Glasgow Alumni cohort

Details of the Glasgow Alumni cohort have been described elsewhere [23], [24]. Briefly, students attending Glasgow University between 1948 and 1968 were invited to participate in a health screening, including a questionnaire and clinical examination. Data collected included socio-economic background, health behaviours, and medical history. Height, weight, and blood pressure were also recorded. A total of 15,322 students (11,755 men and 3,567 women) participated in the study. Students born before 1925 or aged over 25 years at examination were excluded from the analyses. Two students were excluded because of data entry errors. Participants with missing values in the birth year and any of following confounders were also excluded: body height, body mass index, father's socio-economic background and cigarette smoking. Adjusted systolic blood pressure (SBP) and diastolic blood pressure (DBP) were obtained in 9,337 and 9,314 men, respectively, with the adjustment of all confounders. Similarly, adjusted SBP and DBP were obtained in 3,211 and 3,204 women, respectively.

Record linkage and follow-up of the Glasgow Alumni Cohort was under the ethics approval by the Multicentre Research Ethics Committees in the UK: MREC/99/0/9, “Influence of early life nutritional status, adolescent and adult diet on cancer incidence and mortality: a retrospective cohort study of Glasgow University students”, approved in March 2000. There was no consent collected at the time, as it is a historical cohort started 60 years ago; but this deemed acceptable by the ethics committees, if data used in an anonymised form - as they are throughout the analysis.

Partial least squares (PLS) regression

PLS seeks to select components t that maximise the covariance between the outcome (SBP or DBP in this study) and t [20]–[22]. For p variables, x₁, x₂,…, x_p, each PLS component t_i, is a weighted composite of p covariates:(1)Like principal component analysis (PCA), variables with small variances are penalised in the extraction of t, and therefore x_i in equation (1) is usually scaled to have unit variance and zero mean.

In contrast to PCA, PLS extracts components by taking into account their relationships with the outcomes. In PCA, the extraction of components is independent of the outcome variables, whereas in PLS, components are extracted explicitly for their association with the outcomes. The extraction of PLS components operates under the same constraints as with PCA: (i) the sum of the squared weights is unity, i.e. ; and (ii) the correlations amongst all components are zero. The number of t_i that can be extracted is equal to the dimension (i.e. the rank) in the covariate matrix consisting of x_i. For instance, in this study, there are only two dimensions in the data matrix consisting of age, the year at examination, and the year of birth; consequently, only two PLS components can be extracted from the three variables.

PLS components are ordered according to the amount of variance in the outcome that is explained by them, i.e. the first PLS component has greater covariance with the outcome than the second PLS component, and the second has greater covariance than the third, etc. In PLS, the first PLS component explains most of the outcome variance.

The PLS regression coefficient for each x_i is then derived from the sum of products of the regression coefficients for PLS components and the weight for each x_i. For example, when the outcome SBP is regressed on the two PLS components, the equation is given as:where β₁ and β₂ are the regression coefficients for PLS components 1 and 2, respectively, and ε is the residual error term. The PLS regression coefficient for age is therefore .

Note that if all PLS components are used as new covariates, the results from the PLS regression, such as regression coefficients and R², are equivalent to those from PCA regression (and also ordinary least square regression, when the covariate matrix is full-rank). The advantage of PLS over PCA is that the first few components explain most of the covariance between the outcome and covariates.

PLS and perfect collinearity

From a mathematical perspective, identification is a problem for the age-period-cohort analysis using ordinary least squares regression and related methods, because the inverse of the covariate matrix does not exist. However, whilst the inverse of a matrix without full rank does not exist, for a matrix without full rank, a mathematical technique, namely singular value decomposition (SVD), can still be used to obtain unique components of original variables, which are weighted compositions of original covariates [25]–[27]. In short, PCA is related to SVD of correlation/covariance matrix for the covariates, whilst PLS with one outcome is related to successive SVD of the vector for the correlations/covariances between the outcome and covariates [28]. This is why PCA and PLS have been widely used in bioinformatics where the number of variables exceeds the number of observations (which also gives rise to identification problem) [19], [22]. In PCA, three collinear variables with two dimensions (such as age, period, and cohort) are projected into two new latent variables, which are linear combinations of the original three variables; these new latent variable are then used as covariates for the regression analysis [23]. PLS may be viewed as a variant of PCA, where the two latent variables are rotated so that the first latent variable has the largest covariance with the outcome [29]. A technical explanation about how PCA and PLS work for perfectly collinear variables can be found in the Appendix S1. Briefly, it is well known that a linear model with a non-full rank covariate matrix (also called design matrix) has an infinite number of solutions for the choice of regression coefficients, and a constraint is therefore necessary to obtain a unique solution [30]–[32]. PLS implicitly imposes an inherent constraint in its algorithms that “naturally” arises from the intrinsic mathematical relationship: Age + Cohort  =  Period. The application of SVD effectively “inherits” this constraint in the estimation of the PLS regression coefficients [33], [34]. PLS does not intentionally impose this constraint; it emerges only due to the mathematical relationship of APC data. It can be shown that the imposed constraint is different when original or scaled variables are used in PLS, giving rise to different results. It is our view that the implicit constraint made by PLS regression seems to be a reasonable one, as it is a natural consequence of the intrinsic mathematic relationship amongst age, cohort and period. More explanation is found in the online Appendix S1.

Selection of PLS component

To employ PLS is to maximise the covariance between the outcome and new composites, so it is justifiable to use the increments in the explained variance in the outcome (e.g. changes in R²) as a criterion for selecting PLS components. This gives us a measure of predictive ability, the predictive residual error sum of squares (PRESS) [35], [36]. To obtain this, the data are first split into a number of groups. For each, a prediction is obtained using the model derived from all other groups. For example, one observation is left out of the model, and we use the remaining observations to predict the outcome. PRESS is calculated as the sum of squares of the differences between the prediction for each observation (when it is left out of the model) and the observed value of the dependent variables.

Data analysis

We first undertook sex-specific linear PLS regression for SBP and DBP by including the age at examination (16 to 25), the year of examination (1948 to 1968) and the year of birth (1925 to 1950) as continuous covariates. As PLS penalizes variables with small variances (e.g. age at examination), covariates are scaled to have unit variance [37]–[39]. Restricted cubic splines PLSR was then undertaken to explore nonlinear associations [40]. Three knots were placed for the year at examination (year 1954, 1959 and 1964) and four knots for the year of birth (year 1930, 1935, 1940, 1945 and 1950). In the final analysis, we created dummy variables for the three continuous variables to compare the results to those from linear and restricted cubic splines PLS regression. No arbitrary constraint on the dummy variables was required for PLS regression. All analyses were undertaken using a free data-mining software Tanagra (version 1.4.36, http://chirouble.univ-lyon2.fr/~ricco/tanagra/en/tanagra.html) with 1000 nonparametric bootstraps to obtain 95% confidence intervals.

Interpretation of age, period and cohort effects in PLS analysis

Research aiming at solving the collinearity problem in age-period-cohort analysis has generated an extensive literature, and most approaches have tried to accommodate the collinearity problem within the scope of traditional regression analyses. From a statistical viewpoint, an additional constraint can be made to make the effects of age, period, and cohort estimable, but the problem is that there are too many potential constraints. Hence, the more pertinent issue is rather that of interpretation with regard to the chosen constraint. As explained in the Appendix S1, an implicit constraint is imposed in PLS estimation, and this corresponds to the intrinsic mathematic relationship amongst age, period, and cohort. We therefore feel that the constraint imposed by PLS is both justifiable and interpretable.

The cohort effect is usually attributed to the impact of early environment, such as nutrition in pregnancy and early childhood [5], [6]. According to the developmental origins of health and disease hypothesis, early growth environment may have an important impact on health outcomes in later life [41], [42]. When the foetus and infant makes predictive adaptive responses to the environment in the early developmental process, adaptations chosen to cope with the unfavourable environment may have adverse effects for health in later phases of the lifecourse. It has also been suggested that early childhood conditions, such as dehydration, may be associated with blood pressure in later life [43], [44]. On the other hand, changes in nutrition and diet, such as reduction in salt intake and increased consumption of vegetable and fruits in the first half of the last century in the UK [23], [45], may have contributed to the decreasing trend in blood pressure across the year of birth found in the previous and the present studies. Nevertheless, the negative associations between blood pressure and year of examination found in the present study also suggest that the improved nutrition and living environment in the UK might have a continuing impact on population health in adolescence and early adulthood.

In this study, we also found that there seemed to be differences in trends for DBP, where the decreasing trends in exam year and birth year were less notable in women than in men, but for SBP, men and women had similar trends. Women had on average had lower DBP than men by about 10 mmHg, and whilst healthy diets or other factors were associated with improved blood pressure, there might be a physiological limit on how much reduction in blood pressure can be attained due to such factors.

Many studies in the age-period-cohort analysis literature aim to resolve the identification problem in order to estimate the “unique” contribution of the three components [2]. However, as one recent study argued, the conceptual definitions of such effects are not always clear and therefore require further elaboration [5]. The cohort effect, such as that represented by the year of birth in this study, is usually interpreted as the effect of early life experience, e.g. early nutrition in epidemiological research. The period effect is interpreted as exposure or events in later life. From a lifecourse perspective, the impact of environment and its interactions with biological factors continues throughout the developmental process. The demarcation of lifecourse experience into different phases such as cohort versus period, or early versus later life, is a conceptual framework for research, but the underlying biological process is nevertheless continuous. Age, period and cohort are not only mathematically related but also conceptually connected. PLS analysis partitions their joint lifecourse effects according to their covariance structure with each other and the outcome. Results from PLS yield the partitioning of the total effects of age, period and cohort, which has meaning and utility. This is a similar idea to that suggested recently by O'Brien of the partitioning of the total variance [46]. Whilst it may be tempting to interpret the PLS regression coefficients as the “independent” contributions of age, period and cohort, it is more appropriate to view them as their “relative” importance in contribution. In view of this perspective of interpretation, cohort and period had similar effects on blood pressure in this study.

Comparisons between PLS and other modelling strategies in the literature

The major difference between PLS and other modelling strategies is that it is straightforward to incorporate all perfectly collinear variables into the same model. Some approaches proposed in the literature can only be applied to aggregated data [3], [7], [8], but PLS can be used to analyse both individual data such as those in this study and aggregated data such as mortality rates for different age groups in different years. For example, a commonly used approach is to plot the trends in the outcome against age groups for different birth cohorts, and period effects are inferred from the differences in trends between cohorts [4]. PLS is therefore complementary to those approaches. For wider applications of PLS in epidemiology, further developments are required to extend PLS to generalised linear models [47]–[49]. Nevertheless, PLS already provides a flexible modelling strategy for age-period-cohort analysis.

Concluding remarks

There is an extensive literature in epidemiology and social sciences as to how to estimate age, cohort and period effects. Whilst some of the debates and controversies focused upon the identification issue in the estimation [4]–[7], some are more concerned with the meaning and interpretation of those effects [1]–[3], [8]. In this study, we propose to use PLS to address the former, but whilst identification is no longer a computation issue for PLS, more effort is required to clarify the substantive and conceptual issues regarding the definitions and interpretations of age, period and cohort effects. Those conceptual questions may be even harder to answer.