imdtools 📦

Tools for Indices of Multiple Deprivation (IMD) data in R.

Installation and usage

To install:

pak::pak("The-Strategy-Unit/imdtools")

library(imdtools)

To generate a lookup table of IMD2025 ranks and deciles for all English LSOAs:

get_imd_lookup()

This table is created from “File 1” on the GOV.UK IMD2025 webpage.

To generate a table of the LSOA-level transformed scores for each of the 7 IMD domains:

get_transformed_scores()

This table is created from “File 9” on the GOV.UK IMD2025 webpage.

To re-create the published IMD scores by weighting and combining the transformed domain scores, use the calculate_imd_score() function:

get_transformed_scores() |>
  calculate_imd_score(domains = choose_domains())

choose_domains() helps you select a subset of the seven domains should you wish to do so, in order to calculate an amended version of the IMD. This follows the guidance in the IMD Research Report (PDF), Appendix B:

It is possible to use the component domains to produce alternative measures of deprivation at LSOA, based on different domain weights than are used in the IMD.

For example, health researchers may want to use the IMD as a factor to help explain the variation in health outcomes across a sample of areas or individuals. To exclude the effect of the Health Deprivation and Disability Domain, they may want to use a modified measure of deprivation in their statistical analysis, with the Health Deprivation and Disability Domain weight set to zero.

This exclusion can be achieved by doing:

get_transformed_scores() |>
  calculate_imd_score(domains = choose_domains(include_health = FALSE))

This is equivalent to (but more straightforward than) doing:

new_weights <- c(rep(0.225, 2), 0.135, 0, rep(0.093, 3)) # element 4 set to 0
get_transformed_scores() |>
  calculate_imd_score(weights = new_weights)

(NB the vector must stay length 7 - replace a value by 0; don’t omit it).

Further functions may be added in future if they are thought to be helpful.

Discrepancy between calculated and published scores and ranks

If we do:

calculated_data <- get_transformed_scores() |>
  calculate_imd_score()

overall_scores <- get_overall_scores()
common_cols <- c("lsoa21cd", "lsoa21nm", "lad24cd", "lad24nm")
imd_lookup <- get_imd_lookup() |>
  dplyr::left_join(overall_scores, common_cols) |>
  dplyr::arrange(.data[["imd_rank"]])

sum(imd_lookup[["lsoa21cd"]] != calculated_data[["lsoa21cd"]])

[1] 3249

sum(imd_lookup[["imd_decile"]] != calculated_data[["imd_decile"]])

[1] 15

we can see that 3249 LSOAs are in a different row/rank to what is expected from the published ranks.

And in 15 cases this even causes the LSOA to be allocated to a different decile.

Published instructions for calculating an overall score from transformed domain scores and weights

This approach ought to be the correct one, following the published guidance in the IMD Research Report, which makes clear:

The standardised domain scores have been standardised by ranking and then transformed to an exponential distribution. These standardised domain scores have been published to be used as the basis for users to combine the domains together using different weights.
[emphasis added]
[Reconstituting the IMD overall score] can be achieved… using the following equation:

Income Deprivation Domain x domain-weight +
Employment Deprivation Domain x domain-weight + … etc.

Simply: each transformed (standardised) domain score is multiplied by its appropriate domain weight, then all the results are summed together.

This is the approach implemented in our function calculate_imd_score().

Explaining the discrepancy - and correcting it?

One thought is to replace the published value for the weightings for domains 5, 6 and 7 with a slightly amended value: (0.28 / 3) instead of 0.093 exactly.

This would make the seven weights add up to 100% exactly, and it seems not an unreasonable assumption that for this reason the fractional value might have been the one actually used by the creators of the IMD.

However, amending these three weights makes the problem worse instead of better:

# use 0.28/3 instead of 0.093
new_weights <- c(rep(0.225, 2), rep(0.135, 2), rep(0.28 / 3, 3))
new_calculated_data <- get_transformed_scores() |>
  calculate_imd_score(weights = new_weights)

sum(imd_lookup[["lsoa21cd"]] != new_calculated_data[["lsoa21cd"]])

[1] 30415

The lesson here is that the documentation is actually accurate!

The IMD scores are published to an accuracy of 3 decimal places. The values we get from calculate_imd_score() are not rounded. In the majority of cases, if we round our calculated scores to 3dp, they are within 0.001 of the published values.

calc2 <- calculated_data |>
  dplyr::rename_with(\(x) sub("^imd", "calc_imd", x)) |>
  # round to 3dp
  dplyr::mutate(dplyr::across("calc_imd_score", \(x) round(x, 3)))

# Number of LSOAs where rounded calculated score is equal to published score
imd_lookup |>
  dplyr::left_join(calc2, common_cols) |>
  dplyr::filter(imd_score == calc_imd_score) |>
  nrow()

[1] 30547

# Number of LSOAs where rounded calculated score within 0.001 of published score
imd_lookup |>
  dplyr::left_join(calc2, common_cols) |>
  dplyr::filter(abs(imd_score - calc_imd_score) <= 0.001) |>
  nrow()

[1] 32238

This is a great reassurance that our calculation process is fundamentally correct! There are just some discrepancies. And it’s hard to tell where these arise.

Might there be an issue with how the published IMD scores have been rounded?

The value of 3249 seems suspiciously not too far off 10% of the total number of LSOAs in England.

If the published values have been rounded to 3dp from more precise calculated values, we would expect about 10% of those more precise values to have a 4th dp value of 5.

In the R round function, a 5 is “rounded to even”, not simply rounded up. That means that

round(0.1235, 3)

[1] 0.124

produces 0.124 in the way that rounding 5s up would produce. However,

round(0.1225, 3)

[1] 0.122

doesn’t return 0.123, but is instead “rounded to even”, returning 0.122.

If the IMD creators used different software that uses rounding up, could this explain our discrepancies?

Unfortunately not; not based on our calculated values, anyway. There is no pattern that suggests rounding up instead of rounding to even would explain the discrepancies.

What about the ranking of LSOAs with tied scores? If we have 2 (or more) LSOAs with the same overall score (when expressed to 3dp), perhaps residual ordering from an original table of LSOAs might be retained in the production of overall ranks. Whereas ranking based on our more precise (more than 3dp) calculated values might break the apparent ties in a different way.

Again, however, on inspection there seems to be no reason to think that an initial ordering of LSOAs can explain the discrepancies.

It might be possible to hack our calculated values so that they more closely instantiate the published ranks

Although we believe our calculation function is correctly implementing the published method, we may be able to

Through some semi-automated trial and error, we find that a bi-directional adjustment value of 0.000167 can be applied programmatically. This correction reduces the number of LSOAs with a discrepancy in rank compared to their official published ranks.

fix_calculated_data <- function(calculated_data, imd_lookup, adj = 0.000167) {
  common_cols <- c("lsoa21cd", "lsoa21nm", "lad24cd", "lad24nm")
  calculated_data |>
    dplyr::rename_with(\(x) sub("^imd", "calc_imd", x)) |>
    dplyr::left_join(imd_lookup, common_cols) |>
    dplyr::mutate(dplyr::across("calc_imd_score", \(x) {
      dplyr::case_when(
        # if our calculated rank is too low, that means our score was too high
        .data[["calc_imd_rank"]] < .data[["imd_rank"]] ~ x - adj,
        # if our calculated rank is too high, that means our score was too low
        .data[["calc_imd_rank"]] > .data[["imd_rank"]] ~ x + adj,
        .default = x
      )
    })) |>
    dplyr::select(!tidyselect::starts_with("imd")) |>
    dplyr::arrange(dplyr::desc(.data[["calc_imd_score"]])) |>
    dplyr::mutate(
      calc_imd_rank = dplyr::row_number(),
      calc_imd_decile = as.factor(dplyr::ntile(n = 10))
    ) |>
    dplyr::rename_with(\(x) sub("^calc_imd", "imd", x))
}

fixed_calculated_data <- fix_calculated_data(calculated_data, imd_lookup)

sum(imd_lookup[["lsoa21cd"]] != fixed_calculated_data[["lsoa21cd"]])

[1] 330

After applying our illicit adjustment hack, we can now see that just 330 LSOAs have a different rank to what is expected.

rounded_data <- fixed_calculated_data |>
  dplyr::rename_with(\(x) sub("^imd", "calc_imd", x)) |>
  dplyr::mutate(dplyr::across("calc_imd_score", \(x) round(x, 3))) |>
  dplyr::left_join(imd_lookup, common_cols) |>
  dplyr::filter(imd_rank != calc_imd_rank) |>
  dplyr::filter(imd_score != calc_imd_score)


# rounded_data |>
#   dplyr::mutate(dplyr::across(tidyselect::ends_with("imd_score"), as.character)) |>
#   View()

And of those still with incorrect ranks, there are now 39 rows with differences between the calculated scores (when rounded to 3dp) and the published scores.

Conclusions

1. Scores calculated by calculate_imd_score() are in general close enough (within 0.001) of the published overall score to validate that the calculation process in the function is correctly implemented.
2. If the aim is to reproduce the published overall LSOA scores and ranks, the functions here do not succeed in completely accurately generating the expected scores. A correcting function, fix_calculated_data(), which has no theoretical basis, may be applied to the calculated data. This function is currently only found in the README file, not within the package namespace. This function reduces the error rate in calculated scores from approximately 10% to approximately 1%. Most importantly, the correcting function reduces the magnitude of the discrepancies so that in this context the number of LSOAs assigned to the wrong decile was reduced from 15 to zero. However this function relies on the published table of ranks as the basis for comparison in the application of the adjustment quantity, and is therefore only useful when trying to replicate the overall scores and ranks. In which case one might as well just use the original data! If calculating scores using a custom domain selection or custom weights, fix_calculated_data() is not applicable, because we have no authoritative pre-calculated table to compare calculated results against. In these cases, the user should just trust that calculate_imd_score() will provide “as good as can be expected” results in what is after all a process (production of the IMD) that inevitably contains a number of assumptions and approximations.
3. It remains unclear why the ~10% discrepancy rate between published and calculated overall scores exists. It seems clear that the default weightings stored in imd_weights() are accurate. If IMD overall scores (and/or transformed scores) were available with a precision greater than 3 decimal places, we might be able to investigate in a more detailed way where the discrepancies are being introduced, and how.

Reporting problems

Please use GitHub Issues to report any problems or make suggestions for further development.