(Mis)use of Factor Analysis in the Study of Insulin Resistance Syndrome

Debbie A. Lawlor , Shah Ebrahim, Margaret May and George Davey Smith

From the Department of Social Medicine, University of Bristol, Bristol, United Kingdom.

Received for publication May 5, 2003; accepted for publication January 5, 2004.


    ABSTRACT
 TOP
 ABSTRACT
 INTRODUCTION
 FACTOR ANALYSIS
 USE OF FACTOR ANALYSIS...
 USE OF FACTOR ANALYSIS...
 ADDITIONAL ISSUES
 CONCLUSIONS
 REFERENCES
 
Over the last decade, factor analysis has been used increasingly to describe patterns of simultaneous occurrence of the central components of the insulin resistance syndrome. In this paper, the authors describe factor analysis, review studies that have used factor analysis to examine the insulin resistance syndrome, and explore how factor analysis might be used to increase our understanding of this syndrome. Most studies that they reviewed gave vague reasons for using factor analysis and did not demonstrate an understanding of the use and limitations of this statistical method. Confirmatory factor analysis based on sound theoretical concepts and a clear understanding of the statistical methods may provide some insights into the pathophysiology of the syndrome. However, to date none of the studies has adopted this approach, and other statistical approaches and study designs are likely to provide greater understanding of the syndrome.

factor analysis, statistical; metabolic syndrome X


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 FACTOR ANALYSIS
 USE OF FACTOR ANALYSIS...
 USE OF FACTOR ANALYSIS...
 ADDITIONAL ISSUES
 CONCLUSIONS
 REFERENCES
 
The term "insulin resistance syndrome" refers to the clustering of a number of coronary heart disease risk factors including hyperinsulinemia, glucose intolerance, obesity, dyslipidemia, and hypertension, believed to be linked by a common pathophysiologic process, most likely insulin resistance (14). There is debate about the nature of the syndrome, including which risk factors belong to it and what pathologic process is central to its occurrence (3). We suggest that three research questions need to be addressed when considering the existence and clinical importance of a syndrome; these are summarized in table 1.


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TABLE 1. Research questions that need to be addressed when considering the existence and importance of a clinical syndrome
 
Over the last decade, factor analysis has been used increasingly to describe the patterns of simultaneous occurrence of the central components of the insulin resistance syndrome, with 22 publications identified using a MEDLINE search (see table 2 for search strategy) (526). None of these studies describes whether it is using factor analysis for hypothesis-deriving (exploratory) or hypothesis-testing purposes (see below), and rarely does any publication state a priori how factor analysis will improve our understanding of the existence or importance of the insulin resistance syndrome. In this paper, we review these studies and discuss the potential role of factor analysis in increasing our understanding of the insulin resistance syndrome.


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TABLE 2. Search strategy for factor analysis studies of the insulin resistance syndrome
 

    FACTOR ANALYSIS
 TOP
 ABSTRACT
 INTRODUCTION
 FACTOR ANALYSIS
 USE OF FACTOR ANALYSIS...
 USE OF FACTOR ANALYSIS...
 ADDITIONAL ISSUES
 CONCLUSIONS
 REFERENCES
 
Factor analysis is a mathematical model that explains the covariance (or correlation) between a large set of observed variables in terms of a smaller set of unobserved latent variables (known as factors) (2730). An important assumption of factor analysis is that there exist (latent) variables that have not been directly measured. Further, the observed (measured) variables are dependent upon these latent variable(s) but are also subject to random variation (27, 28).

The general factor analysis model assumes that there are m underlying latent factors (where m is smaller than the total number of observed variables included in the analysis). Factor scores (estimates of an individual’s latent factors) can be constructed by taking appropriate linear combinations of the variables in the analysis (2730). The factor analysis model is represented as follows:

x1 = µ1 + ß11 f1 + ß12 f2 + ... ß1m fm + e1

x2 = µ2 + ß21 f1 + ß22 f2 + ... ß2m fm + e2

.

.

xp = µp + ßp1 f1 + ßp2 f2 + ... ßpm fm + ep,

where x1, x2, ... xp are the first, second, and pth observed variables in the analysis, with p being the total number of observed variables and f1, f2, ... fm being the m latent factors. ß11 ... ßpm are called the factor loadings, representing the association between each variable and each latent factor, and the ej values represent the residual variance specific to each variable xj (i.e., that are not explained by the latent factors). The µ values are constants as are the factor loadings (ß), whereas the factors (latent variables f) and the residuals (e) are random variables. The minimum set of assumptions about these random variables is that the residuals (e1, e2, ... ep) are not correlated with each other or with the factors (f1, f2, ... fm).

Factor analysis usually involves three procedures: 1) extraction of factors; 2) rotation of factors to help interpretation; and 3) naming and interpretation of each factor based on the estimated values for the factor loadings. A number of methods can be used for extracting factors. In psychology, maximum likelihood procedures are largely used. However, all of the studies of the insulin resistance syndrome have used principal components analysis to extract factors. The main advantage of using maximum likelihood procedures is that it provides a test of the hypothesis that m common factors are sufficient to describe the data against possible alternative hypotheses of m + k common factors’ providing a better fit of the data.


    USE OF FACTOR ANALYSIS TO DETERMINE WHETHER THE INSULIN RESISTANCE SYNDROME EXISTS AND TO PROVIDE EVIDENCE ABOUT ITS UNDERLYING PATHOPHYSIOLOGY
 TOP
 ABSTRACT
 INTRODUCTION
 FACTOR ANALYSIS
 USE OF FACTOR ANALYSIS...
 USE OF FACTOR ANALYSIS...
 ADDITIONAL ISSUES
 CONCLUSIONS
 REFERENCES
 
A syndrome refers to the simultaneous occurrence in an individual of a set of signs and symptoms that points to a single disease or pathophysiologic condition as the cause. The question of whether the insulin resistance syndrome truly exists or not can be assessed by determining whether the central components occur simultaneously in individuals more than would be expected by chance. One way of determining this is to compare observed to expected frequencies of combinations of the components assuming a binomial distribution. If the components do indeed occur simultaneously more than expected by chance (i.e., if they are not independent), then the observed to expected ratio of the frequencies of having none and large numbers of the components will be high, whereas the observed number of individuals with just one component will be considerably lower than expected (i.e., if components occur simultaneously more than expected by chance, then individuals are likely to have none or many components and are unlikely to have an isolated component). A number of studies of the insulin resistance syndrome, reviewed elsewhere, have used this approach and provide convincing evidence that the components do occur simultaneously more than would be expected by chance (3). The disadvantage of this method is that continuous variables are dichotomized using arbitrary thresholds. Most studies using this approach have used standard clinical thresholds for defining hypertension, obesity, dyslipidemia, and other components of the syndrome. Further, although the absolute values of observed and expected frequencies will differ with different thresholds, the pattern of the ratios will not. For example, in one study, there were a greater than expected occurrence of individuals with two or more components occurring simultaneously and a deficit of individuals with an isolated component when a cutoff of either the 70th or the 90th percentile was used to define each risk factor (31).

Factor analysis is potentially a way of advancing this body of research by explaining associations between components of the syndrome. In epidemiology, we generally distinguish between hypothesis-generating studies and those that are used to confirm an a priori hypothesis. Factor analysis can similarly be used for hypothesis generation (exploratory analysis) or hypothesis testing (confirmatory analysis). In exploratory factor analysis, the number of latent factors is essentially unknown and has to be determined from the data. Confirmatory factor analysis can be used to assess the robustness of results from an explanatory factor analysis in independent data sets or to directly test a priori hypotheses from other research sources. The essential assumption of factor analysis is that there are unmeasured latent variables underlying the associations between the observed measured variables. With respect to the insulin resistance syndrome, knowledge from biologic studies could be used to determine a priori the number of unmeasured (latent) variables that represent the syndrome and thus the number of factors one would expect to extract. For example, if pathophysiologic studies suggested one pathway linking all components of the syndrome, then one would expect a priori that one factor would be a good description of the covariance among the observed variables. Confirmatory maximum likelihood factor analysis could be used to test the hypothesis that one latent factor is a good fit of the data.

None of the studies using factor analysis to explore the insulin resistance syndrome has stated whether it considers its approach to be exploratory or confirmatory. Since none has formally tested the goodness of fit of its model against an a priori hypothesis, we assume that all the studies are exploratory, although a recent review of these studies concluded that they provided valuable insights into the pathophysiology of the syndrome (32). That review concluded that the studies shared three common findings: 1) the identification of between two and four factors; 2) the loading of some measure of insulin resistance or hyperinsulinemia on more than one factor; and 3) a separate factor for blood pressure (32). The interpretation of these findings was that insulin resistance most likely unites different components of the syndrome through different pathophysiologic processes and that hypertension may not be part of the syndrome (5, 32). The suggestion that insulin resistance unites different components of the syndrome through different pathophysiologic processes points to a theoretical model with a number (not specified) of underlying latent factors, in which case investigators should use evidence from biologic (cellular level) studies to determine a priori how many factors one would expect and to test this theory using confirmatory factor analyses. Alternatively, in the absence of such biologic evidence, the findings of current (exploratory) factor analysis studies should be formally tested in confirmatory studies using independent data sets, but including the same component variables and applying the same factor analysis procedures. Ultimately, the factor structure would have to be shown to be consistent with biologic understanding of the syndrome.

A number of approaches are used to estimate the parameters in a factor analysis model, such as principal factor analysis and maximum likelihood estimation. The vast majority of the studies of the insulin resistance syndrome have used principal factor analysis. Principal components analysis transforms the original variables into new ones that are uncorrelated and account for decreasing proportions of the variance of the data (29). The new variables, the principal components, are defined as linear functions of the original variables. Each principal component is associated with an "eigenvalue" that can be used to calculate the variance in the original variables explained by the component (with each original variable standardized to have a variance of 1) (29). When principal components analysis is used to extract factors in a principal factor analysis, only those factors that explain most of the variance in the data are extracted. A common criterion for deciding which of the principal components to include in the factor analysis is that each component must have an eigenvalue of at least 1.0, that is, those that explain at least as much of the variance in the data as any original single variable (29). Using a different eigenvalue criterion to extract factors may produce a different number of factors. This is demonstrated by two publications (20, 21) that were conducted on the same study population. In the first publication (20), the eigenvalue criterion used to extract factors was 1.0, and five "insulin resistance syndrome" factors were extracted; in the second publication (21), the eigenvalue criterion used to extract factors was 2.0, and just two factors were extracted. In one study of the insulin resistance syndrome, the eigenvalue criterion was relaxed from 1.0 to 0.9 in men only so that factors similar to those in women could be extracted (18). In one study (16), an a priori decision was made to extract four factors. However, no rationale was provided for this choice.

The suggestion that hypertension may not be linked to the insulin resistance syndrome to the same extent as other components (because in many studies systolic and diastolic blood pressures load together on one factor that does not include other components of the syndrome) (5) reflects a misunderstanding of factor analysis. These findings reflect the fact that systolic and diastolic blood pressures are more strongly associated with each other than they are with other components of the insulin resistance syndrome, something which most clinicians would expect. One could think of this as the factor analysis identifying a latent variable "tendency to hypertension," which explains much of the variation in systolic and diastolic blood pressures. This is analogous to the problem of collinearity in linear regression models and is therefore not new to epidemiologists. In one study (13), the investigators argued that it was inappropriate to include both diastolic and systolic blood pressures together in a factor analysis, and they repeated their main analysis with the exclusion of diastolic blood pressure. They found that, in women only, systolic blood pressure loaded with insulin resistance, obesity, and dyslipidemia. Similar results were found in a second study in which diastolic blood pressure was not included, and systolic blood pressure loaded on the insulin/glucose factor. However, when the analysis was repeated including diastolic blood pressure, it loaded with systolic blood pressure on a separate factor (26). Therefore, the evidence for inclusion or exclusion of hypertension in the definition of the syndrome is based on whether one or two blood pressure measurements are included in the model rather than on any sound clinical or pathophysiologic reasoning.


    USE OF FACTOR ANALYSIS TO ASSESS THE CLINICAL IMPORTANCE OF THE INSULIN RESISTANCE SYNDROME
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 ABSTRACT
 INTRODUCTION
 FACTOR ANALYSIS
 USE OF FACTOR ANALYSIS...
 USE OF FACTOR ANALYSIS...
 ADDITIONAL ISSUES
 CONCLUSIONS
 REFERENCES
 
One difficulty of assessing the possible clinical importance of the insulin resistance syndrome is the problem of how to define the syndrome. In attempts to resolve the issue of various definitions of the syndrome, both the World Health Organization (33) and the National Cholesterol Education Program Expert Panel on Detection, Evaluation, and Treatment of High Blood Cholesterol in Adults (34) have proposed working definitions of the syndrome for use in research and clinical practice. Studies examining the association between the syndrome and disease outcomes, such as diabetes and cardiovascular disease, have used a variety of different definitions. Four studies have conducted factor analysis and then assessed the associations between the estimated values of the factor scores for individuals and coronary heart disease (10, 11, 16, 18). In all four of these studies (two cross-sectional and two prospective), factors representing some aspect of the insulin resistance syndrome have been found to be associated with coronary heart disease. None of these studies assessed whether the magnitude of the associations between the factor scores and disease outcomes is greater than that between any of the individual components, which could be done and would address the question of how clinically important identifying the presence of the syndrome might be. With growing interest in assessing the prognostic ability or predictive power of different combinations of explanatory variables (35), one could compare the predictive ability of a definition of the insulin resistance syndrome based on factor scores with either individual components of the syndrome or a standard definition of the syndrome, but to date no studies have done this. A more straightforward method of determining whether the syndrome predicts disease outcomes by a greater magnitude than individual components of the syndrome would be to assess additive and multiplicative effects using standard (linear, logistic, Poisson, or Cox) regression models. Cross-sectional results from the Atherosclerosis Risk in Communities (ARIC) Study suggest that there is marked synergism (multiplicative effects) between components of the insulin resistance syndrome in their association with atherosclerosis (36). However, although not specifically commented upon, one prospective study found that the magnitudes of the odds ratios between individual components of the syndrome and coronary heart disease were similar to those for the World Health Organization-defined syndrome (37). We have found similar results in a cross-sectional analysis, in which the odds of coronary heart disease were similar for insulin resistance, obesity, and dyslipidemia as they were for the syndrome defined using criteria from either the World Health Organization or the National Cholesterol Education Program Expert Panel (38).

A further problem of using factor analysis to assess the association between the insulin resistance syndrome and disease outcomes is that factor scores are not easy to obtain in a clinical setting, and therefore the results of such studies are difficult to apply to clinical practice. Randomized controlled trials are required to determine whether clinical outcomes would be improved by screening for other components of the syndrome, among individuals with at least one component, and by directing treatment specifically toward the underlying pathophysiology. Factor analysis is unlikely to be important for assessing this aspect of the clinical relevance of the syndrome.


    ADDITIONAL ISSUES
 TOP
 ABSTRACT
 INTRODUCTION
 FACTOR ANALYSIS
 USE OF FACTOR ANALYSIS...
 USE OF FACTOR ANALYSIS...
 ADDITIONAL ISSUES
 CONCLUSIONS
 REFERENCES
 
Once factors have been extracted, each original variable will have provisional factor loadings contributing to the estimate of each factor. The number of parameters to be estimated in a factor analysis is therefore equal to the number of original variables in the analysis multiplied by m, the number of factors, plus the variances of the specific residuals e for each variable. In the situation where m is greater than 1, there is no unique solution to the factor analysis model. In order to produce a unique solution, additional constraints are placed on the data by a process known as rotation. A number of different rotational methods are possible. All of the factor analysis studies of insulin resistance have undertaken orthogonal rotations. Mathematically, any orthogonal transformation of the factor loadings will give a valid solution, so it is common practice to use a transformation to obtain a mathematically equivalent solution, which is more interpretable in terms of labeling the factors. The aim of the rotation is, therefore, to obtain a simple structure that can be easily interpreted. A general rule is that a loading pattern has a simple structure if variables can be divided into groups such that the loadings within each group are high on a single factor, moderate or low on one other factor, and negligible on the remaining factors. The standard orthogonal rotations that are implemented in statistical packages, such as varimax, quartimax, or promax rotations, use indices of simplicity based on this general idea. With respect to interpreting the results of different factor analysis studies, the type of rotation used might influence which variables load on particular factors. In confirmatory factor analysis, one should be able to replicate findings from an exploratory study using the same rotation method together with other analysis methods being identical.

The magnitudes of factor loadings for each variable are used in the final interpretation of the factor analysis. Conventionally, variables that have a factor loading of 0.4 or greater (or <=–0.4) within a particular factor are considered to be its major components, and factors are usually given names relating to their major components (29). Studies of the insulin resistance syndrome have used a range of loading thresholds from 0.25 to 0.40 or greater. The lower the threshold, the more likely a variable is to load on a particular factor. The choice of 0.4 as the cutoff for deciding that a variable loads on a particular factor is arbitrary and could be considered analogous to the arbitrary choice of the 0.05 cutoff for defining statistical significance (39). This highlights the importance of applying clinical knowledge to the interpretation of the factors and not simply allowing them to be data driven.


    CONCLUSIONS
 TOP
 ABSTRACT
 INTRODUCTION
 FACTOR ANALYSIS
 USE OF FACTOR ANALYSIS...
 USE OF FACTOR ANALYSIS...
 ADDITIONAL ISSUES
 CONCLUSIONS
 REFERENCES
 
The research approach to any clinical syndrome requires a clear appreciation of the questions being addressed and an understanding of the most appropriate study design and statistical methods to answer these questions. Our review of studies using factor analysis to model the insulin resistance syndrome highlights four main problems: 1) studies to date have been exploratory rather than being grounded in (biologic) theory; 2) investigators have not clearly specified or examined whether the linear factor model is a good representation of the theory about the nature of the syndrome; 3) there has been a lack of attention by investigators to the potential for highly correlated variables (such as diastolic and systolic blood pressures) to create distinct factors; and 4) the process of factor rotation and selection of thresholds for factor loading is arbitrary but can importantly affect the results and their interpretation. Most studies that have used factor analysis to study the insulin resistance syndrome have presented only vague aims, most commonly to describe the pattern of simultaneous occurrence of the components of the syndrome. The clinical or public health importance of this is unclear. Others have suggested that using factor analysis has provided useful insights into understanding the pathophysiology of the syndrome (5, 32). Confirmatory factor analysis based on sound theoretical concepts and a clear understanding of the statistical methods might provide additional insights in this area, but, to date, none of the studies has adopted this approach. Ultimately, epidemiologic studies are not the most appropriate design for determining the pathophysiology of a disease. We suggest that future studies using factor analysis to study insulin resistance provide clear aims and justifications for how this particular statistical method will improve understanding of the syndrome.


    NOTES
 
Correspondence to Dr. D. A. Lawlor, Department of Social Medicine, University of Bristol, Canynge Hall, Whiteladies Road, Bristol BS8 2PR, United Kingdom (e-mail: d.a.lawlor{at}bristol.ac.uk). Back


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 REFERENCES
 

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