Author: Hume Winzar

# LikertMakeR vignette

#### Hume Winzar

#### February 2026

LikertMakeR (Winzar, 2025) lets you create synthetic Likert-scale, or related rating-scale, data.
Set the mean, standard deviation, and correlations or model coefficients, 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, regression or ANOVA coefficients, 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 core 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 covariation, scale boundaries, and number of items in a scale. Researchers can also use LikertMakeR to create dummy data 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 have bounds and discrete measurement intervals

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. In practice, all measures are bounded by the constraints of the measurement tool, meaning that they also have upper and lower boundaries and discrete units of measurement, which means that:

3. 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, implying that parametric statistics are quite acceptable.

### A single 1-5 rating scale is NOT a Likert scale - it may be an Likert-scale item.

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

### Most rating scales are skewed

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.

- makeScales() is a wrapper function for lfast() and lcor() to generate items or summated scales with predefined first and second moments and a predefined correlation matrix. This function replaces makeItems() and now includes multi-item measures.

- 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.

- makeRepeated() generates a dataframe of `k` correlated columns based on summary data from a repeated-samples ANOVA.

- makeScalesRegression() generates a dataframe based on results of output from multiple-regression - R2, standardised betas, and IV correlations (if available).

- 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.

  - reliability() Computes internal consistency reliability estimates for a single-factor scale, including Cronbach’s alpha, McDonald’s omega (total), and optional ordinal (polychoric-based) variants and Confidence intervals

# 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 891 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 2099 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 187 iterations
    #> best solution in 2911 iterations
    #> best solution in 950 iterations

The first six observations from this dataframe are:

    #>    x1  x2  x3
    #> 1 3.0 1.4 1.4
    #> 2 2.6 5.0 3.6
    #> 3 3.6 5.0 2.6
    #> 4 3.2 2.2 1.8
    #> 5 1.8 3.6 1.8
    #> 6 2.6 4.8 2.4

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

    #>         x1  x2    x3
    #> mean 2.500 3.0 3.498
    #> sd   0.748 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 -0.10 -0.06
    #> x2 -0.10  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  1.8 1.0 1.4
    #> 2  2.6 3.8 3.6
    #> 3  2.0 1.2 2.4
    #> 4  2.0 2.6 3.0
    #> 5  2.8 3.6 2.0
    #> 6  1.6 1.4 3.8
    #> 7  3.4 4.2 4.2
    #> 8  4.0 5.0 4.2
    #> 9  2.4 1.4 4.2
    #> 10 2.8 4.0 4.4

And the new dataframe 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 = ‘0.1’. A value of ‘0’ produces a matrix where all off-diagonal correlations are equal. Setting ‘variance = 0.25’ or more may be infeasible, so the function should gracefully adjust the `variance` parameter downwards to achieve PD status.

- alpha_noise: Controls random variation in the target Cronbach’s alpha before the correlation matrix is constructed.

When `alpha_noise = 0` (default), the requested alpha is reproduced deterministically (subject to numerical tolerance).

When `alpha_noise > 0`, a small amount of random variation is added to the requested alpha prior to constructing the matrix. This can be useful in teaching or simulation settings where slightly different reliability values are desired across repeated runs.

Internally, noise is added on the Fisher z-transformed scale to ensure the resulting alpha remains within valid bounds (0, 1).

Typical guidance:

- 0.00 — deterministic alpha (default)
- 0.02 — very small variation
- 0.05 — moderate variation
- 0.10 — substantial variation (caution)

Larger values increase the spread of achieved alpha across runs.

- diagnostics: Logical. If `TRUE`, returns a list containing the correlation matrix and a diagnostics list (target/achieved alpha, average inter-item correlation, eigenvalues, PD flag, and key arguments). If `FALSE` (default), returns the correlation matrix only.

#### makeCorrAlpha() examples

##### Four variables, alpha = 0.85

    ## define parameters
    items <- 4
    alpha <- 0.85

    ## apply makeCorrAlpha() function
    set.seed(42)

    cor_matrix_4 <- makeCorrAlpha(items, alpha)

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

    #>        item01 item02 item03 item04
    #> item01  1.000  0.565  0.667  0.697
    #> item02  0.565  1.000  0.484  0.506
    #> item03  0.667  0.484  1.000  0.598
    #> item04  0.697  0.506  0.598  1.000

##### test output with Helper functions

    ## using helper function alpha()

    alpha(cor_matrix_4)
    #> [1] 0.8499981

    ## using helper function eigenvalues()

    eigenvalues(cor_matrix_4, 1)

    #> cor_matrix_4  is positive-definite
    #> [1] 2.7665013 0.5480310 0.4036204 0.2818473

#### makeCorrAlpha() with diagnostics output

    ## apply makeCorrAlpha() with diagnostics
    set.seed(42)

    cor_matrix_5 <- makeCorrAlpha(
      items = 6,
      alpha = 0.90,
      diagnostics = TRUE
    )

##### diagnostics output

    ## output
    cor_matrix_5$R |> round(2)
    #>        item01 item02 item03 item04 item05 item06
    #> item01   1.00   0.59   0.72   0.76   0.73   0.66
    #> item02   0.59   1.00   0.50   0.52   0.50   0.45
    #> item03   0.72   0.50   1.00   0.64   0.61   0.55
    #> item04   0.76   0.52   0.64   1.00   0.64   0.58
    #> item05   0.73   0.50   0.61   0.64   1.00   0.55
    #> item06   0.66   0.45   0.55   0.58   0.55   1.00

    cor_matrix_5$diagnostics
    #> $items
    #> [1] 6
    #>
    #> $alpha_input
    #> [1] 0.9
    #>
    #> $alpha_target_effective
    #> [1] 0.9
    #>
    #> $alpha_achieved
    #> [1] 0.8999999
    #>
    #> $average_r
    #> [1] 0.5999997
    #>
    #> $min_eigenvalue
    #> [1] 0.192857
    #>
    #> $variance_input
    #> [1] 0.1
    #>
    #> $internal_variance_used
    #> [1] 0.1
    #>
    #> $alpha_noise
    #> [1] 0

## 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

### makeScales()

makeScales() generates a dataframe of random discrete values so the data replicate a set of scale items or summated rating scales, and are correlated close to a predefined correlation matrix.

Generally, means, standard deviations, and correlations are correct to two decimal places.

makeScales() 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. default = ‘1’

- upperbound: a vector of length `k` of values for the upper bound of each variable. Default = ‘5’

- items: a vector of length `k` of the number of items in each variable. Default = ‘1’.

- cormatrix: a target correlation matrix with `k` rows and `k` columns.

### makeScales() examples

#### makeScales() example #1. four correlated items

    ## 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
    )

    var_names <- c("var1", "var2", "var3", "var4")
    colnames(corMat) <- var_names
    rownames(corMat) <- var_names

    ## apply makeScales() function
    df <- makeScales(
      n = n,
      means = dfMeans,
      sds = dfSds,
      lowerbound = lowerbound,
      upperbound = upperbound,
      cormatrix = corMat
    )
    #> Variable  1 :  var1  -
    #> reached maximum of 16384 iterations
    #> Variable  2 :  var2  -
    #> reached maximum of 16384 iterations
    #> Variable  3 :  var3  -
    #> best solution in 2246 iterations
    #> Variable  4 :  var4  -
    #> reached maximum of 16384 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

##### Structure of new dataframe

    ## test the function
    str(df)
    #> 'data.frame':    128 obs. of  4 variables:
    #>  $ var1: num  1 2 3 2 3 2 3 2 2 2 ...
    #>  $ var2: num  3 3 4 2 3 3 4 3 4 4 ...
    #>  $ var3: num  2 2 5 1 4 2 5 4 3 4 ...
    #>  $ var4: num  3 3 4 3 4 3 4 4 4 4 ...

##### Means should be correct to two decimal places

    ### 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
    #>       var1  var2  var3  var4
    #> 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

    ### correlations should be correct to two decimal places
    cor(df) |> round(3)
    #>       var1 var2 var3  var4
    #> var1 1.000 0.25 0.35 0.448
    #> var2 0.250 1.00 0.70 0.750
    #> var3 0.350 0.70 1.00 0.850
    #> var4 0.448 0.75 0.85 1.000

#### makeScales() example #2. four Likert scales

- Brand Trust (BT) - The confidence a consumer has in a brand’s reliability and honesty.
- Brand Satisfaction (BS) - Overall affective evaluation of the brand experience.
- Brand Love (BL) - Deep emotional attachment toward the brand.
- Brand Loyalty (BLY) - Intention to repurchase and recommend the brand.

    ## define parameters
    n <- 256
    dfMeans <- c(3.9, 4.1, 3.6, 4.0)
    dfSds <- c(0.6, 0.5, 0.8, 0.7)
    lowerbound <- rep(1, 4)
    upperbound <- rep(5, 4)
    items <- c(4, 3, 4, 3)

    corMat <- matrix(
      c(
        1.00, 0.75, 0.60, 0.70,
        0.75, 1.00, 0.65, 0.72,
        0.60, 0.65, 1.00, 0.68,
        0.70, 0.72, 0.68, 1.00
      ),
      nrow = 4, ncol = 4
    )

    scale_names <- c("BT", "BS", "BL", "BLY")
    rownames(corMat) <- scale_names
    colnames(corMat) <- scale_names

    ## apply makeScales() function
    df <- makeScales(
      n = n,
      means = dfMeans,
      sds = dfSds,
      lowerbound = lowerbound,
      upperbound = upperbound,
      items = items,
      cormatrix = corMat
    )
    #> Variable  1 :  BT  -
    #> best solution in 754 iterations
    #> Variable  2 :  BS  -
    #> best solution in 104 iterations
    #> Variable  3 :  BL  -
    #> best solution in 972 iterations
    #> Variable  4 :  BLY  -
    #> best solution in 1372 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

    ## test the function
    head(df)
    #>    BT       BS   BL      BLY
    #> 1 3.5 4.000000 2.75 3.000000
    #> 2 4.5 5.000000 4.75 5.000000
    #> 3 4.0 4.000000 2.25 4.000000
    #> 4 3.5 4.000000 3.75 3.666667
    #> 5 2.5 2.666667 1.75 3.666667
    #> 6 4.0 4.000000 3.50 4.666667
    tail(df)
    #>       BT       BS   BL      BLY
    #> 251 4.50 4.333333 4.00 4.333333
    #> 252 3.25 3.000000 1.50 3.333333
    #> 253 3.75 3.333333 3.50 2.666667
    #> 254 4.50 4.666667 4.50 5.000000
    #> 255 4.50 4.666667 4.75 5.000000
    #> 256 4.25 4.000000 3.00 2.666667

    ### 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
    #>         BT    BS    BL BLY
    #> mean 3.901 4.102 3.599 4.0
    #> sd   0.600 0.499 0.799 0.7

    ### correlations should be correct to two decimal places
    cor(df) |> round(3)
    #>       BT    BS   BL   BLY
    #> BT  1.00 0.750 0.60 0.700
    #> BS  0.75 1.000 0.65 0.719
    #> BL  0.60 0.650 1.00 0.680
    #> BLY 0.70 0.719 0.68 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 makeScales() 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)

    ## display correlation matrix
    myCorr |> round(3)
    #>        item01 item02 item03 item04 item05 item06
    #> item01  1.000  0.477  0.597  0.632  0.603  0.536
    #> item02  0.477  1.000  0.392  0.414  0.395  0.352
    #> item03  0.597  0.392  1.000  0.519  0.495  0.440
    #> item04  0.632  0.414  0.519  1.000  0.523  0.466
    #> item05  0.603  0.395  0.495  0.523  1.000  0.444
    #> item06  0.536  0.352  0.440  0.466  0.444  1.000

    ### checking Cronbach's Alpha
    alpha(cormatrix = myCorr)
    #> [1] 0.8499932

### 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 <- makeScales(
      n = n, means = myMeans, sds = mySds,
      lowerbound = lowerbound, upperbound = upperbound,
      items = 1,
      cormatrix = myCorr
    )
    #> Variable  1 :  item01  -
    #> best solution in 288 iterations
    #> Variable  2 :  item02  -
    #> best solution in 2220 iterations
    #> Variable  3 :  item03  -
    #> best solution in 1058 iterations
    #> Variable  4 :  item04  -
    #> best solution in 2125 iterations
    #> Variable  5 :  item05  -
    #> best solution in 452 iterations
    #> Variable  6 :  item06  -
    #> best solution in 12575 iterations
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

    ## resulting dataframe
    head(myItems)
    #>   item01 item02 item03 item04 item05 item06
    #> 1      3      2      2      4      4      5
    #> 2      3      3      4      4      4      5
    #> 3      1      3      3      3      3      4
    #> 4      3      4      3      3      4      3
    #> 5      3      4      3      3      3      5
    #> 6      3      3      3      3      3      5
    tail(myItems)
    #>     item01 item02 item03 item04 item05 item06
    #> 251      3      2      4      3      3      3
    #> 252      3      4      3      4      4      5
    #> 253      3      2      3      2      3      5
    #> 254      2      2      4      4      3      2
    #> 255      3      2      1      2      3      5
    #> 256      3      3      3      3      3      4

    ## means and standard deviations
    myMoments <- data.frame(
      means = apply(myItems, 2, mean) |> round(3),
      sds = apply(myItems, 2, sd) |> round(3)
    ) |> t()
    myMoments
    #>       item01 item02 item03 item04 item05 item06
    #> means  2.750  3.000  3.000  3.250  3.500  3.500
    #> sds    0.998  0.751  0.998  0.998  1.002  1.498

    ## Cronbach's Alpha of dataframe
    alpha(NULL, myItems)
    #> [1] 0.8500638

#### Summary plots of new dataframe

Summary of dataframe from makeScales() 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, 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’

- summated: (logical) If TRUE, the scale is treated as a summed score (e.g., 4–20 for four 5-point items). If FALSE, it is treated as an averaged score (e.g., 1–5 in 0.25 increments). Default = TRUE.

- progress: (logical) If TRUE (default is `interactive()`), a progress bar is displayed during optimisation. Set to FALSE for slightly faster execution.

#### makeItemsScale() Example:

#### create items with makeItemsScale()

    summatedScale <- c(11, 8, 14, 12, 9, 12, 10, 9, 8, 18, 16, 16, 14, 8, 15, 12)

    ## apply makeItemsScale() function

    newItems <- makeItemsScale(
      scale = summatedScale,
      lowerbound = 1,
      upperbound = 5,
      items = 4
    )
    #> generate 16 rows
    #> rearrange 4 values within each of 16 rows
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    #> reached maximum of 512 iterations
    #> Complete!
    #> desired Cronbach's alpha = 0.8 (achieved alpha = 0.8002)

    ###  observations and summated scale
    head(cbind(newItems, summatedScale), 16)
    #>    X1 X2 X3 X4 summatedScale
    #> 1   2  4  3  2            11
    #> 2   2  2  2  2             8
    #> 3   4  3  4  3            14
    #> 4   3  3  3  3            12
    #> 5   4  2  2  1             9
    #> 6   4  2  3  3            12
    #> 7   2  2  3  3            10
    #> 8   3  2  1  3             9
    #> 9   1  3  2  2             8
    #> 10  4  4  5  5            18
    #> 11  3  5  4  4            16
    #> 12  4  4  4  4            16
    #> 13  3  4  4  3            14
    #> 14  2  2  2  2             8
    #> 15  4  3  3  5            15
    #> 16  3  3  3  3            12

    ### correlation matrix
    cor(newItems) |> round(2)
    #>      X1   X2   X3   X4
    #> X1 1.00 0.14 0.47 0.50
    #> X2 0.14 1.00 0.73 0.50
    #> X3 0.47 0.73 1.00 0.65
    #> X4 0.50 0.50 0.65 1.00

    ### default Cronbach's alpha = 0.80
    alpha(data = newItems) |> round(4)
    #> [1] 0.8002

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

    #> cor(newItems)  is positive-definite
    #> [1] 2.538 0.889 0.385 0.188

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

    ## apply makeItemsScale() function
    newItems_2 <- makeItemsScale(
      scale = summatedScale,
      lowerbound = 1,
      upperbound = 5,
      items = 4,
      alpha = 0.9
    )
    #> generate 16 rows
    #> rearrange 4 values within each of 16 rows
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    #> reached maximum of 512 iterations
    #> Complete!
    #> desired Cronbach's alpha = 0.9 (achieved alpha = 0.9091)

    ### observations and summated scale
    head(cbind(newItems_2, summatedScale), 16)
    #>    X1 X2 X3 X4 summatedScale
    #> 1   3  3  2  3            11
    #> 2   2  2  2  2             8
    #> 3   3  4  3  4            14
    #> 4   3  3  3  3            12
    #> 5   2  2  3  2             9
    #> 6   3  3  3  3            12
    #> 7   2  3  3  2            10
    #> 8   2  2  3  2             9
    #> 9   2  2  2  2             8
    #> 10  5  5  4  4            18
    #> 11  4  4  4  4            16
    #> 12  4  4  4  4            16
    #> 13  3  3  4  4            14
    #> 14  3  2  2  1             8
    #> 15  4  4  3  4            15
    #> 16  3  2  3  4            12

    ### correlation matrix
    cor(newItems_2) |> round(2)
    #>      X1   X2   X3   X4
    #> X1 1.00 0.85 0.61 0.72
    #> X2 0.85 1.00 0.66 0.73
    #> X3 0.61 0.66 1.00 0.71
    #> X4 0.72 0.73 0.71 1.00

    ### requested Cronbach's alpha = 0.90
    alpha(data = newItems_2) |> round(4)
    #> [1] 0.9091

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

    #> cor(newItems_2)  is positive-definite
    #> [1] 3.147 0.441 0.265 0.148

## 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 = -2.2008, df = 44.113, p-value = 0.03303
    #> alternative hypothesis: true difference in means is not equal to 0
    #> 95 percent confidence interval:
    #>  -0.95783664 -0.04216336
    #> sample estimates:
    #> mean of x mean of y
    #>       2.5       3.0

### 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
    #> Arranging 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  3.33 3 3 2.5 1.67 ...
    #>  $ X2: num  1.5 3.67 3.5 2.83 3.83 ...

    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 2.508 3.000
    #> sd   0.754 0.858

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

    ## run a paired-sample t-test

    paired_t <- t.test(x = pairedDat$X1, y = pairedDat$X2, paired = TRUE)

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

    paired_t
    #>
    #>  Paired t-test
    #>
    #> data:  pairedDat$X1 and pairedDat$X2
    #> t = -2.4411, df = 19, p-value = 0.0246
    #> alternative hypothesis: true mean difference is not equal to 0
    #> 95 percent confidence interval:
    #>  -0.91322541 -0.07010793
    #> sample estimates:
    #> mean difference
    #>      -0.4916667

## Create a dataframe for Repeated-Measures ANOVA

### makeRepeated()

makeRepeated() Reconstructs a synthetic dataset and inter-timepoint correlation matrix from a repeated-measures ANOVA result, based on reported means, standard deviations, and an F-statistic.

This function estimates the average correlation between repeated measures by matching the reported F-statistic, under one of three assumed correlation structures:

- `"cs"` (Compound Symmetry): Compound Symmetry assumes that all repeated measures are equally correlated with each other. That is, the correlation between time 1 and time 2 is the same as between time 1 and time 3, and so on. This structure is commonly used in repeated-measures ANOVA by default. It’s mathematically simple and reflects the idea that all timepoints are equally related. However, it may not be realistic for data where correlations decrease as time intervals increase (e.g., memory decay or learning effects).

- `"ar1"` (First-Order Autoregressive): first-order autoregressive, assumes that measurements closer together in time are more highly correlated than those further apart. For example, the correlation between time 1 and time 2 is stronger than between time 1 and time 3. This pattern is often realistic in longitudinal or time-series studies where change is gradual. The correlation drops off exponentially with each time step.

- `"toeplitz"` (Linearly Decreasing): Toeplitz structure is a more flexible option that allows the correlation between measurements to decrease linearly as the time gap increases. Unlike AR(1), where the decline is exponential, the Toeplitz structure assumes a straight-line drop in correlation.

#### makeRepeated() usage

    makeRepeated(
      n,
      k,
      means,
      sds,
      f_stat,
      df_between = k - 1,
      df_within = (n - 1) * (k - 1),
      structure = c("cs", "ar1", "toeplitz"),
      names = paste0("time_", 1:k),
      items = 1,
      lowerbound = 1, upperbound = 5,
      return_corr_only = FALSE,
      diagnostics = FALSE,
      ...
    )

#### makeRepeated() arguments

- n Integer. Sample size used in the original study.
- k Integer. Number of repeated measures (timepoints).
- means Numeric vector of length `k`. Mean values reported for each timepoint.
- sds Numeric vector of length `k`. Standard deviations reported for each timepoint.
- f_stat Numeric. The reported repeated-measures ANOVA F-statistic for the within-subjects factor.
- df_between, Degrees of freedom between conditions (default: `k - 1`).
- df_within, Degrees of freedom within-subjects (default: `(n - 1) * (k - 1)`).
- structure Character. Correlation structure to assume: `"cs"`, `"ar1"`, or `"toeplitz"` (default).
- names Character vector of length `k`. Variable names for each timepoint (default: `"time_1"` to `"time_k"`).
- items Integer. Number of items used to generate each scale score (passed to `link{lfast}`). (default: `1`).
- lowerbound, Integer. Lower bounds for Likert-type response scales (default: `1`).
- upperbound, Integer. upper bounds for Likert-type response scales (default: `5`).
- return_corr_only Logical. If `TRUE`, return only the estimated correlation matrix.
- diagnostics Logical. If `TRUE`, include diagnostic summaries such as feasible F-statistic range and effect sizes.

#### makeRepeated() examples

    out1 <- makeRepeated(
      n = 128,
      k = 3,
      means = c(3.1, 3.5, 3.9),
      sds = c(1.0, 1.1, 1.0),
      items = 4,
      f_stat = 4.87,
      structure = "cs",
      diagnostics = FALSE
    )
    #> Warning in makeRepeated(n = 128, k = 3, means = c(3.1, 3.5, 3.9), sds = c(1, :
    #> Optimization may not have converged. Check results carefully.
    #> best solution in 1237 iterations
    #> best solution in 144 iterations
    #> best solution in 355 iterations

    head(out1$data)
    #>   time_1 time_2 time_3
    #> 1   2.75   2.50    5.0
    #> 2   4.50   4.00    2.0
    #> 3   3.00   4.25    3.5
    #> 4   4.25   4.25    2.0
    #> 5   1.75   3.50    5.0
    #> 6   1.25   5.00    5.0
    out1$correlation_matrix
    #>            time_1     time_2     time_3
    #> time_1  1.0000000 -0.4899454 -0.4899454
    #> time_2 -0.4899454  1.0000000 -0.4899454
    #> time_3 -0.4899454 -0.4899454  1.0000000

    out2 <- makeRepeated(
      n = 32, k = 4,
      means = c(2.75, 3.5, 4.0, 4.4),
      sds = c(0.8, 1.0, 1.2, 1.0),
      f_stat = 16,
      structure = "ar1",
      items = 5,
      lowerbound = 1, upperbound = 7,
      return_corr_only = FALSE,
      diagnostics = TRUE
    )
    #> best solution in 63 iterations
    #> reached maximum of 1024 iterations
    #> reached maximum of 1024 iterations
    #> best solution in 862 iterations

    print(out2)
    #> $data
    #>    time_1 time_2 time_3 time_4
    #> 1     1.6    3.2    2.0    3.0
    #> 2     4.6    4.6    5.4    3.8
    #> 3     2.6    2.6    5.0    3.2
    #> 4     3.6    3.0    4.4    4.6
    #> 5     2.2    3.0    5.2    5.2
    #> 6     2.4    2.2    1.8    5.6
    #> 7     1.8    2.8    2.4    3.0
    #> 8     3.8    4.0    3.4    5.2
    #> 9     2.2    2.6    4.4    5.4
    #> 10    2.2    1.4    2.4    3.6
    #> 11    1.8    2.6    3.4    4.6
    #> 12    3.0    4.0    3.6    3.4
    #> 13    3.4    3.4    2.2    3.2
    #> 14    2.2    4.6    5.2    4.0
    #> 15    2.6    3.4    2.0    2.6
    #> 16    2.8    5.4    4.6    5.4
    #> 17    2.2    3.4    5.8    6.0
    #> 18    3.4    2.6    4.8    4.0
    #> 19    1.2    2.6    4.0    3.0
    #> 20    2.4    3.8    5.6    5.2
    #> 21    2.2    4.4    4.4    5.8
    #> 22    3.4    3.6    4.0    5.0
    #> 23    3.4    3.2    4.2    4.6
    #> 24    1.8    2.2    3.4    4.0
    #> 25    2.8    4.4    6.4    5.4
    #> 26    1.8    4.4    4.6    5.0
    #> 27    3.4    4.4    3.4    5.6
    #> 28    3.6    2.6    3.4    4.2
    #> 29    3.6    5.4    5.0    3.0
    #> 30    2.8    3.8    3.2    4.8
    #> 31    3.8    5.2    3.6    4.0
    #> 32    3.4    3.2    4.6    5.4
    #>
    #> $correlation_matrix
    #>            time_1    time_2    time_3     time_4
    #> time_1 1.00000000 0.3910032 0.1528835 0.05977794
    #> time_2 0.39100319 1.0000000 0.3910032 0.15288350
    #> time_3 0.15288350 0.3910032 1.0000000 0.39100319
    #> time_4 0.05977794 0.1528835 0.3910032 1.00000000
    #>
    #> $structure
    #> [1] "ar1"
    #>
    #> $feasible_f_range
    #>       min       max
    #>  9.353034 39.481390
    #>
    #> $recommended_f
    #> $recommended_f$conservative
    #> [1] 10.21
    #>
    #> $recommended_f$moderate
    #> [1] 11.91
    #>
    #> $recommended_f$strong
    #> [1] 30.29
    #>
    #>
    #> $achieved_f
    #> [1] 15.99983
    #>
    #> $effect_size_raw
    #> [1] 0.3792188
    #>
    #> $effect_size_standardised
    #> [1] 0.3717831

    out3 <- makeRepeated(
      n = 32, k = 4,
      means = c(2.0, 2.5, 3.0, 2.8),
      sds = c(0.8, 0.9, 1.0, 0.9),
      items = 4,
      f_stat = 24,
      structure = "toeplitz",
      diagnostics = TRUE
    )
    #> Warning in makeRepeated(n = 32, k = 4, means = c(2, 2.5, 3, 2.8), sds = c(0.8,
    #> : Optimization may not have converged. Check results carefully.
    #> best solution in 253 iterations
    #> reached maximum of 1024 iterations
    #> reached maximum of 1024 iterations
    #> reached maximum of 1024 iterations

    str(out3)
    #> List of 8
    #>  $ data                    :'data.frame':    32 obs. of  4 variables:
    #>   ..$ time_1: num [1:32] 3 1.25 2.75 1.25 1 2.75 3.25 2.5 2.25 1 ...
    #>   ..$ time_2: num [1:32] 4 2.25 3 2 2 4.25 3 3.5 2.25 1 ...
    #>   ..$ time_3: num [1:32] 4.5 2.25 3.75 3.25 2.25 4.75 3.25 4 3.5 1.75 ...
    #>   ..$ time_4: num [1:32] 3.75 2 4.25 3 4 4.25 2.25 3 2.25 2.25 ...
    #>  $ correlation_matrix      : num [1:4, 1:4] 1 0.66 0.33 0 0.66 ...
    #>   ..- attr(*, "dimnames")=List of 2
    #>   .. ..$ : chr [1:4] "time_1" "time_2" "time_3" "time_4"
    #>   .. ..$ : chr [1:4] "time_1" "time_2" "time_3" "time_4"
    #>  $ structure               : chr "toeplitz"
    #>  $ feasible_f_range        : Named num [1:2] 5.57 8.64
    #>   ..- attr(*, "names")= chr [1:2] "min" "max"
    #>  $ recommended_f           :List of 3
    #>   ..$ conservative: num 5.59
    #>   ..$ moderate    : num 5.62
    #>   ..$ strong      : num 7.64
    #>  $ achieved_f              : num 9.95
    #>  $ effect_size_raw         : num 0.142
    #>  $ effect_size_standardised: num 0.174

## Generate rating-scale data from multiple regression results

### makeScalesRegression()

Generates synthetic rating-scale data that replicates reported regression results: standardised betas, `R^2`, and correlation matrix of independent variables (if available).

#### makeScalesRegression() usage

    makeScalesRegression <- (
       n,
       beta_std,
       r_squared,
       iv_cormatrix = NULL,
       iv_cor_mean = 0.3,
       iv_cor_variance = 0.01,
       iv_cor_range = c(-0.7, 0.7),
       iv_means,
       iv_sds,
       dv_mean,
       dv_sd,
       lowerbound_iv,
       upperbound_iv,
       lowerbound_dv,
       upperbound_dv,
       items_iv = 1,
       items_dv = 1,
       var_names = NULL,
       tolerance = 0.005
    )

#### makeScalesRegression() arguments

- n sample size.
- beta_std a vector of length `k` (number of independent variables) of standardised betas.
- r_squared model `R^2`
- iv_cormatrix independent variables correlation matrix. Default= `NULL`
- iv_cor_mean if no iv_cormatrix, average IV correlations. Default = `0.3`
- iv_cor_variance if no iv_cormatrix, variation in iv_cormatrix. Default = `0.01`
- iv_cor_range if no iv_cormatrix, range in iv_cormatrix. Default = `c(-0.7, 0.7)`
- iv_means a vector of length `k` of IV mean values
- iv_sds a vector of length `k` of IV standard deviations
- dv_mean mean of Dependent Variable (DV)
- dv_sd standard deviation of DV
- lowerbound_iv a vector of length `k` of lowerbounds for IV’s
- upperbound_iv a vector of length `k` of upperbounds for IV’s
- lowerbound_dv lowerbound for DV
- upperbound_dv upperbound for DV
- items_iv a vector of length `k` of number of items in the IV’s. Default = `1`.
- items_dv number of items in DV. Default = `1`.
- var_names a vector of variable names (Independent Variables first then the Dependent Variable). Default = `NULL`
- tolerance close to target R-squared. Default = `0.005`

#### makeScalesRegression() examples

##### Example 1: With provided IV correlation matrix

    set.seed(123)
    iv_corr <- matrix(c(1.0, 0.3, 0.3, 1.0), nrow = 2)

    result1 <- makeScalesRegression(
      n = 64,
      beta_std = c(0.4, 0.3),
      r_squared = 0.35,
      iv_cormatrix = iv_corr,
      iv_means = c(3.0, 3.5),
      iv_sds = c(1.0, 0.9),
      dv_mean = 3.8,
      dv_sd = 1.1,
      lowerbound_iv = 1,
      upperbound_iv = 5,
      lowerbound_dv = 1,
      upperbound_dv = 5,
      items_iv = 4,
      items_dv = 4,
      var_names = c("Attitude", "Intention", "Behaviour")
    )

    print(result1)
    head(result1$data)

##### Example 2: With optimisation (no IV correlation matrix)

    set.seed(456)
    result2 <- makeScalesRegression(
      n = 64,
      beta_std = c(0.3, 0.25, 0.2),
      r_squared = 0.40,
      iv_cormatrix = NULL, # Will be optimised
      iv_cor_mean = 0.3,
      iv_cor_variance = 0.02,
      iv_means = c(3.0, 3.2, 2.8),
      iv_sds = c(1.0, 0.9, 1.1),
      dv_mean = 3.5,
      dv_sd = 1.0,
      lowerbound_iv = 1,
      upperbound_iv = 5,
      lowerbound_dv = 1,
      upperbound_dv = 5,
      items_iv = 4,
      items_dv = 5
    )

    # View optimised correlation matrix
    print(result2$target_stats$iv_cormatrix)
    print(result2$optimisation_info)

## 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)

    #### 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 makeScales() function
    Att_1 <- makeScales(
      n = n, means = means_1, sds = sds_1,
      lowerbound = rep(lower, 4), upperbound = rep(upper, 4),
      items = 1,
      cormatrix = cor_1
    )
    #> Variable  1 :  item01  -
    #> reached maximum of 16384 iterations
    #> Variable  2 :  item02  -
    #> best solution in 71 iterations
    #> Variable  3 :  item03  -
    #> best solution in 51 iterations
    #> Variable  4 :  item04  -
    #> 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)

    #### 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 makeScales() function
    Att_2 <- makeScales(
      n, means_2, sds_2,
      rep(lower, 5), rep(upper, 5),
      items = 1,
      cor_2
    )
    #> Variable  1 :  item01  -
    #> reached maximum of 16384 iterations
    #> Variable  2 :  item02  -
    #> best solution in 54 iterations
    #> Variable  3 :  item03  -
    #> reached maximum of 16384 iterations
    #> Variable  4 :  item04  -
    #> best solution in 352 iterations
    #> Variable  5 :  item05  -
    #> 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)

    #### 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 makeScales() function
    Att_3 <- makeScales(
      n, means_3, sds_3,
      rep(lower, 6), rep(upper, 6),
      items = 1,
      cor_3
    )
    #> Variable  1 :  item01  -
    #> reached maximum of 16384 iterations
    #> Variable  2 :  item02  -
    #> best solution in 155 iterations
    #> Variable  3 :  item03  -
    #> best solution in 384 iterations
    #> Variable  4 :  item04  -
    #> reached maximum of 16384 iterations
    #> Variable  5 :  item05  -
    #> reached maximum of 16384 iterations
    #> Variable  6 :  item06  -
    #> best solution in 1290 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()
    #> best solution in 4398 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
    #>       item01 item02 item03 item04
    #> 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)
    #>        item01 item02 item03 item04
    #> item01   1.00   0.58   0.60   0.50
    #> item02   0.58   1.00   0.49   0.41
    #> item03   0.60   0.49   1.00   0.42
    #> item04   0.50   0.41   0.42   1.00

    ### Attitude #1 cronbach's alpha
    alpha(cor(Att_1)) |> round(3)
    #> [1] 0.799

    ## 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
    #>       item01 item02 item03 item04 item05
    #> 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)
    #>        item01 item02 item03 item04 item05
    #> item01   1.00   0.55   0.61   0.41   0.56
    #> item02   0.55   1.00   0.63   0.42   0.58
    #> item03   0.61   0.63   1.00   0.47   0.64
    #> item04   0.41   0.42   0.47   1.00   0.43
    #> item05   0.56   0.58   0.64   0.43   1.00

    ### Attitude #2 cronbach's alpha
    alpha(cor(Att_2)) |> round(3)
    #> [1] 0.849

    ## 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
    #>       item01 item02 item03 item04 item05 item06
    #> 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)
    #>        item01 item02 item03 item04 item05 item06
    #> item01   1.00   0.49   0.54   0.43   0.53   0.54
    #> item02   0.49   1.00   0.67   0.53   0.66   0.67
    #> item03   0.54   0.67   1.00   0.58   0.72   0.74
    #> item04   0.43   0.53   0.58   1.00   0.57   0.58
    #> item05   0.53   0.66   0.72   0.57   1.00   0.72
    #> item06   0.54   0.67   0.74   0.58   0.72   1.00

    ### Attitude #2 cronbach's alpha
    alpha(cor(Att_3)) |> round(3)
    #> [1] 0.9

    ## 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   2.999

##### 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 2 3 3 2 3 3 2 3 2 ...
    #>  $ A1_2 : num  2 2 3 3 3 4 4 2 2 2 ...
    #>  $ A1_3 : num  2 3 3 4 1 3 3 2 4 3 ...
    #>  $ A1_4 : num  3 4 4 4 4 4 4 4 3 4 ...
    #>  $ A2_1 : num  2 3 2 2 2 4 4 3 2 3 ...
    #>  $ A2_2 : num  3 3 1 2 2 4 4 2 2 3 ...
    #>  $ A2_3 : num  3 4 2 3 2 5 4 3 3 4 ...
    #>  $ A2_4 : num  3 5 4 3 2 3 3 2 3 3 ...
    #>  $ A2_5 : num  3 3 3 4 2 4 4 4 4 4 ...
    #>  $ A3_1 : num  2 3 2 4 4 2 2 2 3 2 ...
    #>  $ A3_2 : num  2 2 3 3 3 3 2 2 3 3 ...
    #>  $ A3_3 : num  2 3 2 4 3 4 3 4 4 3 ...
    #>  $ A3_4 : num  3 3 3 4 3 4 3 4 4 4 ...
    #>  $ A3_5 : num  3 4 4 5 4 4 4 3 4 4 ...
    #>  $ A3_6 : num  3 4 4 5 4 4 3 3 5 3 ...
    #>  $ Int_1: num  5 1 1 6 1 5 4 1 7 2 ...

    ## 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] 6.93 2.26 1.14 0.75 0.68 0.61 0.53 0.50 0.43 0.41 0.36 0.34 0.29 0.28 0.26
    #> [16] 0.22

    #### 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] 6.77 2.23 0.87 0.69 0.65 0.58 0.51 0.49 0.43 0.36 0.34 0.31 0.29 0.26 0.22

## 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.

- reliability presents a table of internal consistency statistics

### 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

### reliability()

reliabiity() Computes internal consistency reliability estimates for a single-factor scale, including Cronbach’s alpha, McDonald’s omega (total), and optional ordinal (polychoric-based) variants and Confidence intervals.

### reliability() examples

    ## create dataset
    my_cor <- LikertMakeR::makeCorrAlpha(
      items = 4,
      alpha = 0.80
    )

    my_data <- LikertMakeR::makeScales(
      n = 64,
      means = c(2.75, 3.00, 3.25, 3.50),
      sds = c(1.25, 1.50, 1.30, 1.25),
      lowerbound = rep(1, 4),
      upperbound = rep(5, 4),
      cormatrix = my_cor
    )
    #> Variable  1 :  item01  -
    #> Variable  2 :  item02  -
    #> Variable  3 :  item03  -
    #> Variable  4 :  item04  -
    #>
    #> Arranging data to match correlations
    #>
    #> Successfully generated correlated variables

    ## run function
    reliability(my_data)
    #>    coef_name estimate n_items n_obs                notes
    #>        alpha    0.798       4    64 Pearson correlations
    #>  omega_total    0.869       4    64 1-factor eigen omega

    reliability(
      my_data,
      include = c("lambda6", "polychoric")
    )
    #>            coef_name estimate n_items n_obs
    #>                alpha    0.798       4    64
    #>          omega_total    0.869       4    64
    #>              lambda6    0.755       4    64
    #>        ordinal_alpha    0.757       4    64
    #>  ordinal_omega_total    0.846       4    64
    #>                                                notes
    #>                                 Pearson correlations
    #>                                 1-factor eigen omega
    #>                                       psych::alpha()
    #>                              Polychoric correlations
    #>  Polychoric correlations | Ordinal CIs not requested

    ## bootstrapped Confidence intervals can be slow!
    reliability(
      my_data,
      include = "polychoric",
      ci = TRUE,
      n_boot = 64
    )
    #>            coef_name estimate ci_lower ci_upper n_items n_obs
    #>                alpha    0.798    0.692    0.838       4    64
    #>          omega_total    0.869    0.814    0.893       4    64
    #>        ordinal_alpha    0.757    0.617    0.791       4    64
    #>  ordinal_omega_total    0.846    0.778    0.866       4    64
    #>                                                notes
    #>                                 Pearson correlations
    #>                                 1-factor eigen omega
    #>                              Polychoric correlations
    #>  Polychoric correlations | Ordinal CIs via bootstrap

# 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 latentFactoR package is ideal for generating multi-factor items.
`latentFactoR::simulate_factors()` generates data based on latent factor models, which in turn can be adjusted to continuous, polytomous, dichotomous, or mixed. Skews, cross-loadings, wording effects, population errors, and local dependencies can be added.
High recommended!

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/

# References

D’Alessandro, S., H. Winzar, B. Lowe, B.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. https://doi.org/10.32614/RJ-2016-034

Winzar, H. (2025). LikertMakeR (V 1.4.0): Synthesise and correlate Likert scale and related rating-scale data with predefined first and second moments. CRAN: https://CRAN.R-project.org/package=LikertMakeR