Achievable means and SDs behind a reported pair

grimmer_values() returns every pair of a mean and an SD that a sample of n integers can have and that would be reported as x and sd. It is the counterpart of grim_values() for grimmer(), which is TRUE exactly if there is at least one such pair.

grimmer_closest() returns the single achievable pair whose SD is closest to sd, like grim_closest() for means. If x and sd are consistent, it is one of the pairs that grimmer_values() returns. Otherwise, it is closest among the pairs whose mean would be reported as x, or, if there are none, whose mean is the one grim_closest() returns.

Each pair comes with the whole-number sum and sum of squares behind it. Statistics computed from several groups, such as a t-value, can be compared exactly through these integers.

Usage

grimmer_values(
  x,
  sd,
  n,
  digits_x,
  digits_sd,
  items_averaged,
  data_type,
  sd_meaning = c("sd_sample", "sd_pop", "sd_sample_or_pop", "se"),
  rounding = "up_or_down",
  threshold = 5,
  symmetric = FALSE
)

grimmer_closest(
  x,
  sd,
  n,
  digits_x,
  digits_sd,
  items_averaged,
  data_type,
  sd_meaning = c("sd_sample", "sd_pop", "sd_sample_or_pop", "se"),
  rounding = "up_or_down",
  threshold = 5,
  symmetric = FALSE
)

Arguments

x: Numeric. The reported mean.
sd: Numeric. The reported standard deviation.
n: Integer. The reported sample size.
digits_x: Integer. The number of decimal places in x, including trailing zeros. There is no default, since a numeric value does not keep them (1.40 is 1.4).
digits_sd: Integer. The number of decimal places in sd, including trailing zeros. There is no default, since a numeric value does not keep them (1.40 is 1.4).
items_averaged: Integer. As in grim(), for x and sd.
data_type: String. "integer" if the values are whole numbers, as in grimmer(). There is no default, so that every call states this assumption. "continuous" is an error: such data can have any value that would be reported as the one given, so there are none to list. Under NA, a value set is undecidable unless no data can have it.
sd_meaning: String. As in grimmer(), but "sd_sample_or_pop" is not allowed: the same sum of squares gives a different SD under each premise, so call grimmer_values() once for each. Under "se", the sd column holds the standard error.
rounding: String. The rounding method assumed to have produced x. Default is "up_or_down", which accepts either direction; see Defaults in grim(). The others are "up_from_or_down_from", "ceiling_or_floor", "up", "down", "even", "up_from", "down_from", "ceiling", "floor", "trunc", "anti_trunc", "ties_up", "ties_down", "ties_away", "ties_zero", and "ties_even". See vignette("rounding-options", package = "roundwork").
threshold: Numeric. For rounding = "up_from", "down_from", or "up_from_or_down_from", the point within a step at which rounding switches direction, in tenths of a step. Default is 5. Ignored otherwise.
symmetric: Logical. If TRUE, negative numbers are rounded like their absolute values (-2.5 like 2.5) under "up", "down", and the "*_from" methods. Default is FALSE. Must not be TRUE with a "ties_*" rounding method, which already fixes the direction; see roundwork::reround().

Value

A tibble with one row per achievable pair. case is the index of the value set, as in grim_values(). x and sd are the achievable values, n the reported sample size, items_averaged as given, and sum and sum_squares the whole numbers behind them. An inconsistent value set has no rows, and one that cannot be decided has one row of NA but for n and items_averaged, as in grimmer().

grimmer_closest() returns the same columns with exactly one row per value set: of NA but for n and items_averaged if a value is missing or the value set cannot be decided.

Details

The sums are the whole numbers that grim_values() divides by n * items_averaged. For each, the sums of squares are the whole numbers within the rounding bounds of sd that n integers with that sum can have. With items_averaged > 1, the integers are the respondents' totals across the items, so the sums are those of the totals.

The number of pairs grows with n, items_averaged, and sd, and shrinks with digits_x and digits_sd. More than a million pairs for one value set is an error, since the output has one row for each. So is a sum of squares of 2^53 or more, about 9e15, which a double can't hold exactly; grimmer() tests such value sets all the same. grimmer_closest() stops at 2^48, about 3e14, beyond which it can miss the closest pair. With n from 2 to 4, both stop at 2^40, about 1.1e12, as grimmer() does; grimmer_closest() stops there even if x fails GRIM.

Under rounding = "even", the pairs include ties that base::round() may round away from x or sd, as in grim_values().

In grimmer_closest(), pairs whose SDs are equally close go by the mean closer to x, then by the smaller sum, then by the smaller SD. A negative sd is closest to the least SD there is. The number of pairs doesn't matter there, but that of the sums does: for a consistent value set, it tries one sum per remainder modulo n, and more than 100,000 of them is an error. That takes an n of more than about 10^(digits_x + 5).

Examples

# A mean of 5.23 and an SD of 2.55 are consistent with a sample size of 31,
# and there is exactly one way to get there:
grimmer_values(
  x = 5.23,
  sd = 2.55,
  n = 31,
  digits_x = 2,
  digits_sd = 2,
  items_averaged = 1,
  data_type = "integer"
)
#> # A tibble: 1 × 7
#>    case     x    sd     n items_averaged   sum sum_squares
#>   <int> <dbl> <dbl> <int>          <int> <dbl>       <dbl>
#> 1     1  5.23  2.55    31              1   162        1042

# With one decimal place, several pairs round to the same values:
grimmer_values(
  x = 5.3,
  sd = 0.7,
  n = 30,
  digits_x = 1,
  digits_sd = 1,
  items_averaged = 1,
  data_type = "integer"
)
#> # A tibble: 7 × 7
#>    case     x    sd     n items_averaged   sum sum_squares
#>   <int> <dbl> <dbl> <int>          <int> <dbl>       <dbl>
#> 1     1  5.27 0.691    30              1   158         846
#> 2     1  5.27 0.740    30              1   158         848
#> 3     1  5.3  0.651    30              1   159         855
#> 4     1  5.3  0.702    30              1   159         857
#> 5     1  5.3  0.750    30              1   159         859
#> 6     1  5.33 0.661    30              1   160         866
#> 7     1  5.33 0.711    30              1   160         868

# None at all if the value set is inconsistent:
grimmer_values(
  x = 5.23,
  sd = 2.55,
  n = 35,
  digits_x = 2,
  digits_sd = 2,
  items_averaged = 1,
  data_type = "integer"
)
#> # A tibble: 0 × 7
#> # ℹ 7 variables: case <int>, x <dbl>, sd <dbl>, n <int>, items_averaged <int>,
#> #   sum <dbl>, sum_squares <dbl>

# ...but there is a pair close to it:
grimmer_closest(
  x = 5.23,
  sd = 2.55,
  n = 35,
  digits_x = 2,
  digits_sd = 2,
  items_averaged = 1,
  data_type = "integer"
)
#> # A tibble: 1 × 7
#>    case     x    sd     n items_averaged   sum sum_squares
#>   <int> <dbl> <dbl> <int>          <int> <dbl>       <dbl>
#> 1     1  5.23  2.54    35              1   183        1177

Usage

Arguments

Value

Details

See also

Examples