Many epidemiologic studies show that birth size (usually birth weight) has an inverse relationship with health outcomes in later life—known as the fetal-origins-of-adult- diseases hypothesis.1,2 A recent theory (the developmental- origins-of-health-and-disease hypothesis)3 suggests that small birth size in conjunction with compensatory rapid growth in childhood (rather than small birth size per se) might lead to a greater risk of developing chronic adult diseases. Several recent articles have presented evidence that people who develop adult chronic diseases, such as coronary heart disease4 and diabetes,5,6 tend to have had a distinct growth trajectory. By using z-scores to track relative body size over time, it was found that individuals who developed chronic adult diseases were below average in birth size and weight. These people remained small in their early childhood, but started to “catch up” and gain weight rapidly as they grew older; by the time of puberty, their body weight and body mass index exceeded the average for their age. However, identification of growth trajectories by tracing the mean z-scores for subjects with chronic diseases in later life may not be the most appropriate analytic strategy, because the data could be affected by regression to the mean,7 potentially biasing the analysis. Alternative approaches based on modeling individual growth curves have been proposed, and results suggest that early rapid growth might increase the risk of chronic diseases in later adulthood.8,9 Both the fetal origins and developmental origins hypotheses may be accommodated within the framework of the growth acceleration hypothesis,10,11 because small babies are more likely to show rapid postnatal growth.
One challenge in the statistical analyses of the developmental origins hypothesis is to distinguish the effects of birth size, growth in body size, and current body size.12–14 When the only measurements available are birth size and current body size (and, hence, growth), it is not feasible, using ordinary least squares multiple regression, to estimate the independent contributions of the 3 variables conditional on the other two. This is because growth in body size, defined as the difference in current body size from birth size, yields perfect collinearity among the 3 variables, such that only 2 of the 3 variables can be entered into ordinary least squares regression. Regardless of which pair of variables is used, mathematical relationships exist between the pairs of regression coefficients.12–14 Moreover, interpretation of the conditional relationships in ordinary least squares regression remains controversial because a small negative association between birth size and the later-life health outcome usually becomes much larger after adjusting for current body size.15,16
The problem of perfect collinearity in distinguishing the effects of these 3 parameters will not be overcome by increasing the number of body-size measurements throughout the life course, even though this provides more information on growth during life course. We propose a novel approach to estimating the effects on later-life health outcomes of body size at various stages of the life course using partial least squares regression.17–20 Partial least squares was developed in 1970s and 1980s, and since then has been widely used in chemometrics to overcome estimation problems in ordinary least squares regression, such as with highly collinear covariates and where the number of covariates exceeds the number of independent observations.17–20 Although partial least squares has become a popular tool for data analysis in bioinformatics,21,22 it is rarely used in medical statistics and epidemiology. In this study, we use a birth cohort to demonstrate how partial least squares regression can overcome the problem of perfect collinearity in ordinary least squares regression. Specifically, we show how partial least squares may be used to estimate the effects of body size throughout life course on later health outcomes.
To illustrate the application of partial least squares regression to life course data analysis, we examined a prospective study in which participants were recruited from Metropolitan Cebu, an area of the central Philippines.23–25 Data were obtained from the website of the University of North Carolina Population Center (http://www.cpc.unc.edu/projects/cebu/datasets.html); a detailed description of the study can be found on their website and elsewhere.24,25 In summary, all pregnant residents of 33 randomly selected Metro Cebu communities were invited to participate; index-child participants include 3080 singletons born during a 1-year period beginning in April 1983. Prenatal data were collected during the sixth to seventh months of pregnancy. In this study, we used data on body weights measured immediately after birth and at the ages of 1, 2, 8, 15, and 19 years. The later-life health outcomes were average systolic and diastolic blood pressure (BP) measured at the age of 19 years. Raw body weights were transformed to obtain sex-specific z-scores by subtracting sample average weights from individual weights and then dividing by sample standard deviations at each age. Participants without complete data were excluded from the sample. In our demonstration, we used data from 960 boys who had all measurements of body weight or blood pressure for statistical analysis. Growth in body weight was defined as the change in weight z-scores between consecutive measurements; for example, growth in body weight between birth and age 1 (zwt1–0 = zwt1 − zwt0) is the difference in weight z-scores between birth (zwt0) and age 1 (zwt1).
Ordinary Least Squares Regression
We first regressed systolic blood pressure (SBP) and diastolic blood pressure (DBP) on z-score birth weight (zwt0), z-score current weight (zwt19), and changes in weight z-scores between birth and age 19 (z19–0) separately and in pairs. Then, SBP and DBP were regressed on zwt0, changes in weight z-scores between birth and age 1 (zwt1–0), changes in weight z-scores between ages 1 and 2 (zwt2–1), changes in weight z-scores between ages 2 and 8 (zwt8–2), changes in weight z-scores between ages 8 and 15 (zwt15–8), and changes in weight z-scores between ages 15 and 19 (zwt19–15). In a separate analysis, SBP and DBP were regressed on the 5 successive weight z-scores increments and final weight z-score. The ordinary least squares regression analyses were performed using STATA version 10.1 (StataCorp, College Station, TX).
Partial Least Squares Regression
Partial least squares analysis17–22 can be viewed as an extension of principal component analysis.26,27 Along with principal component analysis regression and ordinary least squares regression, partial least squares is a member of the family of continuum regression.28 Like principal component analysis, partial least squares is a data-dimension-reduction method; when there are many collinear covariates in the model, it is sometimes necessary to reduce the number of covariates (because of the small number of observations) by selecting those with greater predictive power or by combining the original covariates into new variables (known as components in principal component analysis and partial least squares). One important advantage of partial least squares over ordinary least squares regression is that partial least squares can estimate the effects of covariates with perfect collinearity. A concise introduction to partial least squares methodology related to our application in this study can be found in the eAppendix (http://links.lww.com/EDE/A380).
For partial least squares regression analysis, SBP and DBP were first regressed on zwt0, zwt19, and zwt19–0. Then SBP and DBP were regressed on zwt0, the 5 successive weight z-scores increments, and zwt19. We carried out these regression analyses using the software package Tanagra version 1.4.33 (http://eric.univ-lyon2.fr/~ricco/tanagra/en/tanagra.html). Partialleast squares regression analysis can also be performed using standard statistical software packages such as SAS (SAS Institute Inc., Cary, NC) and the library pls in R (R Foundation for Statistical Computing, Vienna, Austria). Because no distributional assumption is made for these regression coefficients, we obtained their confidence intervals (CIs) using the bootstrap method with 500 replications.
The body weight and blood pressure measurements for the 960 Cebu boys in this study are summarized in Table 1.
Ordinary Least Squares Regression
In univariable regression, zwt0 had no association with SBP (−0.0 mm Hg per 1 unit increase in z-score [95% CI, −0.7 to 0.7]) or with DBP (−0.1 [−0.7 to 0.6]); both zwt19–0 and zwt19 had a positive association with SBP and DBP (Table 2). In multivariable regression, zwt0 had an inverse association (−0.8 [−0.5 to −0.1]) with both SBP and DBP when zwt19 was adjusted for and a positive association (2.8 [2.0 to 3.6]) when zwt19–0 was adjusted for.
When SBP and DBP were regressed on zwt0 and the 5 successive weight z-scores increments, all covariates had a positive association with the outcomes, but the regression coefficient for zwt0 was the smallest (2.5 [1.7 to 3.4] for SBP and 2.2 [1.4 to 2.9] for DBP). When SBP and DBP were regressed on the 5 successive weight z-scores increments and zwt19, the covariate zwt19 had the largest association with SBP (2.5 [1.7 to 3.4]) and DBP (2.2 [1.4 to 2.9]), and its regression coefficients were identical to those for zwt0 in the previous model (Table 3). Figure 1 presents graphical representations of the regression coefficients for the 2 multivariable regression models in Table 3.
Partial Least Squares Regression
When SBP and DBP were regressed on the first partial least squares component extracted from the covariates zwt0, zwt19–0, and zwt19, associations between both outcomes and each covariate were similar to those in the bivariate ordinary least squares regression. Table 4 shows that both zwt19 and zwt19–0 had strong positive associations with both outcomes, whereas zwt0 showed no association. When 2 components were extracted and used as covariates in partial least squares regression, zwt0 had a small positive association, whereas changes in the regression coefficients for zwt19 and zwt19–0 were relatively small. More than 97% of the covariance between the outcomes and the 3 variables were explained by the first partial least squares component. Therefore, the analysis based on a single component is appropriate.
When SBP and DBP were regressed on the first partial least squares component of zwt0, the 5 successive weight z-scores increments, and zwt19, zwt19 had the largest positive associations with both outcomes (2.6 [2.0 to 3.0] for SBP and 2.0 [1.4 to 2.4] for DBP), whereas zwt0 showed no associations (Table 4). Early changes in weight z-scores had relatively small positive effects on the outcomes compared with later growth in weight z-scores. The inclusion of additional components in the model had little effect on the coefficients, whereas the zwt0 coefficient remained close to 0. Growth from birth to age 2 had smaller positive associations with SBP (for zwt1–0, 0.6 [0 to 1.0] and for zwt2–1, 0.9 [−0.2 to 1.6] in 1-component model) than with later growth from age 8 to 19 (for zwt1–0, 1.2 [0.2 to 2.3] and for zwt2–1, 1.8 [0.7 to 2.7] in 1-component model). In contrast, growth from birth to age 2 had similar or larger positive relationships with DBP compared with later growth, from age 8 to 19, although the confidence intervals were generally large (Table 4). Most of the covariance between the outcomes and the 7 variables were explained by the first component, and the coefficients for the 7 covariates in the model with 2 components showed only very small changes, as additional components were added. This suggests that there are only 2 dimensions to the data. Figure 2 presents graphical representations of the partial least squares regression coefficients of models for various numbers of components for SBP and DBP. For SBP, the effect of growth increases with age (Fig. 2A) whereas for DBP the growth in early life seemed similar to or slightly greater than later growth (Fig. 2B). Nevertheless, current body size had the greatest effect in predicting blood pressure.
Relation Between Ordinary Least Squares and Partial Least Squares Regression Coefficients
Partial least squares regression estimates the individual contribution of birth, growth, and current body size to the prediction of blood pressure, and there is a mathematical relationship between partial least squares results with full dimension (the maximum number of components retained) and ordinary least squares regression coefficients. For example, when SBP is regressed on zwt0, zwt19–0, and zwt19, the equation for the partial least squares model with 2 components is given as:
These results are shown in Table 4.
Simple rearrangements can give rise to ordinary least squares regression models with 2 of the 3 covariates:
These equations present the ordinary-least-square multivariable regression 2 in Table 2. The other 2 multivariable regression models in Table 2 can be derived in a similar way.
For SBP regressed on zwt0, the 5 successive weight z-scores increments, and zwt19, simple rearrangements by replacing zwt19 with (zwt0 + zwt1–0 + zwt2–1 + zwt8–2 + zwt15–8 + zwt19–15) can give rise to ordinary least squares multivariable regression 1 in Table 3. Similarly, by replacing zwt0 with (zwt19 − zwt1–0 − zwt2–1 − zwt8–2 − zwt15–8 − zwt19–15) can lead to ordinary least squares multivariable regression 2. The same relationships apply to the results for DBP.
We demonstrate the application of partial least squares regression to estimate the effects of body weights and growth in weights at various stages throughout childhood and adolescence. This analysis showed that birth weight had a negligible association with both systolic and diastolic BP, and current body weight had the largest association. Growth in later childhood and adolescence had stronger associations with systolic BP than earlier growth, but growth in early life had a similar or slightly stronger association on diastolic BP than later growth. However, because the confidence intervals for the associations of growth with body size were generally large, these differences in effect size should be interpreted cautiously.
When only one partial least squares component was extracted as a covariate in the regression analysis, associations between blood pressure at age 19 and the original covariates were in the same direction as with standard ordinary least squares regression, although with smaller values. It is because of this covariate effect-size reduction that partial least squares regression is considered a shrinkage regression technique.22 Although ordinary least squares regression coefficients are unbiased estimates of the effect size, their confidence intervals increase with the degree of collinearity among covariates. In contrast, partial least squares regression provides “biased” (shrunken) estimates with smaller confidence intervals. The use of shrinkage regression methods (eg, partial least squares, principal component analysis, and ridge regression) provides a trade-off between bias and precision.20–22 In our study, the first partial least squares component explained most of the covariance between blood pressure and life course body size measurements. Figure 2, which may be interpreted as alternative representations of the life course plot,29,30 shows that the curves from models with one partial least squares component are a good approximation to those curves from models with 2 or 6 components, and differences between curves from 2 or 6 component models are very small. The partial least squares model with 2 components may therefore be considered as a parsimonious representation of each child's growth curve related to later blood pressure.
When there is perfect collinearity among covariates, there are, as shown in our example, simple relationships between partial least squares and ordinary least squares regression coefficients when the maximal number of components are extracted in partial least squares. Although the ordinary least squares regression coefficients can be obtained from partial least squares analysis, it is not possible to obtain partial least squares results from ordinary least squares analysis, because the latter is unable to estimate the individual contribution of perfectly collinear variables. Moreover, the traditional ordinary least squares regression analysis with the adjustment of multiple body sizes will inadequately allocate the effects of life course growth on later health. For example, the much larger positive effects for birth and growth in multivariable regression 1 in Tables 2 and 3 are caused by attributing the effects of current body size to the preceding body size measurements.
Partial least squares may be viewed as an approach to distributing the overall contribution of body size and growth measurements to the outcomes according to the correlations among covariates and outcome. For example, current body size had the strongest association with both systolic and diastolic BP in our data set, and it is therefore not surprising that it had the largest regression coefficients in partial least squares analysis—but these coefficients are smaller than the ordinary least squares estimates. This is because some of the effects of current body size on blood pressure may come from rapid growth throughout the life course. Partial least squares regression coefficients for later growth (such as zwt19–15) were also smaller than the ordinary least squares estimates, indicating that some of the effects in ordinary least squares regression might be explained by boys with greater growth during these ages becoming bigger adults. Because body growth is a continuous process, measurements of body size at different ages or growth in different phases during the life course are inevitably correlated, and these correlations are large when time intervals between measurements are short. Therefore, it is intrinsically difficult to disentangle the life course effects of growth on later health—which is why alternative methods such as partial least squares are required.
Partial least squares analysis is a dimension-reduction method. In general, not all partial least squares components are selected as covariates for regression analysis; typically, the first few partial least squares components explain most of the covariance between the outcome and covariates in the ordinary least squares regression. Otherwise, results from partial least squares regression will be the same as ordinary least squares regression, unless some of the original covariates are perfectly collinear, as in this study (eAppendix, http://links.lww.com/EDE/A380). However, the more components selected as covariates, the greater the explained variance in y (ie, the greater the model R2); it is therefore essential to determine the appropriate complexity of the model (ie, to balance gain in model prediction power against potential overfitting).20 From a statistical viewpoint, it seems sufficient in this study to select the first component because the remaining 5 components explain <1% of total model R2 for SBP and 2% for DBP; this is confirmed by statistical indices such as predictive residual sum of squares and Q2 (cross-validated R2).20,31 The major advantage of using partial least squares rather than ordinary least squares regression in life course epidemiology is its ability to estimate the effects of growth in different phases on health in later life. Thus, by overcoming the problem of perfect collinearity, our selection of the number of components is not based on increases in the model R2 or other similar indices.
Given the potential complexity in the growth process and growth patterns throughout the life course, statistical methods (no matter how sophisticated) can unravel only a portion of the intricacy of growth. Statistical techniques that facilitate modeling of repeated measurements, such as multilevel modeling,32 functional data analysis,33 and structural equation modeling,23,34,35 are useful because characteristics of growth patterns are estimated, and their associations with disease outcomes are explored. In the social sciences, 2 recently developed methods have been used to identify trajectories for subgroups: (1) group-based modeling, which has been widely used in criminology to identify the trajectory in delinquency for repeated offenders in adulthood36; and (2) growth mixture modeling,37 which is an extension of latent growth curve models. Although these methods are powerful tools in identifying distinct growth patterns, these methods do not always provide stable results due to problems in model convergence around local maxima/minima.38 It would also be more difficult to compare the growth patterns uncovered in different data sets using these methods than to compare partial least squares regression coefficients, and therefore more difficult to validate or corroborate results across different studies and populations. Furthermore, unlike any likelihood-based methods, such as structural equation modeling, partial least squares algorithms always achieve convergence and are free from problems of inadmissible results, known as Heywood cases in the literature.39 This makes partial least squares a useful tool in exploring life course effects of growth in body size on later-life health outcomes. Recent developments in partial least squares have been extended to generalized linear models.40
This study used z-score birth weight, z-score current weight, and 5 successive z-score increments in partial least squares analysis. An alternative approach would be to undertake partial least squares analysis on the original 6 z-score weights, which can be viewed as the partial least squares version of life course plots.29,30 Because the 6 variables are highly correlated (up to 0.87 in Table 1), collinearity is a potential problem for ordinary least squares regression, as shown in our additional analysis in which there were negative associations of blood pressure with birth weight and weight at 1 year or 2 years, as well as large regression coefficients for weight at 18 years15,16 (Fig. 3). Partial least squares analysis shows that only 2 components were required to achieve most of the model R2, whereas the indices for component selection, such as Q2 and predictive residual sum of squares, suggest that the model with the first component is the parsimonious model. The association between body weights and blood pressure increased with age, whereas early body weights had negative effects in models with 2 or more components (eTable 1, http://links.lww.com/EDE/A380). The most radical partial least squares analysis is to include the 6 body weight z-scores and 15 changes in weight z-scores throughout the life course. As results in the figures can be derived from the model with 21 variables for body weights and growth, the 21-variable model can be considered the “saturated model.” Results show that the effects of body size or growth in size on systolic BP, in general, increase with age (eTable 2, http://links.lww.com/EDE/A380), whereas the effects of body size or growth in size on diastolic BP remained similar throughout the life course. The indices for component selection indicate that models with only 1 component are parsimonious.
There are several limitations in this study, and the interpretation of our analyses needs to consider these carefully. First, to simplify our demonstration on how to apply partial least squares to life course data analysis, we excluded subjects with incomplete data; this may cause bias. Incomplete data is a common problem in life course epidemiology, and has to be dealt with appropriately before undertaking partial least squares analysis. Some partial least squares software packages provide crude methods for imputing missing values, but more needs to be done. Second, although we coded the weight measurements according to age, not all participants were measured at the same ages.24 There might be up to 2 years of difference in the ages between the youngest and oldest participants in the latter period of observation in the Cebu cohort. Third, changes in weight z-scores were derived from body-size measurements, and therefore, changes in weight z-scores contain more measurement errors. The impact of measurement errors on partial least squares life course analysis merits further research. Fourth, body-size measures were sparse in the Cebu cohort after the age of 2, and therefore, it was not possible to estimate precisely the critical phase of growth in the early life course period. However, the performance of partial least squares in the analysis of life course data with intensively repeated measurements remains to be investigated. For instance, if body size was measured every 6 months for 18 years after birth, the resulting 37 measurements would be highly correlated. Whether all measurements should then be included in the partial least squares analysis—and how informative the results would be—requires more theoretical and empirical investigations. In contrast, methods such as growth mixture modeling might provide a more holistic approach by estimating distinct growth curves. Given the complexity of life course analysis, it is likely that more than one approach is necessary.
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