LikertMakeR synthesises and correlates Likert-scale and related rating-scale data. You decide the mean and standard deviation, and (optionally) the correlations among vectors, and the package will generate data with those same predefined properties.
The package generates a column of values that simulate the same properties as a rating scale. If multiple columns are generated, then you can use LikertMakeR to rearrange the values so that the new variables are correlated exactly in accord with a user-predefined correlation matrix.
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 apparent understanding of scale distributions, and their impact on scale properties. Hopefully, this tool will help researchers, teachers, and other reviewers, to better think about rating-scale distributions, and the effects of variance, scale boundaries, and number of items in a scale.
A Likert scale is the mean, or sum, of several ordinal rating scales. Typically, they are bipolar (usually “agree-disagree”) responses to propositions that are determined to be moderately-to-highly correlated and that capture some facet of a theoretical 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. If the mean is not in the middle of a scale, and if the standard deviation is any more than about 1/4 of the scale range, then the data will be always skewed.
lfast() quickly generate a vector of values with approximate predefined moments.
lcor() takes a dataframe of rating-scale values and rearranges the values in each column so that the columns are correlated to match a predefined correlation matrix.
makeCorrAlpha constructs a random 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.
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.
> ```
>
> install.packages("LikertMakeR")
> library(LikertMakeR)
>
> ```
> ```
>
> library(devtools)
> install_github("WinzarH/LikertMakeR")
> library(LikertMakeR)
>
> ```
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
The previous version of LikertMakeR had a function, lexact(), which was very slow and no more accurate than lfast(). So, lexact() is now deprecated.
lexact() Deprecated. lexact() is now simply a wrapper for lfast().
The function, lcor(), 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 parameters:
data: a starter data set of rating-scales. Number of columns must match the dimensions of the target correlation matrix.
target: the target correlation matrix.
Let’s generate some data: three 5-point Likert scales, each with five items.
## generate uncorrelated synthetic data
n <- 128
lowerbound <- 1
upperbound <- 5
items <- 5
mydat3 <- data.frame(
x1 = lfast(n, 2.5, 0.75, lowerbound, upperbound, items),
x2 = lfast(n, 3.0, 1.50, lowerbound, upperbound, items),
x3 = lfast(n, 3.5, 1.00, lowerbound, upperbound, items)
)
#> [1] "best solution in 994 iterations"
#> [1] "best solution in 763 iterations"
#> [1] "best solution in 1724 iterations"
The first six observations from this data-frame are:
#> x1 x2 x3
#> 1 2.6 3.4 4.4
#> 2 2.8 1.0 3.2
#> 3 2.2 5.0 3.2
#> 4 4.0 1.4 3.8
#> 5 1.6 5.0 3.8
#> 6 2.2 5.0 2.0
And the first and second moments (to 3 decimal places) are:
#> x1 x2 x3
#> mean 2.502 3.0 3.500
#> sd 0.750 1.5 0.999
We can see that the data have first and second moments very close to what is expected.
The synthetic data have low correlations:
#> x1 x2 x3
#> x1 1.000 0.038 -0.01
#> x2 0.038 1.000 -0.13
#> x3 -0.010 -0.130 1.00
Now, let’s define a target correlation matrix:
## describe a target correlation matrix
tgt3 <- matrix(
c(
1.00, 0.85, 0.75,
0.85, 1.00, 0.65,
0.75, 0.65, 1.00
),
nrow = 3
)
So now we have a data-frame with desired first and second moments, and a target correlation matrix.
The first column of the new data-frame will not change, but values of the other columns are rearranged.
The first six observations from this data-frame are:
#> V1 V2 V3
#> 1 4.2 5 5.0
#> 2 1.0 1 1.4
#> 3 1.4 1 1.6
#> 4 3.6 5 4.8
#> 5 4.2 5 5.0
#> 6 1.2 1 1.8
And the new data frame is correlated close to our desired correlation matrix; here presented to 3 decimal places:
#> V1 V2 V3
#> V1 1.00 0.85 0.75
#> V2 0.85 1.00 0.65
#> V3 0.75 0.65 1.00
makeCorrAlpha(), constructs a random correlation matrix of given dimensions and predefined Cronbach’s Alpha.
To create the desired correlation matrix, the user must define the following parameters:
items: or “k” - the number of rows and columns of the desired correlation matrix.
alpha: the target value for Cronbach’s Alpha
variance: a notional variance coefficient to affect the spread of values in the correlation matrix. Default = ‘0.5’. A value of ‘0’ produces a matrix where all off-diagonal correlations are equal. Setting ‘variance = 1.0’ gives a wider range of values.
Random values generated by makeCorrAlpha() are highly volatile. makeCorrAlpha() may not generate a feasible (positive-definite) correlation matrix, especially when
variance is high relative to
desired Alpha, and
desired correlation dimensions
makeCorrAlpha() will inform the user if the resulting correlation matrix is positive definite, or not.
If the returned correlation matrix is not positive-definite, a feasible solution may be still possible, and often is. The user is encouraged to try again, possibly several times, to find one.
## define parameters
items <- 4
alpha <- 0.85
# variance <- 0.5 ## by default
## apply makeCorrAlpha() function
set.seed(42)
cor_matrix_4 <- makeCorrAlpha(items, alpha)
#> correlation values consistent with desired alpha in 2261 iterations
#> The correlation matrix is positive definite
makeCorrAlpha() produced the following correlation matrix (to three decimal places):
#> [,1] [,2] [,3] [,4]
#> [1,] 1.000 0.445 0.481 0.577
#> [2,] 0.445 1.000 0.632 0.647
#> [3,] 0.481 0.632 1.000 0.735
#> [4,] 0.577 0.647 0.735 1.000
#> cor_matrix_4 is positive-definite
#>
#> Eigenvalues:
#> 2.771181 0.5922674 0.3851331 0.2514183
#> [1] 2.7711812 0.5922674 0.3851331 0.2514183
## define parameters
items <- 12
alpha <- 0.90
variance <- 1.0
## apply makeCorrAlpha() function
set.seed(42)
cor_matrix_12 <- makeCorrAlpha(items, alpha, variance)
#> correlation values consistent with desired alpha in 26735 iterations
#> The correlation matrix is positive definite
makeCorrAlpha() produced the following correlation matrix (to two decimal places):
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
#> [1,] 1.00 -0.31 -0.26 -0.25 -0.17 -0.16 -0.08 -0.05 0.04 0.07 0.08 0.10
#> [2,] -0.31 1.00 0.10 0.17 0.19 0.19 0.20 0.20 0.21 0.25 0.34 0.34
#> [3,] -0.26 0.10 1.00 0.35 0.36 0.36 0.37 0.41 0.42 0.42 0.43 0.45
#> [4,] -0.25 0.17 0.35 1.00 0.45 0.46 0.46 0.46 0.48 0.49 0.49 0.52
#> [5,] -0.17 0.19 0.36 0.45 1.00 0.52 0.53 0.53 0.54 0.58 0.59 0.60
#> [6,] -0.16 0.19 0.36 0.46 0.52 1.00 0.61 0.62 0.64 0.65 0.66 0.67
#> [7,] -0.08 0.20 0.37 0.46 0.53 0.61 1.00 0.67 0.67 0.70 0.71 0.72
#> [8,] -0.05 0.20 0.41 0.46 0.53 0.62 0.67 1.00 0.74 0.77 0.79 0.81
#> [9,] 0.04 0.21 0.42 0.48 0.54 0.64 0.67 0.74 1.00 0.83 0.88 0.91
#> [10,] 0.07 0.25 0.42 0.49 0.58 0.65 0.70 0.77 0.83 1.00 0.92 0.92
#> [11,] 0.08 0.34 0.43 0.49 0.59 0.66 0.71 0.79 0.88 0.92 1.00 0.95
#> [12,] 0.10 0.34 0.45 0.52 0.60 0.67 0.72 0.81 0.91 0.92 0.95 1.00
#> cor_matrix_12 is positive-definite
#>
#> Eigenvalues:
#> 6.6755 1.414525 0.935596 0.6735671 0.5536723 0.5100483 0.3867301 0.3380953 0.2605829 0.1603447 0.05985932 0.03147918
#> [1] 6.675 1.415 0.936 0.674 0.554 0.510 0.387 0.338 0.261 0.160 0.060 0.031
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.
Generally, means, standard deviations, and correlations are correct to two decimal places.
makeItems() is a wrapper function for
lfast(), which takes repeated samples selecting 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.
To create the desired dataframe, the user must define the following parameters:
n: number of observations
dfMeans: a vector of length ‘k’ of desired means of each variable
dfSds: a vector of length ‘k’ of desired standard deviations of each variable
lowerbound: a vector of length ‘k’ of values for the lower bound of each variable (For example, ‘1’ for a 1-5 rating scale)
upperbound: a vector of length ‘k’ of values for the upper bound of each variable (For example, ‘5’ for a 1-5 rating scale)
cormatrix: a target correlation matrix with ‘k’ rows and ‘k’ columns.
## define parameters
n <- 128
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, 0.45,
0.25, 1.00, 0.70, 0.75,
0.35, 0.70, 1.00, 0.85,
0.45, 0.75, 0.85, 1.00
),
nrow = 4, ncol = 4
)
## apply makeItems() function
df <- makeItems(
n = n,
means = dfMeans,
sds = dfSds,
lowerbound = lowerbound,
upperbound = upperbound,
cormatrix = corMat
)
#> [1] "best solution in 16384 iterations"
#> [1] "best solution in 16384 iterations"
#> [1] "best solution in 727 iterations"
#> [1] "best solution in 16384 iterations"
## test the function
head(df); tail(df)
#> V1 V2 V3 V4
#> 1 2 1 1 2
#> 2 2 1 1 2
#> 3 4 5 5 5
#> 4 4 5 5 5
#> 5 2 1 1 2
#> 6 3 3 4 4
#> V1 V2 V3 V4
#> 123 2 2 1 3
#> 124 2 3 1 3
#> 125 3 4 4 4
#> 126 4 3 2 4
#> 127 1 4 2 4
#> 128 1 3 3 4
apply(df, 2, mean) |> round(3)
#> V1 V2 V3 V4
#> 2.5 3.0 3.0 3.5
apply(df, 2, sd) |> round(3)
#> V1 V2 V3 V4
#> 1.004 1.004 1.501 0.753
cor(df) |> round(3)
#> V1 V2 V3 V4
#> V1 1.000 0.25 0.35 0.448
#> V2 0.250 1.00 0.70 0.750
#> V3 0.350 0.70 1.00 0.850
#> V4 0.448 0.75 0.85 1.000
This is a two-step process:
apply makeCorrAlpha() to generate a correlation matrix from desired alpha,
apply makeItems() to generate rating-scale items from the correlation matrix and desired moments
So required parameters are:
k: number items/ columns
alpha: a target Cronbach’s Alpha.
n: number of observations
lowerbound: a vector of length ‘k’ of values for the lower bound of each variable
upperbound: a vector of length ‘k’ of values for the upper bound of each variable
means: a vector of length ‘k’ of desired means of each variable
sds: a vector of length ‘k’ of desired standard deviations of each variable
## define parameters
k <- 6
alpha <- 0.85
## generate correlation matrix
set.seed(42)
myCorr <- makeCorrAlpha(k, alpha)
#> correlation values consistent with desired alpha in 2425 iterations
#> The correlation matrix is positive definite
## display correlation matrix
myCorr |> round(3)
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1.000 0.123 0.314 0.394 0.422 0.433
#> [2,] 0.123 1.000 0.453 0.455 0.480 0.486
#> [3,] 0.314 0.453 1.000 0.531 0.537 0.639
#> [4,] 0.394 0.455 0.531 1.000 0.657 0.671
#> [5,] 0.422 0.480 0.537 0.657 1.000 0.690
#> [6,] 0.433 0.486 0.639 0.671 0.690 1.000
### checking Cronbach's Alpha
alpha(myCorr)
#> [1] 0.8499957
## define parameters
n <- 256
myMeans <- c(2.75, 3.00, 3.00, 3.25, 3.50, 3.5)
mySds <- c(1.00, 0.75, 1.00, 1.00, 1.00, 1.5)
lowerbound <- rep(1, k)
upperbound <- rep(5, k)
## Generate Items
myItems <- makeItems(n, myMeans, mySds, lowerbound, upperbound, myCorr)
#> [1] "best solution in 2617 iterations"
#> [1] "best solution in 30 iterations"
#> [1] "best solution in 2383 iterations"
#> [1] "best solution in 271 iterations"
#> [1] "best solution in 10740 iterations"
#> [1] "best solution in 4776 iterations"
## resulting data frame
head(myItems)
#> V1 V2 V3 V4 V5 V6
#> 1 3 1 2 1 1 1
#> 2 4 4 4 5 5 5
#> 3 5 5 5 5 5 5
#> 4 1 1 2 1 1 1
#> 5 5 5 5 5 5 5
#> 6 1 1 1 1 1 1
tail(myItems)
#> V1 V2 V3 V4 V5 V6
#> 251 3 4 2 4 2 5
#> 252 2 4 4 3 3 5
#> 253 3 3 4 4 4 5
#> 254 2 3 4 4 3 1
#> 255 3 4 3 4 5 5
#> 256 3 3 4 3 4 4
## means and standard deviations
myMoments <- data.frame(
means = apply(myItems, 2, mean) |> round(3),
sds = apply(myItems, 2, sd) |> round(3)
) |> t()
myMoments
#> V1 V2 V3 V4 V5 V6
#> means 2.750 3.000 3.000 3.250 3.500 3.5
#> sds 0.998 0.751 1.002 0.998 1.002 1.5
## Cronbach's Alpha of data frame
alpha(NULL, myItems)
#> [1] 0.8498222
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, a report on whether the correlation matrix is positive definite, and produces 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.
## define parameters
df <- data.frame(
V1 = c(4, 2, 4, 3, 2, 2, 2, 1),
V2 = c(3, 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, 0.75,
0.35, 1.00, 0.65, 0.55,
0.45, 0.65, 1.00, 0.65,
0.75, 0.55, 0.65, 1.00
),
nrow = 4, ncol = 4
)
## apply function examples
alpha(cormatrix = corMat)
#> [1] 0.8395062
alpha(data = df)
#> [1] 0.8026658
alpha(NULL, df)
#> [1] 0.8026658
alpha(corMat, df)
#> Warning: Both cormatrix and data present.
#>
#> Using cormatrix by default.
#> [1] 0.8395062
eigenvalues() calculates eigenvalues of a correlation matrix, reports on whether the matrix is positive-definite, and optionally produces a scree plot.
## define parameters
correlationMatrix <- matrix(
c(
1.00, 0.25, 0.35, 0.45,
0.25, 1.00, 0.70, 0.75,
0.35, 0.70, 1.00, 0.85,
0.45, 0.75, 0.85, 1.00
),
nrow = 4, ncol = 4
)
## apply function
evals <- eigenvalues(cormatrix = correlationMatrix)
#> correlationMatrix is positive-definite
#>
#> Eigenvalues:
#> 2.748499 0.8122627 0.3048151 0.1344231
print(evals)
#> [1] 2.7484991 0.8122627 0.3048151 0.1344231
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
sampling with a predetermined probability distribution
marginal model specification
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 Heinz (2021) for an excellent and short example using the following packages:
See also the rLikert() function from the responsesR package, Lalovic (2021), for an approach using optimal discretization and skew-normal distribution.
Marginal model specification extends the idea of a predefined probability distribution to multivariate and correlated data-frames.
SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types on CRAN.
lsasim: Functions to Facilitate the Simulation of Large Scale Assessment Data on CRAN. See Matta et al. (2018)
SimCorMultRes: Simulates Correlated Multinomial Responses on CRAN. See Touloumis (2016)
covsim: VITA, IG and PLSIM Simulation for Given Covariance and Marginals on CRAN. See Grønneberg et al. (2022)
Grønneberg, S., Foldnes, N., & Marcoulides, K. M. (2022). covsim: An R Package for Simulating Non-Normal Data for Structural Equation Models Using Copulas. Journal of Statistical Software, 102(1), 1–45. doi:10.18637/jss.v102.i03
Heinz, A. (2021), Simulating Correlated Likert-Scale Data In R: 3 Simple Steps (blog post) https://glaswasser.github.io/simulating-correlated-likert-scale-data/
Lalovic, M. (2021), responsesR: Simulate Likert scale item responses (on GitHub) https://github.com/markolalovic/responsesR
Matta, T.H., Rutkowski, L., Rutkowski, D. & Liaw, Y.L. (2018), lsasim: an R package for simulating large-scale assessment data. Large-scale Assessments in Education 6, 15. doi:10.1186/s40536-018-0068-8
Touloumis, A. (2016), Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package, The R Journal 8:2, 79-91. doi:10.32614/RJ-2016-034