PHS and diagnosis. Linear regression was used to compute the relation between PHS and the mediators, whereas a logistic regression was used to compute the one between PHS and diagnosis, since the diagnosis was a categorical variable with four levels. In particular, we binarized the diagnosis variable in three different ways, performing different diagnostic comparisons: NL vs. non-NL, NL+EMCI vs. LMCI+AD, non-AD vs. AD. The mediation analysis process is shown in Figure 2.

Fig. 2.

Mediation Analysis pipeline.

Each one of the 13 PET imaging parameters was tested as a mediator separately, in order to investigate whether there are some QTs that have a stronger mediating effect than others and find possible mediation patterns in the different diagnostic comparisons. Table I shows a legend of the 13 QTs. More information can be found in the official ADNI documentiation.

TABLE I.

imaging Quantitative traits legend

QT1	SPAP_GLOBAL_SUVR
QT2	SPAP_FRONTAL_SUVR
QT3	SPAP_TEMPORAL_SUVR
QT4	SPAP_ANTERIOR_CINGULATE_SUVR
QT5	SPAP_POSTERIOR_aNGULATE_SUVR
QT6	SPAP_PARIETAL_SUVR
QT7	SPAP_PRECUNEUS_SUVR
QT8	AVID_STAGE_4_GLOBAL_SUVR
QT9	AVID_STAGE_4_FRONTAL_MEDIAL_ORBITAL_SUVR
QT10	AVID_STAGE_4_TEMPORAL_SUVR
QT11	AVID_STAGE_4_PARIETAL_SUVR
QT12	AVID_STAGE_4_PRECUNEUS_SUVR
QT13	AVID_STAGE_4_ANTERIOR_QNGULATE_SUVR

Mediation analysis produces several different parameters as outputs; the main ones are:

• ACME (Average Casual Mediation Effect), i.e. the indirect effect of the independent variable on the dependent variable that goes through the mediator.

• ADE (Average Direct Effect), i.e. the direct effect of the independent variable on the dependent variable.

• Total Effect, i.e. the combination of the mediated effect and the direct effect.

When both ACME and the Total Effect are significant but the ADE is not significant, a so-called Total Mediation is present, as the relation between the independent and the dependent variable goes entirely through the mediator. If also the ADE is significant, the mediation is referred to as a Partial Mediation. Table II reports the results for the first comparison, i.e. healthy vs. non-healthy subjects. Table III reports the relative P-Values. It can be noticed that all parameters are statistically significant for each one of the mediators, i.e. a partial mediation is always present. Quantitatively speaking, the mediating and direct effects are also very similar among all mediators.

TABLE II.

ACME, ADE and Total Effect for each mediator considering Normal vs. non-normal patients.

	ACME	ADE	Total_effect
QT1	0.0441	0.1164	0.1605
QT2	0.0298	0.1296	0.1594
QT3	0.0419	0.1185	0.1604
QT4	0.0316	0.1286	0.1602
QT5	0.0326	0.1275	0.1601
QT6	0.0399	0.1202	0.1600
QT7	0.0454	0.1146	0.1600
QT8	0.0476	0.1104	0.1580
QT9	0.0319	0.1271	0.1590
QT10	0.0391	0.1221	0.1612
QT11	0.0416	0.1178	0.1594
QT12	0.0462	0.1131	0.1593
QT13	0.0352	0.1255	0.1606

TABLE III.

P-Values of ACME, ADE and Total Effect for each mediator considering Normal vs. non-normal patients.

	ACME_P_Val	ADE_P_Val	TotEff_P_Val
QT1	<0.001	<0.001	<0.001
QT2	0.0220	<0.001	<0.001
QT3	0.0020	<0.001	<0.001
QT4	0.0140	<0.001	<0.001
QT5	<0.001	<0.001	<0.001
QT6	<0.001	<0.001	<0.001
QT7	<0.001	<0.001	<0.001
QT8	<0.001	<0.001	<0.001
QT9	0.0160	<0.001	<0.001
QT10	<0.001	<0.001	<0.001
QT11	<0.001	<0.001	<0.001
QT12	<0.001	<0.001	<0.001
QT13	0.0020	<0.001	<0.001

The second diagnostic comparison has been performed aggregating the four groups into two categories: NL+EMCI and LMCI+AD. With this comparison it is possible to investigate whether the imaging QTs can have a mediating effect in the determination of different disease stages. The mediation results are shown in Table IV and the P-Values are visible in Table V.

TABLE IV.

ACME, ADE and Total Effect for each mediator considering Normal and EMCI vs. LMCI and AD patients.

	ACME	ADE	Total_eff
QT1	0.0690	0.0596	0.1286
QT2	0.0693	0.0600	0.1293
QT3	0.0720	0.0575	0.1295
QT4	0.0610	0.0684	0.1294
QT5	0.0485	0.0813	0.1297
QT6	0.0776	0.0525	0.1300
QT7	0.0806	0.0483	0.1289
QT8	0.0669	0.0629	0.1298
QT9	0.0763	0.0530	0.1293
QT10	0.0554	0.0746	0.1300
QT11	0.0778	0.0523	0.1301
QT12	0.0774	0.0504	0.1278
QT13	0.0721	0.0568	0.1289

TABLE V.

P-Values of ACME, ADE and Total Effect for each mediator considering Normal and EMCI vs. LMCI and AD patients. Non significant values are in bold

	ACME_P_Val	ADE_P_Val	Totef_P_Val
QT1	<0.001	0.0340	<0.001
QT2	<0.001	0.0240	<0.001
QT3	<0.001	0.0500	<0.001
QT4	<0.001	0.0180	<0.001
QT5	<0.001	0.0040	<0.001
QT6	<0.001	0.0640	<0.001
QT7	<0.001	0.0800	<0.001
QT8	<0.001	0.0220	<0.001
QT9	<0.001	0.0580	<0.001
QT10	<0.001	0.0120	<0.001
QT11	<0.001	0.0560	<0.001
QT12	<0.001	0.0880	<0.001
QT13	<0.001	0.0400	<0.001

In this case it can be noticed that the ADE related to some of the QTs are not significant, as the P-Values are greater than 0.05 (highlighted in bold in the table). In this case, considering that both the ACME and the Total Effect are significant, there is a total mediation.

The final diagnostic comparison has been the one between AD diagnosed subject and all other subjects. The results of the mediation analysis are reported in Table VI and the P-Values in Table VII. In particular, looking at the P-Values, it can be noticed that nine mediators out of 13 perform a total mediation, as the direct effects are not significant, whereas the mediated and total effects are.

TABLE VI.

ACME, ADE and Total Effect for each mediator considering AD patients vs. all other groups.

	ACME	ADE	Total_effect
QT1	0.0171	0.0228	0.0400
QT2	0.0169	0.0232	0.0400
QT3	0.0125	0.0265	0.0391
QT4	0.0144	0.0250	0.0394
QT5	0.0139	0.0255	0.0394
QT6	0.0196	0.0202	0.0398
QT7	0.0168	0.0229	0.0397
QT8	0.0134	0.0262	0.0395
QT9	0.0173	0.0221	0.0394
QT10	0.0134	0.0258	0.0392
QT11	0.0159	0.0234	0.0393
QT12	0.0134	0.0257	0.0391
QT13	0.0136	0.0257	0.0393

TABLE VII.

P-Values of ACME, ADE and Total Effect for each mediator considering AD patients vs. non-AD patients. Non significant values are in bold.

	ACME_P_Val	ADE_P_Val	TotEff_P_Val
QT1	0.0060	0.0800	<0.001
QT2	0.0060	0.0820	<0.001
QT3	0.0440	0.0380	<0.001
QT4	0.0040	0.0540	<0.001
QT5	0.0060	0.0520	0.0020
QT6	<0.001	0.1280	<0.001
QT7	0.0060	0.0780	0.0020
QT8	0.0220	0.0440	<0.001
QT9	<0.001	0.0980	0.0020
QT10	0.0140	0.0300	0.0020
QT11	0.0040	0.0520	<0.001
QT12	0.0260	0.0600	<0.001
QT13	0.0080	0.0640	<0.001

Generally speaking, a mediation, either partial or total, is always present. Total mediation happens for some of the QTs in separating patients with a late stage of the disease, whereas mediation is partial in all cases considering the comparison between healthy and non-healthy subjects.

B. Chow test Results

Mediation analysis confirmed that the imaging phenotypes can be mediators in the relation between genetic data and diagnosis, i.e., they can help better characterize the biological mechanisms that lead from genetic risk to disease outcomes. Given this result, it is possible to investigate more in detail the intra-group differences in the associations between PHS and imaging phenotypes. To this end, we performed a Chow test on linear regression models developed for each QT, using the PHS as independent variable and each QT as dependent variable. In particular, the test verifies whether the coefficients of four different regressions, one for each diagnostic groups, are significantly different. Figure 3 shows the P-Values of the Chow test performed for QT12, that has been selected as an example. The null hypothesis of the test is that coefficients are not different, therefore significant differences are not present between healthy and EMCI subjects and between LMCI and AD subjects, whereas statistical significance occurs in the other comparisons. A change in the regression slope therefore probably occurs when the disease progresses from EMCI to LMCI. All other QTs showed similar results.

Fig. 3.

Chow test P-Values considering the relation between PHS and QT12 for each diagnostic group.

Splitting the dataset into two groups instead of four, i.e. NL+EMCI vs. LMCI+AD and performing another Chow test, the P-Value is really low (1.45 · 10⁻¹⁰), thus confirming that two different models can be used to describe patients with a disease stage pre- and post-LMCI.

C. Mixed-effects Models Results

Similarly to the Chow test, Mixed-Effects Models were used to better characterize the variability of the relation between PHS and imaging phenotypes across different diagnostic groups. This approach differs from the Chow test mainly because the relations in the different groups are not considered separately, but a hierarchical model is built instead, as global population effects (fixed effects) and individual group effects (random effects) are both part of the model formulation. Since according to the Chow test not all groups had statistically different coefficients, we created a model that assumes that all groups have the same β coefficients, but a random intercept is added to each group. From a mathematical point of view, the equation can be written as follows:

$$Q{T}^i={\beta }_0^i+{\beta }_1^i\cdot PHS+(1\mid \text{ Diagnosis })+ϵ$$ (5)

where i indicates the i-th imaging QT, β₀ and β₁ are the fixed effects and (1|Diagnosis) indicates the random intercept, that is different for each one of the four diagnostic groups. Tables VIII and IX contain the result of the model created for QT9, that was selected as an example. Also in this case, results were really similar for all QTs. In particular, Table VIII shows the fixed effects, whereas Table IX shows the random intercepts for each group.

TABLE VIII.

Fixed effects of the linear mixed effect model for QT9

Fixed Effect	Value	2.5% CI	97.5% CI
Intercept	1.039	0.969	1.115
PHS	0.136	0.112	0.161

TABLE IX.

Random effects of the linear mixed effect model for QT9

Group	Random Intercept
Healthy	−0.06
EMCI	−0.034
LMCI	0.41
AD	0.055

In order to compute model significance, we calculated the P-Value of an ANOVA test associated to the same group subdivision, which was close to zero. From these results, it appears that the relation between PHS and each imaging QT can be described with a standard linear model, and a different intercept can be defined for each group, with higher values correspondent to higher disease stages. This result is visible also in 4, that represents a scatter plot of the data with interpolation lines color-coded according to the diagnostic group.

It can be interesting also to notice that the separation between the EMCI and LMCI lines is wider than all the other lines, and the random intercept is negative for healthy and EMCI subject and positive for LMCI and AD patients. This goes in agreement with the Chow test results, that showed that separating the diagnostic groups into two categories aggregating healthy and EMCI in one group and LMCI and AD in another groups generates two different models.

V. Discussion

Alzheimer’s diasease is a complex condition that results from the combination of a lot of genetic and epigenetic risk factors, plus certain environmental and pharmaceutical exposures. The underlying mechanisms that lead to the disease progression are still partially unknown. Genomic research has contributed to the identification of several genetic features that increase the risk of developing AD, and research is still ongoing to better clarify how these mechanisms work and how they can relate to other variables (e.g. molecular, imaging, fluid, cognitive and environmental measures) to create specific clinical outcomes [23], [24]. Improving our understanding of these mechanisms can lead to the design of prevention strategies or targeted medications to treat the disease, which represents a seriously impairing condition and causes high healthcare costs [25].

In the last years, GWAS studies have been crucial in finding susceptible loci and genes, while imaging techniques have contributed to improve the diagnosis and characterization of disease phenotypes. Nevertheless, the causal link between genetic risk and imaging phenotypes is not always simple to understand, as the disease can take many years to develop. In this paper, we proposed an approach that can help identify stage-specific disease patterns taking into consideration both genetic and imaging phenotypes data.

Mediation analysis was used to assess whether imaging QTs mediate the relation between genetic risk and disease stage, whereas Chow test and linear Mixed-Effect models were used to observe how the relation between genome and phenotype changes in relation to the cognitive impairment stage. A polygenic hazard score obtained by the combination of the effect sizes of numerous SNPs was used to quantify genetic risk. General results show that Amyloid deposition PET images tend to mediate the relation between genetic risk and disease stage mainly when subjects progress from an EMCI to a LMCI stage, at a point where two different models can be used to express the relation between genetic risk and phenotype for patients that belong to the EMCI or normal categories and those that have a more advance disease stage, as demonstrated by the Chow test.

Nevertheless, Mixed-Effects models allow to formalize this relation for all four groups, showing how a specific additional intercept can be defined for each diagnostic category. This result is not entirely surprising, as it is already known that higher quantities of amyloid deposition correspond to more advanced disease stages, but the ability to model how genetic risk combines to image phenotypes to define these stages poses the bases for the construction of advanced predictive models that can serve as a powerful diagnostic tool.

It should be noted that data sample size, especially related to the number of AD patients (only 24 out of 559), can be a limitation in this study, as it can contribute to increasing the difficulty of mathematically separating AD patients from LMCI subjects. More tests should be performed on larger datasets and with different polygenic risk scores and imaging parameters, nevertheless this approach shows promising results in beginning to discover casual imaging genetics patterns in the different stages of AD.

VI. Conclusions

In this study we showed a novel approach to unravel the biological pathways that lead from genetic risk and intermediate imaging phenotype evaluation to different stages of Alzheimer’s Disease using a combination of methods, i.e. mediation analysis, Chow test and Mixed-Effects Models. Amyloid PET scans are generally good mediators between genetic predisposition and disease progression, especially for patients that progress from an early to a late cognitive impairment stage. Thanks to the Chow test and Mixed Models, it is also possible to model how the relation between genetic risk and phenotype outcome changes in different patient groups according to their diagnosis.

Fig. 4.

Scatter plot of the relation between PHS and QT9, with different lines and colors representing the four different diagnostic groups