Our focus for the data analyses was to find one or more of the following patterns in the data: (1) individual microbes which were either correlated or anti-correlated with AD, or (2) combinations of microbes that were correlated or anti-correlated with AD, given the number of bacteria observed. In other words, we were not interested in not only whether a bacterium was intrinsically pathogenic but also whether pathogenicity derived from a polymicrobial interaction acting within or between ecosystems.

We took different analytical approaches to address these types of patterns. Despite their differences, any similar findings between the two methods can provide mutual support for their respective findings and serve as strong evidence that the findings were inherently reproducible.

To find relationships between individual bacterial species and AD, we first investigated the differences in relative abundance of individual taxa between AD and control groups. We used a hierarchical Bayesian modeling approach based on a Dirichlet-multinomial model (DMM) (Harrison et al., 2020). This procedure was supervised as it used information about whether or not the samples came from AD subjects. Its results showed which individual bacteria are associated with AD.

To find relationships between combinations of bacterial species and AD that could be utilized as evidence supportive of a bacterial component in AD etiology, we used an approach, called Latent Dirichlet Allocation (LDA) (Blei et al., 2003; Griffiths and Steyvers, 2004), that first found relationships among the bacteria without using information about the disease state of the subject at all, i.e., it was unsupervised. This approach enabled us to group the bacteria and their abundances into different subsets of bacteria and their abundances, called classes, and then relate the classes to AD.

The brain microbiome data are characterized by sample heterogeneity, huge variance in counts (overdispersion), sparseness, and compositionality (see Supplement Figure SF1 ). These characteristics are often observed in microbiome data (Weiss et al., 2017; Lutz et al., 2022) and are very challenging for the statistical analysis, which explain our choice of particular analytical approaches. Table 1 shows the top 30 genera ordered by their abundances in the data set. The list contains both species and genera where we have broken out species for several high abundance genera. In this paper, we will often refer to Cutibacterium, Acinetobacter, and Comamonas as the principal bacteria mainly because of their overall abundance and prevalence, but in the case of Comamonas, because of its abundance and prevalence within a single class not associated with AD.

As samples vary in total read number, low-yield samples could introduce substantial noise, so the samples with less than 100 total reads were removed from the dataset. Four blank extraction controls (composed of all reagents used during sample processing but without sample input) were processed in the same way as the true biological samples to allow identification of any contamination from reagents or during sample processing. Potential contaminant OTUs were detected based on their occurrence in biological samples vs. negative controls using a prevalence-based method (IsNotContaminant function) from the R package Decontam (Davis et al., 2018). To qualify as contaminant, an OTU had to have a score ≥0.5 or a higher mean relative abundance in the negative controls than the biological samples ( Supplementary Table S1 ). Contaminant OTUs were then removed from the dataset. The phyloseq R package (McMurdie and Holmes, 2013) was used for handling OTU counts, taxonomy, and sample metadata.

Exploratory data analysis has been done using compositional data analysis (CoDA) methods which are based on log-ratio transformation of the data (Gloor et al., 2017). OTU counts were normalized using the centered log-ratio (clr) transformation by taking the log of the ratio of each OTU counts to the geometric mean of all OTUs in a sample. A pseudocount of 1 was applied to zero entries in the OTU count table before taking the log to the base 2. A positive clr value for a given OTU indicates a relatively higher amount than the overall composition mean, and a negative value indicates a relatively lower amount. Clr-transformed data were used to produce a heatmap with the pheatmap R package and to perform a principal component analysis (PCA) using the prcomp function from the R Stats package (R Core Team, 2018).

To test for differences in the relative abundances of individual OTUs between AD and control sampling groups, the OTU count data were analyzed using a hierarchical Bayesian model that relies on the Dirichlet and multinomial distributions as described in Harrison et al. (2020). Briefly, these two distributions are used to construct a hierarchical model which estimates relative abundances among samples and sampling groups and enables the detection of differences in relative abundances for each OTU between AD and control groups (see Supplementary Methods for a full description of the model). The Dirichlet-multinomial model (DMM) is relevant for the compositional structure of microbiome data because analyses are performed on proportions and there is an interdependency between parameters of the Dirichlet and the multinomial distributions, permitting identification of differences not easily found with the frequentist methods. Moreover, DMM quantifies and propagates the uncertainty associated with the OTU abundances in the parameter estimates, which make multiple comparison corrections unnecessary (Fordyce et al., 2011; Holmes et al., 2012; Harrison et al., 2020).

Like the original implementation of LDA which used words as the data, we needed a discrete form of the data to work with. The microbial objects that combined microbial name and abundance bin, described above, sufficed, but with a few changes.

For many of the lower-occurring and lower-abundance objects, we surmised that we needed even less abundance resolution, so we reduced the resolution to two bins instead of 14. A number of very low-occurring objects were also merged together under a general name instead of the individual bacterial names. Some taxa were grouped to species and some to genus levels. We did not group more by going further up the phylogenetic tree because we did not wish to lose behavioral information. We did not know a priori that this would work but had prior experience analyzing the gut microbiomes of ~7,000 subjects. This merging was performed with well-defined rules, primarily based on abundance resolution or occurrence, to maintain objectivity. We review the heuristic optimization procedures we used in the Supplementary Methods .

These changes to the data binning improved the similarity between pairs of samples that was limited by the sparsity of microbial measurements and large width of the microbial abundance distributions. For example, if two samples both had a high but different abundance of a particular species, they now both contained the same object, “bacteria-name-hi”. Overall, this scheme reduced the sparsity and allowed the algorithm to perform better.

In order to gain a sense for how the LDA algorithm works and how its results are expressed, we first describe its use for documents and then its adaptation to microbiome data.

The LDA (Blei et al., 2003; Griffiths and Steyvers, 2004) algorithm works by classifying words, i.e., assigning classes (topics in the literature) to each word. It does not directly classify a document. Documents are “classified” by statistically summarizing the fraction of their words assigned to each class. Similarly, unique words are “classified” by summarizing the fraction of times they are assigned to each class over the entire set of documents. This word classification summary is how LDA reveals that a particular word may have more than one meaning. Classes (topics) are summarized by the fraction of times each word in the set of documents is assigned to a class. Class summaries are also how LDA reveals how a set of words has a common meaning forming a topic. LDA works by locating word co-occurrence within a document. Another way of saying this is that LDA finds the context of words.

It will become clear shortly why it is important to stress that document classifications, word classifications, and class summaries are not words. They are statistical summaries of word classifications.

Our implementation of LDA’s words, microbial objects, carry information about both microbial behavior (i.e., its identified name) and an abundance which is the result of underlying microscopic ecosystems summed over the sample. We thought that if we used the LDA algorithm to classify these objects, biological meaning could be revealed by a relationship between class and the occurrence of AD.

These microbial objects are assigned one of a preset number of classes using the LDA algorithm. A cartoon version of the algorithm is described at a high level in Figure 1 and described in detail in the Supplementary Methods .

The LDA results data are tabulated in two tables. The sample distribution table has a row for each sample, and its columns are labeled by class. Its entries are the number of times the sample’s objects were classified with a particular class. The rows can be normalized to show the fraction of objects classified in a class. The object distribution table has a row for each unique object with columns labeled by class. One microbe may have multiple rows corresponding to different abundance bins. The entries are the number of times objects were classified with a particular class for the entire data set. The object class distribution is computed by normalizing across the rows to show the fraction of times each object is assigned to a class. In either representation, the rows should be thought of as vector-valued classifications. In other words, samples and objects have multi-valued classifications.

If the object table were normalized by column instead, the composition of each class is revealed by microbial object. In other words, the normalized column is the computed microbiome. Its elements are the fractional number of times each object is classified into a particular class and what we call the rigorous microbiome.

The existence of a class structure within a data set provides an opportunity to uncover patterns that can be missed by methods that ignore class. In fact, ignoring class structure implicitly averages out the very evidence that is sought.

In order to make analysis and discussion easier, we label the components of the classifications with colors. From this, we define the color of a sample or object to be the color of its largest component. For example, a red sample’s largest component is the red component. We also use the concept of color to approximate or describe microbiomes. In this example, it can be thought of as the set of microbial objects that occur in samples of a given color.

Because the number of classes is much less than the number of objects or the number of bacterial species, LDA results in a dimensionality reduction of the data. The number of classes is determined by an optimization process discussed in the Supplementary Methods . This should not be confused with the dimensionality reduction achieved by PCA. Typically, PCA evidences sample clustering by plotting two different linear combinations of abundances for each sample, the principal components, in two dimensions. These clusters are not the same as LDA’s. Other dimensionality reduction schemes, e.g., t-SNE and UMAP (van der Maaten and Hinton, 2008; McInnes et al., 2018), work in a different way, and it is not clear that they can easily allow objects (or measurements) to have multiple meanings although the common objects within the samples of a sample cluster in low dimensions could represent common meaning the way a class does in LDA. More information is provided in the Supplementary Methods , but a detailed discussion is beyond the scope of this work.

We emphasize that LDA’s results are not abundances themselves or even linear combinations of abundances. Rather, they are statistical summaries of microbial object classifications, which are measures of the co-occurrence or context of multiple objects that are described by their class composition.

Our adaptation of LDA to small microbiome data sets also involved several other procedures, which are discussed in detail in the Supplementary Methods . From here on, we will refer to our implementation as modified LDA or MLDA.

The next step in the methodology is to construct statistics of the MLDA results to infer information about microbial spatial distributions at the microscopic cellular and macroscopic brain levels, how they change over time, and what their relationship is with AD.

Since we know the sample spatial positions, we can look at how class varies with position. This is not the same as looking at how a particular microbe’s abundance varies with position because the same microbial object may have significant components in more than one class and their spatial positions could be different. For example, we determine in which lobes possible pathogenic classes predominate. Other statistics derived from the occurrence of class by subject can be determined whether the spatial distribution of classes is random or regionally clustered, suggesting where it is located with respect to the brain’s physiological structures and the paths taken to the brain.

Second, we can look at how abundance occurrence is distributed within class for particular bacteria, especially the principals. The width of this distribution is associated with the microscopic ecosystem spatial arrangements. Narrower distribution suggests more microscopic homogeneity within the sample, and wider distributions are associated with more heterogeneity, both of which suggest how far apart different principal bacteria are from one another.

Similarities among the classes of samples and the assumption that health precedes disease suggests how to order them in time. Finally, relationships between sample classes with subject suggests possible pathogenic microbiomes. In other words, the higher-order statistics permits identification of the temporal-spatial aspects of relationships that could be evidence for a bacterial component in the etiology of Alzheimer’s disease.

The patterns found by MLDA are sometimes difficult to understand so we developed graphical visualization techniques to assist us. Additional details can be found in the Supplementary Methods (Also see Results) section).

We used MLDA to compute five distinct color classes of samples. The results of our computations are summarized in a graph shown in Figure 5 , which is the sum of five different stochastic runs. We chose five classes as the optimum for our computation with the methods described previously.

The set of microbial objects resulting from a statistical summary of microbial objects from the samples in each color cluster will be called the class microbiome in the following paragraphs, which approximates the rigorous microbiome. The observed microbial objects derive from the summation of one or more ecosystems at the cellular scale during the physical sampling process. We will show how to characterize these ecosystems and how they determine the sample measurements in the discussion section.

In the graph (see Figure 5 ) of these five homogeneously colored clusters, two are very distinct, the green and orange, whereas the other three are merged together: blue, red, and magenta. This means the green and orange microbiomes are both different from one another and different from the blue, red, and magenta supercluster microbiomes. On the other hand, the latter are more related to one another since they are closer to one another. Each cluster represents a different underlying microbiome with a different set of microbial objects statistics. The green and orange sets must therefore have very different objects whereas the blue, red, and magenta have objects in common, specifically blue with red and red with magenta although not blue with magenta. As we have pointed out before, common objects between classes could be suggestive of multiple biological meanings of an object. This is one of the peculiarities and benefits of MLDA, that identical measurements mean different things in different contexts.

The statistical results shown in Table S3 , Table S4 , and Table S5 show these microbiome summaries for each of the color classes for objects with abundance bin ≥10 (0.3% abundance) and counts ≥5 in Table S3 and ≥2 in Table S4 and Table S5 . Different tables show different abundance combinations for particular microbes to show the importance of various microbes. Table S7 shows the same information in a different form. Each object is shown with their approximate microbiome computed from the occurrence count of the object in the sample color class.

For each color class, we also show the fraction of samples that come from AD subjects. Diamond-shaped nodes are from AD subjects and circles from the controls. It does not necessarily follow that this number is an estimate of the pathogenicity of the underlying microbiome, which will be explained in Theme 4 below.

Note that several nodes fall outside the clusters. This occurs because MLDA uses a stochastic algorithm which creates statistical fluctuations in class assignments. When the two largest class components are close, sometimes one ends up a little bigger and sometimes the other. The sample will be pulled from the cluster if one of these ends up being too large.

In the Figure 6 graphs, we show the samples in which the most abundant species are found, combining their objects in abundance ranges 12–14. The samples that contain these objects are shown by enlarging the sample/nodes containing these microbial objects. C. acnes is most prevalent in the blue–red–magenta supercluster with some in the orange. The blue is dominated by the 14 abundance range, the red has a combination of 14 and 13 abundance ranges, and the magenta has a combination of 13, 12, and 11 abundance ranges. Acinetobacter junii is mainly found in the green and orange classes and Comamonas jiangduensis is found almost exclusively in the green class. These underlying details can be found in Table S3 , Table S4 , Table S5 , Table S7 , and Table 4 .

To understand the low-count microbial objects, it is best to look at high-abundance and low-abundance objects separately. Because we are spreading low-count objects over five classes, it is difficult to find statistically significant patterns among them. In fact, we were unable to find any statistically sound patterns among the low-count and low-abundance objects. We did, however, find a fundamentally important pattern for high-abundance low-count objects that occurs mainly in the magenta and red classes, although signs of it can be traced to the other classes too; see Table 4 . Specifically, we noticed that samples with C. acnes with abundances 11–13 in the red and magenta classes correlated with a set of low-count bacteria with abundance level 14. In most cases, there was only one that occurred per sample. In Figure 7 , we show two different ways of defining this set.

In Figure 7 , we illustrate this point in a few different ways. Graphs a–c show low-count high-abundance objects in the sample with various occurrence rates. The right shows C. acnes-(11-13) objects to demonstrate that they co-occur in many samples with the low-count high-abundance objects. From here on, we will refer to the abundance 14 objects of (c), occurring in the red and magenta classes, as the M+ set because the most prevalent member of the set is Methylobacterium. In fact, of the 28 samples where these C. acnes objects occur, 22 contain M+ objects. The nine M+ objects that occur two or more times are Bacillus, Bradyrhizobium, Caulobacter, Delftia, Kocuria, Methylobacterium, Nitrosospira, Sediminibacterium, and Variovorax. Their objects can also be found in Table S4 .

Three of the color classes have a heterogeneous mix of AD and control samples (orange, blue, and red). The other two, green and magenta, are nearly homogeneous in the disease state, comprising almost entirely samples from either AD or control subjects. We initially thought that we should observe clusters whose disease state was nearly homogeneous, meaning that either a microbiome was pathogenic or not. The reason for the heterogeneity will become clear with the results of Theme 4 below.

We thought that these statistics suggested that there may be within-class differences in the microbiomes of the AD and control samples that could split the orange, blue, and red classes into homogeneous color clusters with the right input and object merging parameters. Indeed, there were within-class differences between the cohorts, but when we tried to split the clusters by using a larger number of input classes with these adjustments, the clusters would not split. The object differences between samples were just not large enough to support entirely new color classes.

We will look at three aspects of these results in light of the MLDA results. First, we will compare the PCA method’s capability of identifying microbiome clusters; second, we will look at the bacteria that were identified as being correlated or anti-correlated with AD; and last, we will look at a contamination issue.

In Figure 8 , we show the same chart as Figure 2 except that the nodes are colored by the class colors of the MLDA results. The green and orange samples are distinctly clustered whereas the blue, red, and magenta are mixed together. While the orange are clustered, there is not an obvious way to distinguish the colors. The MLDA plus graph does show the blue, red, and magenta samples in a super cluster; however, these color clusters are distinct and defined by the maximum class component whereas they are interspersed in the PCA results. A complete reconciliation of the PCA and MLDA results is beyond the scope of this paper.

For the following, see Table S3 , Table S4 , and Table S5 . The DMM method reveals Cutibacterium acnes as associated with AD. DMM has no class structure, so the method essentially averages over it. As we discuss both in this section and the next, this association is likely due to the prevalence of C. acnes in abundance levels 11–13 in the red and magenta clusters being high enough not to be washed out by its presence elsewhere where it is frequently found in the controls.

The method also found Comamonas jiangduensis to be correlated with the controls, which is also found with the MLDA method. Regarding Acinetobacter, one species A. junii was found to be anti-correlated with AD and another A. tjernbergiae was correlated with AD. A. junii objects are found in both the green and orange classes, with green not being associated with AD and orange somewhat associated with AD. A. tjernbergiae objects are mainly found in the orange class. Overall, since DMM is essentially averaging over the classes, these results appear to be consistent with the MLDA results.

Last, but importantly, the Sediminibacterium and Methylobacterium species identified by DMM as associated with AD is consistent with the MLDA findings. These two are among the M+ bacterial set that is found in the red and magenta classes. Methylobacterium is mainly found in the samples of the magenta class, which is most associated with AD. Other findings of DMM can be reconciled with MLDA by referring to Table S3 , Table S4 , and Table S5 . Overall, correlation of the results with AD is more complicated than this discussion and requires an understanding of the subject level results.

The first method found that Staphylococcus epidermidis was associated with AD. The main OTUs in this species were, however, removed in the background removal process for the second method because all of the OTUs present in the negative controls were removed even if there was only a small amount as was the case for this OTU. A post-MLDA analysis found that S. epidermidis was present in 45 samples ≥ abundance level 11 and in 39 samples ≥ abundance level 12. In this analysis, we were able to estimate the class distribution of S. epidermis objects and found that their class distributions were fairly flat, which is consistent with a contaminant. Furthermore, its objects with abundances ≥12 come from AD samples 59% of the time partially accounting for the DMM result.

In summary, the results of the DMM analysis are what happens when class is not considered in an analysis. As it is a confounding variable, ignoring it can sometimes skew results, although not always.

A fundamental result of the graph visualization of the MLDA results is the appearance of homogeneously colored clusters. This occurs because the value of the maximum MLDA component (color) of the node classification vector is ⪆0.4, showing that a particular class dominates in each sample. Furthermore, each color cluster contains samples from many subjects suggesting similarities in microbiome across subjects. Even though each subject has been undersampled (two to five samples/brain), the class structure suggests picturing the physical sampling process as coming from two virtual brains, one with AD and one without AD, each with about 60 samples/brain. This grouping can inform the large-scale macroscopic distribution of the individual microbiome classes.

By examining the actual underlying objects in each color class, we can learn even more. In Table 4 , we present the abundance statistics of each of the principal objects in each class, specifically C. acnes, A. junii, C. jianduensis, and M+. See the Supplementary Methods for a detailed description of this table. The principal C. acnes, A. junii, C. jianduensis, and M+ object abundance averages and distributions within a class provide information about the microscopic ecosystem structure, specifically their density and their spatial homogeneity on sample scales. In other words, these statistics provide information regarding the structure within the sample from which we can infer a microscopic structure.

We show the occurrence of each object in each class and then characterize the width of the abundance distribution over bins. The average abundance is related to the density of underlying ecosystems, and the width of the abundance is related to the spatial homogeneity of ecosystems within a sample. More details of this interpretation are provided in the discussion. Generally, when the abundance average is ~14, we call it high; between 13 and 14, we call medium; and everything else is called low. Also, when the distribution is concentrated in one or two bins, we call it narrow, mainly two or three bins with some peaking, medium, and greater than or equal to three bins, we call wide. We will explain in the discussion how to predict the spatial distribution of underlying ecosystem mixtures from these results, which is not obvious because it depends on the ecosystem model. The sample data are presented in Table S6 .

In presenting the results in this manner, we also need to call attention to an important equivalence principle that we use to understand these results. Furthermore, we emphasize that we are assuming that the microbial objects used in the computations result from summing over physically sampled mixtures of ecosystems, but we do not know much about the ecosystems yet. Therefore, we are assuming that each class microbiome results from a different mixture of ecosystems. The principle is as follows.

The sum of the virtual sampling of ecosystem mixtures equals the physical sampling of the sum of the ecosystem mixtures

In other words, we can treat the results of Table 4 as what we would obtain had we been able to individually sample a single ecosystem class mixture. We can then use these results to derive something about the nature of the individual class ecosystem mixtures. This is done in the discussion section.

A strong statistical relationship between pairs of color clusters, which might indicate a temporal relationship, should involve samples that contain principal microbial objects whose abundances are the same or differ by one. These are situations where it is likely that one microbe is just beginning to outcompete others or the reverse.

To visualize this, we constructed graphs where the samples of class pairs that meet these criteria are enlarged. In Figure 9 , we show results for the highly occurring species of the Cutibacterium, Acinetobacter, and Comamonas genera as well as the low-count high-abundance objects referred to as M+. Refer to Table 4 for more details. For less frequently occurring objects, the prevalences are too low to be useful.

This analysis found that the green–orange, orange–blue, green–blue, blue–red, and red–magenta pairs showed the strongest relationships utilizing the above evaluation standards. Refer to the Supplementary Methods for detailed discussion of the comparisons.

Now that we have established relationships between pairs of classes, it is straightforward to order them in time. To do this, we need a beginning and an end which is provided by AD statistics and the reasonable assumption that health precedes disease. Earlier, we cautioned about the use of these statistics because it does not necessarily follow that the bacteria in all of the samples of an AD subject are necessarily responsible for the disease. This is most likely not the case, however, for the green and magenta classes.

When the class AD statistic is either close to 0% or close to 100%, we are looking at situations where the population either never came from a diseased subject (green) or almost always came from a diseased subject (magenta). In these cases, the former would most likely not be pathogenic, or the subject would have AD. As the latter is almost always associated with disease, it is a reasonable hypothesis that it is pathogenic. Given that all the other sample colors are associated with both AD and controls, unless the physical sampling of the subject brains somehow missed a part of the brain that contained other pathogenic ecosystems not in any of our data, the magenta class most likely contains pathogenic ecosystems.

Refer to Figure 10 . The only strong relationship to green is orange although there might be a minor link to blue. At the other end, the only relationship to magenta is red. The only one we are left with is blue to red which must go in between the former two. Now we have a time-ordered class network. Another way of saying this is that we have derived a temporal variable where class color is the variable.

At this point, we observe that the network essentially reveals temporal dynamics for all of the principal bacteria, but most importantly, it has revealed the dynamics of C. acnes. It begins in the green class with a low average abundance and a broad distribution. It evolves in the orange to a medium average abundance that is still quite broad. In the blue class, it reaches a high narrow peak. In the red class, the abundance begins to fall and broaden. It falls further in the magenta class to a medium level with an even broader distribution.

In the green class, A. junii displays a medium average and narrow width. It stays roughly the same in the orange class. It then rapidly diminishes in blue and continues its decline to low abundance for red and magenta. The C. jiangduensis dynamic is more pronounced. It begins with a medium average abundance and width in green and then drops precipitously and broadens in orange. It is essentially not present in blue, red, or magenta.

On closer inspection, there seems to be another inter-object dynamic, an anti-correlation with C. acnes as seen in Table 4 . In the green, C. acnes is either non-existent, as seen in half the samples or at very low abundances. Conversely, this is the class where the Comamonas and Acinetobacter species have the highest abundances and are the most prevalent. The orange class has the next lowest level of C. acnes where it is present in almost all of the samples but with only a small peak in abundance (at 13) observed with seven samples and displaying a very wide distribution ranging from 7 to 14. Curiously, there is virtually no Comamonas, but Acinetobacter persists at high levels.

In the blue class, the levels of C. acnes are the highest among all of the classes. There is virtually no Comamonas, and A. junii levels are low; the latter occurring in only about half the samples with a very wide abundance distribution from 9 to 13. In the red class, there is a high level of C. acnes but A. junii is further diminished being present in less than a quarter of the samples over a wide abundance distribution of 10–13. Even with somewhat lower levels of C. acnes in magenta compared with red, A. junii is still only present in fewer than a quarter of the samples over a distribution from 12 to 14.

Thus overall, it appears that when C. acnes is not present or is present only at low levels, we observe both Comamonas spp. and Acinetobacter spp. However, as the abundance of C. acnes increases, first the Comamonas spp. is lost and then the Acinetobacter spp. As the C. acnes abundance decreases from the red class to the magenta class, a new non-specific dynamic enters the mix. The high-abundance low-count microbes, M+, appear mainly in the upper red and magenta classes. They increase in the magenta class as the C. acnes abundance falls, resulting in an antiparallel dynamic. There is something very curious here that we will take up again in discussion of the ecosystems and the biology. The C. acnes distributions in the magenta class look similar to those in the orange class, but there are major differences between the classes otherwise. This is an example of objects with multiple meanings described in the Methods section that MLDA can identify. The orange has a lot of Acinetobacter spp. whereas magenta does not, and the magenta has a lot of M+ whereas the orange class does not. Clearly, the ecosystem evolution that goes along with C. acnes’ rise and fall is neither the same nor reversible. Time could provide the explanation. Some of the more prevalent M+ are present at low abundances and prevalence in the earliest stages, orange and green (see Table 4 ). Curiously, there is only one instance of Methylobacterium in orange and two in upper red. Given long enough, however, many seem to be able to take over, even if C. acnes is present, either by increasing from earlier lower levels, or coming onto the scene later, like Methylobacterium.

Overall, these findings will constrain the underlying ecosystem structure that is presented in the discussion section.

The disease state statistics of each class, which summarize the fraction of samples from AD subjects, provide the first clues about whether a particular microbiome is pathogenic or not, and we have used these clues to set the temporal order of classes. However, as we pointed out earlier, these numbers are not estimates of the actual pathogenicities of the classes for every class. These statistics summarize the fraction of samples from a class that come from a subject who has AD.

If it were assumed that class AD statistics were an estimate of class pathogenicity, this would be tantamount to assuming a stochastic pathogenicity mechanism where sometimes a microbiome is responsible for AD and sometimes not. It might be reasonable if we were observing AD-control mixes of 80–20 or even 70–30 where we might be able to speculate that they derived from individual differences such as immune system capability. The AD statistics for blue, orange, and red were, however, just too close to 50–50 for comfort. Furthermore, since we were unable to split the color clusters through parameter adjustment, we were confident in their robustness and therefore their microbiomes. Therefore, we needed to find a more parsimonious explanation than accepting stochastic pathogenicity. We found one in the subject color class statistics. It is important to remember that the class results emerge from an analysis of bacterial data only. The disease statistics from the inset of Figure 5 are a statement of results only, i.e., which samples in each class came from AD or control brains and are not computed by fitting an a priori hypothesis.

Because the disease state variable is an attribute of a subject, not a sample, it was necessary to relate the subjects’ sample color class distributions to disease state in order to find a relationship between class and AD. In Figure 11 , we display the color class of each sample for all the subjects.

A glance at this figure indicates that the occurrence of a single magenta sample accounts for a subject’s AD in almost all cases

This suggests that the ecosystems underlying the other classes are not pathogenic even though many of their samples come from subjects who have AD. In order to do a more rigorous computation, we defined a subject-level classification as the normalized sum of the classifications of each of its samples. The sum was unweighted as we had no a priori way to assign weights. Using the resultant mixture vectors as independent variables in a logit regression with a cutoff of 0.5 (Wolfram Research Inc, 2008), we were able to obtain a high accuracy prediction (about 88%) of AD or lack of AD. True Positive, False Positive, True Negative, and False Negative rates were found to be 88%, 13%, 87%, and 13%, respectively. See the Supplementary Methods for additional details.

There are insufficient data to say anything definitive about location, but we offer one set of results. There is a hint that when a magenta infection occurs in an AD subject, it is likely to be found at least in the frontal lobe. Out of 13 AD subjects with magenta infections, nine have at least one frontal lobe magenta sample whereas only two have temporal lobe magenta samples without a frontal lobe magenta sample. Of these 13 AD subjects, five have temporal lobe magenta samples that also have magenta frontal lobe infections. For the three subjects that had samples in the entorhinal cortex, only one had samples in the temporal or frontal cortex and, in this case, it was both. The other two had neither.

Thus, at least with this small set of data, there is a suggestion that subjects with the magenta class in their frontal lobe are more likely to have the diminished cognitive abilities that result in AD diagnoses.