Nutritional values probability theory
When analysing the nutritional values of groups as products, such as categories, one likes to know what the expected value and spread around that values of that group is. This might for instance help to detect subgroups and unusual products. Plotting the individual values of products in the group is already very useful, but one likes to do more.
The usual tools for determining an average and distribution is to assume that the probability distribution of the nutritional values follows a normal gaussian distribution. Using a gaussian distribution assumes that the values can get any value: from minus infinity to plus infinity. For nutritional values this is not the case. Nutritional values are fractions that go from 0 to an upper limit. For Energy this upper limit is 3700 kJ and for the nutrients this is 1. The Energy limit is reached if 100% of the nutrients consist of oil for instance.
That the normal distribution cannot be used is very well demonstrated by the Violin plots. Underlying the graph is the assumption of a normal distribution. This in turn leads to a plot that can go below zero.
If a normal distribution cannot be used, what should be used then? Some other probability distribution should be used.
Note: this page limits itself to statistical considerations for nutritional values. There is much material available on Internet, which describes things much better. The page page is more about what should and can be applied to nutritional values.
Probability distribution
There exist many probability distributions, so which one is applicable to nutritional values. What we are looking for lies in the category of continuous distributions, which means that the nutritional values are Real numbers. And the distribution should be on a bounded interval. The most general of these distributions is the Beta distribution.
Beta distribution
The formula for the beta distribution is give in the figure below.
In the formula the values x are between 0 and 1. The α and β determine the shape of the function. The Γ(z) is the Gamma function. The parameters α and β are positive integers (not 0).
Probably α and β will roughly equal for most nutritional values. If the nutritional value is close to 0, α=1 and β is large; if the nutritional value is close to 1, α is large and β=1.
The mean is define by α/(α+β) and the variance by αβ/[(αβ)2(α+β+1)].
Applicability
Is this distribution applicable to the nutritional values of the groups of products? And when is it not? And how would be able to detect that? If there are within a group two different subgroups that have clearly different nutritional values, then it is obvious this distribution can not be applied. And the subgroups should be analysed separately.
Scaling
The Beta distribution assumes that the x-values are between 0 and 1. The nutritional values published on products are either in grams per 100 gram or in grams per serving. The Energy values are in KJ (or kcal) per 100 grams (or per serving). All these numbers need to be scaled, so that they are between 0 and 1.
We have now two possibilities:
- the nutrients of a product are analysed by weight, i.e. what weight fraction of a product is due a certain nutrient;
- the nutrients of a product are analysed by energy, i.e. what energy fraction of a product is due to a certain nutrient
The energy must be scaled differently and the maximum allowed value of 3700 kJ should be used to scale the values.
Is this scaling sufficient? The minimum value is always 0. But the maximum value is not always 1, but that depends on other the existence of other nutrients and components. If we use another maximum value the formula above should be adapted.
Multivariate considerations
The beta function described above assumes that the nutritional values are uniformly distributed between 0 and 1. This unfortunately is not always the case. If a product consists of one nutrient this is correct. But if there are more than one nutrient, only the sum of all nutrients is between 0 and 1. A single nutrient has than a maximum that is smaller than 1. But we do not know that maximum beforehand.
The published macronutrients do not represent 100% of the weight of a product. For instance water represents a large fraction by weight of a product, but its value is unknown. Even not all macronutrients are always available (fiber).
Dirichlet distribution
Thus there is an interdependence between the maxima of each nutrient, which is set by the total. A Dirichlet distribution would be more applicable.
This Beta distribution is a special case of the Dirichlet distribution. The Dirichlet distribution is described by 5 axes (fat, carbohydrates, fiber, protein, salt). Each of the axes has its own parameter αi.
The sum of all the α's gives the αtotal. This is used the mean and variance for each axis, i.e. Meani= αi/αtotal. And the variance: αi(αtotal-αi)/(α02(αtotal-1)).
If we look only at a single nutrient, the corresponding Beta-distribution is defined by: Beta(αi, αtot-αi), thus the β of the Beta distribution is αtot-αi
Energy
The Dirichlet distribution can only be used for the nutrients, as the sum of the nutrients is 1 (100% of the product weight). Thus we cannot include the Energy in the formula. Something that is more generalised is required.
The first thing to look at is the Generalized Dirichlet which doubles the parameters for each axis. But is not clear if this would be helpful. No useful examples can be found. A critique of the Dirichlet distribution is the restricted covariance matrix.
Adding Energy to the set of variables to be analysed introducing another problem: the variables are no longer independent. There exist linear relations between some nutrients (fat, carbohydrates, fiber, proteins) and energy. Any multivariate distribution should take this in account.
Another possibility is to use a Copula. Have to look further into this to see whether it might be useful.
Sub-nutrients
For some macronutrients, there are also sub-nutrients. For instance for fat, saturated fat. And for carbohydrates the sugars. Could the play a role in a probability distribution? A keyword might be Nested Dirichlet distributions.
Estimates
The theory above only describes the formulas describing the distributions. In that case we know already all parameters α and β. We want to determine these α and β from actual data. This requires finding the best estimate for these parameters. There is no straightforward way (a formula) to go from the data to these values. It is required to solve some complex equations by numerical analysis.
Beta distribution two parameter case
In this case we assume that the data is distributed between 0 and 1. This implies that we are only looking for the α and β. The wikipedia page for the Beta distribution describes 3 approaches to finding these α and β. There are three methods to determine the α and β. The explanation is quite dense and not easy to transform to a practical approach.
For the Maximum Likelihood method a more easily accessible approach is available. In short this method tries to find the maximum of the distribution, i.e. where the second partial derivatives are zero.
The Excel file published on this site has been transformed into a Numbers spreadsheet. To use these spreadsheets you need to input the normalised data and define an initial α and β. At first it is not clear what to use as initial values. Some experimentation is required as not all initial values will converge to a solution. Using the solution for α and β we can now calculate mean and variance. And we can create a plot of the solution together with a distribution of the data:
Beta distribution three parameter case
Should an upper limit of 1 be used Or should this upper limit be estimated as well? We should implement an numerical estimation, but that requires doing some of the math first. The wikipedia page can serve as inspiration.
Beta distribution examples
It necessary to investigate a bit more what the acceptable values for alpha and beta are for nutritional values. Examples of data are gathered on this page.
Upper limit values
Note that any value in a nutritional table that is indicated by an upper limit (i.e. <) should be replace by zero (0). However this might bias the actual distribution to 0. And if we use the limits as actual values we might bias the actual distribution to a too high level.