Alzheimer’s disease (AD) is a neurological illness with early cognitive and behavioral disruption. The early cognitive deficits are frequently in the domain of memory, especially retentive memory. However, researchers and clinicians are now appreciating the behavioral heterogeneity of AD cognitive deficits and recognize that early and isolated deficits in domains of language, visuospatial abilities, executive function, and even mood may represent nascent AD [

1]. Mild Cognitive Impairment (MCI) is a recently described diagnostic entity which may represent a transition state between normal aging and AD [

2]. MCI has been defined as memory complaint with objective memory impairment in the context of normal general cognitive function and intact activities of daily living. As in AD, non-mnemonic subtypes of MCI are recognized, including those primarily affecting language [

3], visuoperception [

4], and executive abilities [

5].

It is difficult to discern if different groups of patients employ the same cognitive processes while performing standardized neuropsychological tests. This is tied to understanding what areas of cognition are impaired in AD and MCI and how these impairments are revealed through neuropsychological testing. Additionally, while the diversity of cognitive deficits among individuals with AD and MCI can make clinical diagnosis difficult, a more fundamental challenge arises from the use of neuropsychological tests to assess single domains of cognitive function. Most standardized tests used by clinical neuropsychologists rely on multiple cognitive capacities for successful completion of each task, and the inferences derived from test performance should be tempered by an understanding of the component processes involved.

As a formal, data-driven method, factor analysis has been applied to neuropsychological tests (for a survey, [

6–

9]) to derive underlying neuropsychological dimensions. Our approach to better understanding the functional cognitive structure underlying neuropsychological test performance employs Principal Components Analysis (PCA), which reduces a correlation matrix of test measures to a few factors that are implicit in the data [

10–

11]. (In this article we will use the term “factor” instead of “component” as they are nearly analogous). As Harman [

12] pointed out, following the principle of parsimony that is common to all scientific theory, a law or model should be simpler than the data upon which it is based. Thus, the number of factors should be less than the number of variables (test measures), and, in the linear description of each variable, the complexity should be low.

PCA condenses a wide array of measures that are based on different metrics and cognitive functions into simpler and interpretable factors. When derived from neuropsychological tests, a factor can be considered as a cognitive dimension (e.g., a general memory dimension). PCA provides both factor loadings (which relate test measures to the cognitive dimensions) and factor scores (which pertain to an individual’s performance on those cognitive dimensions). The factor loadings describe properties of the system in terms of weights (correlations) of neuropsychological test measures on the underlying factors. In other words, the factor loadings represent the tests’ varying contributions to each dimension while the factor scores represent an individual’s performance on each dimension. Each factor can be identified by its pattern of these test measure loadings. Larger loadings (more distant from zero) are more salient in this identification and interpretation. Additionally, PCA achieves data reduction, which can be an important advantage in subsequent analyses where degrees of freedom limitations may be present.

PCA is an additive factor model, where the performance measure of an individual on a test is the summation of each factor’s contribution to that test measure. The contribution is the product of the factor loading on that test measure (which is a static structure) and the factor score (which varies with the individual). As said above, a neuropsychological test may involve multiple cognitive capacities that are difficult to separate. Through PCA, this distinction may be quantitatively expressed as a test measure’s loadings on the associated factors for those cognitive dimensions.

While there are obvious quantitative differences among AD, MCI, and normal individuals in neuropsychological test performance, an important question remains concerning qualitative differences among these groups. These differences refer to the underlying dimensionality of test performance including the relations among test variables. Understanding these relations may reveal how these test variables measure cognitive performance in AD, MCI, and normal elderly. In terms of factor analysis, factor invariance between two factor structures derived from different groups should be established before meaningful comparisons with factor scores can be made [

13]. This issue of studying the underlying cognitive structure of AD, MCI, and normal cognition has been addressed by other researchers [

8–

9,

14–

16]. Siedlecki et al. [

9] studied neuropsychological invariance using exploratory and confirmatory factor analyses and determined that, generally, there was structural (but not metric) invariance among AD, MCI, and normal elderly but with unanswered questions pertaining to how delayed memory might be different in AD versus MCI and normal elderly.

There are difficulties inherent in trying to compare factors across solutions generated from different subject groups. There is an arbitrariness to factor rotation that can result in two factor solutions occupying the same factor space with markedly different orientations, which can result in misleadingly small congruence coefficients ([

17]). One would not expect the same factors to appear in the same order in each solution. Additionally, selecting the number of factors to retain is inherently an arbitrary process (one in which we combine the Eigenvalue > 1 rule with interpretability). Particularly for the factors whose Eigenvalues are clustered around 1, due to noise there will be fluctuations concerning which are retained and which are not. Thus, it is not likely the exact same set of factors will appear in each data set. The method we employ in this paper allows better comparisons of factor solutions than generating an independent, orthogonal solution from each subject group and attempting to determine if each contains the same set of factors. By rotating a replication factor structure to a target factor structure, we can directly measure how similar the replication structure is to the target by the congruence coefficients without the difficulties sometimes encountered when using confirmatory factor analysis [

13,

18–

19]. When used in conjunction with statistical tests of fit, orthogonal Procrustes rotation leads to the acceptance of models that are replicable and the rejection of those that are not [

18].

We will explore the invariance issue with a different approach that utilizes PCA and orthogonal Procrustes rotation to reveal factor similarities. We will test the invariance between the factor structures of our subject sets by statistically evaluating three types of congruence coefficients through a bootstrap procedure that randomly permutes the factor matrices [

13,

18]. We administered a battery of neuropsychological tests to elderly participants and will compare the underlying cognitive structures of three elderly sets of subjects (an AD-Set, an MCI-Set, and a Control-Set) with that of a Combined-Set (comprised of different AD, MCI, and Control subjects). We will determine whether each set shares a similar factor structure and how strong this similarity is by using the Combined-Set’s structure as a target and rotating the other sets to best match this target. The important result of such a comparison lies in the common metric derived from the Combined-Set. A common metric ensures that the same measurement axes can be used for all the groups involved in the metric’s creation. If the factor structures of the AD-Set, MCI-Set, and Control-Set are equivalent to that of the Combined-Set (within the bounds of sampling error), then it is reasonable to build a single factor structure simultaneously from all of these groups. Then, the factor scores created by PCA for each individual can determine group membership by placing the test performance tied to those scores upon the axes provided by the common metric. This could be a formal way of assessing where a novel patient belongs along those axes, as part of the AD group, as part of a Control group, or some place in the middle where the MCI group may lie. Such a diagnostic method could possibly aid in early detection of both AD and MCI.