LikertMakeR vignette

from CRAN

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

development version from GitHub.

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

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 256 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 7723 iterations

Example: likelihood-of-purchase scale

---

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 812 iterations
#> best solution in 7553 iterations
#> best solution in 385 iterations

The first six observations from this dataframe are:

#>    x1  x2  x3
#> 1 1.4 1.0 5.0
#> 2 2.8 5.0 4.2
#> 3 3.4 1.8 2.0
#> 4 2.0 4.8 4.4
#> 5 3.6 1.0 3.4
#> 6 2.2 2.8 4.0

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

#>         x1    x2    x3
#> mean 2.500 3.002 3.498
#> sd   0.752 1.501 1.001

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 -0.02 0.03
#> x2 -0.02  1.00 0.00
#> x3  0.03  0.00 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 1.2 2.8
#> 2  3.4 5.0 4.2
#> 3  1.6 1.2 2.8
#> 4  3.2 4.8 4.8
#> 5  4.2 5.0 3.2
#> 6  2.6 2.8 4.8
#> 7  1.4 1.2 2.6
#> 8  2.0 1.8 3.0
#> 9  2.2 3.2 3.6
#> 10 3.0 3.4 3.6

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

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.

## 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

#>       [,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

## 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

## 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

#>        [,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

## 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

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

makeCorrLoadings() examples

## 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 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

## 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

## 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

## 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

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

### 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 0.740
#> [3,] 0.350 0.70 1.000 0.836
#> [4,] 0.448 0.74 0.836 1.000

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

## 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

---

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

## 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

## 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

## 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

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!

## test function output
str(pairedDat)
#> 'data.frame':    20 obs. of  2 variables:
#>  $ X1: num  3.17 3.5 3.67 3.17 1.33 ...
#>  $ X2: num  3.5 3.5 2.67 1.67 2.17 ...

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

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 = -2.4455, 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

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). Use this if you assume stable relationships between all repeated measures.

• "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. Use this structure if you believe the relationship between repeated measures weakens steadily over time.

• "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 — a gentle fade over time. This may be useful in studies where changes across time are more gradual or irregular, but not strictly exponential. It’s a good middle ground when neither compound symmetry nor AR(1) seems quite right.

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,
  ...
)

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 1114 iterations
#> best solution in 661 iterations
#> best solution in 1021 iterations

head(out1$data)
#>   time_1 time_2 time_3
#> 1   1.50   4.25   5.00
#> 2   3.25   2.25   4.75
#> 3   4.50   2.25   4.00
#> 4   3.50   2.00   5.00
#> 5   3.75   4.25   3.00
#> 6   3.75   2.50   4.25
out1$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 110 iterations
#> best solution in 673 iterations
#> best solution in 336 iterations
#> reached maximum of 1024 iterations

print(out2)
#> $data
#>    time_1 time_2 time_3 time_4
#> 1     1.8    1.8    3.2    5.0
#> 2     1.8    1.6    2.8    5.0
#> 3     3.0    3.2    2.4    4.2
#> 4     3.2    3.4    4.0    5.0
#> 5     1.6    2.0    2.2    4.0
#> 6     3.4    4.2    4.4    2.6
#> 7     3.0    4.0    3.8    5.0
#> 8     3.6    4.6    4.8    4.8
#> 9     2.4    3.8    1.4    5.0
#> 10    2.2    3.0    2.0    2.8
#> 11    2.4    2.8    5.0    6.2
#> 12    3.0    3.8    3.8    5.8
#> 13    1.4    4.0    5.4    4.0
#> 14    2.4    5.8    5.4    5.0
#> 15    2.0    3.2    4.0    2.0
#> 16    4.4    4.6    6.0    5.6
#> 17    2.8    2.4    4.8    5.2
#> 18    2.6    4.0    4.6    5.6
#> 19    2.0    4.0    6.0    5.2
#> 20    3.0    2.4    4.8    4.0
#> 21    2.6    3.8    5.2    4.8
#> 22    1.4    2.6    3.8    4.4
#> 23    3.6    3.2    4.6    4.6
#> 24    4.0    5.6    4.0    5.0
#> 25    3.0    3.2    4.8    4.2
#> 26    2.2    3.4    3.0    3.0
#> 27    2.0    3.6    3.4    4.0
#> 28    3.4    4.6    2.4    3.2
#> 29    4.4    3.0    3.0    3.6
#> 30    3.4    2.2    3.0    3.0
#> 31    3.0    4.4    5.2    5.0
#> 32    3.0    3.8    4.8    4.2
#> 
#> $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.
#> reached maximum of 1024 iterations
#> best solution in 721 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] 3.25 1.75 3.25 1.75 1 2.5 1.5 1 2.5 1 ...
#>   ..$ time_2: num [1:32] 2.75 2.5 4.25 2 1.25 2.25 1.5 1.5 3 1.5 ...
#>   ..$ time_3: num [1:32] 2 1.75 4.75 3.25 3 1.5 2 4 3.25 2.5 ...
#>   ..$ time_4: num [1:32] 2.25 1.75 4 2.75 3.75 1.75 1.5 3.75 1.5 2 ...
#>  $ 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

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

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 14116 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 1070 iterations
#> Variable  3
#> best solution in 68 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 22469 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 106 iterations
#> Variable  3
#> reached maximum of 16384 iterations
#> Variable  4
#> best solution in 290 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 603 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 343 iterations
#> Variable  3
#> best solution in 42 iterations
#> Variable  4
#> reached maximum of 16384 iterations
#> Variable  5
#> reached maximum of 16384 iterations
#> Variable  6
#> best solution in 1230 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 3059 iterations
names(intent) <- "int"

## 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 0.18 0.27 0.43
#> [2,] 0.18 1.00 0.63 0.71
#> [3,] 0.27 0.63 1.00 0.77
#> [4,] 0.43 0.71 0.77 1.00

### Attitude #1 cronbach's alpha
alpha(cor(Att_1)) |> round(3)
#> [1] 0.8

## 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 0.06 0.31 0.44 0.50
#> [2,] 0.06 1.00 0.57 0.63 0.65
#> [3,] 0.31 0.57 1.00 0.68 0.68
#> [4,] 0.44 0.63 0.68 1.00 0.80
#> [5,] 0.50 0.65 0.68 0.80 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 0.33 0.36 0.42 0.47 0.48
#> [2,] 0.33 1.00 0.50 0.53 0.57 0.63
#> [3,] 0.36 0.50 1.00 0.65 0.72 0.78
#> [4,] 0.42 0.53 0.65 1.00 0.82 0.83
#> [5,] 0.47 0.57 0.72 0.82 1.00 0.85
#> [6,] 0.48 0.63 0.78 0.83 0.85 1.00

### Attitude #2 cronbach's alpha
alpha(cor(Att_3)) |> round(3)
#> [1] 0.898


## 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

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

## data structure
str(my_correlated_scales)
#> 'data.frame':    128 obs. of  16 variables:
#>  $ A1_1 : num  2 2 3 2 5 2 2 2 2 2 ...
#>  $ A1_2 : num  4 3 1 2 3 3 4 2 3 2 ...
#>  $ A1_3 : num  4 2 2 3 4 4 4 2 4 3 ...
#>  $ A1_4 : num  4 3 3 3 4 4 4 3 4 3 ...
#>  $ A2_1 : num  1 2 2 2 4 4 4 2 3 2 ...
#>  $ A2_2 : num  2 2 1 3 3 4 2 1 2 2 ...
#>  $ A2_3 : num  2 2 3 2 4 3 3 2 3 3 ...
#>  $ A2_4 : num  2 3 2 3 3 4 4 2 3 3 ...
#>  $ A2_5 : num  2 3 3 3 4 5 4 2 4 3 ...
#>  $ A3_1 : num  4 2 2 2 4 3 2 3 4 2 ...
#>  $ A3_2 : num  3 1 3 3 2 3 1 3 4 2 ...
#>  $ A3_3 : num  4 3 4 2 3 3 2 3 5 2 ...
#>  $ A3_4 : num  4 3 4 1 4 3 3 3 5 2 ...
#>  $ A3_5 : num  4 3 4 2 4 4 3 3 5 3 ...
#>  $ A3_6 : num  4 3 4 2 4 4 3 4 5 2 ...
#>  $ Int_1: num  9 3 0 5 2 3 4 0 3 6 ...

## 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] 7.10 2.29 1.13 1.01 0.84 0.73 0.58 0.51 0.41 0.34 0.32 0.22 0.16 0.14 0.13
#> [16] 0.10

#### 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] 6.94 2.26 1.03 0.86 0.79 0.68 0.56 0.44 0.34 0.34 0.22 0.17 0.15 0.13 0.10

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

evals <- eigenvalues(correlationMatrix, 1)

#> correlationMatrix  is positive-definite
print(evals)
#> [1] 2.7484991 0.8122627 0.3048151 0.1344231