# Aggregation

This vignette describes the process of aggregating indicators, in COINr.

# Introduction

Aggregation is the operation of combining multiple indicators into one value. Many composite indicators have a hierarchical structure, so in practice this often involves multiple aggregations, for example aggregating groups of indicators into aggregate values, then aggregating those values into higher-level aggregates, and so on, until the final index value.

Aggregating should almost always be done on normalised data, unless the indicators are already on very similar scales. Otherwise the relative influence of indicators will be very uneven.

Of course you don’t have to aggregate indicators at all, and you might be content with a scoreboard, or perhaps aggregating into several aggregate values rather than a single index. However, consider that aggregation should not substitute the underlying indicator data, but complement it.

Overall, aggregating indicators is a form of information compression - you are trying to combine many indicator values into one, and inevitably information will be lost (this recent paper may be of interest). As long as this is kept in mind, and indicator data is presented and made available along side aggregate values, then aggregate (index) values can complement indicators and be used as a useful tool for summarising the underlying data, and identifying overall trends and patterns.

## Weighting

Many aggregation methods involve some kind of weighting, i.e. coefficients that define the relative weight of the indicators/aggregates in the aggregation. In order to aggregate, weights need to first be specified, but to effectively adjust weights it is necessary to aggregate.

This chicken and egg conundrum is best solved by aggregating initially with a trial set of weights, perhaps equal weights, then seeing the effects of the weighting, and making any weight adjustments necessary.

## Approaches

### Means

The most straightforward and widely-used approach to aggregation is the weighted arithmetic mean. Denoting the indicators as $$x_i \in \{x_1, x_2, ... , x_d \}$$, a weighted arithmetic mean is calculated as:

$y = \frac{1}{\sum_{i=1}^d w_i} \sum_{i=1}^d x_iw_i$

where the $$w_i$$ are the weights corresponding to each $$x_i$$. Here, if the weights are chosen to sum to 1, it will simplify to the weighted sum of the indicators. In any case, the weighted mean is scaled by the sum of the weights, so weights operate relative to each other.

Clearly, if the index has more than two levels, then there will be multiple aggregations. For example, there may be three groups of indicators which give three separate aggregate scores. These aggregate scores would then be fed back into the weighted arithmetic mean above to calculate the overall index.

The arithmetic mean has “perfect compensability”, which means that a high score in one indicator will perfectly compensate a low score in another. In a simple example with two indicators scaled between 0 and 10 and equal weighting, a unit with scores (0, 10) would be given the same score as a unit with scores (5, 5) – both have a score of 5.

An alternative is the weighted geometric mean, which uses the product of the indicators rather than the sum.

$y = \left( \prod_{i=1}^d x_i^{w_i} \right)^{1 / \sum_{i=1}^d w_i}$

This is simply the product of each indicator to the power of its weight, all raised the the power of the inverse of the sum of the weights.

The geometric mean is less compensatory than the arithmetic mean – low values in one indicator only partially substitute high values in others. For this reason, the geometric mean may sometimes be preferred when indicators represent “essentials”. An example might be quality of life: a longer life expectancy perhaps should not compensate severe restrictions on personal freedoms.

A third type of mean, in fact the third of the so-called Pythagorean means is the weighted harmonic mean. This uses the mean of the reciprocals of the indicators:

$y = \frac{\sum_{i=1}^d w_i}{\sum_{i=1}^d w_i/x_i}$

The harmonic mean is the the least compensatory of the the three means, even less so than the geometric mean. It is often used for taking the mean of rates and ratios.

### Other methods

The weighted median is also a simple alternative candidate. It is defined by ordering indicator values, then picking the value which has half of the assigned weight above it, and half below it. For ordered indicators $$x_1, x_2, ..., x_d$$ and corresponding weights $$w_1, w_2, ..., w_d$$ the weighted median is the indicator value $$x_m$$ that satisfies:

$\sum_{i=1}^{m-1} w_i \leq \frac{1}{2}, \: \: \text{and} \sum_{i=m+1}^{d} w_i \leq \frac{1}{2}$

The median is known to be robust to outliers, and this may be of interest if the distribution of scores across indicators is skewed.

Another somewhat different approach to aggregation is to use the Copeland method. This approach is based pairwise comparisons between units and proceeds as follows. First, an outranking matrix is constructed, which is a square matrix with $$N$$ columns and $$N$$ rows, where $$N$$ is the number of units.

The element in the $$p$$th row and $$q$$th column of the matrix is calculated by summing all the indicator weights where unit $$p$$ has a higher value in those indicators than unit $$q$$. Similarly, the cell in the $$q$$th row and $$p$$th column (which is the cell opposite on the other side of the diagonal), is calculated as the sum of the weights unit where $$q$$ has a higher value than unit $$p$$. If the indicator weights sum to one over all indicators, then these two scores will also sum to 1 by definition. The outranking matrix effectively summarises to what extent each unit scores better or worse than all other units, for all unit pairs.

The Copeland score for each unit is calculated by taking the sum of the row values in the outranking matrix. This can be seen as an average measure of to what extent that unit performs above other units.

Clearly, this can be applied at any level of aggregation and used hierarchically like the other aggregation methods presented here.

In some cases, one unit may score higher than the other in all indicators. This is called a dominance pair, and corresponds to any pair scores equal to one (equivalent to any pair scores equal to zero).

The percentage of dominance pairs is an indication of robustness. Under dominance, there is no way methodological choices (weighting, normalisation, etc.) can affect the relative standing of the pair in the ranking. One will always be ranked higher than the other. The greater the number of dominance (or robust) pairs in a classification, the less sensitive country ranks will be to methodological assumptions. COINr allows to calculate the percentage of dominance pairs with an inbuilt function.

# Coins

We now turn to how data sets in a coin can be aggregated using the methods described previously. The function of interest is Aggregate(), which is a generic with methods for coins, purses and data frames. To demonstrate COINr’s Aggregate() function on a coin, we begin by loading the package, and building the example coin, up to the normalised data set.

library(COINr)

# build example up to normalised data set
coin <- build_example_coin(up_to = "Normalise")
#> iData checked and OK.
#> iMeta checked and OK.
#> Written data set to .$Data$Raw
#> Written data set to .$Data$Denominated
#> Written data set to .$Data$Imputed
#> Written data set to .$Data$Screened
#> Written data set to .$Data$Treated
#> Written data set to .$Data$Normalised

Consider what is needed to aggregate the normalised data into its higher levels. We need:

• The data set to aggregate
• The structure of the index: which indicators belong to which groups, etc.
• Weights to assign to indicators
• Specifications for aggregation: an aggregation function (e.g. the weighted mean) and any other parameters to be passed to that function

All of these elements are already present in the coin, except the last. For the first point, we simply need to tell Aggregate() which data set to use (using the dset argument). The structure of the index was defined when building the coin in new_coin() (the iMeta argument). Weights were also attached to iMeta. Finally, specifications can be specified in the arguments of Aggregate(). Let’s begin with the simple case though: using the function defaults.

# aggregate normalised data set
coin <- Aggregate(coin, dset = "Normalised")
#> Written data set to .$Data$Aggregated

By default, the aggregation function performs the following steps:

• Uses the weights that were attached to iMeta
• Aggregates hierarchically (with default method of weighted arithmetic mean), following the index structure specified in iMeta and using the data specified in dset
• Creates a new data set .$Data$Aggregated, which consists of the data in dset, plus extra columns with scores for each aggregation group, at each aggregation level.

Let’s examine the new data set. The columns of each level are added successively, working from level 1 upwards, so the highest aggregation level (the index, here) will be the last column of the data frame.

dset_aggregated <- get_dset(coin, dset = "Aggregated")

nc <- ncol(dset_aggregated)
# view aggregated scores (last 11 columns here)
dset_aggregated[(nc - 10) : nc] |>
signif(3)
#>   ConEcFin Environ Instit   P2P Physical Political Social SusEcFin Conn Sust
#> 1     12.6    31.9   52.4 39.50     34.8      52.5   71.9     55.7 38.4 53.2
#> 2     26.2    69.5   77.5 54.10     41.1      78.2   72.8     62.9 55.4 68.4
#> 3     48.2    53.0   75.6 43.30     72.0      80.8   86.2     50.1 64.0 63.1
#> 4     13.3    81.7   26.5  5.85     22.9      32.4   27.5     64.6 20.2 57.9
#> 5     24.6    55.7   75.9 27.10     28.4      67.5   53.3     61.7 44.7 56.9
#>   Index
#> 1  45.8
#> 2  61.9
#> 3  63.5
#> 4  39.0
#> 5  50.8

Here we see the level 2 aggregated scores created by aggregating each group of indicators (the first eight columns), followed by the two sub-indexes (level 3) created by aggregating the scores of level 2, and finally the Index (level 4), which is created by aggregating the “Conn” and “Sust” sub-indexes.

The format of this data frame is not hugely convenient for inspecting the results. To see a more user-friendly version, use the get_results() function.

## COINr aggregation functions

Let’s now explore some of the options of the Aggregate() function. Like other coin-building functions in COINr, Aggregate() comes with a number of inbuilt options, but can also accept any function that is passed to it, as long as it satisfies some requirements. COINr’s inbuilt aggregation functions begin with a_, and are:

• a_amean(): the weighted arithmetic mean
• a_gmean(): the weighted geometric mean
• a_hmean(): the weighted harmonic mean
• a_copeland(): the Copeland method (note: requires by_df = TRUE)

By default, the arithmetic mean is called but we can easily change this to the geometric mean, for example. However here we run into a problem: the geometric mean will fail if any values to aggregate are less than or equal to zero. So to use the geometric mean we have to re-do the normalisation step to avoid this. Luckily this is straightforward in COINr:

coin <- Normalise(coin, dset = "Treated",
global_specs = list(f_n = "n_minmax",
f_n_para = list(l_u = c(1,100))))
#> Written data set to .$Data$Normalised
#> (overwritten existing data set)

Now, since the indicators are scaled between 1 and 100 (instead of 0 and 100 as previously), they can be aggregated with the geometric mean.

coin <- Aggregate(coin, dset = "Normalised",
f_ag = "a_gmean")
#> Written data set to .$Data$Aggregated
#> (overwritten existing data set)

## External functions

All of the four aggregation functions mentioned above have the same format (try e.g. ?a_gmean), and are built into the COINr package. But what if we want to use another type of aggregation function? The process is exactly the same.

In this section we use some functions from other packages: the matrixStats package and the Compind package. These are not imported by COINr, so the code here will only work if you have these installed. If this vignette was built on your computer, we have to check whether these packages are installed:

ms_installed <- requireNamespace("matrixStats", quietly = TRUE)
ms_installed
#>  TRUE

ci_installed <- requireNamespace("Compind", quietly = TRUE)
ci_installed
#>  TRUE

If either of these have returned FALSE, in the following code chunks you will see some blanks. See the online version of this vignette to see the results, or install the above packages and rebuild the vignettes.

Now for an example, we can use the weightedMedian() function from the matrixStats package. This has a number of arguments, but the ones we will use are x and w (with the same meanings as COINr functions), and na.rm which we need to set to TRUE.

# RESTORE above eval=ms_installed
library(matrixStats)

# aggregate using weightedMedian()
coin <- Aggregate(coin, dset = "Normalised",
f_ag = "weightedMedian",
f_ag_para = list(na.rm = TRUE))

The weights w do not need to be specified in f_ag_para because they are automatically passed to f_ag unless specified otherwise.

The general requirements for f_ag functions passed to Aggregate() are that:

1. The input to the function is a numeric vector x, possibly with missing values
2. The function returns a single (scalar) aggregated value
3. If the function accepts a vector of weights, this vector (of the same length of x) is passed as function argument w. If the function doesn’t accept a vector of weights, we can set w = "none" in the arguments to Aggregate(), and it will not try to pass w.
4. Any other arguments to f_ag, apart from x and w, should be included in the named list f_ag_para.

Sometimes this may mean that we have to create a wrapper function to satisfy these requirements. For example, the ‘Compind’ package has a number of sophisticated aggregation approaches. The “benefit of the doubt” uses data envelopment analysis to aggregate indicators, however the function Compind::ci_bod() outputs a list. We can make a wrapper function to use this inside COINr:

# RESTORE ABOVE eval= ci_installed