Correlations: simple correlations and complex modeling with correlations
Historically, the simple correlation method was the first method to model a symptom cluster and its consequences (see Table 1). Researchers selected a possible cluster clinically or theoretically, and provided the evidence for a cluster using correlation among symptoms . Correlation examines the covariation of two symptoms with the correlation coefficient quantifying the direction (same or opposite) and strength of the covariation. Studies have reported low-to-moderate correlations between symptoms without justifying the strengths of the relationships to be a part of a cluster [1,12].
Partial correlation can be used to quantify the residual relationships between two symptoms after controlling for the other variable(s) in the dataset. By expanding the directional relationships of partial correlations, Beck et al. tested a mediation model for a symptom cluster. They found that pain had an indirect effect on fatigue via sleep disturbance in addition to its direct effect. Correlation methods can provide the mathematical evidence of a selected cluster, but they have limitations when investigating cluster solutions in which symptoms have complex interrelationships .
Common factor analysis
In terms of empirical identification of symptom clusters, factor analysis has been used frequently in oncology [5▪,13,14]. Factor analysis is a class of statistical methods explaining observed associations among many variables by using unobserved variables, called factors. It assumes that the linear combinations of some unknown source variables create the observed variables . Common factors are the common underlying sources for multiple observed variables and thus, induce correlations among variables . By identifying common factors, the structures or dimensions that underlie a set of observed variables can be identified. Two or more symptoms that share one common factor are considered to form a symptom cluster.
Factor analysis is a helpful method when complex relationships among many symptoms may exist. Its limitations include the subjectivity in determining the final solution, the need for a large sample, and the need for specialized approaches for categorical data. In a recent study, theoretically modeled symptom clusters were confirmed by confirmatory factor analysis [17▪].
Principal component analysis
Principal component analysis (PCA) is often described as a type of factor analysis, however, it mathematically has nothing to do with unobserved variables. PCA searches for one or more components that best reproduce variances in a set of data . It assumes that all variances of a set of variables can be summarized into components. A component is simply a combination of correlated variables whereas a common factor induces correlations among observed variables [16,19]. A study by Jimenez et al.[6▪] is an exemplar of oncologic symptom clustering using PCA and identified four symptom clusters in a large sample of patients with various types of advanced cancer.
PCA and factor analysis are superficially similar; in that, both approaches result in a smaller number of dimensions reflecting information from a larger pool of items (e.g., symptom indicators). The decision between using PCA or factor analysis should be guided by the purpose of the analysis, as well as theoretical considerations about the relations among the symptoms . PCA is a data-reduction technique that involves no assumptions about the relations among the variables. It is of use when the goal of the analysis is to reduce a large number of variables to a smaller, more manageable number, perhaps in the context of a modest sample size. PCA is particularly useful when the symptom indicators are minimally correlated. Factor analysis, in contrast, is an explicit model of the relations among the indicators. It can be especially informative when there is a conceptualized common variable that might underlie several observed symptoms. Factor analysis is likely to yield poor results when indicators are not well correlated, as that suggests that no common factor exists. Practically, PCA and factor analysis can yield different symptom clusters in oncology due to the small number of symptoms and lower correlation levels among these symptoms [15,20]. In fact, PCA and factor analysis created different oncologic symptom clusters in the same sample [21▪▪].
Cluster analysis of symptoms and patients
Cluster analysis is a class of graphical and statistical classification methods used to group units into homogeneous subgroups based on relative similarities among the units on a set of attributes . A subgroup is referred to as a cluster, which is a statistical term and should be differentiated from the conceptual term of symptom cluster. Clustering strives for mutual exclusiveness in its classification: a unit belongs to only one cluster. Although the similarity of units can be determined through various methods, standardized distance measures are now the standard and relay information on both the rank order and level of differences between units regarding the attributes of interest. Cluster analysis can categorize symptoms that occur in a similar pattern across patients and thus, can be used to identify symptom clusters. For example, Kirkova et al. conducted cluster analysis with a large sample of advanced cancer patients and identified three symptom clusters stable over two samples (see Table 1). The main advantage of cluster analysis is that it enables the examination of a large number of symptoms with limited sample sizes. However, determining the final solution is subjective and the handling of missing data can be tedious.
Cluster analysis can also categorize patients based on their similarities regarding a set of attributes (e.g., symptoms). Note that empirically identified symptom clusters cannot be guaranteed to exist in patients. Cluster analysis of patients can be a method to identify patients who experience a particular cluster, that is, a phenotype of symptom experience. Several studies have examined whether cancer patients experience a theoretically chosen or empirically identified symptom cluster, and have investigated subgroups of patients with unique symptom cluster experience [8▪,23].
Latent class analysis/latent profile analysis
Latent class analysis (LCA) is a new technique for grouping patients in oncology. It is similar to cluster analysis in purpose but its use is limited to patient classification. LCA is a latent variable method and assumes there is a latent nominal variable categorizing patients into subgroups. The observed indicator variables display the nature of the subgroups . A latent variable summarizes the correlation between indicator variables as a common factor in factor analysis. LCA uses an iterative process through the expectation maximization algorithm to find the best solution. It provides stronger model fit statistics than cluster analysis. The replication of a solution can be tested and thus, stability of the findings in the subsamples can be established. The method was originally developed for binary indicators but has been extended to quantitative variables, and this is referred to as latent profile analysis (LPA), which has been commonly used in social and behavioral science. LPA was used recently to examine the phenotypes of symptom cluster experience and its association with cytokine genes [9▪▪]. LCA and LPA require careful model construction and selection.
Structural equation modeling
Structural equation modeling enables researchers to identify subsets of measured variables that are assumed to collectively represent a higher order latent construct, such as a symptom cluster. It evaluates the extent to which a subset of measured variables represents a higher-order latent construct and examines the relationships of the latent variable to other variables of interest (either independent or outcome variables) . By doing so, a complex symptom cluster can be identified and simultaneous evaluations of the relationships between multiple variables, mediators, and outcome variables can be performed.
Structural equation modeling requires the specification of a theoretical model (e.g., a higher-order latent construct), the variables used to operationalize these constructs, and the nature and direction of relationships among the variables. This technique was used to examine the temporal changes in symptoms before death .
METHODOLOGICAL ISSUES IN MODELING TECHNIQUES
Several issues need to be considered in order to establish a symptom cluster study's validity. First, the assumptions governing any statistical method should be considered when choosing a method. For example, factor analysis is preferred to PCA when identifying symptom clusters induced by common factors and conceptual interpretability is paramount . Conversely, PCA is better if the goal is to identify clusters on the basis of the variance minimization principle and thus optimize the (mathematical) homogeneity of derived clusters. Studies indeed often ignore whether the statistical assumptions or conceptual framework for a particular symptom cluster is congruent with the selected analytical method. This shortcoming of not matching assumptions, frameworks, and analytics characterized many early symptom cluster studies. For instance, Jimenez et al.[6▪] chose PCA without explaining what kind of symptom cluster they were exploring and how PCA could better identify such clusters over factor analysis. Future studies should emanate more clearly from conceptual frameworks and match the conceptual dynamics with the analytical approaches.
Second, the domain of symptoms (i.e., the number and type of symptoms that are included in an analysis) should be considered. Earlier studies often used secondary data that, in the primary database, included a comprehensive symptom inventory. However, researchers should be cautious when selecting the symptom domain. At the core, this is a sampling issue – not of patients, but of the symptoms covered by inventories and other data collection methods. When the domain of symptoms is under-represented or over-represented, symptom clusters can be underidentified or overidentified . The measurement time point is important, and studies should be timed with respect to the illness trajectory and the nature of the target population as these can change the domains of symptoms. When the same domain was used, even with different statistical methods, cluster analysis and PCA can yield the same symptom cluster [see the reference  for exemplar]. However, the use of different domains can yield different clusters even with the same method.
Third, the use of a single item from a comprehensive symptom inventory for a particular symptom can increase measurement error for some symptom constructs. On the contrary, inclusion of multiple items to measure the same symptom construct may bias findings due to relatively higher correlations between the items of the same construct. Thus, measures should be carefully selected.
Fourth, many multivariate statistical methods involve subjective decision-making about various steps in the analysis and in determining the final solution. Therefore, theoretically and practically sound criteria to reach final solutions should be established and, most importantly, researchers should specify and justify each and every one of them.
Fifth, the stability of a solution needs to be confirmed. The structural stability of the solution is the generalizability of the findings from the overall sample into subsamples. Often, the sample is heterogeneous in terms of disease stage, type of cancer, treatment, or any unmeasured characteristics. Analysis of heterogeneous data can lead to false findings. One study  examined the structural stability in a given sample. The stability over time is related to whether findings at one time point can be replicated at another time point. Two exemplar studies examined the stability of a symptom cluster over time [5▪,13]. The stability over time issue has been limited to qualitative evaluation. Longitudinal data analysis may enable researchers to better quantify the nature of symptom experience over time and permit quantitative evaluation of the stability over time.
Lastly, sample size has been recognized as an issue and recent studies are using larger samples. A small sample size threatens the external validity of study findings. Sample size calculations should be performed in consideration of the design and analytics used.
STATISTICAL METHODS TO EXAMINE MECHANISMS OF SYMPTOM CLUSTERING
From the identification of symptom clusters, we are now moving toward a new area, the mechanisms underlying symptom clustering. All aforementioned methods apply to cross-sectional data. However, cross-sectional studies are limited in their ability to generate evidence about directional or causal relationships between symptoms and their possible underlying mechanisms. Longitudinal study designs and relevant statistical methods allow for the construction and testing of complex symptom cluster models with possible underlying mechanisms. Several statistical approaches for longitudinal study design are introduced but their use in symptom cluster study is very limited at this point.
Regression approach for longitudinal data
The regression framework can be used to analyze longitudinal data by incorporating an indicator of ‘time’ in the statistical model. Doing so must be done with care, as the dependence among observations can result in violations of the assumptions of conventional linear regression techniques, often biasing estimates of standard errors and therefore significance tests. Several technical solutions are available to handle these issues. One approach is to compute standard errors using approaches that relax these assumptions, such as ‘robust’ standard errors or generalized least squares estimation. Another method is the inclusion of a person-specific error term either as a random variable or a fixed constant in the model (i.e., random effect models or fixed effect models). In these models, time itself and lagged biological and/or psychological predictors of symptom cluster experiences can be included. By doing so, the causal influences of these predictors can be clarified, providing stronger evidence on the mechanism of symptom cluster experience. No symptom cluster studies used this regression approach.
Growth modeling, growth curve analysis, and growth mixture analysis
The growth modeling technique, also called growth curve analysis, can model changes in a ‘measured’ continuous or categorical-dependent variable over time using continuous ‘latent’ growth factors [27,28]. Growth modeling uses the structural equation modeling framework in which growth factors are treated as quantitative latent variables with random coefficients. Thus, coefficients for growth factors (i.e., the means and variances of slopes and intercepts in individual growth) are estimated as regression coefficients in regression analysis.
Growth mixture modeling is a variation of growth modeling where the data are expected to come from unobserved sub-populations and group memberships are unknown . Growth curve analysis in oncology has been used to identify a group of symptoms changing over time in a similar pattern [11▪▪] and to examine the patterns of fatigue over time . Growth modeling requires extensive prior knowledge to modeling the patterns and complicated procedures in order to find the final solution. Growth modeling can also include biological/psychological predictors and examine how these predictors are associated with symptom cluster experience over time. By doing so, growth modeling can provide stronger evidence for the causal relationships between symptoms and their predictors.
Latent transition analysis
Latent transition analysis (LTA) models change over time in a latent class variable, just as growth curves model changes over time in a continuous variable. LTA focuses on the ‘transition patterns’ of subject memberships in latent classes over two consecutive time points – e.g., membership in a given latent class at Time 1 of a longitudinal study is associated with certain probabilities of membership in the possible latent classes at Time 2 . The changing patterns are quantified in a matrix of transition probabilities between two time points. Covariates, which can include a manipulated treatment, can be included to determine their influence on the transition matrix using multinomial logistic regression links. By doing so, it can provide evidence for the directional influence of covariates (e.g., possible underlying mechanisms) on outcome variables (e.g., symptom cluster experience).
In social and behavioral science, the developmental stages have been modeled with LTA . Although LTA has never been used in oncology, it certainly has implications in this field. A particular strength of LTA is that the measurement of the latent class can differ on different occasions – for example, in an etiologic study, some symptom indicators may not be applicable at the earliest measurement occasions. Of note, transitions can only be quantified over two adjacent time points and thus, examining transitions over multiple time points can be tedious, though higher-order LTA techniques (e.g., classes of typical transition patterns) can mitigate this drawback.
In designing longitudinal studies, several issues should be considered. First, when changes in a single dependent variable are examined, a strategy for summarizing multiple symptoms within a cluster into a single variable should be determined. When scaling differs across symptoms, weights of each symptom will also differ when combining these symptoms. Using the same scales over multiple symptom measures should be carefully considered as simple standardization procedure can impose a variance restriction over time. Second, the selection of measurement timing, frequency, and time window should evolve from a theoretical framework that describes the changing patterns of a particular symptom cluster. Appropriate statistical techniques should be chosen based on this framework.
We reviewed several statistical approaches that have been used to identify and model symptom clusters in oncology: correlation method, factor analysis, PCA, cluster analysis of symptoms and subjects, LCA, and structural equation modeling. We also discussed several longitudinal data analysis techniques to examine the underlying mechanisms of symptom clusters. Some methods use only observed variables and some use latent variables to explain the observed phenomena. All latent variable methods have advantages regarding the accommodation of covariates and the consequences of symptom cluster experience; thus, these methods could be implemented for testing a complex symptom cluster experience theory.
Conflicts of interest
There is no conflict of interest.
This study was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2012R1A1A1009672). I.A. was supported as Director of the Academic Fellowship Program in Clinical Outcomes and Comparative Effectiveness Research funded by the Bureau of Health Professions, US Department of Health and Human Services, through the Arizona Health Education Centers Program.
REFERENCES AND RECOMMENDED READING
Papers of particular interest, published within the annual period of review, have been highlighted as:
- ▪ of special interest
- ▪▪ of outstanding interest
Additional references related to this topic can also be found in the Current World Literature section in this issue (pp. 119–120).
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Keywords:© 2013 Lippincott Williams & Wilkins, Inc.
methods; statistics; symptom clusters