Purpose

The package should be useful for teaching in the Social Sciences, and for scholars who wish to “replicate” rating-scale data for further analysis and visualisation when only summary statistics have been reported.

I was prompted to write the functions in LikertMakeR after reviewing too many journal article submissions where authors presented questionnaire results with only means and standard deviations (often only the means), with no understanding of the real distributions. Hopefully, this tool will help researchers, teachers, and other reviewers, to better think about rating-scale distributions, and the effects of variance, boundaries and number of items in a scale.

Rating scale properties

A Likert scale is the mean, or sum, of several ordinal rating scales. They are bipolar (usually “agree-disagree”) responses to propositions that are determined to be moderately-to-highly correlated and capturing various facets of a construct.

Rating scales, such as Likert scales, are not continuous or unbounded.

For example, a 5-point Likert scale that is constructed with, say, five items (questions) will have a summed range of between 5 (all rated ‘1’) and 25 (all rated ‘5’) with all integers in between, and the mean range will be ‘1’ to ‘5’ with intervals of 1/5=0.20. A 7-point Likert scale constructed from eight items will have a summed range between 8 (all rated ‘1’) and 56 (all rated ‘7’) with all integers in between, and the mean range will be ‘1’ to ‘7’ with intervals of 1/8=0.125.

Rating-scale boundaries define minima and maxima for any scale values. If the mean is close to one boundary then data points will gather more closely to that boundary and the data will always be skewed.

Alternative methods & packages

LikertMakeR is intended for synthesising & correlating rating-scale data with means, standard deviations, and correlations as close as possible to predefined parameters. If you don’t need your data to be close to exact, then other options may be faster or more flexible.

Different approaches include:

• sampling from a truncated normal distribution. Data are sampled from a normal distribution, and then truncated to suit the rating-scale boundaries, and rounded to set discrete values as we see in rating scales. See Heiz (2021) for an excellent and short example using the following packages:

  • truncnorm

  • faux

  • See also responsesR, by Marko Lalovic on GitHub, for an approach using optimal discretization and skew-normal distribution.

• sampling with a predetermined probability distribution

  • for example, the following code will generate a vector of values with approximately the given probabilities.

     
     n <- 128
     sample(1:5, n, replace = TRUE,
       prob = c(0.1, 0.2, 0.4, 0.2, 0.1)
     )

• Marginal model specification as in Touloumis (2016) and Grønneberg et al. (2022) using:

  • SimCorMultRes

  • covsim

Generate synthetic rating-scale data

To synthesise a rating scale with LikertMakeR, the user must input the following parameters:

• n: sample size

• mean: desired mean

• sd: desired standard deviation

• lowerbound: desired lower bound

• upperbound: desired upper bound

• items: number of items making the scale - default = 1

• seed: optional seed for reproducibility

LikertMakeR offers two different functions for synthesising a rating scale: lfast() and lexact()

## a five-item, seven-point Likert scale

x <- lfast(
  n = 512,
  mean = 4.0,
  sd = 1.0,
  lowerbound = 1,
  upperbound = 7,
  items = 5
)

## an 11-point likelihood-of-purchase scale

x <- lfast(256, 2, 2, 0, 10, seed = 42)

x <- lexact(
  n = 64,
  mean = 5.0,
  sd = 1.0,
  lowerbound = 1,
  upperbound = 7,
  items = 5
)
#> 
#> ***** summary of DEoptim object ***** 
#> best member   :  28 21 20 23 30 21 26 22 27 21 26 27 26 22 28 34 26 27 30 27 22 33 21 28 23 27 29 24 15 29 33 25 20 25 17 33 21 13 31 19 21 26 35 31 24 27 32 25 17 22 28 29 34 26 25 25 19 31 24 24 18 18 21 18 
#> best value    :  0.02519 
#> after         :  134 generations 
#> fn evaluated  :  86400 times 
#> *************************************

x <- lexact(64, 2, 1.8, 0, 10, seed = 42)
#> 
#> ***** summary of DEoptim object ***** 
#> best member   :  3 1 0 5 1 1 0 1 4 2 0 2 2 4 1 2 2 6 0 5 3 4 4 1 4 1 0 4 1 0 2 2 2 8 1 1 4 0 2 2 0 2 3 2 1 3 2 0 1 3 0 1 0 1 4 0 7 0 1 2 1 1 3 2 
#> best value    :  0.87831 
#> after         :  110 generations 
#> fn evaluated  :  71040 times 
#> *************************************

x <- lexact(
  n = 64,
  mean = 1.2,
  sd = 1.00,
  lowerbound = -3,
  upperbound = 3,
  items = 4
)
#> 
#> ***** summary of DEoptim object ***** 
#> best member   :  10 8 0 8 12 4 7 7 3 7 6 9 1 1 6 6 2 12 0 8 2 7 7 7 9 -2 2 5 8 7 4 8 -3 -3 8 6 7 6 4 -1 8 8 2 2 4 11 -2 8 -5 5 1 5 8 4 7 10 5 9 0 1 10 0 2 -1 
#> best value    :  0.60951 
#> after         :  93 generations 
#> fn evaluated  :  60160 times 
#> *************************************

Correlating vectors of synthetic rating scales

LikertMakeR offers another function, lcor(), which rearranges the values in the columns of a data-set so that they are correlated at a specified level. It does not change the values - it swaps their positions within each column so that univariate statistics do not change, but their correlations with other vectors do.

lcor() systematically selects pairs of values in a column and swaps their places, and checks to see if this swap improves the correlation matrix. If the revised data-frame produces a correlation matrix closer to the target correlation matrix, then the swap is retained. Otherwise, the values are returned to their original places. This process is iterated across each column.

To create the desired correlated data, the user must define the following data-frames:

• data: a starter data set of rating-scales

• target: the target correlation matrix

n <- 32

# set.seed(42)

x1 <- lexact(n, 2.5, 0.75, 1, 5, 5)
#> 
#> ***** summary of DEoptim object ***** 
#> best member   :  12 8 13 9 10 14 9 13 13 8 12 14 6 22 15 11 12 10 8 15 20 14 14 19 16 14 16 11 12 6 14 10 
#> best value    :  0.69599 
#> after         :  55 generations 
#> fn evaluated  :  17920 times 
#> *************************************
x2 <- lexact(n, 3.0, 1.50, 1, 5, 5)
#> 
#> ***** summary of DEoptim object ***** 
#> best member   :  24 21 11 24 22 11 23 5 24 14 8 25 24 7 20 6 9 23 6 8 15 19 5 12 6 5 24 22 7 24 14 12 
#> best value    :  0.23398 
#> after         :  45 generations 
#> fn evaluated  :  14720 times 
#> *************************************
x3 <- lexact(n, 3.5, 1.00, 1, 5, 5)
#> 
#> ***** summary of DEoptim object ***** 
#> best member   :  21 22 17 21 20 16 11 22 13 20 15 21 15 22 21 6 22 18 16 12 24 11 21 14 16 22 20 11 6 25 22 17 
#> best value    :  0.10413 
#> after         :  5 generations 
#> fn evaluated  :  1920 times 
#> *************************************

mydat3 <- cbind(x1, x2, x3) |> data.frame()

#>        x1  x2  x3
#> par1  2.4 4.8 4.2
#> par2  1.6 4.2 4.4
#> par3  2.6 2.2 3.4
#> par4  1.8 4.8 4.2
#> par5  2.0 4.4 4.0
#> par6  2.8 2.2 3.2
#> par7  1.8 4.6 2.2
#> par8  2.6 1.0 4.4
#> par9  2.6 4.8 2.6
#> par10 1.6 2.8 4.0

#>  x1  x2  x3 
#> 2.5 3.0 3.5

#>    x1    x2    x3 
#> 0.748 1.501 0.999

#>       x1    x2   x3
#> x1  1.00 -0.34 0.10
#> x2 -0.34  1.00 0.15
#> x3  0.10  0.15 1.00

## describe a target correlation matrix

tgt3 <- matrix(
  c(
    1.00, 0.80, 0.75,
    0.80, 1.00, 0.90,
    0.75, 0.90, 1.00
  ),
  nrow = 3
)

## apply lcor function

new3 <- lcor(mydat3, tgt3)

#>      x1  x2   x3
#> x1 1.00 0.8 0.75
#> x2 0.80 1.0 0.90
#> x3 0.75 0.9 1.00

#>  x1  x2  x3 
#> 2.5 3.0 3.5

#>    x1    x2    x3 
#> 0.748 1.501 0.999

n <- 128

x1 <- lfast(n, 2.50, 1.50, 1, 7, 1)
x2 <- lfast(n, 4.50, 1.00, 1, 7, 1)
x3 <- lfast(n, 4.50, 1.50, 1, 7, 1)
x4 <- lfast(n, 5.25, 0.70, 1, 7, 1)
x5 <- lfast(n, 5.25, 1.25, 1, 7, 1)

mydat5 <- cbind(x1, x2, x3, x4, x5) |> data.frame()

#>    x1    x2    x3    x4    x5 
#> 2.406 4.469 4.469 5.258 5.258

#>    x1    x2    x3    x4    x5 
#> 1.498 1.011 1.474 0.713 1.365

#>       x1    x2    x3    x4    x5
#> x1  1.00  0.11 -0.03  0.12 -0.02
#> x2  0.11  1.00 -0.07 -0.02  0.09
#> x3 -0.03 -0.07  1.00  0.02 -0.04
#> x4  0.12 -0.02  0.02  1.00  0.15
#> x5 -0.02  0.09 -0.04  0.15  1.00

## describe a target correlation matrix

tgt5 <- matrix(
  c(
    1.00, 0.80, 0.75, 0.25, -0.75,
    0.80, 1.00, 0.70, 0.25, -0.70,
    0.75, 0.70, 1.00, 0.25, -0.50,
    0.25, 0.25, 0.25, 1.00, -0.40,
    -0.75, -0.70, -0.50, -0.40, 1.00
  ),
  nrow = 5
)

## apply lcor function

new5 <- lcor(mydat5, tgt5)

#>       x1    x2    x3    x4    x5
#> x1  1.00  0.80  0.73  0.25 -0.62
#> x2  0.80  1.00  0.70  0.25 -0.70
#> x3  0.73  0.70  1.00  0.25 -0.50
#> x4  0.25  0.25  0.25  1.00 -0.40
#> x5 -0.62 -0.70 -0.50 -0.40  1.00