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README

(V 1.1.0 May 2025)

Synthesise and correlate Likert scales, and similar rating-scale data, with predefined first & second moments (mean and standard deviation), Cronbach’s Alpha, Factor Loadings, and other summary statistics.

LikertMakeR synthesises rating-scale data. Such scales are constrained by upper and lower bounds and discrete increments.

The package is intended for:

1. “reproducing” or “reverse-engineering” rating-scale data for further analysis and visualisation when only summary statistics have been reported,

2. teaching. Helping researchers and students to better understand the relationships among scale properties, sample size, number of items, etc. …

3. checking the feasibility of scale moments with given scale and correlation properties.

Functions in this version of LikertMakeR are:

• lfast() applies a simple Evolutionary Algorithm, based on repeated random samples from a scaled Beta distribution, to approximate predefined first and second moments.

• lcor() rearranges the values in the columns of a dataframe so that they are correlated to match a predefined correlation matrix.

• makeCorrAlpha constructs a random item correlation matrix of given dimensions and predefined Cronbach’s Alpha.

• makeItems() is a wrapper function for lfast() and lcor() to generate synthetic rating-scale data with predefined first and second moments and a predefined correlation matrix.

• makeCorrLoadings constructs a item correlation matrix based on factor loadings and factor correlations as might be reported in Exploratory Factor Analysis (EFA) or Structural Equation Modelling (SEM).

• makeItemsScale() Generate a dataframe of rating scale items from a summative scale and desired Cronbach’s Alpha.

• makePaired() Generate a dataset from paired-sample t-test summary statistics.

• correlateScales() generates a multidimensional dataframe by combining several dataframes of rating-scale items so that their summated scales are correlated according to a predefined correlation matrix.

• alpha() calculates Cronbach’s Alpha from a given correlation matrix or a given dataframe

• eigenvalues() calculates eigenvalues of a correlation matrix, reports on positive-definite status of the matrix and, optionally, displays a scree plot to visualise the eigenvalues

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 among each other, and capturing various facets of a theoretical construct.

Summated rating 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.

Technically, because Likert scales, and similar rating scales are bounded with discrete intervals, parametric statistics (such as mean, standard deviation, and correlation) should not be applied to summated rating scales. In practice, however, such parametric statistics are commonly used in the social sciences because:

1. they are in common usage and easily understood,

2. results and conclusions drawn from technically-correct non-parametric statistics are (almost) always the same as for parametric statistics for such data.
   D’Alessandro et al. (2020) argue that a summated scale, made with multiple items, “approaches” an interval scale measure.

Likert-scale items, such as responses to a single 1-to-5 agree-disagree question, should not be analysed by professional or responsible researchers. There is too much random error in a single item. Rensis Likert (1932) designed the scale with the logic that a random overstatement on one item is likely to be compensated by a random understatement on another item, so that, when multiple items are combined, we get a reasonably consistent, internally reliable, measure of the target construct.

Typically, a researcher will synthesise rating-scale data by sampling with a predetermined probability distribution.
For example, the following code will generate a vector of values for a single Likert-scale item, 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)
      )

This approach is good for testing Likert items but it does not help when working on complete Likert scales, or for when we want to specify means and standard deviations as they might be reported in published research.

The function lfast() allows the user to specify exact univariate statistics as they might ordinarily be reported. lcor() will take multiple scales created with lfast() and rearrange values so that the vectors are correlated.

makeCorrAlpha() generates a correlation matrix from a predefined Cronbach’s Alpha(), enabling the user to apply makeItems() to generate scale items that produce an exact Cronbach’s Alpha. makeCorrLoadings() generates a correlation matrix from factor loadings data, enabling the user to apply makeItems() to generate multidimensional data.

makeItems() will generate synthetic rating-scale data with predefined first and second moments and a predefined correlation matrix.

makeItemsScale() generate a dataframe of rating scale items from a summative scale and desired Cronbach’s Alpha. correlateScales() generates a multidimensional dataframe by combining several dataframes of rating-scale items so that their summated scales are correlated according to a predefined correlation matrix.

To download and install the package, run the following code from your R console.

From CRAN:

 install.packages('LikertMakeR')
 

The latest development version is available from the author’s GitHub repository.

 library(devtools)
 install_github("WinzarH/LikertMakeR")
 
 

• lfast() generates a vector of synthetic values with predefined first and second moments. It should be accurate to two decimal places.

lfast(n, mean, sd, lowerbound, upperbound, items = 1, precision = 0)

• n: sample size

• mean: desired mean

• sd: desired standard deviation

• lowerbound: desired lower bound (e.g. ‘1’ for a 1-5 rating scale)

• upperbound: desired upper bound (e.g. ‘5’ for a 1-5 rating scale)

• items: number of items making the scale. Default = ‘1’

• precision: can relax the level of accuracy of moments. Default = ‘0’

 x <- lfast(
   n = 128, 
   mean = 4.5, 
   sd = 1.0, 
   lowerbound = 1, 
   upperbound = 7, 
   items = 5
   )

 x <- lfast(
   n = 128, 
   mean = 1.0, 
   sd = 1.0, 
   lowerbound = -3, 
   upperbound = 3, 
   items = 4
   )

 x <- lfast(
   n = 256, 
   mean = 3.25, 
   sd = 1.0, 
   lowerbound = 1, 
   upperbound = 5, 
   items = 5,
   precision = 4
   )
  

 x <- lfast(256, 2.5, 2.5, 0, 10)
 

The function, lcor(), applies a simple evolutionary algorithm to rearrange the values in the columns of a data set so that they are correlated at a specified level. lcor() does not change the values - it swaps their positions in each column so that univariate statistics do not change, but their correlations with other columns do.

  lcor(data, target)

• data: a starter data set of rating-scales

• target: the target correlation matrix

  n <- 64
 x1 <- lfast(n, 3.5, 1.00, 1, 5, 5) 
 x2 <- lfast(n, 1.5, 0.75, 1, 5, 5) 
 x3 <- lfast(n, 3.0, 1.70, 1, 5, 5) 
 x4 <- lfast(n, 2.5, 1.50, 1, 5, 5)   
 
 mydat4 <- data.frame(x1, x2, x3, x4) 
 
 head(mydat4)
 cor(mydat4) |> round(3)
 

 tgt4 <- matrix(
 c(
   1.00, 0.55, 0.60, 0.75,
   0.55, 1.00, 0.25, 0.65,
   0.60, 0.25, 1.00, 0.80,
   0.75, 0.65, 0.80, 1.00
 ),
 nrow = 4
 )
 

 new4 <- lcor(data = mydat4, target = tgt4)
 
 cor(new4) |> round(3)

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

 tgt3 <- matrix(
   c(
      1.00, -0.50, -0.85,
     -0.50,  1.00,  0.60,
     -0.85,  0.60,  1.00
   ),
   nrow = 3
 )
 

 new3 <- lcor(mydat3, tgt3) 
 
 cor(new3) |> round(3)

makeCorrAlpha(), constructs a random correlation matrix of given dimensions and predefined Cronbach’s Alpha.

  makeCorrAlpha(items, alpha, variance = 0.5, precision = 0)

• items: ‘k’, dimensions (number of rows & columns) of the desired correlation matrix

• alpha: target Cronbach’s Alpha (usually positive, must be greater than ‘-1’ and less than ‘+1’)

• variance: standard deviation of values sampled from a normally-distributed log transformation. Default = ‘0.5’. A value of ‘0’ makes all values in the correlation matrix the same, equal to the mean correlation needed to produce the desired Alpha. A value of ‘2’, or more, risks producing a matrix that is not positive-definite, so not feasible.

• precision: a value between ‘0’ and ‘3’ to add some random variation around the target Cronbach’s Alpha. Default = ‘0’. A value of ‘0’ produces the desired Alpha, generally exact to two decimal places. Higher values produce increasingly random values around the desired Alpha.

Random values generated by makeCorrAlpha() are volatile. makeCorrAlpha() may not generate a feasible (positive-definite) correlation matrix, especially when variance is high relative to

• desired Alpha, and
• desired correlation dimensions (number of items)

makeCorrAlpha() will inform the user if the resulting correlation matrix is positive definite, or not.

If the returned correlation matrix is not positive-definite, because solutions are so volatile, a feasible solution still may be possible, and often is. The user is encouraged to try again, possibly several times, to find one.

items <- 4
alpha <- 0.85

apply makeCorrAlpha() function

cor_matrix_4 <- makeCorrAlpha(items, alpha, variance)

test output with Helper functions

alpha(cor_matrix_4)
eigenvalues(cor_matrix_4, 1)

items <- 8
alpha <- 0.95
variance <- 1.0

apply makeCorrAlpha() function

cor_matrix_8 <- makeCorrAlpha(items, alpha, variance)

test output

alpha(cor_matrix_8)
eigenvalues(cor_matrix_8, 1)

precision <- 2

apply makeCorrAlpha() function

cor_matrix_8a <- makeCorrAlpha(items, alpha, variance, precision)

test output

alpha(cor_matrix_8a)
eigenvalues(cor_matrix_8a, 1)

makeCorrLoadings() generates a correlation matrix from factor loadings and factor correlations as might be seen in Exploratory Factor Analysis (EFA) or a Structural Equation Model (SEM).

  makeCorrLoadings(loadings, factorCor = NULL, uniquenesses = NULL, nearPD = FALSE)

• loadings: ‘k’ (items) by ‘f’ (factors) matrix of standardised factor loadings. Item names and Factor names can be taken from the row_names (items) and the column_names (factors), if present.

• factorCor: ‘f’ x ‘f’ factor correlation matrix. If not present, then we assume that the factors are uncorrelated (orthogonal), which is rare in practice, and the function applies an identity matrix for factor_cor.

• uniquenesses: length ‘k’ vector of uniquenesses. If NULL, the default, compute from the calculated communalities.

• nearPD: (logical) If TRUE, then the function calls the nearPD function from the Matrix package to transform the resulting correlation matrix onto the nearest Positive Definite matrix. Obviously, this only applies if the resulting correlation matrix is not positive definite. (It should never be needed.)

“Censored” loadings (for example, where loadings less than some small value (often ‘0.30’), are removed for ease-of-communication) tend to severely reduce the accuracy of the makeCorrLoadings() function. For a detailed demonstration, see the file, makeCorrLoadings_Validate.pdf in the package website on GitHub.

Example loadings

factorLoadings <- matrix(
  c(
      0.05, 0.20, 0.70,
      0.10, 0.05, 0.80,
      0.05, 0.15, 0.85,
      0.20, 0.85, 0.15,
      0.05, 0.85, 0.10,
      0.10, 0.90, 0.05,
      0.90, 0.15, 0.05,
      0.80, 0.10, 0.10
   ),
   nrow = 8, ncol = 3, byrow = TRUE
)

row and column names

rownames(factorLoadings) <- c("Q1", "Q2", "Q3", "Q4", "Q5", "Q6", "Q7", "Q8")
colnames(factorLoadings) <- c("Factor1", "Factor2", "Factor3")

Factor correlation matrix

factorCor <- matrix(
c(
  1.0,  0.5, 0.4,
  0.5,  1.0, 0.3,
  0.4,  0.3, 1.0
 ),
nrow = 3, byrow = TRUE
)

itemCorrelations <- makeCorrLoadings(factorLoadings, factorCor)

round(itemCorrelations, 3)

itemCors <- makeCorrLoadings(factorLoadings)

round(itemCors, 3)

makeItems() generates a dataframe of random discrete values from a scaled Beta distribution so the data replicate a rating scale, and are correlated close to a predefined correlation matrix.

makeItems() is a wrapper function for:

• lfast(), which generates a vector that best fits the desired moments, and

• lcor(), which rearranges values in each column of the dataframe so they closely match the desired correlation matrix.

makeItems(n, means, sds, lowerbound, upperbound, cormatrix)

• n: number of observations to generate

• means: target means: a vector of length ‘k’ of mean values for each scale item

• sds: target standard deviations: a vector of length ‘k’ of standard deviation values for each scale item

• lowerbound: vector of length ‘k’ (same as rows & columns of correlation matrix) of values for lower bound of each scale item (e.g. ‘1’ for a 1-5 rating scale)

• upperbound: vector of length ‘k’ (same as rows & columns of correlation matrix) of values for upper bound of each scale item (e.g. ‘5’ for a 1-5 rating scale)

• cormatrix: target correlation matrix: a ‘k’ x ‘k’ square symmetric matrix of values ranging between ‘-1 ’and’+1’, and ‘1’ in the diagonal.

n <- 16
dfMeans <- c(2.5, 3.0, 3.0, 3.5)
dfSds <- c(1.0, 1.0, 1.5, 0.75)
lowerbound <- rep(1, 4)
upperbound <- rep(5, 4)

corMat <- matrix(
c(
 1.00, 0.25, 0.35, 0.40,
 0.25, 1.00, 0.70, 0.75,
 0.35, 0.70, 1.00, 0.80,
 0.40, 0.75, 0.80, 1.00
 ),
 nrow = 4, ncol = 4
)

df <- makeItems(
   n = n,
   means = dfMeans,
   sds = dfSds,
   lowerbound = lowerbound,
   upperbound = upperbound,
   cormatrix = corMat
 )

print(df)

apply(df, 2, mean) |> round(3)

apply(df, 2, sd) |> round(3)

cor(df) |> round(3)

• makeItemsScale() generates a dataframe of rating-scale items from a summated rating scale and desired Cronbach’s Alpha.

makeItemsScale(scale, lowerbound, upperbound, items, 
alpha = 0.8, variance = 0.5)

• scale: a vector or dataframe of the summated rating scale. Should range from (‘lowerbound’ * ‘items’) to (‘upperbound’ * ‘items’)

• lowerbound: lower bound of the scale item (example: ‘1’ in a ‘1’ to ‘5’ rating)

• upperbound: upper bound of the scale item (example: ‘5’ in a ‘1’ to ‘5’ rating)

• items: k, or number of columns to generate

• alpha: desired Cronbach’s Alpha. Default = ‘0.8’

• variance: quantile for selecting the combination of items that give summated scores. Must lie between ‘0’ (minimum variance) and ‘1’ (maximum variance). Default = ‘0.5’.

n <- 64
mean <- 3.5
sd <- 1.00
lowerbound <- 1
upperbound <- 5
items <- 4

meanScale <- lfast(
  n = n, mean = mean, sd = sd,
  lowerbound = lowerbound, upperbound = upperbound,
  items = items 
)

summatedScale <- meanScale * items

newItems_1 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lowerbound, 
  upperbound = upperbound,
  items = items
)

cor(newItems_1) |> round(2)
alpha(data = newItems_1)
eigenvalues(cor(newItems_1), 1)

newItems_2 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lowerbound, 
  upperbound = upperbound,
  items = items,
  alpha = 0.9
)

cor(newItems_2) |> round(2)
alpha(data = newItems_2)
eigenvalues(cor(newItems_2), 1)

newItems_3 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lowerbound, 
  upperbound = upperbound,
  items = items,
  alpha = 0.6,
  variance = 0.7
)   

cor(newItems_3) |> round(2)
alpha(data = newItems_3)
eigenvalues(cor(newItems_3), 1)

makePaired() generates a dataset from paired-sample t-test summary statistics.

makePaired() generates correlated values so the data replicate rating scales taken, for example, in a before and after experimental design. The function is effectively a wrapper function for lfast() and lcor() with the addition of a t-statistic from which the between-column correlation is inferred.

Paired t-tests apply to observations that are associated with each other. For example: the same people before and after a treatment; the same people rating two different objects; ratings by husband & wife; etc.

makePaired(n, means, sds, t_value, lowerbound, upperbound, items = 1, precision = 0)

• n sample size
• means a [1:2] vector of target means for two before/after measures
• sds a [1:2] vector of target standard deviations
• t_value desired paired t-statistic
• lowerbound lower bound (e.g. ‘1’ for a 1-5 rating scale)
• upperbound upper bound (e.g. ‘5’ for a 1-5 rating scale)
• items number of items in the rating scale. Default = 1
• precision can relax the level of accuracy required. Default = 0, which generally gives results correct within two decimal places. A value of ‘1’ generally creates a vector with moments correct within ‘0.025’; ‘2’ generally within ‘0.05’.

n <- 20
means <- c(2.5, 3.0)
sds <- c(1.0, 1.5)
lowerbound <- 1
upperbound <- 5
items <- 6
t <- -2.5

pairedDat <- makePaired(n = n, means = means, sds = sds, t_value = t, lowerbound = lowerbound, upperbound = upperbound, items = items)

str(pairedDat)
cor(pairedDat) |> round(2)
pairedMoments <- data.frame(
  mean = apply(newDat, MARGIN = 2, FUN = mean) |> round(3),
  sd = apply(newDat, MARGIN = 2, FUN = sd) |> round(3)
) |> t()
pairedMoments

t.test(pairedDat$X1, pairedDat$X2, paired = TRUE)

correlateScales() takes several dataframes of rating-scale items and rearranges their rows so that the scales are correlated according to a predefined correlation matrix. Univariate statistics for each dataframe of rating-scale items do not change, and inter-item correlations within a dataframe do not change, but their correlations with rating-scale items in other dataframes do change.

correlateScales(dataframes, scalecors)

• dataframes: a list of ‘k’ dataframes to be rearranged and combined

• scalecors: target correlation matrix - a symmetric k*k positive-semi-definite matrix, where ‘k’ is the number of dataframes

n <- 64
lower <- 1
upper <- 5

cor_1 <- makeCorrAlpha(items = 3, alpha = 0.85)
means_1 <- c(2.5, 2.5, 3.0)
sds_1 <- c(0.9, 1.0, 1.0)
Att_1 <- makeItems(
  n, means_1, sds_1,
  rep(lower, 4), rep(upper, 4),
  cor_1
)

cor_2 <- makeCorrAlpha(items = 3, alpha = 0.80)
means_2 <- c(2.5, 3.0, 3.5)
sds_2 <- c(1.0, 1.5, 1.0)
Att_2 <- makeItems(
  n, means_2, sds_2,
  rep(lower, 5), rep(upper, 5),
  cor_2
)

cor_3 <- makeCorrAlpha(items = 3, alpha = 0.75)
means_3 <- c(2.5, 3.0, 3.5)
sds_3 <- c(1.0, 1.5, 1.0)

Att_3 <- makeItems(
  n, means_3, sds_3,
  rep(lower, 6), rep(upper, 6),
  cor_3
)

scale_cors <- matrix(
  c(
    1.0, 0.6, 0.5,
    0.6, 1.0, 0.4, 
    0.5, 0.4, 1.0
  ),
  nrow = 3
)

data_frames <- list("A1" = Att_1, "A2" = Att_2, "A3" = Att_3)

my_correlated_scales <- correlateScales(
  dataframes = data_frames,
  scalecors = scale_cors
)

str(my_correlated_scales)

cor(my_correlated_scales) |> round(2)

eigenvalues(cormatrix = cor(my_correlated_scales), scree = TRUE) |> 
round(2)

likertMakeR() includes two additional functions that may be of help when examining parameters and output.

• alpha() calculates Cronbach’s Alpha from a given correlation matrix or a given dataframe

• eigenvalues() calculates eigenvalues of a correlation matrix, and reports on whether the correlation matrix is positive definite and an optional scree plot

alpha() accepts, as input, either a correlation matrix or a dataframe. If both are submitted, then the correlation matrix is used by default, with a message to that effect.

alpha(cormatrix = NULL, data = NULL)

• cormatrix: correlation matrix for examination: a square symmetrical matrix with values ranging from ‘-1’ to ‘+1’ and ‘1’ in the diagonal

• data: a data frame or data matrix

df <- data.frame(
 V1  =  c(4, 2, 4, 3, 2, 2, 2, 1),
 V2  =  c(4, 1, 3, 4, 4, 3, 2, 3),
 V3  =  c(4, 1, 3, 5, 4, 1, 4, 2),
 V4  =  c(4, 3, 4, 5, 3, 3, 3, 3)
)

corMat <- matrix(
 c(
  1.00, 0.35, 0.45, 0.70,
  0.35, 1.00, 0.60, 0.55,
  0.45, 0.60, 1.00, 0.65,
  0.70, 0.55, 0.65, 1.00
 ),
 nrow = 4, ncol = 4
)

alpha(cormatrix = corMat)

alpha(data = df)

alpha(NULL, df)

alpha(corMat, df)

eigenvalues() calculates eigenvalues of a correlation matrix, reports on whether the matrix is positive-definite, and optionally produces a scree plot.

eigenvalues(cormatrix, scree = FALSE) 

• cormatrix: a correlation matrix.

• scree: (logical) default = FALSE. If TRUE (or 1), then eigenvalues() produces a scree plot to illustrate the eigenvalues.

correlationMatrix <- matrix(
 c(
  1.00, 0.25, 0.35, 0.40,
  0.25, 1.00, 0.70, 0.75,
  0.35, 0.70, 1.00, 0.80,
  0.40, 0.75, 0.80, 1.00
 ),
 nrow = 4, ncol = 4
)

evals <- eigenvalues(cormatrix = correlationMatrix)

print(evals)

evals <- eigenvalues(correlationMatrix, 1)

print(evals)

 Winzar, H. (2022). LikertMakeR: Synthesise and correlate Likert-scale 
 and related rating-scale data with predefined first & second moments, 
 Version 1.0.1 (2025),
 The Comprehensive R Archive Network (CRAN),
<https://CRAN.R-project.org/package=LikertMakeR>
    

@software{winzar2022,
title = {LikertMakeR: Synthesise and correlate Likert-scale 
and related rating-scale data with predefined first & second moments},
author = {Hume Winzar},
abstract = {LikertMakeR synthesises Likert scale and related rating-scale data with predefined means and standard deviations, and optionally correlates these vectors to fit a predefined correlation matrix or Cronbach's Alpha.},
journal = {The Comprehensive R Archive Network (CRAN)},
month = {12},
year = {2022},
version = {1.0.1 (2025)}
url = {https://CRAN.R-project.org/package=LikertMakeR},
}

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