# LikertMakeR

### synthesise and correlate rating-scale data

#### April 2024

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.

Functions can be combined to generate synthetic rating-scale data from a predefined Cronbach’s Alpha.

## 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.

## Motivation

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. Researchers can also use LikertMakeR to prepare analyses ahead of a formal survey.

## Rating scale properties

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.

## LikertMakeR functions

• lfast() generate a vector of values with predefined mean and standard deviation.

• 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 from a 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.

• makeItemsScale() generates a random dataframe of scale items based on a predefined summated scale, such as created by the lfast() function.

• correlateScales() creates a dataframe of correlated summated scales as one might find in completed survey questionnaire and possibly used in a Structural Equation model.

• 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.

# Using LikertMakeR

### from CRAN

> 
>
> install.packages("LikertMakeR")
> library(LikertMakeR)
>
> 

### development version from GitHub.

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

## Generate synthetic rating-scale data

### lfast()

• lfast() applies a simple evolutionary algorithm which draws repeated random samples from a scaled Beta distribution. It produces a vector of values with mean and standard deviation correct to two decimal places.

To synthesise a rating scale with lfast(), 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 slow and no more accurate than the latest version of lfast(). So, lexact() is now deprecated.

#### lfast() example

##### a four-item, five-point Likert scale
x1 <- lfast(
n = 512,
mean = 2.5,
sd = 0.75,
lowerbound = 1,
upperbound = 5,
items = 4
)
#> [1] "best solution in 32 iterations"

##### an 11-point likelihood-of-purchase scale
###### lfast()
x3 <- lfast(256, 3, 2, 0, 10)
#> [1] "best solution in 1057 iterations"

### lexact()

lexact() Deprecated. lexact() is now simply a wrapper for lfast().

## Correlating rating scales

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()

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.

### lcor() example

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 1695 iterations"
#> [1] "best solution in 4760 iterations"
#> [1] "best solution in 1222 iterations"

The first six observations from this data-frame are:

#>    x1  x2  x3
#> 1 3.0 2.0 3.2
#> 2 2.0 2.6 4.0
#> 3 2.0 2.4 4.0
#> 4 2.4 4.4 2.8
#> 5 3.0 4.8 4.6
#> 6 2.8 5.0 4.2

And the first and second moments (to 3 decimal places) are:

#>         x1    x2    x3
#> mean 2.500 3.002 3.500
#> sd   0.752 1.500 1.002

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 0.065 0.087
#> x2 0.065 1.000 0.011
#> x3 0.087 0.011 1.000

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.

## apply lcor() function

new3 <- lcor(mydat3, tgt3)

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 1.2  1 1.2
#> 2 1.2  1 1.2
#> 3 1.2  1 1.4
#> 4 4.8  5 5.0
#> 5 4.0  5 5.0
#> 6 4.0  5 5.0

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

## Generate a correlation matrix from Cronbach’s Alpha

### makeCorrAlpha()

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. Setting ‘variance = 2.0’, or above, may be feasible but increases the likelihood of a non-positive-definite matrix.

### makeCorrAlpha() is volatile

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.

#### makeCorrAlpha() examples

##### Four variables, alpha = 0.85, variance = default
## 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 59 iterations
#> The correlation matrix is positive definite

makeCorrAlpha() produced the following correlation matrix (to three decimal places):

#>       [,1]  [,2]  [,3]  [,4]
#> [1,] 1.000 0.425 0.433 0.507
#> [2,] 0.425 1.000 0.693 0.694
#> [3,] 0.433 0.693 1.000 0.766
#> [4,] 0.507 0.694 0.766 1.000
##### test output with Helper functions
## using helper function alpha()

alpha(cor_matrix_4)
#> [1] 0.8500063
## using helper function eigenvalues()

eigenvalues(cor_matrix_4, 1)

#> cor_matrix_4  is positive-definite
#> [1] 2.7842025 0.6581071 0.3291732 0.2285172

#### twelve variables, alpha = 0.90, variance = 1

## 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 4312 iterations
#> Correlation matrix is not yet positive definite
#>
#> Working on it
#> improved at swap - 12
#> improved at swap - 67
#> improved at swap - 79
#> improved at swap - 80
#> improved at swap - 115
#> improved at swap - 121
#> improved at swap - 128
#> improved at swap - 130
#> improved at swap - 134
#> improved at swap - 137
#> improved at swap - 146
#> improved at swap - 151
#> improved at swap - 160
#> improved at swap - 162
#> improved at swap - 166
#> improved at swap - 174
#> improved at swap - 183
#> improved at swap - 188
#> improved at swap - 191
#> improved at swap - 208
#> improved at swap - 263
#> improved at swap - 304
#> improved at swap - 399
#> improved at swap - 400
#> improved at swap - 402
#> improved at swap - 445
#> improved at swap - 485
#> improved at swap - 542
#> stopped at swap - 542
#> 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 -0.51 -0.67 -0.32 -0.30 -0.29 -0.27 -0.14 -0.07 -0.04 -0.03  0.00
#>  [2,] -0.51  1.00  0.06  0.31  0.43  0.26  0.28  0.20  0.26  0.06  0.25  0.34
#>  [3,] -0.67  0.06  1.00  0.61  0.36  0.62  0.57  0.47  0.45  0.46  0.47  0.33
#>  [4,] -0.32  0.31  0.61  1.00  0.48  0.50  0.60  0.36  0.39  0.53  0.64  0.59
#>  [5,] -0.30  0.43  0.36  0.48  1.00  0.42  0.56  0.62  0.62  0.62  0.56  0.63
#>  [6,] -0.29  0.26  0.62  0.50  0.42  1.00  0.81  0.66  0.70  0.70  0.70  0.70
#>  [7,] -0.27  0.28  0.57  0.60  0.56  0.81  1.00  0.57  0.71  0.72  0.72  0.73
#>  [8,] -0.14  0.20  0.47  0.36  0.62  0.66  0.57  1.00  0.71  0.79  0.79  0.78
#>  [9,] -0.07  0.26  0.45  0.39  0.62  0.70  0.71  0.71  1.00  0.80  0.83  0.84
#> [10,] -0.04  0.06  0.46  0.53  0.62  0.70  0.72  0.79  0.80  1.00  0.88  0.89
#> [11,] -0.03  0.25  0.47  0.64  0.56  0.70  0.72  0.79  0.83  0.88  1.00  0.97
#> [12,]  0.00  0.34  0.33  0.59  0.63  0.70  0.73  0.78  0.84  0.89  0.97  1.00
##### test output
alpha(cor_matrix_12)
#> [1] 0.9000045

eigenvalues(cor_matrix_12, 1) |> round(3)

#> cor_matrix_12  is positive-definite
#>  [1] 6.964 1.743 1.087 0.658 0.567 0.377 0.254 0.159 0.127 0.051 0.014 0.001

## Generate a dataframe of rating scales from a correlation matrix and predefined moments

### makeItems()

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.

### makeItems() examples

## 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
)
#> Variable  1[1] "reached maximum of 16384 iterations"
#> Variable  2[1] "reached maximum of 16384 iterations"
#> Variable  3[1] "best solution in 2371 iterations"
#> Variable  4[1] "reached maximum of 16384 iterations"
#>
#> Arranging data to match correlations
#>
#> Successfully generated correlated variables

## test the function
#>   V1 V2 V3 V4
#> 1  4  5  5  5
#> 2  5  5  5  5
#> 3  5  5  5  5
#> 4  3  5  5  5
#> 5  3  5  5  5
#> 6  3  5  5  5
tail(df)
#>     V1 V2 V3 V4
#> 123  2  1  1  2
#> 124  2  4  3  4
#> 125  3  2  5  4
#> 126  3  3  3  4
#> 127  3  2  2  3
#> 128  3  2  3  3
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

## Generate a dataframe from Cronbach’s Alpha and predefined moments

This is a two-step process:

1. apply makeCorrAlpha() to generate a correlation matrix from desired alpha,

2. 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

### Step 1: Generate a correlation matrix

## define parameters
k <- 6
alpha <- 0.85

## generate correlation matrix
set.seed(42)
myCorr <- makeCorrAlpha(k, alpha)
#> correlation values consistent with desired alpha in 15193 iterations
#> The correlation matrix is positive definite

## display correlation matrix
myCorr |> round(3)
#>        [,1]   [,2]  [,3]  [,4]  [,5]  [,6]
#> [1,]  1.000 -0.153 0.116 0.430 0.438 0.473
#> [2,] -0.153  1.000 0.480 0.498 0.528 0.585
#> [3,]  0.116  0.480 1.000 0.602 0.625 0.641
#> [4,]  0.430  0.498 0.602 1.000 0.662 0.677
#> [5,]  0.438  0.528 0.625 0.662 1.000 0.684
#> [6,]  0.473  0.585 0.641 0.677 0.684 1.000

### checking Cronbach's Alpha
alpha(myCorr)
#> [1] 0.8500101

### Step 2: Generate dataframe

## 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)
#> Variable  1[1] "best solution in 972 iterations"
#> Variable  2[1] "best solution in 17 iterations"
#> Variable  3[1] "best solution in 973 iterations"
#> Variable  4[1] "best solution in 4866 iterations"
#> Variable  5[1] "best solution in 336 iterations"
#> Variable  6[1] "best solution in 16769 iterations"
#>
#> Arranging data to match correlations
#>
#> Successfully generated correlated variables

## resulting data frame
#>   V1 V2 V3 V4 V5 V6
#> 1  3  1  1  1  1  1
#> 2  3  2  1  1  1  1
#> 3  2  2  1  1  2  1
#> 4  1  5  5  5  5  5
#> 5  1  5  5  5  5  5
#> 6  1  5  5  5  5  5
tail(myItems)
#>     V1 V2 V3 V4 V5 V6
#> 251  1  4  3  3  3  1
#> 252  2  4  3  2  4  3
#> 253  4  2  2  4  2  4
#> 254  3  3  4  2  4  5
#> 255  4  3  3  4  4  5
#> 256  4  2  4  4  4  5

## 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 3.500
#> sds   0.998 0.751 1.002 0.998 0.998 1.498

## Cronbach's Alpha of data frame
alpha(NULL, myItems)
#> [1] 0.8499588

## Create a multidimensinoal dataframe of correlated scale items

### correlateScales()

Correlated rating-scale items generally are summed or averaged to create a measure of an “unobservable”, or “latent”, construct.

correlateScales() takes several such 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, but their correlations with rating-scale items in other dataframes do.

To run correlateScales(), parameters are:

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

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

As with other functions in LikertMakeR, correlateScales() focuses on item and scale moments (mean and standard deviation) rather than on covariance structure. If you wish to simulate data for teaching or experimenting with Structural Equation modelling, then I recommend the sim.item() and sim.congeneric() functions from the psych package

### correlateScales() examples

#### three attitudes and a behavioural intention

##### create dataframes of Likert-scale items
n <- 32
lower <- 1
upper <- 5

### attitude #1
cor_1 <- makeCorrAlpha(items = 4, alpha = 0.90)
#> correlation values consistent with desired alpha in 174 iterations
#> The correlation matrix is positive definite
means_1 <- c(2.5, 2.5, 3.0, 3.5)
sds_1 <- c(0.9, 1.0, 0.9, 1.0)
Att_1 <- makeItems(
n, means_1, sds_1,
rep(lower, 4), rep(upper, 4),
cor_1
)
#> Variable  1[1] "reached maximum of 1024 iterations"
#> Variable  2[1] "reached maximum of 1024 iterations"
#> Variable  3[1] "reached maximum of 1024 iterations"
#> Variable  4[1] "reached maximum of 1024 iterations"
#>
#> Arranging data to match correlations
#>
#> Successfully generated correlated variables

### attitude #2
cor_2 <- makeCorrAlpha(items = 5, alpha = 0.85)
#> correlation values consistent with desired alpha in 4798 iterations
#> The correlation matrix is positive definite
means_2 <- c(2.5, 2.5, 3.0, 3.0, 3.5)
sds_2 <- c(1.0, 1.0, 0.9, 1.0, 1.5)
Att_2 <- makeItems(
n, means_2, sds_2,
rep(lower, 5), rep(upper, 5),
cor_2
)
#> Variable  1[1] "reached maximum of 1024 iterations"
#> Variable  2[1] "reached maximum of 1024 iterations"
#> Variable  3[1] "reached maximum of 1024 iterations"
#> Variable  4[1] "reached maximum of 1024 iterations"
#> Variable  5[1] "reached maximum of 1024 iterations"
#>
#> Arranging data to match correlations
#>
#> Successfully generated correlated variables

### attitude #3
cor_3 <- makeCorrAlpha(items = 6, alpha = 0.75)
#> correlation values consistent with desired alpha in 2313 iterations
#> The correlation matrix is positive definite
means_3 <- c(2.5, 2.5, 3.0, 3.0, 3.5, 3.5)
sds_3 <- c(1.0, 1.5, 1.0, 1.5, 1.0, 1.5)

Att_3 <- makeItems(
n, means_3, sds_3,
rep(lower, 6), rep(upper, 6),
cor_3
)
#> Variable  1[1] "reached maximum of 1024 iterations"
#> Variable  2[1] "reached maximum of 1024 iterations"
#> Variable  3[1] "reached maximum of 1024 iterations"
#> Variable  4[1] "reached maximum of 1024 iterations"
#> Variable  5[1] "reached maximum of 1024 iterations"
#> Variable  6[1] "reached maximum of 1024 iterations"
#>
#> Arranging data to match correlations
#>
#> Successfully generated correlated variables

### behavioural intention
intent <- lfast(n, mean = 3.0, sd = 3, lowerbound = 0, upperbound = 10) |>
data.frame()
#> [1] "reached maximum of 1024 iterations"
names(intent) <- "int"
###### check properties of item dataframes
## Attitude #1
A1_moments <- data.frame(
means = apply(Att_1, 2, mean) |> round(2),
sds = apply(Att_1, 2, sd) |> round(2)
) |> t()
A1_moments
#>         V1   V2   V3   V4
#> means 2.50 2.50 3.00 3.50
#> sds   0.92 1.02 0.92 1.02
cor(Att_1) |> round(2)
#>      V1   V2   V3   V4
#> V1 1.00 0.55 0.62 0.73
#> V2 0.55 1.00 0.73 0.75
#> V3 0.62 0.73 1.00 0.80
#> V4 0.73 0.75 0.80 1.00

## Attitude #2
A2_moments <- data.frame(
means = apply(Att_2, 2, mean) |> round(2),
sds = apply(Att_2, 2, sd) |> round(2)
) |> t()

A2_moments
#>         V1   V2   V3   V4   V5
#> means 2.50 2.50 3.00 3.00 3.50
#> sds   1.02 1.02 0.92 1.02 1.48
cor(Att_2) |> round(2)
#>      V1   V2   V3   V4   V5
#> V1 1.00 0.25 0.28 0.38 0.41
#> V2 0.25 1.00 0.52 0.56 0.62
#> V3 0.28 0.52 1.00 0.73 0.74
#> V4 0.38 0.56 0.73 1.00 0.77
#> V5 0.41 0.62 0.74 0.77 1.00

## Attitude #3
A3_moments <- data.frame(
means = apply(Att_3, 2, mean) |> round(2),
sds = apply(Att_3, 2, sd) |> round(2)
) |> t()

A3_moments
#>         V1  V2   V3  V4   V5  V6
#> means 2.50 2.5 3.00 3.0 3.50 3.5
#> sds   1.02 1.5 1.02 1.5 1.02 1.5
cor(Att_3) |> round(2)
#>      V1   V2   V3   V4   V5   V6
#> V1 1.00 0.00 0.16 0.23 0.25 0.25
#> V2 0.00 1.00 0.27 0.37 0.36 0.37
#> V3 0.16 0.27 1.00 0.38 0.38 0.40
#> V4 0.23 0.37 0.38 1.00 0.42 0.57
#> V5 0.25 0.36 0.38 0.42 1.00 0.59
#> V6 0.25 0.37 0.40 0.57 0.59 1.00

## Intention

intent_moments <- data.frame(
mean = apply(intent, 2, mean) |> round(2),
sd = apply(intent, 2, sd) |> round(2)
) |> t()

intent_moments
#>       int
#> mean 3.00
#> sd   3.02
##### correlateScales parameters
### target scale correlation matrix
scale_cors <- matrix(
c(
1.0, 0.6, 0.5, 0.3,
0.6, 1.0, 0.4, 0.2,
0.5, 0.4, 1.0, 0.1,
0.3, 0.2, 0.1, 1.0
),
nrow = 4
)

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

#### apply the correlateScales() function

my_correlated_scales <- correlateScales(
dataframes = data_frames,
scalecors = scale_cors
)
#> scalecors  is positive-definite
#> New dataframe successfully created

#### plot the new correlated scale items

###### Check the properties of our derived dataframe
## data structure
str(my_correlated_scales)
#> 'data.frame':    32 obs. of  16 variables:
#>  $A1_1 : num 4 4 1 1 1 1 3 2 3 3 ... #>$ A1_2 : num  4 4 1 1 2 3 3 2 3 3 ...
#>  $A1_3 : num 5 5 2 2 2 3 3 2 3 3 ... #>$ A1_4 : num  5 5 3 2 2 3 4 3 4 4 ...
#>  $A2_1 : num 4 4 1 1 2 2 4 4 2 2 ... #>$ A2_2 : num  5 4 1 2 1 4 2 2 3 3 ...
#>  $A2_3 : num 5 5 2 1 2 4 3 3 3 3 ... #>$ A2_4 : num  5 5 2 1 2 3 3 2 3 3 ...
#>  $A2_5 : num 5 5 1 1 1 5 4 4 5 4 ... #>$ A3_1 : num  2 2 2 1 4 4 2 2 3 4 ...
#>  $A3_2 : num 5 5 1 1 1 1 1 3 4 1 ... #>$ A3_3 : num  5 4 1 2 2 3 5 3 4 3 ...
#>  $A3_4 : num 5 5 1 1 3 4 1 5 2 4 ... #>$ A3_5 : num  5 5 3 2 3 5 5 4 4 3 ...
#>  $A3_6 : num 5 5 1 2 5 5 5 5 4 2 ... #>$ Int_1: num  9 3 0 0 0 0 0 0 0 2 ...
## eigenvalues of dataframe correlations
eigenvalues(cormatrix = cor(my_correlated_scales), scree = TRUE) |> round(2)

#> cor(my_correlated_scales)  is positive-definite
#>  [1] 6.25 1.98 1.53 1.31 0.95 0.75 0.64 0.55 0.53 0.45 0.33 0.23 0.18 0.14 0.12
#> [16] 0.06

## Helper functions

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()

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() examples

## 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)
#> Both cormatrix and data present.
#>
#> Using cormatrix by default.
#> [1] 0.8395062

### eigenvalues()

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

### eigenvalues() examples

## 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

print(evals)
#> [1] 2.7484991 0.8122627 0.3048151 0.1344231
##### eigenvalues() function with optional scree plot
evals <- eigenvalues(correlationMatrix, 1)

#> correlationMatrix  is positive-definite

# 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

• sampling with a predetermined probability distribution

• marginal model specification

### 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 Heinz (2021) for an excellent and short example using the following packages:

### sampling with a predetermined probability distribution

• the following code will generate a vector of values with approximately the given probabilities. Good for simulating a single item.
n <- 128
sample(1:5, n,
replace = TRUE,
prob = c(0.1, 0.2, 0.4, 0.2, 0.1)
)

### marginal model specification

Marginal model specification extends the idea of a predefined probability distribution to multivariate and correlated data-frames.

### Factor Models: Classical Test Theory (CTT)

The psych package has several excellent functions for simulating rating-scale data based on factor loadings. These focus on factor and item correlations rather than item moments. Highly recommended.

Also:

simsem has many functions for simulating and testing data for application in Structural Equation modelling. See examples at https://simsem.org/

### General data simulation

simpr provides a general, simple, and tidyverse-friendly framework for generating simulated data, fitting models on simulations, and tidying model results.

# References

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

Pornprasertmanit, S., Miller, P., & Schoemann, A. (2021). simsem: R package for simulated structural equation modeling https://simsem.org/

Revelle, W. (in prep) An introduction to psychometric theory with applications in R. To be published by Springer. (working draft available at https://personality-project.org/r/book/ )

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