The New England Journal of Statistics in Data Science

Multiple-week diet records require participants to record everything they eat/drink over several weeks. They are considered the gold standard for collecting dietary information because, unlike other methods, they do not rely on memory. The high costs and participant burden have severely limited their use in large-scale epidemiological studies; however, their ability to accurately provide detailed dietary information makes them useful in validation studies of other dietary assessment tools and in monitoring compliance in trials. In addition, recording can worsen as recording days increase [131].

Multiple 24-hour recalls involve reporting all foods and beverages consumed in the previous 24 hours (or in a calendar day) to a trained interviewer in person or over the phone for more than one assessment. Although reliance on the participant’s memory leaves room for measurement error, a skilled interviewer can produce highly detailed and useful nutritional data, (almost) comparable to a diet record [22]. This method has been widely used in clinical nutrition and dietary intervention trials. It is also employed in national surveys to monitor trends in nutritional intakes [131].

A food-frequency questionnaire (FFQ) consists of a structured food list and a frequency response section in which the participant indicates his/her usual frequency of intake of each food or beverage over a certain period in the past, usually one year. Portion sizes, with the indication of a standard portion size (in grams or natural units), are also generally queried [131]. Because of their relying on memory, FFQs are more prone to a biased and/or partial recording of dietary information, compared with 24-hour recalls and diet records administered for many days spread out over the entire reference period of an FFQ [104]. The FFQ is, however, the most common option for measuring intake in extensive observational studies. Indeed, it is easy to administer, has a low participant burden, and well captures usual long-term dietary intake. These features also allow for repeated assessments of dietary habits via FFQ over time. This is crucial for capturing longer-term diet variation [131].

Country-specific, complete, and up-to-date food composition databases are essential to convert food consumption (as collected with any of the previous dietary assessment tools) into macro-nutrients, micro-nutrients, bioactive substances (e.g., polyphenols), and non-nutrient (e.g., ethanol or contaminants) intakes. The difference between macro- and micro-nutrients is based on their different function: the former create energy and then promote the organism’s growth, and the latter contribute to other (i.e., optimal cell) functions. The former are expressed in grams, the latter in milligrams or micrograms. Estimating nutrient composition from food intake data poses additional challenges. Among others, a food’s nutrient content varies with production location, season, growing conditions, storage, processing, and cooking techniques. Some of these factors are unaccounted for in food composition databases, generating estimates of nutritional content that can be affected by errors. However, the degree to which this is problematic differs from nutrient to nutrient. Food composition databases represent the current composition of foods consumed by a given population as far as they are regularly updated. Continuous updates allow targeting and covering newly manufactured pre-packaged products, whose formulation and nutrient composition is swiftly changing due to policies, customers’ demand, and marketing strategies. The incomplete coverage of foods and the lack of information on some nutrients or bioactive substances of health interest in a food composition database might be potential sources of error. While these sources of error do not substantially compromise the ability to rank individuals based on nutrient intake and, therefore, to evaluate associations with health outcomes [131], estimating nutrient composition from food intake data requires continuous improvements in the accuracy of food composition databases, especially given the changing food landscape [128].

When participants provide biological specimens, researchers can also assay biomarkers to integrate information on foods/beverages from the dietary assessment tool or nutrients from the food composition databases. Examples of biomarkers include doubly labeled water (for total energy intake), urinary nitrogen (for protein intake), 24-hour urinary sodium and potassium, blood lipid profiles, and serum and plasma folate. In principle, biomarkers provide objective intake measurements without bias due to self-reporting. In practice, the use of biomarkers in investigating nutrient-disease relations has been limited to nested case-control studies, small trials, and validation studies of dietary assessment tools. This is due to their inherent limitations, including a lack of sensitive or specific biomarkers for many foods and nutrients, their expensive collection, and assessment errors from multiple sources. In addition, biomarkers may not be indicators of individual long-term intake [104].

While the appropriate application of different diet assessment methods, alone or in combination, and of food composition databases allows for a reasonably comprehensive assessment of the diet in free-living populations, awareness of sources/types of measurement error in dietary intake data is crucial in the analysis of the association between diet and non-communicable diseases, including cancer [131, 52].

Usual intake (in the short or the long term) is estimated via the reported intake, as derived from dietary assessment methods. This estimation procedure poses critical challenges. In its simplest form, the following additive model expresses the relation between true, ${T_{i}}$, and reported, ${R_{it}}$, intake: where the reported intake ${R_{it}}$ for the i-th individual on the t-th time (e.g., day) linearly depends on one source of random error, ${\epsilon _{i,t}}$, and three different sources of systematic error, ${\beta _{0}}$, ${\beta _{1}}$, and ${u_{i}}$. In detail, the random measurement error ${\epsilon _{i,t}}$ of the i-th individual on the t-th time models individual consumption variation over time and usually reflects day-to-day changes in consumption; ${\beta _{0}}$ represents systematic error occurring in the same way to all individuals, and therefore the effect is constant; ${\beta _{1}}$ depends on the true intake, ${T_{i}}$, and its magnitude is proportional to or multiplied by the true intake; ${u_{i}}$ is the subject-specific bias, depending on individual characteristics (i.e., age, sex, or education) that lead to systematic under- or over-reporting consumption.

Validation studies allow for assessing the relationship between reported and true intake. In the absence of a “true” gold standard, an alloyed gold standard (i.e., a reference dietary assessment method with random error only) is used to estimate the true intake. This gold standard is generally a multiple-week diet record or a recovery biomarker, such as a specific biological product directly related to intake and not subject to homestasis or substantial inter-individual differences in metabolism. The gold standard is needed to distinguish the systematic error components and correct the intake estimates.

The 24-hour recalls have generally larger random within-person error than an FFQ, but smaller systematic error, when the two tools are compared with a reference recovery biomarker [105]. The random error in 24-hour recalls is mostly driven by day-to-day variation in intake and other random errors that affect reporting from day to day. It can be mitigated by averaging over many repeats. FFQs usually have lower random within-person variation than the other dietary assessment methods, because they are designed to assess the usual average intake over a longer time period. In FFQs, the error is driven by inaccuracies associated with recalling long-term intakes and features of the instrument, such as the finite food list and the (relative) lack of detail about foods consumed. These biases are systematic and are not mitigated by averaging across repeated measures. On average, at the population or group level, the Observing Protein and Energy Nutrition [105] and other validation studies suggest serious energy underreporting, of approximately 10% when using 24-hour recalls and about 30% when using FFQs. This underreporting arises from sources of group-level bias (${u_{i}}$), constant additive bias ${\beta _{0}}$, and intake-related bias ${\beta _{1}}$.

Although FFQs may suffer from greater measurement errors, they have been shown to have acceptable validity when compared to reference measures [12, 116]: typical correlation coefficients for individual nutrients or foods range from 0.4 to 0.7 [131]. Along with repeated FFQs, adjustment for total energy intake in long-term prospective cohort studies further improves these validity coefficients. When biomarkers are available together with dietary records, triangulation methods can be used to obtain improved estimates of correlations of FFQ intake with true intake [91]. These validity coefficients can be used to correct for measurement error in epidemiologic analyses, and the application of these measurement error correction methods is increasingly being extended to more complex analyses [97, 102, 112]. These techniques have allowed for valid inferences to be drawn from large cohort studies with the use of FFQ data.

In conclusion, the considerable progress made – especially with the use of repeated measures of diet over time – has enabled nutritional epidemiologists to increase reliability in collecting and using dietary information at the individual and population levels. However, continued improvements in dietary assessment methodology and measurement error correction are needed to support the current understanding of the relationship between diet and non-communicable diseases, including cancer [104].

One of the major remarks against nutritional epidemiology is that it mostly relies on observational data, which is deemed inferior to experimental data in determining causality. While randomized controlled trials (RCTs) with hard clinical endpoints occupy the highest position in this hierarchy, RCTs are usually not the most suitable/feasible study design to answer nutritional epidemiologic questions regarding long-term effects on health/disease (e.g., cancer) of specific foods or nutrients [104]. Unlike classic drug trials, in RCTs of dietary interventions typically [104, 108]:

In the absence of evidence from large RCTs on hard endpoints, nutritional epidemiologists typically rely on prospective cohort studies, the strongest available observational design, to infer causality. Being prospective, cohort studies are less affected by the typical biases (i.e., reverse causation, recall bias, and selection bias) of retrospective or cross-sectional study designs.

Reverse causation describes the situation in which the outcome affects the exposure rather than the other way around. This is a common concern with cross-sectional and retrospective case-control studies as they assess exposure and outcome simultaneously (although in case-control studies, exposure information concerns the past). Prospective cohort studies can minimize the possibility of reverse causation because participants are followed forward in time; these studies can also examine the extent of reverse causation from subclinical disease by lagged analyses [104].

Compared to retrospective case-control studies, prospective cohort studies begin with a disease-free population at baseline that is followed up to ascertain incident cases that develop over time and can minimize both selection bias (controls not being representative of the underlying population that gave rise to cases) and recall bias (knowledge of disease status affecting recall of diet) [104].

A major challenge when working with observational data is confounding. A confounder is a variable associated with both the exposure (i.e., diet) and the outcome (i.e., cancer), and, when unaccounted for, introduces bias into the exposure-outcome relation. The main reason why randomized trials are considered superior in inferring causality is that, as far as the sample size is large enough, the random allocation of participants to treatment groups nullifies measured and unmeasured confounding.

To account for this bias in an observational study design, researchers must identify all relevant confounders based on existing evidence/theory on the association between dietary habits and the disease (e.g., cancer type) under consideration. Once data are collected, the investigator can statistically adjust for confounders in a multiple regression model, including the main exposure (i.e., diet) and/or restrict the analysis to a specific subgroup to minimize residual confounding. Sensitivity analyses further strengthen results by suggesting the magnitude of unmeasured confounding needed to neutralize an effect completely [104]. A prospective design additionally allows for up-to-date tracking of confounders and, in this way, limits the risk of residual confounding. While updated information may reduce measurement errors in the assessment of confounders, additional information can be collected when needed.

Although there are several ways to account for confounding in prospective cohort studies, the critical assumption of no unmeasured or residual confounding needed to infer causality cannot be empirically verified in observational epidemiology [53]. For this reason, prospective cohort studies are often seen as providing statistical associations but not causality. However, when satisfied, the Hill criteria [54] supports the possibility of inferring causality from observational data when randomized trials of hard endpoints are not feasible [104].

In addition, evidence from prospective cohort studies should be integrated with results from randomized trials of intermediate responses, in vitro and in vivo studies, to arrive at a consensus on diet and a health/disease (e.g., cancer). The inference of causality is strengthened when these different types of studies provide consistent evidence in the context of the larger evidence base.

In conclusion, when randomized trials of hard endpoints are unfeasible, well-conducted prospective cohort studies can be used to infer causality with a high degree of certainty. Sources of bias, including confounding, can be minimized by relying on high-quality study design, careful statistical analysis and interpretation, and replications of the findings across different populations. Corroborating data from multiple study types and populations can enhance the weight of evidence [104].

The aim of PCA and EFA is to reduce the dimensionality of the data by transforming an original, more extensive set of correlated food groups/nutrients into a smaller and more easily interpretable set of uncorrelated variables, called principal components or factors. Both approaches answer the following question: “What are the major components/factors in a population under study, i.e., those contributing most to the variation of nutrient/food group intakes reported by study participants?” [70].

To answer this question, PCA uses the singular value decomposition and identifies principal components based on the covariance/correlation matrix of the input variables (nutrients or food groups). The resulting components are linear combinations of the original variables with suitable weights (loadings) that explain as much of the variation in the original variables as possible [47]. EFA starts from the same covariance/correlation matrix and shares the data reduction rationale of PCA, but it is based on a statistical model where the random vector of observations (i.e., individual’s dietary data) is explained in terms of some latent common and specific factors. The definition of a statistical model allows rotation of the factor loading matrix, improving the interpretation of the identified factors. EFA may use different estimation methods for model parameter estimation, including PCA and maximum likelihood. Therefore, a principal component factor analysis is defined as an EFA where the PCA method is adopted for parameter estimation. Principal component factor analysis is the more common method used so far to derive a posteriori dietary patterns in nutritional epidemiology [33].

The most followed approaches to select the appropriate number of principal components/factors are eigenvalue greater than 1 criterion (when the correlation matrix is adopted), visual inspection of the scree-plot, and a sensible interpretation of the dietary patterns. A fixed threshold (generally 5%) can also be decided, and only the components/factors whose explained variance exceeds the chosen threshold are incorporated in the analysis [47].

Following both approaches, individuals are ranked based on the degree to which their diets conform to each of the identified factors; this is done by adopting continuously scaled scores either obtained by simple matrix algebra (PCA) or by different estimating procedures (EFA).

Scores are further entered into a regression model for disease (e.g., cancer) risk estimation. In this case, scores categorization into quantiles improves results interpretation because scores, although continuous, have a restricted scale and no measurement unit. Unlike most a priori indexes, any one principal component/factor does not represent the entire eating pattern for any individual or group because the principal components/factors are not mutually exclusive. However, each person’s overall eating pattern can be inferred by assessing his/her multiple principal component/factor scores [70]. The simultaneous inclusion of all dietary patterns in the same regression model without additional multicollinearity issues is granted by scores being uncorrelated by design (PCA) or by additional orthogonal rotation (EFA). This implies that the effect on disease (e.g., cancer) risk of each dietary pattern can be easily adjusted for the remaining dietary patterns.

In disentangling the overall diet into a few summarized profiles, PCA and EFA provide an immediate representation of how food groups/nutrients interact within each dietary pattern. This happens through the principal component or factor loadings. Indeed, the magnitude of each loading for a given component/factor measures the importance of the corresponding food group/nutrient to that component/factor [33, 47]. If a few loadings are high (in absolute value) on a pattern, this pattern will be mostly characterized by pairwise interactions of the corresponding food group/nutrients. Results are generally meaningful, interpretable, associated with health outcomes, including cancer [33, 6, 45], and show modest but raw reproducibility across populations [4, 77, 86].

PCA and EFA have also limitations and challenges. Subjectivity is introduced in constructing food groups or selecting nutrients, in preprocessing input variables, in choosing which data matrix to work on (e.g., covariance or correlation matrix, separate analyses for known subgroups in the data or not) or the number of factors to retain, in the opportunity for factor rotation and which rotation to choose (in EFA only), and in the identification of some criteria for labelling the factors. Unless data collection methods are comparable, input variable choice/preprocessing is standardized, and the dietary pattern identification method is the same, results are not comparable across studies. Even when the previous aspects are standardized, labelling of principal components/factors is still very subjective. In principle, variables corresponding to “large” loadings are interpreted as being important for describing the original data; variables corresponding to “small” loadings can be discarded. However, such interpretation is complicated by the fact that all component/factor loadings are nonzero. Various cut-off rules, rotation strategies, and other procedures have been developed to simplify interpretation, but these largely ad hoc procedures do not contribute to the transparency or objectivity of PCA/EFA. Alternatives to PCA/EFA that offer more interpretable components/factors by forcing loading patterns to include many loadings exactly equal to zero (i.e., by forcing the identification of “sparse” components) are interesting because they reflect current nutritional knowledge where each pattern is typically described by a small subset of food groups/nutrients [65]. In addition, since the label generally needs to be short, often they do not adequately convey to what the underlying principal component/factor is [108]. This further makes comparison of results across studies more challenging. Few rigorous statistical procedures have been implemented to examine internal consistency and validity of the identified solutions [85], although some efforts have been made to assess reproducibility and validity of PCA-based or EFA-based dietary patterns in more recent years [36].

Confirmatory factor analysis (CFA) [50] has not been so much used in nutritional epidemiology so far [126]. The main difference with EFA is that CFA involves specifying the number of factors, which variables will load on each factor (i.e., by putting some 0s in the factor-loading matrix structure instead of estimating each loading) and which relationship exists between each pair of latent factors [66]. In this way, CFA is able to perform hypothesis testing on the overall factor structure and on factor loadings of food groups/nutrients to estimate rigorously the number of factors and identify food groups/nutrients contributing significantly to those factors [85, 83]. Alternatively, CFA can be employed to test the goodness of fit and validity of the factor structure of dietary patterns in a second step, i.e., after PCA or EFA was performed [92, 103]. It is not scientifically confirmed if the results are more accurate than those obtained in a single-step process [90]. With this motivation, some studies performed CFA as a one-step approach instead of using previous results from PCA or EFA.

The advantage of CFA compared to the two methods previously presented is that a latent structure can be specified and tested, and if the researcher has some additional a priori knowledge, this can be incorporated into the model [5].

While PCA and EFA work on data matrix columns (i.e., food groups/nutrients), CA [9] is traditionally used in nutritional epidemiology to explore data matrix rows (i.e., individuals), to identify groups of participants (clusters) with a specific dietary behavior, based on a pre-specified measure of similarity/difference in food group/nutrient intake among individuals [28]. So, CA replies to the following question: “Are there groups of individuals characterized by distinct dietary patterns?” [70].

On the opposite of PCA and EFA, where the subjects can belong to more than one principal component/factor, CA provides one group belonging indicator for each subject. The group belonging indicator is then entered into a multiple regression model with confounding factors to estimate adjusted disease (e.g., cancer) risk related to specific group belongings.

Hard clustering, where study participants are grouped into mutually exclusive clusters, and individuals only belong to one cluster, is by far the most followed approach in nutritional epidemiology [47]. Among available methods, K-means and Ward’s minimum-variance method are widely used, with one paper only [75] comparing previous approaches with the flexible beta one.

The K-means algorithm partitions observations into K clusters in which each observation belongs to the cluster with the nearest mean (cluster centers or centroids) by minimizing within-cluster variances (squared Euclidean distances). The advantages of K-means clustering include its simple interpretation, low computation complexity, fast calculation speed, and suitability for large samples.

Ward’s minimum-variance method is an agglomerative hierarchical clustering algorithm, with the number of clusters changing at each step [49]. Indeed, at each step, one has to find the pair of clusters that leads to a minimum increase (i.e., a weighted squared distance between cluster centers) in total within-cluster variance after merging. Thus, the calculation is slow, and the Ward’s approach is hard to apply in large samples [135].

Among the advantages of hard CA, distinct subgroups of individuals – where everyone belongs only to one specific dietary pattern group – are easy to interpret and relate to disease (e.g., cancer) risk. The dendrogram from Ward’s method shows the clustering process and results visually [47].

Among limitations, uncertainty in individuals’ classification is removed, and each individual is assigned to a cluster with a probability of 1 or 0 [49]. Second, subjective decisions are required at several steps and include selection of input variables (i.e., nutrients, food groups, or factor scores) and data preprocessing, similarity measure, clustering algorithm, initial values, and number of clusters [47]. While some objective methods for selecting appropriate clustering algorithms and the number of clusters exist, the reproducibility of results cannot ensure their ability in representing actual dietary behavior and its relation with cancer risk [75]. Third, sensitivity of the CA to outliers is also an issue. Fourth, formal comparison of results from different clustering algorithms is not as easy as with PCA/EFA, where congruence coefficients between factor loadings are recognized as the preferred method to compare different solutions [16]. Fifth, the CA output is just a group belonging indicator, and there is no PCA-based or FA-based loading matrix to provide an immediate representation of how food groups/nutrients interact within each dietary pattern. So, when the CA is completed, further analyses to compare dietary, socio-demographic, or socio-economic profiles across clusters must be used to interpret the identified patterns [56]. Sixth, a major drawback of both K-means and Ward’s method is their tendency to create spherical clusters of equal volume [49], which leads to biased clustering solutions when this assumption is not met by the data. In addition, another limitation of Ward’s method is its tendency to create clusters with an equal number of observations, which is an unrealistic scenario in nutritional epidemiology [49]. Seventh, the reference category to estimate the effect of each dietary pattern on disease (e.g., cancer) risk is not as naturally identified as in regression models based on PCA/EFA-based dietary patterns. As one group has to be chosen as the reference group to express disease risk related to the remaining groups, what are the main characteristics this reference group should show? Should it be the bigger one or the one with a more balanced diet? No consensus exists on this issue. Floating absolute risk method has been used once to analyze the association between CA-based patterns and breast and ovarian cancers, to overcome this issue [32].

The term finite mixture model refers to a convex combination of a finite number of probability distributions, each of these commonly designated as mixture component. Mixture models can therefore be viewed as model-based clustering, where clusters are groups of individuals in the data (with a similar dietary behavior) induced by mixture components. Classification uncertainty is taken into account by estimating individual probabilities of belonging to each identified cluster, based on available data. For this reason, this approach provides an example of soft clustering, where subjects are assigned to each class with a weight equal to posterior membership probability for that class. The unknown true parameter vector (including parameters of each mixture component and the mixture weights) as well as the unknown allocation variables are estimated by the maximum likelihood method using the Expectation-Maximization (EM) algorithm [94, 49] in the frequentist framework. As an alternative to the EM algorithm, the mixture model parameters can be deduced using posterior sampling as indicated by Bayes’ theorem. This is still regarded as an incomplete data problem whereby membership of data points is the missing data. A two-step iterative procedure known as Gibbs sampling can be used. In both cases, a “hard” clustering solution can then be obtained by simply assigning each observation to the cluster to which it belongs with the highest probability, following Bayes’ Theorem.

In finite mixture models, the number of components is still pre-specified to be equal to K. Covariates can be accommodated [41, 99] or not [49] within the model to describe dependence of main dietary variables on other lifestyle or anthropometric variables. In the former case we have the general multivariate mixture model, also denoted by regression mixture model. In the latter case, standard mixture models are assumed and fitted. Both approaches have been applied in nutritional epidemiology in the frequentist framework [96, 94, 111, 41, 99, 49, 20, 21].

The paper by Greve et al. [49] shows the higher flexibility of Gaussian mixture models as implemented by Fraley and Raftery [44] in comparison with K-means and Ward’s hierarchical clustering on simulated and real-life data. Even on simulated data with spherical clusters of equal volume, the clustering solutions obtained from this Gaussian mixture model were more similar to the true cluster structure than those obtained from the K-means algorithm or Ward’s method in more than 72% of all simulated data sets. For simulated data sets with clusters of variable volume, shape and orientation, the Gaussian mixture model achieved a higher agreement with the true cluster structure in more than 90% of data sets [49]. The decomposition of the variance-covariance matrix proposed in this approach enables the researchers to place constraints on the geometrical properties of the clusters and thus to specify a desired degree of flexibility in terms of cluster volume, shape, and orientation. The choice of the number of clusters and of the available models (i.e., different parameterizations of the variance-covariance matrix) is transformed into a model selection problem. The final model is then identified according to information criteria after the finite mixture model is fitted by setting different values for the number of clusters or imposing different restrictions on the variance-covariance matrix [49].

Because finite mixture models has many parameters, large samples are generally required, especially when the number of selected clusters is moderate-to-high. Thus, a restricted mixture model is proposed that reduces the number of parameters and is suitable for small-to moderately-sized samples [99]. This regression mixture model has been shown to generalize the one by Fahey et al., also applied in nutritional epidemiology [41], and includes in the same model the Gaussian and binomial distributions for modeling different forms of food group consumption within the exponential family.

The finite mixture model method can also be used to classify the population according to the factor scores derived from EFA-based CFA. This so called two-step classification approach combines the advantages of both finite mixture models on food items to identify mutually exclusive clusters and CFA to understand which foods are eaten in combination [111].

Among major advantages, finite mixture models are more oriented towards capturing real-life dietary behavior because they can account for the within-cluster correlation between dietary variables [49], allow the variances of dietary components to vary within and between clusters, and enable covariate adjustment (e.g., age, sex, non-alcohol or total energy intake) for food intake simultaneously with the fitting process [41, 99, 20, 111].

Among major issues, the observed data may violate the distributional hypotheses, which are per se difficult to adhere to in a multivariate setting. When there are many 0 values – indicating non consumption – the need to deal with them increases the models complexity, as does the high number of parameters to be estimated [41, 99]. In addition, sensitivity to the initial values, convergence to local extremum, and slow convergence speed have been reported for most approaches [135]. Finally, although finite mixture models improve on selection of the optimal number of clusters compared to traditional CA approaches, still this selection is not made simultaneously within an overall parameters’ estimation process [94, 49, 20, 21].

A composite of hierarchical CA and traditional PCA, treelet transform [72] provides an improvement over traditional PCA and an important contribution to clustering methodology. For clustering methodology, it provides a framework which actively searches for the correct underlying correlation structure of the data. Its improvement over PCA happens especially when the correlation matrix is believed to be sparse, as in the analysis of dietary patterns, and is generally worth note [123].

Firstly introduced in nutritional epidemiology in 2011 [48], treelet transform is a dimensionality reduction technique aimed at converting a set of observations of possibly correlated variables into orthogonal components. Similarly to PCA/EFA, identifying the optimal number of retained components is based on scree-plot inspection (and related percentage of explained variance) and interpretability, and interpretation/labelling of components are based on loadings. Scores are determined for each component of the treelet transform and measure adherence to a given component. Unlike PCA scores which are always uncorrelated, treelet transform scores generally have a small degree of correlation.

Treelet transform combines the quantitative pattern extraction of PCA with the interpretational advantages of hierarchical clustering of variables. The two variables showing the highest covariance/correlation are identified in the treelet transform, and a PCA is performed on them. They are then replaced with the score of their first principal component, and a merge is indicated in the cluster tree. This operation is re-iterated until all variables have joined the cluster tree. In this way, the treelet transform produces a hierarchical grouping of variables that may reveal the data structure’s intrinsic characteristics.

By combining PCA and hierarchical clustering, treelet transform introduces sparsity into principal components (i.e., making many loadings equal to 0), thus potentially simplifying the interpretation. While EFA achieves this structure in a post-hoc analysis with the use of factor rotation and loading truncations (in which the factor loadings with absolute values smaller than an arbitrary threshold are ignored) [65], treelet transform directly derives sparse components that, similarly to PCA components, account for a large part of the variation in the original data and can be used analogously; treelet transform leads to an associated cluster tree that provides a concise visual representation of loading sparsity patterns and the general dependency structure of the data [48].

Alongside the cluster tree, treelet transform yields a coordinate system for the data at each level of the cluster tree. Selecting a cluster tree level (cut-level) amounts to choosing the level of detail desired in the data dimensionality reduction [2]. As pointed out in one of the Discussions in the original paper, the use of treelet transform leads to a trade-off between the amount of variability explained and sparsity. The objective is to “make the results as sparse as possible but not any sparser” [2]. Treelet transform has been used to derive dietary patterns using nutrients [2], food items [106], and food groups [48, 93].

Treelet transform works best for dimensionality reduction and/or feature selection when sample sizes are relatively small, and the data are sparse, with unknown groupings of correlated variables. Unlike PCA and its subjective choice of components to retain, treelet transform has a fast and efficient algorithm for determining the optimal number of dietary patterns to be retained and assessing dietary pattern internal reproducibility [107]. Each derived component can involve only a small number of input dietary data, so pattern structure is generally simpler than in PCA; the corresponding cluster tree for all variables also supports pattern interpretation. Not only treelet transform provides a concise visual representation of loading sparsity patterns, but it also shows the data’s general dependency structure [48].

In line with other CA techniques, a degree of subjectivity exists in choosing the cut-level of the cluster tree before extracting components. When the cut-level is close to the root, most variables are included in the components, with potential interpretation issues; the information is comparable to PCA output when all variables contribute to treelet components. As the cut-level moves away from the root, the component loadings become sparse (as many are equal to 0), and the components possibly become more interpretable [107]; however, this may lead to components that do not capture dietary complexity and are therefore not informative [2]. Cross-validation can be used to identify an optimal cut-level. Once the cut-level is chosen, the loadings computed are invariant to the number of retained components; hence, the number of components is an a priori parameter to be specified in the cross-validation step. In addition, if the correlation of some nutrients/food groups is too strong, then the sparsity hypothesis may not hold [57]. It also remains debatable whether patterns derived by treelet transform are more effective than those from PCA/EFA or CA methods in exploring the relationship with health outcomes, including cancer [106].

Reduced rank regression was formally introduced in nutritional epidemiology in 2004 to combine the advantages of the a priori and a posteriori approaches [55]. A posteriori pattern method can be applied to investigate if major consumption patterns of a specific population have relevance for health outcomes; a priori patterns can help clarify if adherence to specific benchmark diets is related to reduced disease risk. Reduced rank regression fills a gap as it more directly relates the step of data-driven pattern identification to the health/disease outcome of interest. In detail, after about 20 years, researchers in nutritional epidemiology are still increasingly interested in identifying specific dietary patterns related to established and new pathways in the development of major chronic diseases, including cancer [127]. So, reduced rank regression replies to the following question: “What combinations of dietary components explain the most variation in a set of intermediate health markers? And then, in further analysis, does that pattern explain the disease outcome of interest?” [70].

The method has similarities with PCA, but it uses two different sets of variables: a set of independent variables or predictors, generally dietary variables, and a set of response variables, expected to be associated with the disease under examination based on a priori knowledge [47]. Food groups are generally used as predictors. Nutrient intakes, contaminants, and endogenous biomarkers or intermediate disease phenotypes are generally used as response variables according to the specific research question [135].

The identified dietary patterns are projections of the principal components obtained from the covariance/correlation matrix of the responses onto the space of predictors. They can be interpreted as those linear functions of the original dietary variables that maximally explain variation in the response variables. Reduced rank regression can therefore be interpreted as a PCA applied to responses and subsequent linear regression of principal components on predictors, although it is somewhat more efficient and sophisticated than this two-step procedure [127].

Reduced rank regression starts from a linear function of responses called response score that will then be projected onto the space of predictors to produce a factor score, that is, a linear function of predictors. Both scores form an inseparable pair reflecting the same latent variable in different sets of original variables. Because the first aim of this method is to explain a high proportion of response variation, the evaluation of factors extracted by reduced rank regression should be based on response scores rather than factor scores. However, similarly to PCA and EFA, factor scores represent the comprehensive variables used in subsequent statistical analysis, to assess the association between the identified dietary patterns and disease outcome [55].

Given that linear functions of predictors are defined as principal components of the responses, there will be as many dietary patterns as were selected responses. This is different from PCA, in which the analysis cab identify as many principal components as predictors. Still, similar to PCA, only a subset of patterns might be informative [47]. In reduced rank regression, only one or a few principal components of responses might account for most of the responses’ variation, and the corresponding pattern would be the strongest candidate to be selected [127].

The exploratory nature of the approach hampers the generalization of the results from one population to others. However, compared to PCA and EFA, reduced rank regression allows the use of the same set of response variables in different study populations and thus improves results comparison across different populations [33]. In addition, the so-called simplified patterns (i.e., scores are calculated based on only those food items strongly contributing to the pattern, to which all contribute with equal weight) ease the step of external validation of pattern-disease associations. Such validation is highly recommended given that pattern identification (using informative responses) is usually carried out within the same study in which the pattern-disease association is evaluated [127].

Major challenges in performing reduced rank regression include the choice of predictors and response variables to work on, the number of response variables to consider and their relationship, the number of principal components to retain, and their labelling. Most studies try to cover the whole diet including all available food items or food groups and very few studies investigated the influences of the selection or building of food groups on the extraction of dietary patterns [127]. The value of the application of reduced rank regression is considerably dependent on a good selection of response variables. If responses with no or little intercorrelation are selected for analysis, they will unlikely be well reflected by a single response score. Instead, it is more likely that only a fraction of responses is accounted for – in the worst-case scenario, single response scores (and consequently single reduced rank regression patterns) reflect only a single response variable. In this case, more than a dietary pattern have to be retained and these dietary patterns might reflect different pathways relevant to disease risk [127]. Confounding for responses is a second crucial issue. The derived pattern may considerably differ depending on an adequate consideration of confounders [127]. So far, medication and anthropometric characteristics have been considered in some studies as confounding factors, but there could be scenarios where the role of confounding factors is less obvious.

A method similar to reduced rank regression is partial least squares, a regression model of multiple predictor variables on multiple response variables, sometimes used in nutritional epidemiology in comparison with PCA and reduced rank regression (e.g., [55, 29, 81]).

The partial least squares method is a compromise between PCA and reduced rank regression [55, 29, 81]. Indeed, while PCA selects factors that explain as much predictor variation as possible and reduced rank regression extracts factors that explain as much response variation as possible, partial least squares balances the two goals of explaining predictor variation and explaining response variation [92]. So, partial least squares replies to the following question: “What combinations of dietary components explain the most variation in dietary components and in a set of intermediate health markers? And then, in further analysis, does that pattern explain the disease outcome of interest?” [70].

The three methods are similar in terms of their mathematical foundation and their technique of deriving factors. For each method, the coefficient vectors of the extracted linear functions are eigenvectors of a covariance matrix. PCA uses the covariance matrix of predictors, whereas reduced rank regression starts from the covariance matrix of responses. Partial least squares uses the matrix of covariances between predictors and responses. However, as the number of factors cannot be greater than the rank of the corresponding covariance matrix, partial least squares can extract as many dietary patterns as were the selected predictor variables, like in PCA, but differently from reduced rank regression. The eigenvalue belonging to an eigenvector quantifies the percentage of variation explained by the corresponding linear function of the original variables (i.e., predictors or responses depedenign on the method). The factors obtained by PCA, partial least squares, and reduced rank regression usually are sorted by decreasing eigenvalues. While the first factor of PCA is the linear function of predictors that maximizes the explained variation in predictors, it is, in general, not optimal in terms of response variation. In contrast, the first factor of reduced rank regression explains more variation in response than any other linear function of predictors, but possibly explains only a moderate fraction of predictor variation. The first factor from partial least squares maximizes the covariance between linear combinations of predictors and responses [55].

Due to the orthogonality of eigenvectors, successive extracted factors from all three methods are uncorrelated. Therefore, the variation in the original variables (i.e., predictors or responses, depending on the method), can be decomposed into percentages of variation explained by the obtained factors. These uncorrelated factors can simultaneously be chosen as independent variables in a regression model for predicting disease (e.g., cancer) risk without confounding each other [55].

Similarly to reduced rank regression, major challenges in performing partial least squares include the choice of both predictors and response variables to work on, the number of response variables and food groups to consider and their relationships, as well as the number of factors to retain, and their labelling.

CA has a parallel methodology that defines distinct subgroups in a population while making full use of information on a response variable. This is “classification and regression tree” analysis [11]. Classification and regression tree is a non-parametric decision tree procedure (i.e., selection of nodes with successive splitting to produce different subgroups) that identifies mutually exclusive and exhaustive subgroups of subjects sharing common characteristics that are associated with the response variable of interest [73].

Compared to reduced rank regression, decision tree analysis uses one response variable only. This may be a disease risk factor, including also overall measures of diet quality [51], or the disease outcome [13]. Therefore classification and regression tree could be used to answer the question: “What combinations of dietary components explain the most variation in one response variable, like the disease outcome or a selected risk factor for the disease?” [70]. The response variable can be either categorical (in the so called classification tree analysis) or continuous (in the so called regression tree analysis), whereas independent variables can be any combination of categorical and continuous variables. No data assumptions are required [73, 92].

Decision tree analysis ends up with a graphical output that is a multi-level structure that resembles branches of a tree, with the root node and several leaf nodes. A classification rule is a path from the root node to a leaf node associated with the response variable. The results can thus be interpreted as “hierarchical” dietary patterns. For example, one might find that in predicting a health outcome the most important variable is the amount of added sugars consumed. If intake of added sugars is high, the next most important factor in predicting the health outcome might be the amount of, say, solid fats, but, if added sugars are low, the next most important factor might be fruit intake. The terminal nodes show the specific pattern features of the sub-populations in percentage, including the number of participants and the probability or mean values of the response variable in the terminal node [119].

Until now, decision tree analysis was seldom applied to derive dietary patterns [51] or risk-related patterns, including dietary and other risk factors [13].

Among major advantages, classification and regression tree can be used to reveal heterogeneity in the dietary behavior of a population, when present, and to develop preventive measures tailored to specific sub-populations. In principle, the output is very intuitive and, due to its transparent nature, users can trace back through the generated model. When selected rules involve a few variables, this approach may allow to demonstrate the effect (on disease risk) associated to modifications of single dietary habits, prompting to an individualized approach to public health messages [71]. Decision tree analysis is also able to generate new aetiological hypotheses, without prior assumptions on potential risk factors. It might be particularly suitable in identifying disease risk based on a combination of food groups and other non-dietary risk factors. In this sense, it is crucial that this approach allows for independent variables of any kind (i.e., categorical or continuous variables).

Major disadvantages have likely avoided a wider diffusion of classification and regression tree in nutritional epidemiology. In particular, one key variable can dominate the model [135], misclassification can be rather large [70], and overfitting may be a serious issue [51]. If many classification rules are generated, the selection of meaningful rules will require considerable professional knowledge. Rules containing many variables can be long and/or complex even if they are meaningful, making it difficult to translate them into simple health recommendations [135].

Other data mining techniques, such as random forest, artificial neural networks, and Naïve Bayes Classifiers, have also been used to analyze the relationship between dietary patterns and diseases in a few applications [51, 31, 8].

In nutritional epidemiology, multicollinearity may be strong between nutrients, as well as between food groups. In observational studies, several confounding factors potentially measure slightly different variants of the same lifestyle characteristics (e.g., physical activity or alcohol drinking consumption). Interactions may also exist between dietary exposures and confounding factors (e.g., between dietary patterns and alcohol consumption or education). This makes fitting a standard regression model for assessing disease risk challenging.

Least absolute shrinkage and selection operator (LASSO) is a regression-based methodology that allows a large number of covariates to be included in the model and penalizes the absolute value of the regression coefficients, thus, regulating the impact a coefficient may have on the overall regression. The greater the penalization, the greater the shrinkage of coefficients (some reaching 0), thus automatically removing unnecessary/uninfluential covariates. Thus the LASSO minimizes regression coefficients in order to reduce the likelihood of overfitting. The algorithm shrinks the sum of the absolute value of regression coefficients, producing coefficients that are exactly 0, and thus selecting the nonzero variables to remain in the model. The shrinkage amount is controlled by the shrinkage parameter λ.

A critical choice in the LASSO method is selecting the appropriate amount of shrinkage, as controlled by λ.

We identified two papers [134, 80] applying LASSO in nutritional epidemiology. The former paper [80] adopts the logistic LASSO in the National Health and Nutrition Examination Survey (NHANES) 1999-2010 to estimate the association between a composite risk pattern, including diet and other risk factors, and self-reported breast cancer (binary variable, no/yes), in females $\ge 50$ years. Based on 29 variables, including 21 macro- (density method) and micro-nutrients, alcohol, and coffee, the following variables: age, parity, vitamin B12, caffeine, and alcohol remain in the model, as far as the penalty parameter λ increases in the LASSO. This paper adopts the cross-validation technique to choose the optimal λ value.

In the latter paper, LASSO is not directly used to identify a disease-related risk profile, but as an alternative approach for assessing disease risk. Based on FFQ data from the NHANES 2005-2006, including healthy US adults $(n=2609)$, ten PCA-based dietary patterns (65% of the total variation) are associated with cardiovascular disease risk factors via the LASSO method. It is shown that the LASSO method outperforms the traditional linear regression model, by better predicting the levels of triglycerides, LDL cholesterol, HDL cholesterol, and total cholesterol (LASSO adjusted ${R^{2}}=0.861$, versus traditional linear model adjusted ${R^{2}}=0.163$). The study adjusts for confounding factors such as age and body mass index. The authors also test the prediction accuracy of the model performance using an independent test set.