Imputation is the process of estimating missing data points. To get started with imputation, a reasonable first step is to see how much missing data we have in the data set. We begin by building the example coin, up the point of assembling the coin, but not any further:

To check missing data, the `get_data_avail()`

function can
be used. It can output to either the coin or to a list – here we output
to a list to readily display the results.

The output list has data availability by unit:

```
head(l_avail$Summary)
#> uCode N_missing N_zero N_miss_or_zero Dat_Avail Non_Zero
#> 31 AUS 0 3 3 1.0000000 0.9387755
#> 1 AUT 0 2 2 1.0000000 0.9591837
#> 2 BEL 0 2 2 1.0000000 0.9591837
#> 32 BGD 6 1 7 0.8775510 0.9767442
#> 3 BGR 0 0 0 1.0000000 1.0000000
#> 33 BRN 10 2 12 0.7959184 0.9487179
```

The lowest data availability by unit is:

We can also check data availability by indicator. This is done by
calling `get_stats()`

:

```
df_avail <- get_stats(ASEM, dset = "Raw", out2 = "df")
head(df_avail[c("iCode", "N.Avail", "Frc.Avail")], 10)
#> iCode N.Avail Frc.Avail
#> 1 LPI 51 1.000
#> 2 Flights 51 1.000
#> 3 Ship 51 1.000
#> 4 Bord 51 1.000
#> 5 Elec 51 1.000
#> 6 Gas 51 1.000
#> 7 ConSpeed 43 0.843
#> 8 Cov4G 51 1.000
#> 9 Goods 51 1.000
#> 10 Services 51 1.000
```

By indicator, the minimum data availability is:

With missing data, several options are available:

- Leave it as it is and aggregate anyway (there is also the option for data availability thresholds during aggregation - see Aggregation)
- Consider removing indicators that have low data availability (this has to be done manually because it affects the structure of the index)
- Consider removing units that have low data availability (see Unit Screening)
- Impute missing data

These options can also be combined. Here, we focus on the option of imputation.

The `Impute()`

function is a flexible function that
imputes missing data in a data set using any suitable function that can
be passed to it. In fact, `Impute()`

is a *generic*,
and has methods for coins, data frames, numeric vectors and purses.

Let’s begin by examining the data frame method of
`Impute()`

, since it is easier to see what’s going on. We
will use a small data frame which is easy to visualise:

```
# some data to use as an example
# this is a selected portion of the data with some missing values
df1 <- ASEM_iData[37:46, 36:39]
print(df1, row.names = FALSE)
#> Pat CultServ CultGood Tourist
#> 23.7 0.13405 NA 11.519
#> 583.5 2.20754 16.182 24.040
#> 3.6 0.05780 0.985 6.509
#> 249.8 1.79800 NA 17.242
#> NA NA NA 3.315
#> 64.2 1.15292 7.555 26.757
#> 0.3 0.00266 0.046 0.404
#> NA 0.08905 NA 2.907
#> 46.5 0.34615 1.213 3.370
#> 7.2 0.03553 1.256 0.966
```

In the simplest case, imputation can be performed column-wise, i.e. by imputing each indicator one at a time:

```
Impute(df1, f_i = "i_mean")
#> Pat CultServ CultGood Tourist
#> 37 23.70 0.1340500 4.5395 11.519
#> 38 583.50 2.2075400 16.1820 24.040
#> 39 3.60 0.0578000 0.9850 6.509
#> 40 249.80 1.7980000 4.5395 17.242
#> 41 122.35 0.6470778 4.5395 3.315
#> 42 64.20 1.1529200 7.5550 26.757
#> 43 0.30 0.0026600 0.0460 0.404
#> 44 122.35 0.0890500 4.5395 2.907
#> 45 46.50 0.3461500 1.2130 3.370
#> 46 7.20 0.0355300 1.2560 0.966
```

Here, the “Raw” data set has been imputed by substituting missing
values with the mean of the non-`NA`

values for each column.
This is performed by setting `f_i = "i_mean"`

. The
`f_i`

argument refers to a function that imputes a numeric
vector - in this case the built-in `i_mean()`

function:

```
# demo of i_mean() function, which is built in to COINr
x <- c(1,2,3,4, NA)
i_mean(x)
#> [1] 1.0 2.0 3.0 4.0 2.5
```

The key concept here is that the simple function
`i_mean()`

is applied by `Impute()`

to each
column. This idea of passing simpler functions is used in several key
COINr functions, and allows great flexibility because more sophisticated
imputation methods can be used from other packages, for example.

In COINr there are currently four basic imputation functions which impute a numeric vector:

`i_mean()`

: substitutes missing values with the mean of the remaining values`i_median()`

: substitutes missing values with the median of the remaining values (this will be more robust to outliers)`i_mean_grp()`

: substitutes missing values with the mean of a subset of the remaining values, defined by a separate grouping vector`i_median_grp()`

: substitutes missing values with the mean of a subset of the remaining values, defined by a separate grouping vector

These are very simple imputation methods, but more sophisticated options can be used by calling functions from other packages. The group imputation functions above are useful in an indicator context: for example in country-level indicator analysis we can substitute missing values by the mean/median within the same GDP/capita group, which is often a better approach than a flat mean across all countries. Obviously this is context-dependent however.

For now let’s explore the options native to COINr. We can also apply
the `i_median()`

function in the same way to substitute with
the indicator median. Adding a little complexity, we can also impute by
mean or median, but within unit (row) groups. Let’s assume that the
first five rows in our data frame belong to a group “a”, and the
remaining five to a different group “b”. In practice, these could be
e.g. GDP, population or wealth groups for countries - we might
hypothesise that it is better to replace `NA`

values with the
median inside a group, rather than the overall median, because countries
within groups are more similar.

To do this on a data frame we can use the `i_median_grp()`

function, which requires an additional argument `f`

: a
grouping variable. This is passed through `Impute()`

using
the `f_i_para`

argument which takes any additional parameters
top `f_i`

apart from the data to be imputed.

```
# row grouping
groups <- c(rep("a", 5), rep("b", 5))
# impute
dfi2 <- Impute(df1, f_i = "i_median_grp", f_i_para = list(f = groups))
# display
print(dfi2, row.names = FALSE)
#> Pat CultServ CultGood Tourist
#> 23.70 0.134050 8.5835 11.519
#> 583.50 2.207540 16.1820 24.040
#> 3.60 0.057800 0.9850 6.509
#> 249.80 1.798000 8.5835 17.242
#> 136.75 0.966025 8.5835 3.315
#> 64.20 1.152920 7.5550 26.757
#> 0.30 0.002660 0.0460 0.404
#> 26.85 0.089050 1.2345 2.907
#> 46.50 0.346150 1.2130 3.370
#> 7.20 0.035530 1.2560 0.966
```

The `f_i_para`

argument requires a named list of
additional parameter values. This allows functions of any complexity to
be passed to `Impute()`

. By default, `Impute()`

applies `f_i`

to each column of data, so `f_i`

is
expected to take a numeric vector as its first input, and specifically
have the format `function(x, f_i_para)`

where `x`

is a numeric vector and `...`

are further arguments. This
means that the first argument of `f_i`

*must* be
called “x”. To use functions that don’t have `x`

as a first
argument, you would have to write a wrapper function.

Other than imputing by column, we can also impute by row. This only
really makes sense if the indicators are on a common scale, i.e. if they
are normalised first (or perhaps if they already share the same units).
To impute by row, set `impute_by = "row"`

. In our example
data set we have indicators on rather different scales. Let’s see what
happens if we impute by row mean but *don’t* normalise:

```
Impute(df1, f_i = "i_mean", impute_by = "row", normalise_first = FALSE)
#> Pat CultServ CultGood Tourist
#> 37 23.700000 0.13405 11.784350 11.519
#> 38 583.500000 2.20754 16.182000 24.040
#> 39 3.600000 0.05780 0.985000 6.509
#> 40 249.800000 1.79800 89.613333 17.242
#> 41 3.315000 3.31500 3.315000 3.315
#> 42 64.200000 1.15292 7.555000 26.757
#> 43 0.300000 0.00266 0.046000 0.404
#> 44 1.498025 0.08905 1.498025 2.907
#> 45 46.500000 0.34615 1.213000 3.370
#> 46 7.200000 0.03553 1.256000 0.966
```

This imputes some silly values, particularly in “CultGood”, because “Pat” has much higher values. Clearly this is not a sensible strategy, unless all indicators are on the same scale. We can however normalise first, impute, then return indicators to their original scales:

```
Impute(df1, f_i = "i_mean", impute_by = "row", normalise_first = TRUE, directions = rep(1,4))
#> Pat CultServ CultGood Tourist
#> 37 23.70000 0.134050 2.850908 11.519
#> 38 583.50000 2.207540 16.182000 24.040
#> 39 3.60000 0.057800 0.985000 6.509
#> 40 249.80000 1.798000 10.163326 17.242
#> 41 64.72133 0.246215 1.828412 3.315
#> 42 64.20000 1.152920 7.555000 26.757
#> 43 0.30000 0.002660 0.046000 0.404
#> 44 39.42134 0.089050 1.128411 2.907
#> 45 46.50000 0.346150 1.213000 3.370
#> 46 7.20000 0.035530 1.256000 0.966
```

This additionally required to specify the `directions`

argument because we need to know which direction each indicator runs in
(whether they are positive or negative indicators). In our case all
indicators are positive. See the vignette on Normalisation for more details on indicator
directions.

The values imputed in this way are more realistic. Essentially we are replacing each missing value with the average (normalised) score of the other indicators, for a given unit. However this also only makes sense if the indicators/columns are similar to one another: high values of one would likely imply high values in the other.

Behind the scenes, setting `normalise_first = TRUE`

first
normalises each column using a min-max method, then performs the
imputation, then returns the indicators to the original scales using the
inverse transformation. Another approach which gives more control is to
simply run `Normalise()`

first, and work with the normalised
data from that point onwards. In that case it is better to set
`normalise_first = FALSE`

, since by default if
`impute_by = "row"`

it will be set to `TRUE`

.

As a final point on data frames, we can set
`impute_by = "df"`

to pass the entire data frame to
`f_i`

, which may be useful for more sophisticated
multivariate imputation methods. But what’s the point of using
`Impute()`

then, you may ask? First, because when imputing
coins, we can impute by indicator groups (see next section); and second,
`Impute()`

performs some checks to ensure that
non-`NA`

values are not altered.

Imputing coins is similar to imputing data frames because the coin
method of `Impute()`

calls the data frame method. Please read
that section first if you have not already done so. However, for coins
there are some additional function arguments.

In the simple case we impute a named data set `dset`

using
the function `f_i`

: e.g. if we want to impute the “Raw” data
set using indicator median values:

```
ASEM <- Impute(ASEM, dset = "Raw", f_i = "i_mean")
#> Written data set to .$Data$Imputed
ASEM
#> --------------
#> A coin with...
#> --------------
#> Input:
#> Units: 51 (AUS, AUT, BEL, ...)
#> Indicators: 49 (Goods, Services, FDI, ...)
#> Denominators: 4 (Area, Energy, GDP, ...)
#> Groups: 4 (GDP_group, GDPpc_group, Pop_group, ...)
#>
#> Structure:
#> Level 1 Indicator: 49 indicators (FDI, ForPort, Goods, ...)
#> Level 2 Pillar: 8 groups (ConEcFin, Instit, P2P, ...)
#> Level 3 Sub-index: 2 groups (Conn, Sust)
#> Level 4 Index: 1 groups (Index)
#>
#> Data sets:
#> Raw (51 units)
#> Imputed (51 units)
```

Here, `Impute()`

extracts the “Raw” data set as a data
frame, imputes it using the data frame method (see previous section),
then saves it as a new data set in the coin. Here, the data set is
called “Imputed” but can be named otherwise using the
`write_to`

argument.

We can also impute by group using a grouped imputation function.
Since unit groups are stored within the coin (variables labelled as
“Group” in `iMeta`

), these can be called directly using the
`use_group`

argument (without having to specify the
`f_i_para`

argument):

```
ASEM <- Impute(ASEM, dset = "Raw", f_i = "i_mean_grp", use_group = "GDP_group", )
#> Written data set to .$Data$Imputed
#> (overwritten existing data set)
```

This has imputed each indicator using its GDP group mean.

Row-wise imputation works in the same way as with a data frame, by
setting `impute_by = "row"`

. However, this is particularly
useful in conjunction with the `group_level`

argument. If
this is specified, rather than imputing across the entire row of data,
it splits rows into indicator groups, using the structure of the index.
For example:

```
ASEM <- Impute(ASEM, dset = "Raw", f_i = "i_mean", impute_by = "row",
group_level = 2, normalise_first = TRUE)
#> Written data set to .$Data$Imputed
#> (overwritten existing data set)
```

Here, the `group_level`

argument specifies which
level-grouping of the indicators to use. In the ASEM example here, we
are using level 2 groups, so it is substituting missing values with the
average normalised score within each sub-pillar (in the ASEM example
level 2 is called “sub-pillars”).

Imputation in this way has an important relationship with
aggregation. This is because if we *don’t* impute, then in the
aggregation step, if we take the mean of a group of indicators, and
there is a `NA`

present, this value is excluded from the mean
calculation. Doing this is mathematically equivalent to assigning the
mean to that missing value and then taking the mean of all of the
indicators. This is sometimes known as “shadow imputation”. Therefore,
one reason to use this imputation method is to see which values are
being implicitly assigned as a result of excluding missing values from
the aggregation step.

Last we can see an example of imputation by data frame, with the
option `impute_by = "row"`

. Recall that this option requires
that the function `f_i`

accepts and returns entire data
frames. This is suitable for more sophisticated multivariate imputation
methods. Here we’ll use a basic implementation of the Expectation
Maximisation (EM) algorithm from the Amelia package.

Since COINr requires that the first argument of `f_i`

is
called `x`

, and the relevant Amelia function doesn’t satisfy
this requirement, we have write a simple wrapper function that acts as
an intermediary between COINr and Amelia. This also gives us the chance
to specify some other function arguments that are necessary.

```
# this function takes a data frame input and returns an imputed data frame using amelia
i_EM <- function(x){
# impute
amOut <- Amelia::amelia(x, m = 1, p2s = 0, boot.type = "none")
# return imputed data
amOut$imputations[[1]]
}
```

Now armed with our new function, we just call that from
`Impute()`

. We don’t need to specify `f_i_para`

because these arguments are already specified in the intermediary
function.

```
# impute raw data set
coin <- Impute(coin, dset = "Raw", f_i = i_EM, impute_by = "df", group_level = 2)
```

This has now passed each group of indicators at level 2 as data frames to Amelia, which has imputed each one and passed them back.

Purse imputation is very similar to coin imputation, because by
default the purse method of `Impute()`

imputes each coin
separately. There is one exception to this: if
`f_i = "impute_panel`

, the data sets inside the purse are
imputed using the last available data point, using the
`impute_panel()`

function. In this case, coins are not
imputed individually, but treated as a single data set. In this case,
optionally set `f_i_para = list(max_time = .)`

where
`.`

should be substituted with the maximum number of time
points to search backwards for a non-`NA`

value. See
`impute_panel()`

for more details. No further arguments need
to be passed to `impute_panel()`

.

It is difficult to show this working without a contrived example, so
let’s contrive one. We take the example panel data set
`ASEM_iData_p`

, and introduce a missing value `NA`

in the indicator “LPI” for unit “GB”, for year 2022.

```
# copy
dfp <- ASEM_iData_p
# create NA for GB in 2022
dfp$LPI[dfp$uCode == "GB" & dfp$Time == 2022] <- NA
```

This data point has a value for the previous year, 2021. Let’s see what it is:

Now let’s build the purse and impute the raw data set.

```
# build purse
ASEMp <- new_coin(dfp, ASEM_iMeta, split_to = "all", quietly = TRUE)
# impute raw data using latest available value
ASEMp <- Impute(ASEMp, dset = "Raw", f_i = "impute_panel")
#> Written data set to .$Data$Imputed
#> Written data set to .$Data$Imputed
#> Written data set to .$Data$Imputed
#> Written data set to .$Data$Imputed
#> Written data set to .$Data$Imputed
```

Now we check whether our imputed point is what we expect: we would
expect that our `NA`

is now replaced with the 2021 value as
found previously. To get at the data we can use the
`get_data()`

function.

```
get_data(ASEMp, dset = "Imputed", iCodes = "LPI", uCodes = "GBR", Time = 2021)
#> Time uCode LPI
#> 183 2021 GBR 4.386542
```

And indeed this corresponds to what we expect.