Granularity-related inconsistency of means mapped to error repeats, or GRIMMER, tests whether a reported mean and standard deviation (SD) of integer data are possible given the reported sample size (Anaya 2016; Allard 2018). It extends GRIM (Brown and Heathers 2017): the mean must pass GRIM, and the SD must be achievable by a sample of integers whose sum is one of those that GRIM allows for the mean. scrutiny implements Allard’s Analytic-GRIMMER algorithm.
The GRIMMER functions work like their GRIM counterparts, so this
vignette follows vignette("grim") and is shorter where the
two tests do not differ. It covers the basics;
vignette("grimmer-in-depth") goes further.
Basic GRIMMER testing
Single cases: grimmer()
To test whether a mean of 7.3 and an SD of 2.51 are consistent with a sample size of 12:
grimmer(x = 7.3, sd = 2.51, n = 12, digits_x = 1, digits_sd = 2, items_averaged = 1, data_type = "integer")
#> [1] FALSEdigits_x and digits_sd are the numbers of
decimal places of the reported mean and SD. As in grim(),
they must be given because numeric values lose trailing zeros. So must
items_averaged and data_type; see
vignette("grim").
The SD is taken to be a sample SD, which divides by
(sd_meaning = "sd_sample"). That is what nearly all
software reports by default: R, SPSS, Stata, SAS, Excel, and pandas.
NumPy’s np.std() is the main exception and reports the
population SD, which divides by
.
If the data may have been analyzed with it, set
sd_meaning = "sd_pop"; see
vignette("grimmer-in-depth").
Many cases: grimmer_map()
Enter the reported values into a data frame and call
grimmer_map() on it:
flying_pigs1 <- tibble::tribble(
~x , ~sd , ~n ,
8.9 , 2.81 , 25 ,
2.6 , 2.05 , 25 ,
7.2 , 2.89 , 25 ,
3.6 , 3.11 , 25 ,
9.2 , 7.13 , 25 ,
10.4 , 2.53 , 25 ,
7.3 , 3.14 , 25
)
grimmer_map(flying_pigs1, digits_x = 1, digits_sd = 2, items_averaged = 1, data_type = "integer")
#> # A tibble: 7 × 9
#> # GRIMMER: default settings
#> x sd n items_averaged data_type digits_x digits_sd consistency
#> <dbl> <dbl> <int> <int> <chr> <int> <int> <lgl>
#> 1 8.9 2.81 25 1 integer 1 2 TRUE
#> 2 2.6 2.05 25 1 integer 1 2 FALSE
#> 3 7.2 2.89 25 1 integer 1 2 TRUE
#> 4 3.6 3.11 25 1 integer 1 2 TRUE
#> 5 9.2 7.13 25 1 integer 1 2 TRUE
#> 6 10.4 2.53 25 1 integer 1 2 TRUE
#> 7 7.3 3.14 25 1 integer 1 2 TRUE
#> # ℹ 1 more variable: reason <chr>consistency is the test result, and reason
states which part of the test an inconsistent value set failed. A value
set passes GRIMMER if it passes GRIM and the three tests of Allard (2018):
- The reconstructed sum of squared observations must be an integer, and one that integers with the reconstructed sum can reach. scrutiny adds this second part to Allard’s test: for example, no 11 integers with a sum of 40 have a sum of squares below 148, which is that of seven 4s and four 3s. For fewer than five values, some larger sums of squares are impossible, too.
- The reconstructed SD must match the reported one. Test 1 implies this, so it never appears as a reason.
- The sum of squares must have the same parity (odd or even) as the sum.
Consistent value sets are marked "Passed all".
Scale items
For a multi-item scale, give the number of items as an
items_averaged argument or column, as in
grim_map(). Taken as single items, half of these value sets
from a three-item scale are wrongly flagged:
flying_pigs2 <- tibble::tribble(
~x , ~sd , ~n ,
5.90 , 2.19 , 40 ,
5.71 , 1.42 , 40 ,
3.50 , 1.81 , 40 ,
3.82 , 2.43 , 40 ,
4.61 , 1.92 , 40 ,
5.24 , 2.51 , 40
)
flying_pigs2 |>
grimmer_map(digits_x = 2, digits_sd = 2, items_averaged = 1, data_type = "integer")
#> # A tibble: 6 × 9
#> # GRIMMER: default settings
#> x sd n items_averaged data_type digits_x digits_sd consistency
#> <dbl> <dbl> <int> <int> <chr> <int> <int> <lgl>
#> 1 5.9 2.19 40 1 integer 2 2 TRUE
#> 2 5.71 1.42 40 1 integer 2 2 FALSE
#> 3 3.5 1.81 40 1 integer 2 2 TRUE
#> 4 3.82 2.43 40 1 integer 2 2 TRUE
#> 5 4.61 1.92 40 1 integer 2 2 FALSE
#> 6 5.24 2.51 40 1 integer 2 2 FALSE
#> # ℹ 1 more variable: reason <chr>
flying_pigs2 |>
grimmer_map(digits_x = 2, digits_sd = 2, items_averaged = 3, data_type = "integer")
#> # A tibble: 6 × 9
#> # GRIMMER: default settings
#> x sd n items_averaged data_type digits_x digits_sd consistency
#> <dbl> <dbl> <int> <int> <chr> <int> <int> <lgl>
#> 1 5.9 2.19 40 3 integer 2 2 TRUE
#> 2 5.71 1.42 40 3 integer 2 2 TRUE
#> 3 3.5 1.81 40 3 integer 2 2 TRUE
#> 4 3.82 2.43 40 3 integer 2 2 TRUE
#> 5 4.61 1.92 40 3 integer 2 2 TRUE
#> 6 5.24 2.51 40 3 integer 2 2 TRUE
#> # ℹ 1 more variable: reason <chr>Explaining consistency results
explain_print() explains the results of
grimmer_map() step by step, as for GRIM. Here is the
inconsistent value set from above:
flying_pigs1 |>
grimmer_map(digits_x = 1, digits_sd = 2, items_averaged = 1, data_type = "integer") |>
dplyr::filter(!consistency) |>
explain_print()
#> GRIMMER test: x = 2.6, sd = 2.05, n = 25, items_averaged = 1, data_type = "integer"
#>
#> The mean was reported as 2.6, with 1 decimal place. Before rounding (assuming
#> either rounding up from 5 or down from 5), it could have been anything from
#> 2.55 to 2.65.
#>
#> The mean of 25 integers is their sum divided by 25. So the sum must be from
#> 2.55 * 25 = 63.75 to 2.65 * 25 = 66.25.
#>
#> A sum of integers is itself an integer, and the integers from 64 to 66 are in
#> this range. For example, 65 / 25 = 2.6, which rounds to 2.6.
#>
#> The SD was reported as 2.05, with 2 decimal places. Before rounding, it could
#> have been anything from 2.045 to 2.055.
#>
#> Together, the sum and the SD determine the sum of squares, which is what you
#> get by squaring each of the 25 values and adding them up. It is
#> (25 - 1) * SD^2 + sum^2 / 25. The values are integers, so their squares are
#> integers, too, and so is the sum of squares. What's more, an integer and its
#> square are either both odd or both even. So the sum of squares must be odd if
#> the sum is odd, and even if the sum is even.
#>
#> With a sum of 64, the sum of squares is from
#> 24 * 2.045^2 + 64^2 / 25 = 264.2086 to 24 * 2.055^2 + 64^2 / 25 = 265.1926. The
#> only integer in this range is 265, which is odd, but 64 is even.
#>
#> With a sum of 65, the sum of squares is from
#> 24 * 2.045^2 + 65^2 / 25 = 269.3686 to 24 * 2.055^2 + 65^2 / 25 = 270.3526. The
#> only integer in this range is 270, which is even, but 65 is odd.
#>
#> With a sum of 66, the sum of squares is from
#> 24 * 2.045^2 + 66^2 / 25 = 274.6086 to 24 * 2.055^2 + 66^2 / 25 = 275.5926. The
#> only integer in this range is 275, which is odd, but 66 is even.
#>
#> ==> Inconsistentexplain_map() adds the explanations as an
explanation column instead, and
grimmer_explain() explains a single value set. To see the
test rather than read it, draw one value set with
grimmer_plot_line(); see
vignette("grimmer-in-depth").
Summaries with audit()
flying_pigs1 |>
grimmer_map(digits_x = 1, digits_sd = 2, items_averaged = 1, data_type = "integer") |>
audit()
#> # A tibble: 1 × 6
#> all_cases incon_cases emp_incon fail_grim fail_test1 fail_test3
#> <int> <int> <dbl> <int> <int> <int>
#> 1 7 1 0.1429 0 0 1-
all_cases,incon_cases,emp_incon: the total number of value sets, the number of inconsistent ones, and their ratio. -
fail_grim,fail_test1,fail_test3: the number of value sets that failed GRIM or GRIMMER’s first or third test.
Interpreting results
An inconsistency means that an assumption of the test is false or that a number was reported incorrectly, often for a benign reason. What to rule out before drawing conclusions is discussed in the section on interpreting GRIM results of Hussey (2026). The same reasoning applies to GRIMMER.
Going further
vignette("grimmer-in-depth") covers SDs that are
population SDs or standard errors, bounded scales, the means and SDs
behind a test result with grimmer_values() and
grimmer_plot_line(), plots of many value sets with
grim_plot_grid(), values around the reported ones with
grimmer_map_seq(), and unknown group sizes with
grimmer_map_total_n().