Author: Hume Winzar

# LikertMakeR vignette

#### Hume Winzar

#### April 2025

    library(LikertMakeR)

LikertMakeR (Winzar, 2022) lets you create synthetic Likert-scale, or related rating-scale, data.
Set the mean, standard deviation, and correlations, and the package generates data matching those properties. It can also rearrange existing data columns to achieve a desired correlation structure or generate data based on Cronbach’s Alpha, factor correlations or other summary statistics.

## Purpose

The package should be useful for teaching in the Social Sciences, and for scholars who wish to “replicate” or “reverse engineer” 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 & students, 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.

Technically, because they are bounded and not continuous, parametric statistics, such as mean, standard deviation, and correlation, should not be applied to summated rating scales. In practice, however, 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.
   For example, D’Alessandro et al. (2020) argue that a summated scale, made with multiple items, “approaches” an interval scale measure. This implies that parametric statistics are acceptable.

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, then the data will be always skewed, as shown in the following plots.

Off-centre means always give skewed distribution in bounded rating scales

## 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.

- makeCorrLoadings constructs a random correlation matrix from a given factor loadings matrix, and factor-correlations matrix.

- 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 with a desired Cronbach’s Alpha.

- makePaired() generates a dataframe of two correlated columns based on summary data from a paired-sample t-test.

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

- Helper Functions

  - 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

## Download and Install 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 typically 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

An earlier 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

    nItems <- 4
    mean <- 2.5
    sd <- 0.75

    x1 <- lfast(
      n = 512,
      mean = mean,
      sd = sd,
      lowerbound = 1,
      upperbound = 5,
      items = nItems
    )
    #> best solution in 760 iterations

Example: 4-item, 1-5 Likert scale

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

###### lfast()

    x2 <- lfast(256, 3, 2.5, 0, 10)
    #> best solution in 2310 iterations

Example: likelihood-of-purchase scale

## 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 dataframe 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)
    )
    #> best solution in 2453 iterations
    #> best solution in 375 iterations
    #> best solution in 2261 iterations

The first six observations from this dataframe are:

    #>    x1  x2  x3
    #> 1 2.4 4.0 3.6
    #> 2 2.4 3.4 4.2
    #> 3 3.4 4.2 4.6
    #> 4 2.6 2.2 3.2
    #> 5 3.0 4.0 4.4
    #> 6 2.6 1.0 3.4

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

    #>         x1  x2    x3
    #> mean 2.498 3.0 3.498
    #> sd   0.749 1.5 1.000

We can see that the data have first and second moments are very close to what is expected.

As we should expect, randomly-generated synthetic data have low correlations:

    #>       x1    x2    x3
    #> x1  1.00  0.12 -0.06
    #> x2  0.12  1.00 -0.04
    #> x3 -0.06 -0.04  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 dataframe with desired first and second moments, and a target correlation matrix.

    ## apply lcor() function
    new3 <- lcor(data = mydat3, target = tgt3)

Values in each column of the new dataframe do not change from the original; the values are rearranged.

The first ten observations from this dataframe are:

    #>     X1  X2  X3
    #> 1  3.2 4.8 3.2
    #> 2  2.4 1.6 3.8
    #> 3  3.4 5.0 4.6
    #> 4  1.6 2.2 2.0
    #> 5  3.8 4.8 4.8
    #> 6  2.8 2.8 4.6
    #> 7  2.6 2.4 3.2
    #> 8  1.8 1.8 3.4
    #> 9  2.4 3.4 3.6
    #> 10 2.6 3.2 3.2

And the new data frame is correlated close to our desired correlation matrix; here presented to 3 decimal places:

    #>      X1   X2   X3
    #> X1 1.00 0.85 0.75
    #> X2 0.85 1.00 0.65
    #> X3 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 = items, alpha = alpha, variance = 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

    ## calculate Cronbach's Alpha
    alpha(cor_matrix_12)
    #> [1] 0.9000045

    ## calculate eigenvalues of the correlation matrix
    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 correlation matrix from factor loadings

### makeCorrLoadings

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() usage

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

##### makeCorrLoadings() arguments

- 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.)

##### Note

“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 vignette file, makeCorrLoadings_Validate.

### makeCorrLoadings() examples

#### Typical application from published EFA results

##### define parameters

    ## 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
    )

##### Apply the function

    ## apply makeCorrLoadings() function
    itemCorrelations <- makeCorrLoadings(factorLoadings, factorCor)

    ## derived correlation matrix to two decimal places
    round(itemCorrelations, 2)
    #>      Q1   Q2   Q3   Q4   Q5   Q6   Q7   Q8
    #> Q1 1.00 0.62 0.67 0.48 0.42 0.42 0.43 0.41
    #> Q2 0.62 1.00 0.72 0.43 0.36 0.36 0.44 0.42
    #> Q3 0.67 0.72 1.00 0.50 0.43 0.43 0.46 0.45
    #> Q4 0.48 0.43 0.50 1.00 0.79 0.83 0.65 0.58
    #> Q5 0.42 0.36 0.43 0.79 1.00 0.80 0.54 0.48
    #> Q6 0.42 0.36 0.43 0.83 0.80 1.00 0.59 0.52
    #> Q7 0.43 0.44 0.46 0.65 0.54 0.59 1.00 0.78
    #> Q8 0.41 0.42 0.45 0.58 0.48 0.52 0.78 1.00

##### Test makeCorrLoadings() output

    ## correlated factors mean that eigenvalues should suggest two or three factors
    eigenvalues(cormatrix = itemCorrelations, scree = TRUE)

    #> itemCorrelations  is positive-definite
    #> [1] 4.7679427 1.2254239 0.7641967 0.3799863 0.2668158 0.2237851 0.2073574
    #> [8] 0.1644922

##### Assuming orthogonal factors

    ## orthogonal factors are assumed when factor correlation matrix is not included
    orthogonalItemCors <- makeCorrLoadings(factorLoadings)

    ## derived correlation matrix to two decimal places
    round(orthogonalItemCors, 2)
    #>      Q1   Q2   Q3   Q4   Q5   Q6   Q7   Q8
    #> Q1 1.00 0.58 0.63 0.29 0.24 0.22 0.11 0.13
    #> Q2 0.58 1.00 0.69 0.18 0.13 0.10 0.14 0.17
    #> Q3 0.63 0.69 1.00 0.26 0.22 0.18 0.11 0.14
    #> Q4 0.29 0.18 0.26 1.00 0.75 0.79 0.32 0.26
    #> Q5 0.24 0.13 0.22 0.75 1.00 0.78 0.18 0.14
    #> Q6 0.22 0.10 0.18 0.79 0.78 1.00 0.23 0.18
    #> Q7 0.11 0.14 0.11 0.32 0.18 0.23 1.00 0.74
    #> Q8 0.13 0.17 0.14 0.26 0.14 0.18 0.74 1.00

##### Test orthogonal output

    ## eigenvalues should suggest exactly  three factors
    eigenvalues(cormatrix = orthogonalItemCors, scree = TRUE)

    #> orthogonalItemCors  is positive-definite
    #> [1] 3.2769426 1.8091128 1.4966064 0.4244753 0.2966222 0.2605233 0.2402622
    #> [8] 0.1954553

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

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

    ### means should be correct to two decimal places
    dfmoments <- data.frame(
      mean = apply(df, 2, mean) |> round(3),
      sd = apply(df, 2, sd) |> round(3)
    ) |> t()

    dfmoments
    #>       [,1]  [,2]  [,3]  [,4]
    #> mean 2.500 3.000 3.000 3.500
    #> sd   1.004 1.004 1.501 0.753

    ### correlations should be correct to two decimal places
    cor(df) |> round(3)
    #>       [,1] [,2]  [,3]  [,4]
    #> [1,] 1.000 0.25 0.350 0.448
    #> [2,] 0.250 1.00 0.700 0.750
    #> [3,] 0.350 0.70 1.000 0.822
    #> [4,] 0.448 0.75 0.822 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

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
    myAlpha <- 0.85

    ## generate correlation matrix
    set.seed(42)
    myCorr <- makeCorrAlpha(items = k, alpha = myAlpha)
    #> 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(cormatrix = 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 = n, means = myMeans, sds = mySds,
      lowerbound = lowerbound, upperbound = upperbound,
      cormatrix = myCorr
    )
    #> Variable  1
    #> best solution in 972 iterations
    #> Variable  2
    #> best solution in 17 iterations
    #> Variable  3
    #> best solution in 973 iterations
    #> Variable  4
    #> best solution in 4866 iterations
    #> Variable  5
    #> best solution in 336 iterations
    #> Variable  6
    #> best solution in 16769 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

    ## resulting data frame
    head(myItems)
    #>
    #> 1 4 3 3 4 4 5
    #> 2 2 3 3 3 3 3
    #> 3 3 3 3 4 3 4
    #> 4 4 2 3 2 4 3
    #> 5 2 4 3 4 4 5
    #> 6 4 3 3 3 4 5
    tail(myItems)
    #>
    #> 251 2 2 2 2 3 1
    #> 252 1 2 2 1 3 1
    #> 253 1 2 3 2 2 1
    #> 254 4 3 4 5 5 5
    #> 255 3 3 3 3 4 3
    #> 256 4 4 5 5 5 5

    ## means and standard deviations
    myMoments <- data.frame(
      means = apply(myItems, 2, mean) |> round(3),
      sds = apply(myItems, 2, sd) |> round(3)
    ) |> t()
    myMoments
    #>        [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
    #> 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

#### Summary plots of new data frame

Summary of dataframe from makeItems() function

## Generate a dataframe of rating-scale items from a summated rating scale

### makeItemsScale()

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

To create the desired dataframe, the user must define the following parameters:

- 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’.

#### makeItemsScale() Example:

##### generate a summated scale

    ## define parameters
    n <- 256
    mean <- 3.00
    sd <- 0.85
    lowerbound <- 1
    upperbound <- 5
    items <- 4

    ## apply lfast() function
    meanScale <- lfast(
      n = n, mean = mean, sd = sd,
      lowerbound = lowerbound, upperbound = upperbound,
      items = items
    )
    #> best solution in 900 iterations

    ## sum over all items
    summatedScale <- meanScale * items

Summated scale distribution

#### create items with makeItemsScale()

    ## apply makeItemsScale() function
    newItems_1 <- makeItemsScale(
      scale = summatedScale,
      lowerbound = lowerbound,
      upperbound = upperbound,
      items = items
    )
    #> generate 256 rows
    #> rearrange 4 values within each of 256 rows
    #> Complete!
    #> desired Cronbach's alpha = 0.8 (achieved alpha = 0.8004)

    ### First 10 observations and summated scale
    head(cbind(newItems_1, summatedScale), 10)
    #>    V1 V2 V3 V4 summatedScale
    #> 1   4  1  1  3             9
    #> 2   2  2  2  2             8
    #> 3   5  1  2  4            12
    #> 4   5  4  4  4            17
    #> 5   5  2  3  3            13
    #> 6   5  5  5  4            19
    #> 7   4  1  2  4            11
    #> 8   5  4  4  5            18
    #> 9   5  3  4  5            17
    #> 10  4  1  4  3            12

    ### correlation matrix
    cor(newItems_1) |> round(2)
    #>      V1   V2   V3   V4
    #> V1 1.00 0.56 0.61 0.51
    #> V2 0.56 1.00 0.60 0.33
    #> V3 0.61 0.60 1.00 0.39
    #> V4 0.51 0.33 0.39 1.00

    ### default Cronbach's alpha = 0.80
    alpha(data = newItems_1) |> round(4)
    #> [1] 0.8004

    ### calculate eigenvalues and print scree plot
    eigenvalues(cor(newItems_1), 1) |> round(3)

    #> cor(newItems_1)  is positive-definite
    #> [1] 2.517 0.717 0.403 0.364

#### makeItemsScale() with same summated values and higher alpha

    ## apply makeItemsScale() function
    newItems_2 <- makeItemsScale(
      scale = summatedScale,
      lowerbound = lowerbound,
      upperbound = upperbound,
      items = items,
      alpha = 0.9
    )
    #> generate 256 rows
    #> rearrange 4 values within each of 256 rows
    #> Complete!
    #> desired Cronbach's alpha = 0.9 (achieved alpha = 0.8778)

    ### First 10 observations and summated scale
    head(cbind(newItems_2, summatedScale), 10)
    #>    V1 V2 V3 V4 summatedScale
    #> 1   4  1  2  2             9
    #> 2   3  1  2  2             8
    #> 3   3  3  3  3            12
    #> 4   5  3  5  4            17
    #> 5   4  2  4  3            13
    #> 6   5  4  5  5            19
    #> 7   4  1  4  2            11
    #> 8   5  4  5  4            18
    #> 9   5  4  4  4            17
    #> 10  4  1  4  3            12

    ### correlation matrix
    cor(newItems_2) |> round(2)
    #>      V1   V2   V3   V4
    #> V1 1.00 0.58 0.68 0.64
    #> V2 0.58 1.00 0.58 0.66
    #> V3 0.68 0.58 1.00 0.73
    #> V4 0.64 0.66 0.73 1.00

    ### requested Cronbach's alpha = 0.90
    alpha(data = newItems_2) |> round(4)
    #> [1] 0.8778

    ### calculate eigenvalues and print scree plot
    eigenvalues(cor(newItems_2), 1) |> round(3)

    #> cor(newItems_2)  is positive-definite
    #> [1] 2.929 0.457 0.366 0.248

#### same summated values with lower alpha that may require higher variance

    ## apply makeItemsScale() function
    newItems_3 <- makeItemsScale(
      scale = summatedScale,
      lowerbound = lowerbound,
      upperbound = upperbound,
      items = items,
      alpha = 0.6,
      variance = 0.7
    )
    #> generate 256 rows
    #> rearrange 4 values within each of 256 rows
    #> Complete!
    #> desired Cronbach's alpha = 0.6 (achieved alpha = 0.5989)

    ### First 10 observations and summated scale
    head(cbind(newItems_3, summatedScale), 10)
    #>    V1 V2 V3 V4 summatedScale
    #> 1   1  1  3  4             9
    #> 2   1  4  2  1             8
    #> 3   2  4  4  2            12
    #> 4   3  5  4  5            17
    #> 5   2  5  2  4            13
    #> 6   4  5  5  5            19
    #> 7   1  4  3  3            11
    #> 8   5  5  5  3            18
    #> 9   4  5  5  3            17
    #> 10  2  3  2  5            12

    ### correlation matrix
    cor(newItems_3) |> round(2)
    #>      V1   V2   V3   V4
    #> V1 1.00 0.45 0.45 0.09
    #> V2 0.45 1.00 0.25 0.17
    #> V3 0.45 0.25 1.00 0.22
    #> V4 0.09 0.17 0.22 1.00

    ### requested Cronbach's alpha = 0.70
    alpha(data = newItems_3) |> round(4)
    #> [1] 0.5989

    ### calculate eigenvalues and print scree plot
    eigenvalues(cor(newItems_3), 1) |> round(3)

    #> cor(newItems_3)  is positive-definite
    #> [1] 1.862 0.946 0.742 0.450

## Create a dataframe for a t-test

Generating a data for an independent-samples t-test is trivial with LikertMakeR. But a dataframe for a paired-sample t-test is tricky because the observations are related to each other. That is, we must generate a dataframe of correlated observations.

### Independent-samples t-test

Note that such tests don’t even require the same sample-size.

    ## define parameters
    lower <- 1
    upper <- 5
    items <- 6

    ## generate two independent samples
    x1 <- lfast(
      n = 20, mean = 2.5, sd = 0.75,
      lowerbound = lower, upperbound = upper, items = items
    )
    #> reached maximum of 1024 iterations
    x2 <- lfast(
      n = 30, mean = 3.0, sd = 0.85,
      lowerbound = lower, upperbound = upper, items = items
    )
    #> reached maximum of 1024 iterations

    ## run independent-samples t-test
    t.test(x1, x2)
    #>
    #>  Welch Two Sample t-test
    #>
    #> data:  x1 and x2
    #> t = -2.1599, df = 44.101, p-value = 0.03626
    #> alternative hypothesis: true difference in means is not equal to 0
    #> 95 percent confidence interval:
    #>  -0.95576930 -0.03311959
    #> sample estimates:
    #> mean of x mean of y
    #>  2.500000  2.994444

### makePaired() paired-sample t-test

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 rating the same object before and after a treatment, the same people rating two different objects, ratings by husband & wife, etc.

makePaired() has similar parameters as for the lfast() function with the addition of a value for the desired t-statistic.

- 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.

- precision can relax the level of accuracy required, as in lfast().

### makePaired() examples

    ## define parameters
    n <- 20
    means <- c(2.5, 3.0)
    sds <- c(0.75, 0.85)
    lower <- 1
    upper <- 5
    items <- 6
    t <- -2.5

    ## run the function
    pairedDat <- makePaired(
      n = n, means = means, sds = sds,
      t_value = t,
      lowerbound = lower, upperbound = upper, items = items
    )
    #> Initial data vectors
    #> reached maximum of 1024 iterations
    #> reached maximum of 1024 iterations
    #> Rearrange values to conform with desired t-value
    #> Complete!

###### check properties of new data

    ## test function output
    str(pairedDat)
    #> 'data.frame':    20 obs. of  2 variables:
    #>  $ X1: num  3.17 3.67 1.83 2.33 2.83 ...
    #>  $ X2: num  2.5 3.67 2.83 3.5 4 ...

    cor(pairedDat) |> round(2)
    #>      X1   X2
    #> X1 1.00 0.38
    #> X2 0.38 1.00

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

    pairedMoments
    #>         X1    X2
    #> mean 2.500 2.992
    #> sd   0.759 0.851

###### run a paired-sample t-test with the new data

    ## run a paired-sample t-test
    paired_t <- t.test(pairedDat$X1, pairedDat$X2, paired = TRUE)

    paired_t
    #>
    #>  Paired t-test
    #>
    #> data:  pairedDat$X1 and pairedDat$X2
    #> t = -2.4455, df = 19, p-value = 0.02438
    #> alternative hypothesis: true mean difference is not equal to 0
    #> 95 percent confidence interval:
    #>  -0.91246509 -0.07086825
    #> sample estimates:
    #> mean difference
    #>      -0.4916667

## Create a multidimensional 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 <- 128
    lower <- 1
    upper <- 5

    ### attitude #1

    #### generate a correlation matrix
    cor_1 <- makeCorrAlpha(items = 4, alpha = 0.80)
    #> correlation values consistent with desired alpha in 8915 iterations
    #> The correlation matrix is positive definite

    #### specify moments as vectors
    means_1 <- c(2.5, 2.5, 3.0, 3.5)
    sds_1 <- c(0.75, 0.85, 0.85, 0.75)

    #### apply makeItems() function
    Att_1 <- makeItems(
      n = n, means = means_1, sds = sds_1,
      lowerbound = rep(lower, 4), upperbound = rep(upper, 4),
      cormatrix = cor_1
    )
    #> Variable  1
    #> reached maximum of 16384 iterations
    #> Variable  2
    #> best solution in 2787 iterations
    #> Variable  3
    #> best solution in 56 iterations
    #> Variable  4
    #> reached maximum of 16384 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

    ### attitude #2

    #### generate a correlation matrix
    cor_2 <- makeCorrAlpha(items = 5, alpha = 0.85)
    #> correlation values consistent with desired alpha in 9433 iterations
    #> The correlation matrix is positive definite

    #### specify moments as vectors
    means_2 <- c(2.5, 2.5, 3.0, 3.0, 3.5)
    sds_2 <- c(0.75, 0.85, 0.75, 0.85, 0.75)

    #### apply makeItems() function
    Att_2 <- makeItems(
      n, means_2, sds_2,
      rep(lower, 5), rep(upper, 5),
      cor_2
    )
    #> Variable  1
    #> reached maximum of 16384 iterations
    #> Variable  2
    #> best solution in 1284 iterations
    #> Variable  3
    #> reached maximum of 16384 iterations
    #> Variable  4
    #> best solution in 334 iterations
    #> Variable  5
    #> reached maximum of 16384 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

    ### attitude #3

    #### generate a correlation matrix
    cor_3 <- makeCorrAlpha(items = 6, alpha = 0.90)
    #> correlation values consistent with desired alpha in 37872 iterations
    #> The correlation matrix is positive definite

    #### specify moments as vectors
    means_3 <- c(2.5, 2.5, 3.0, 3.0, 3.5, 3.5)
    sds_3 <- c(0.75, 0.85, 0.85, 1.0, 0.75, 0.85)

    #### apply makeItems() function
    Att_3 <- makeItems(
      n, means_3, sds_3,
      rep(lower, 6), rep(upper, 6),
      cor_3
    )
    #> Variable  1
    #> reached maximum of 16384 iterations
    #> Variable  2
    #> best solution in 461 iterations
    #> Variable  3
    #> best solution in 149 iterations
    #> Variable  4
    #> reached maximum of 16384 iterations
    #> Variable  5
    #> reached maximum of 16384 iterations
    #> Variable  6
    #> best solution in 1779 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

    ### behavioural intention
    intent <- lfast(n, mean = 4.0, sd = 3, lowerbound = 0, upperbound = 10) |>
      data.frame()
    #> reached maximum of 16384 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()

    ### Attitude #1 moments
    A1_moments
    #>       [,1] [,2] [,3] [,4]
    #> means 2.50 2.50 3.00 3.50
    #> sds   0.75 0.85 0.85 0.75

    ### Attitude #1 correlations
    cor(Att_1) |> round(2)
    #>      [,1] [,2] [,3] [,4]
    #> [1,] 1.00 0.20 0.42 0.49
    #> [2,] 0.20 1.00 0.55 0.57
    #> [3,] 0.42 0.55 1.00 0.79
    #> [4,] 0.49 0.57 0.79 1.00

    ### Attitude #1 cronbach's alpha
    alpha(cor(Att_1)) |> round(3)
    #> [1] 0.801

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

    ### Attitude #2 moments
    A2_moments
    #>       [,1] [,2] [,3] [,4] [,5]
    #> means 2.50 2.50 3.00 3.00 3.50
    #> sds   0.75 0.85 0.75 0.85 0.75

    ### Attitude #2 correlations
    cor(Att_2) |> round(2)
    #>      [,1] [,2] [,3] [,4] [,5]
    #> [1,] 1.00 0.31 0.32 0.33 0.50
    #> [2,] 0.31 1.00 0.53 0.54 0.58
    #> [3,] 0.32 0.53 1.00 0.71 0.72
    #> [4,] 0.33 0.54 0.71 1.00 0.77
    #> [5,] 0.50 0.58 0.72 0.77 1.00

    ### Attitude #2 cronbach's alpha
    alpha(cor(Att_2)) |> round(3)
    #> [1] 0.85

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

    ### Attitude #3 moments
    A3_moments
    #>       [,1] [,2] [,3] [,4] [,5] [,6]
    #> means 2.50 2.50 3.00    3 3.50 3.50
    #> sds   0.75 0.85 0.85    1 0.75 0.85

    ### Attitude #3 correlations
    cor(Att_3) |> round(2)
    #>      [,1] [,2] [,3] [,4] [,5] [,6]
    #> [1,] 1.00 0.31 0.37 0.40 0.43 0.43
    #> [2,] 0.31 1.00 0.53 0.61 0.63 0.71
    #> [3,] 0.37 0.53 1.00 0.72 0.71 0.74
    #> [4,] 0.40 0.61 0.72 1.00 0.75 0.80
    #> [5,] 0.43 0.63 0.71 0.75 1.00 0.84
    #> [6,] 0.43 0.71 0.74 0.80 0.84 1.00

    ### Attitude #2 cronbach's alpha
    alpha(cor(Att_3)) |> round(3)
    #> [1] 0.899

    ## Behavioural Intention

    intent_moments <- data.frame(
      mean = apply(intent, 2, mean) |> round(3),
      sd = apply(intent, 2, sd) |> round(3)
    ) |> t()

    ### Intention moments
    intent_moments
    #>        int
    #> mean 4.000
    #> sd   3.004

##### correlateScales parameters

    ### target scale correlation matrix
    scale_cors <- matrix(
      c(
        1.0, 0.7, 0.6, 0.5,
        0.7, 1.0, 0.4, 0.3,
        0.6, 0.4, 1.0, 0.2,
        0.5, 0.3, 0.2, 1.0
      ),
      nrow = 4
    )

    ### bring dataframes into a list
    data_frames <- list("A1" = Att_1, "A2" = Att_2, "A3" = Att_3, "Int" = intent)

#### apply the correlateScales() function

    ### apply 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':    128 obs. of  16 variables:
    #>  $ A1_1 : num  2 3 4 3 3 3 2 2 3 2 ...
    #>  $ A1_2 : num  3 2 2 2 2 2 3 4 4 2 ...
    #>  $ A1_3 : num  3 4 4 3 3 3 3 4 4 3 ...
    #>  $ A1_4 : num  3 4 4 4 4 4 3 4 5 3 ...
    #>  $ A2_1 : num  2 3 4 2 2 2 3 3 3 2 ...
    #>  $ A2_2 : num  2 4 3 3 2 3 2 3 3 2 ...
    #>  $ A2_3 : num  2 4 5 3 3 3 3 4 4 3 ...
    #>  $ A2_4 : num  2 4 4 4 3 3 3 3 3 3 ...
    #>  $ A2_5 : num  2 5 4 4 4 3 3 4 4 4 ...
    #>  $ A3_1 : num  2 2 2 4 2 2 2 3 2 3 ...
    #>  $ A3_2 : num  2 2 3 3 2 3 2 4 2 2 ...
    #>  $ A3_3 : num  2 3 3 4 3 4 3 4 2 3 ...
    #>  $ A3_4 : num  3 3 3 4 2 3 3 5 3 4 ...
    #>  $ A3_5 : num  3 3 4 4 3 4 3 4 3 4 ...
    #>  $ A3_6 : num  3 3 4 4 3 4 4 5 3 4 ...
    #>  $ Int_1: num  4 4 2 9 2 2 3 4 9 4 ...

    ## eigenvalues of dataframe correlations
    Cor_Correlated_Scales <- cor(my_correlated_scales)
    eigenvalues(cormatrix = Cor_Correlated_Scales, scree = TRUE) |> round(2)

    #> Cor_Correlated_Scales  is positive-definite
    #>  [1] 7.05 2.26 1.11 0.93 0.80 0.74 0.62 0.45 0.42 0.35 0.30 0.27 0.23 0.20 0.15
    #> [16] 0.13

    #### Eigenvalues of predictor variable items only
    Cor_Attitude_items <- cor(my_correlated_scales[, -16])
    eigenvalues(cormatrix = Cor_Attitude_items, scree = TRUE) |> round(2)

    #> Cor_Attitude_items  is positive-definite
    #>  [1] 6.89 2.22 0.93 0.90 0.74 0.73 0.48 0.42 0.38 0.32 0.27 0.24 0.20 0.15 0.13

## 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)
    #> Alert:
    #> 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
    print(evals)
    #> [1] 2.7484991 0.8122627 0.3048151 0.1344231

# 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:

- truncnorm

- faux

- See also the rLikert() function from the excellent latent2likert package, Lalovic (2024), for an approach using optimal discretization and skew-normal distribution. latent2likert() converts continuous latent variables into ordinal categories to generate Likert scale item responses.

### 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 dataframes.

- SimCorrMix: Simulation of Correlated Data with Multiple Variable Types Including Continuous and Count Mixture Distributions on CRAN.

- 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)

- GenOrd:Simulation of Discrete Random Variables with Given Correlation Matrix and Marginal Distributions on CRAN.

- 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)

### 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.

- psych::sim.item Generate simulated data structures for circumplex, spherical, or simple structure

- psych::sim.congeneric Simulate a congeneric data set with or without minor factors See Revelle (in prep)

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

D’Alessandro, S., H. Winzar, B. Lowe, Ba.J. Babin, W. Zikmund (2020). Marketing Research 5ed, Cengage Australia. https://cengage.com.au/sem121/marketing-research-5th-edition-dalessandro-babin-zikmund

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 (2024). latent2likert: Converting Latent Variables into Likert Scale Responses. R package version 1.2.2, https://latent2likert.lalovic.io/.

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. 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

Winzar, H. (2020). LikertMakeR: Synthesise and correlate Likert scale and related rating-scale data with predefined first and second moments. CRAN: <doi: 10.32614/CRAN.package.LikertMakeR>