II. Compositional Data Analysis of Alcohol Shares

The initial focus here is on the wine, beer, and spirits share of total alcohol consumption for country i in year t. If these shares are denoted w _i,t, b _i,t, and s _i,t then, as well as the non-negativity constraints w _i,t ≥ 0, b _i,t ≥ 0, and s _i,t ≥ 0, there is also the adding-up constraint w _i,t + b _i,t + s _i,t = 1. If there are N countries and T years of observations, then this data may be represented by the set of N × 3 matrices

$${\bi X}_t = \left[ {\matrix{ {\matrix{ {w_{1,t}} \cr {\matrix{ {w_{2,t}} \cr \vdots \cr {w_{N,t}} \cr}} \cr}} & {\matrix{ {\matrix{ {b_{1,t}} \cr {b_{2,t}} \cr \vdots \cr}} \cr {b_{N,t}} \cr}} & {\matrix{ {\matrix{ {s_{1,t}} \cr {s_{2,t}} \cr \vdots \cr}} \cr {s_{N,t}} \cr}} \cr}} \right] = \left[ {\matrix{ {{\bi w}_t} & {{\bi b}_t} & {{\bi s}_t} \cr}} \right]\, \quad\quad t = 1,2, \cdots, T,$$

where w _t + b _t + s _t = ι = [  1    1|    …    1 ]′, the N × 1 unit vector. The sub-matrix

$${\bi X}_t^{\left( 2 \right)} = \left[ {\matrix{ {{\bi w}_t} & {{\bi b}_t} \cr}} \right],$$

then lies in the two-dimensional simplex 𝒮 ² embedded in three-dimensional real space ℛ ³ with s _t = ι − w _t − b _t being the vector of “fill-up” values and ${\bi X}_t = \left[ {\matrix{ {{\bi X}_t^{\left( 2 \right)}} & {{\bi s}_t} \cr}} \right]$.

There are several difficulties with analysing X_t within the simplex sample space, these being a consequence of the summation constraint rendering standard covariance and correlation analysis invalid. These were foreseen by Pearson (Reference Pearson1897) but were brought to the attention of applied statisticians by Aitchison (Reference Aitchison1982), who subsequently followed this article with his book (Aitchison, Reference Aitchison2003). For example, consider the crude covariance matrix of the shares for a particular year (the subscript t is now omitted for notational convenience)

$${\bi {K}} = \left[ {\matrix{ {\sigma _w^2} & {\sigma _{wb}} & {\sigma _{ws}} \cr {\sigma _{wb}} & {\sigma _b^2} & {\sigma _{bs}} \cr {\sigma _{ws}} & {\sigma _{bs}} & {\sigma _s^2} \cr}} \right],$$

with accompanying crude correlations

$$\rho _{wb} = \sigma _{wb}/\sigma _w\sigma _b\, \quad\quad \rho _{ws} = \sigma _{ws}/\sigma _w\sigma _s\, \quad\quad \rho _{bs} = \sigma _{bs}/\sigma _b\sigma _s.$$

Now consider the covariance between w and w + b + s, σ _w,w+b+s. Because of the unit sum constraint this is the covariance between w and ι, which is clearly zero. However, this then implies that, since $\sigma _{w,w + b + s} = \sigma _w^2 + \sigma _{wb} + \sigma _{ws}$, it must be the case that

(1)$$\sigma _{wb} + \sigma _{ws} = - \sigma _w^2.$$

The right-hand side of equation (1) must be negative except for the trivial situation where the wine share is constant across all countries. Thus at least one of the covariances on the left must be negative or, equivalently, that there must be at least one negative element in the first row of the crude covariance matrix K. The same negative bias will similarly occur in the other rows of K so that all three covariances must be negative.

In terms of the correlations, it would normally be the case that the three correlations ρ _wb, ρ _ws, and ρ _bs could take any values in the interval –1 to + 1 subject to the non-negative definiteness condition

$$1 + 2\rho _{wb}\rho _{ws}\rho _{ws} - \rho _{wb}^2 - \rho _{ws}^2 - \rho _{bs}^2 \ge 0.$$

For example, since 1 + 2ρ ² − 3ρ ³ ≥ 0 for 0 ≤ ρ ≤ 1, it should be perfectly feasible for all three correlations to be equal and positive. Such a set of correlations for a three-part composition is, however, prohibited by the negative bias property.

These difficulties also affect the coefficient of variation employed by HA as one of their indicators of convergence. The coefficients of variation for the three shares are defined as

$$cv\left( w \right) = \sigma _w/\mu _w\, \quad\quad cv\left( b \right) = \sigma _b/\mu _b\,\quad\quad cv\left( s \right) = \sigma _s/\mu _s,$$

where μ _w , μ _b, and μ _s are the mean shares. Given the unit sum constraint it is straightforward to show that since, for example, μ _s = 1 − μ _b − μ _w and $\sigma _s^2 = - \left( {\sigma _{bs} + \sigma _{ws}} \right)$, then

$$cv\left( s \right) = \sqrt { - \left( {\sigma _{bs} + \sigma _{ws}} \right)} /\left( {1 - \mu _b - \mu _w} \right) = \sqrt { - \left( {\sigma _{bs} + \sigma _{ws}} \right)} /\left( {1 - \sigma _bcv\left( b \right) - \sigma _wcv\left( w \right)} \right),$$

and dependencies therefore exist between the three coefficients of variation, thus rendering problematic any conclusions about convergence based on these statistics.

Because of the difficulties in interpreting covariances and correlations calculated from X_t within the simplex sample space, Aitchison (Reference Aitchison1982, Reference Aitchison2003) thus proposed mapping ${\bi X}_t^{\left( 2 \right)} $ from 𝒮 ² to the two-dimensional real space ℛ ² and then examining the statistical properties of the transformed data within ℛ ².Footnote ² Several transformations have been proposed for doing this, the most popular being the additive log-ratio, defined, for the general three-shares composition

$${\bi X}_t = \left[ {\matrix{ {{\bi x}_{1,t}} & {{\bi x}_{2,t}} & {{\bi x}_{3,t}} \cr}} \right] = \left( {\matrix{ {{\bi X}_t^{\left( 2 \right)}} & {{\bi x}_{3,t}} \cr}} \right),$$

where

$${\bi x}_{i,t} = \left( {\matrix{ {x_{i1,t}} & \cdots & {x_{iN,t}} \cr}} \right)^{\prime}\,\quad\quad i = 1,\; 2,\; 3,$$

as

$${\bi Y}_t = \left[ {\matrix{ {{\bi y}_{1,t}} & {{\bi y}_{2,t}} \cr}} \right] = a_2\left( {{\bi X}_t^{\left( 2 \right)}} \right) = \left[ {\matrix{ {log\left( {{\bi x}_{1,t}/{\bi x}_{3,t}} \right)} & {log\left( {{\bi x}_{2,t}/{\bi x}_{3,t}} \right)} \cr}} \right].$$

The inverse transformation, known as the additive-logistic, is

$${\bi X}_t^{\left( 2 \right)} = a_2^{ - 1} \left( {{\bi Y}_{\bi t}} \right) = \left[ {\matrix{ {exp\left( {{\bi y}_{1,t}} \right)/{\bi y}_t} & {\matrix{ {exp\left( {{\bi y}_{2,t}} \right)/{\bi y}_t} & {exp\left( {{\bi y}_{3,t}} \right)/{\bi y}_t} \cr}} \cr}} \right]$$

$${\bi x}_{3,t} = 1/{\bi y}_t,$$

where

$${\bi y}_t = 1 + exp\left( {{\bi y}_{1,t}} \right) + exp\left( {{\bi y}_{2,t}} \right).$$

Although it is important to note that a _d (·) is invariant to the choice of fill-up share, so that nothing in the statistical analysis hinges on which of the shares is chosen for this role, it is possible to avoid choosing a fill-up by using the centred log-ratio transformation, defined as

$${\bi Z}_t = c_d\left( {{\bi X}_t^{\left( 2 \right)}} \right) = \left[ {\matrix{ {log\left( {{\bi x}_{1,t}/g\left( {{\bi X}_t} \right)} \right)} & {log\left( {{\bi x}_{2,t}/g\left( {{\bi X}_t} \right)} \right)} & {log\left( {{\bi x}_{3,t}/g\left( {{\bi X}_t} \right)} \right)} \cr}} \right],$$

where

$$g\left( {{\bi X}_t} \right) = \left[ {\matrix{ {{\left( {x_{11,t}x_{21,t}x_{31,t}} \right)}^{1/3}} \cr {{\left( {x_{12,t}x_{22,t}x_{32,t}} \right)}^{1/3}} \cr {\matrix{ \vdots \cr {{\left( {x_{1N,t}x_{2N,t}x_{3N,t}} \right)}^{1/3}} \cr}} \cr}} \right],$$

is the vector of geometric means.

Although the centred log-ratio transformation introduces a singularity since Z_t ι = 0, which has limited its application in favour of the additive log-ratio transformation, it does have a symmetric, albeit singular, covariance matrix

$${\bi \Gamma} _t = \left[ {\gamma _{ij,t}} \right] = \left[ {cov\left\{ {log\left( {{\bi x}_{i,t}/g\left( {{\bi X}_t} \right)} \right),log\left( {{\bi x}_{\,j,t}/g\left( {{\bi X}_t} \right)} \right)} \right\}:i,j = 1,2,3} \right].$$

In the application considered here this is a major advantage over the additive log-ratio transform, for a measure of the total variability of X_t is given by the trace of Γ_t

$$tr\left( {{\bi \Gamma} _t} \right) = \lambda _1 + \lambda _2,$$

where λ ₁ > λ ₂ are the eigenvalues of Γ_t (Aitchison, Reference Aitchison2003, pp. 190–191, Definition 8.1 and Property 8.1).

III. Convergence of Alcohol Shares through Time

The trace of the centred log-ratio covariance matrix, tr(Γ_t), thus provides a measure of the total variation in alcohol shares in year t, and computation of this statistic across the T  years for which data on alcohol shares are available will therefore allow an assessment of the extent of convergence to the (changing) mean shares through time.

Annual data from 1961 to 2014 on wine, beer, and spirits shares of total alcohol consumption volume is available for up to 53 countries and regions each year in Tables 53(a)–(c) of Anderson and Pinilla (Reference Anderson and Pinilla2017).Footnote ³ For each year t = 1961, · · · , 2014 the centred log-ratios were calculated for the set of country shares available and the covariance matrix Γ_t constructed. From this the eigenvalues were computed and tr(Γ_t) obtained. The sequence of 54 traces are plotted in Figure 1 with the fitted quadratic trend

$$\widehat{{tr\left( {{\bi \Gamma} _t} \right)}} = 9.99 - 0.230t + 0.0015t^2,$$

superimposed to emphasise the decline over time in this measure of total variability.

Figure 1 Trace of Log-Ratio Covariance Matrix, 1961–2014, with Fitted Quadratic Trend Superimposed

It is thus clear that, using this metric of total variation based on the compositional character of shares data, that the variation in alcohol shares around the global mean shares has declined substantially over the half-century from the early 1960s, thus confirming, in a statistically valid way, the findings of HA.