---
title: "LikertMakeR vignette"
author: "Hume Winzar"
date: "December 2025"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{LikertMakeR vignette}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```


```{r setup, echo=FALSE}
library(LikertMakeR)
```


```{r logo, fig.align='center', echo=FALSE, out.width = '25%'}
knitr::include_graphics("img/LikertMakeR_3.png")
```


<!-- ![LikertMakeR](LikertMakeR_hex.png){#id .class width=25%} -->


_**LikertMakeR**_ [(Winzar, 2025)](https://cran.r-project.org/package=LikertMakeR) 
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 variance, 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. <br /> 
For example, [D'Alessandro _et al._ (2020)](https://cengage.com.au/sem121/marketing-research-5th-edition-dalessandro-babin-zikmund) 
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)](https://archive.org/details/likert-1932/mode/2up) 
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. <br />
If the mean is not in the middle of a scale, 
then the data will be always skewed, as shown in the following plots.


```{r skew, fig.align='center', echo=FALSE, out.width = '85%', fig.cap="Off-centre means always give skewed distribution in bounded rating scales"}
knitr::include_graphics("img/skew_chart.png")
```

_____

## _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 - R<sup>2</sup>, 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


```{r lfastExample}
nItems <- 4
mean <- 2.5
sd <- 0.75

x1 <- lfast(
  n = 512,
  mean = mean,
  sd = sd,
  lowerbound = 1,
  upperbound = 5,
  items = nItems
)
```



```{r fig1, fig.height=4, fig.width=5, fig.align='center', echo = FALSE, crop = TRUE, fig.cap="Example: 4-item, 1-5 Likert scale"}
## distribution of x

hist(x1,
  cex.axis = 0.5, cex.main = 0.75,
  breaks = seq(from = (1 - (1 / 8)), to = (5 + (1 / 8)), by = (1 / 4)),
  col = "skyblue", xlab = NULL, ylab = NULL,
  main = paste0("mu=", round(mean(x1), 2), ", sd=", round(sd(x1), 2))
)
```


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

###### _lfast()_

```{r lfastx2}
x2 <- lfast(256, 3, 2.5, 0, 10)
```


```{r fig2, fig.height=4, fig.width=5, fig.align='center', echo = FALSE, crop = TRUE, fig.cap="Example: likelihood-of-purchase scale"}
## generate histogram

hist(x2,
  cex.axis = 0.5, cex.main = 0.75,
  breaks = seq(from = -0.5, to = 10.5, by = 1),
  col = "skyblue", xlab = NULL, ylab = NULL,
  main = paste0("mu=", round(mean(x2), 2), ", sd=", round(sd(x2), 2))
)
```

____

## 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.

```{r lcorExample}
## 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)
)
```

The first six observations from this dataframe are:

```{r mydat3Head, echo = FALSE}
head(mydat3, 6)
```

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

```{r mydat3Moments, echo = FALSE}
moments <- data.frame(
  mean = apply(mydat3, 2, mean) |> round(3),
  sd = apply(mydat3, 2, sd) |> round(3)
) |> t()

moments
```

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:

```{r mydat3Cor, echo = FALSE}
cor(mydat3) |> round(2)
```

Now, let's define a target correlation matrix:

```{r tgt3}
## 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.

```{r new3}
## 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:

```{r new3Head, echo = FALSE}
head(new3, 10)
```

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

```{r new3Cor, echo = FALSE}
cor(new3) |> round(3)
```

____


## Generate a correlation matrix from Cronbach's Alpha

### makeCorrAlpha()

**_makeCorrAlpha()_**, constructs a random correlation matrix of given 
 dimensions and predefined Cronbach's Alpha. 
 
To create the desired correlation matrix, the user must define the 
following parameters:

- **_items_**: or "k" - the number of rows and columns of the 
 desired correlation matrix. 

- **_alpha_**: the target value for Cronbach's Alpha

- **_variance_**: a notional variance coefficient to affect the spread of 
values in the correlation matrix. Default = '0.5'. 
A value of '0' produces a matrix where all off-diagonal correlations 
are equal. 
Setting 'variance = 1.0' gives a wider range of values. 
Setting 'variance = 2.0', or above, may be feasible but increases the 
likelihood of a non-positive-definite matrix.

-  **_precision_**: a value between '0' and '3' to add some random 
  variation around the target _Cronbach's Alpha_.
  Default = '0'.
  A value of '0' produces the desired _Alpha_, generally exact to two 
  decimal places. Higher values produce increasingly random values around 
  the desired _Alpha_.

- **_sort_cors_**: Logical. Default = `FALSE`. If `TRUE`, then runs more 
  quickly, but produces a less natural correlation matrix. 
  
- **_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() is volatile

Random values generated by _makeCorrAlpha()_ are highly volatile.
 _makeCorrAlpha()_ may not generate a feasible (positive-definite)
 correlation matrix, especially when

 * variance is high relative to
 
     - desired Alpha, and
     - desired correlation dimensions

_makeCorrAlpha()_ will inform the user if the resulting correlation
 matrix is positive definite, or not.

If the returned correlation matrix is not positive-definite, 
a feasible solution may be still possible, and often is. 
The user is encouraged to try again, possibly several times, to find one.

#### _makeCorrAlpha()_ examples

##### Four variables, alpha = 0.85, variance = default

```{r cor_matrix_4}
## define parameters
items <- 4
alpha <- 0.85
# variance <- 0.5 ## by default

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

cor_matrix_4 <- makeCorrAlpha(items, alpha)
```

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

```{r cor_matrix_4Print, echo = FALSE}
cor_matrix_4 |> round(3)
```



##### test output with Helper functions

```{r cor_matrix_4Alpha}
## using helper function alpha()

alpha(cor_matrix_4)
```


```{r fig3, fig.height=4, fig.width=5, fig.align='center', echo = TRUE, crop = TRUE}
## using helper function eigenvalues()

eigenvalues(cor_matrix_4, 1)
```


#### twelve variables, alpha = 0.90, variance = 1.0 

High variance will not produce a positive-definite correlation matrix.


```{r cor_matrix_12, message=FALSE}
## 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)
```

######  -

_makeCorrAlpha()_ produced the following correlation matrix 
(to two decimal places):

```{r cor_matrix_12Print, echo = FALSE}
cor_matrix_12 |> round(2)
```


##### test output

```{r fig4, fig.height=4, fig.width=5, fig.align='center', echo = TRUE, crop = TRUE}
## calculate Cronbach's Alpha
alpha(cor_matrix_12)

## calculate eigenvalues of the correlation matrix
eigenvalues(cor_matrix_12, 1) |> round(3)
```


#### makeCorrAlpha() with diagnostics output

```{r cor_matrix_d}
## apply makeCorrAlpha() with diagnostics
set.seed(42)

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

##### diagnostics output

```{r cor_matrix_5_mat}
## output
cor_matrix_5$R |> round(2)

cor_matrix_5$diagnostics
```



____

## 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 


```{r parameters_makeCorrLoadings, echo=TRUE}
## 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

```{r apply_makeCorrLoadings, echo=TRUE}
## apply makeCorrLoadings() function
itemCorrelations <- makeCorrLoadings(factorLoadings, factorCor)

## derived correlation matrix to two decimal places
round(itemCorrelations, 2)
```

##### Test makeCorrLoadings() output

```{r test_makeCorrLoadings, echo=TRUE}
## correlated factors mean that eigenvalues should suggest two or three factors
eigenvalues(cormatrix = itemCorrelations, scree = TRUE)
```


#####  Assuming orthogonal factors

```{r orthogonal_makeCorrLoadings, echo=TRUE}
## orthogonal factors are assumed when factor correlation matrix is not included
orthogonalItemCors <- makeCorrLoadings(factorLoadings)

## derived correlation matrix to two decimal places
round(orthogonalItemCors, 2)
```

##### Test orthogonal output

```{r test_orthogonal, echo=TRUE}
## eigenvalues should suggest exactly  three factors
eigenvalues(cormatrix = orthogonalItemCors, scree = TRUE)
```
____


## 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

#### four correlated items

```{r makeScalesExample1}

## 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
)
```

##### Structure of new dataframe

```{r str_makeScalesExample1}

## test the function
str(df)
```

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

```{r moments_makeScalesExample1}

### 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
```

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

```{r cors_makeScalesExample1}

### correlations should be correct to two decimal places
cor(df) |> round(3)
```

#### 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. 


```{r makeScalesExample2}
## 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
)

## test the function
head(df)
tail(df)

### 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

### correlations should be correct to two decimal places
cor(df) |> round(3)
```


____


## 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

```{r genCorrelation}
## 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)

### checking Cronbach's Alpha
alpha(cormatrix = myCorr)
```


### Step 2: Generate dataframe

```{r gendataframe}
## 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
)

## resulting dataframe
head(myItems)
tail(myItems)

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

## Cronbach's Alpha of dataframe
alpha(NULL, myItems)
```

#### Summary plots of new dataframe

```{r fig5, fig.height=5, fig.width=5, fig.align='center', echo=FALSE, warning=FALSE, crop = TRUE, fig.cap="Summary of dataframe from makeScales() function"}
# Correlation panel

panel.cor <- function(x, y) {
  usr <- par("usr")
  on.exit(par(usr))
  par(usr = c(0, 1, 0, 1))
  r <- round(cor(x, y), digits = 2)
  txt <- paste0(r)
  cex.cor <- 0.8 / strwidth(txt)
  text(0.5, 0.5, txt, cex = 1.25)
}

# Customize upper panel

upper.panel <- function(x, y) {
  points(x, y, pch = 19, col = "#0000ff11")
}

# diagonals

panel.hist <- function(x, ...) {
  usr <- par("usr")
  on.exit(par(usr))
  par(usr = c(usr[1:2], 0, 1.5))
  h <- hist(x, plot = FALSE)
  breaks <- h$breaks
  nB <- length(breaks)
  y <- h$counts
  y <- y / max(y)
  rect(breaks[-nB], 0, breaks[-1], y, col = "#87ceeb66")
}

# Create the plots

pairs(myItems,
  lower.panel = panel.cor,
  upper.panel = upper.panel,
  diag.panel = panel.hist
)
```

____


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

### makeItemsScale()

  -  **_makeItemsScale()_** generates a dataframe of rating-scale items from a summated rating scale and desired _Cronbach's Alpha_.


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

  - **_scale_**: a vector or dataframe of the summated rating scale.
   Should range from ('lowerbound' * 'items') to ('upperbound' * 'items') 
 
  - **_lowerbound_**: lower bound of the scale item
   (example: '1' in a '1' to '5' rating) 
 
  - **_upperbound_**:  upper bound of the scale item
   (example: '5' in a '1' to '5' rating) 
 
  - **_items_**: k, or number of columns to generate
  
  - **_alpha_**: desired Cronbach's Alpha. Default = '0.8'
  
  - **_variance_**: quantile for selecting the combination of items that 
  give summated scores. 
  Must lie between '0' (minimum variance) and '1' (maximum variance). 
  Default = '0.5'.
  
 
#### _makeItemsScale()_ Example:

##### generate a summated scale

```{r generate_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
)

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

```{r summatedScale_histogram, echo=FALSE, fig.height=4, fig.width=5, fig.align='center', crop = TRUE, fig.cap="Summated scale distribution"}
## Histogram of summated scale

hist(summatedScale,
  cex.axis = 0.5, cex.main = 0.75,
  breaks = seq(
    from = ((lowerbound * items) - 0.5),
    to = ((upperbound * items) + 0.5), by = 1
  ),
  col = "skyblue", xlab = NULL, ylab = NULL,
  main = paste0(
    "mu=", round(mean * items, 2), ", sd=", round(sd * items, 2), ", range:",
    (lowerbound * items), ":", (upperbound * items)
  )
)
```


#### create items with _makeItemsScale()_

```{r makeItemsScale_example_1, fig.height=4, fig.width=5, fig.align='center', crop = TRUE}
## apply makeItemsScale() function

newItems_1 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lowerbound,
  upperbound = upperbound,
  items = items
)

### First 10 observations and summated scale
head(cbind(newItems_1, summatedScale), 10)

### correlation matrix
cor(newItems_1) |> round(2)

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

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


####  _makeItemsScale()_ with same summated values and higher _alpha_

```{r makeItemsScale_example_2, fig.height=4, fig.width=5, fig.align='center', crop = TRUE}
## apply makeItemsScale() function
newItems_2 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lowerbound,
  upperbound = upperbound,
  items = items,
  alpha = 0.9
)

### First 10 observations and summated scale
head(cbind(newItems_2, summatedScale), 10)

### correlation matrix
cor(newItems_2) |> round(2)

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

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

####  same summated values with lower _alpha_ may require higher _variance_

```{r makeItemsScale_example_3, fig.height=4, fig.width=5, fig.align='center', crop = TRUE}
## apply makeItemsScale() function
newItems_3 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lowerbound,
  upperbound = upperbound,
  items = items,
  alpha = 0.6,
  variance = 0.7
)

### First 10 observations and summated scale
head(cbind(newItems_3, summatedScale), 10)

### correlation matrix
cor(newItems_3) |> round(2)

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

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

----

## 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.

```{r independent-samples_t-test, echo = TRUE}
## 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
)
x2 <- lfast(
  n = 30, mean = 3.0, sd = 0.85,
  lowerbound = lower, upperbound = upper, items = items
)

## run independent-samples t-test
t.test(x1, x2)
```

### 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

```{r paired-sample_t-test_data, , echo = TRUE}
## 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
)
```

###### check properties of new data

```{r makePaired_output_properties, echo=TRUE}
## test function output
str(pairedDat)

cor(pairedDat) |> round(2)

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

pairedMoments
```


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

```{r paired-sample_t-test, echo=TRUE}
## 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
```


----

## 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

```{r  makeRepeated-test, echo=TRUE}
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
)

head(out1$data)
out1$correlation_matrix


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
)

print(out2)


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
)

str(out3)
```


----


## 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](https://CRAN.R-project.org/package=psych)

### correlateScales() examples

#### three attitudes and a behavioural intention

##### create dataframes of Likert-scale items
 
```{r correlateScales_dataframes, echo = TRUE}
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
)

### 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
)

### 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
)

### behavioural intention
intent <- lfast(n, mean = 4.0, sd = 3, lowerbound = 0, upperbound = 10) |>
  data.frame()
names(intent) <- "int"
```

###### check properties of item dataframes

```{r dataframe_properties}
## 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

### Attitude #1 correlations
cor(Att_1) |> round(2)

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

## 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

### Attitude #2 correlations
cor(Att_2) |> round(2)

### Attitude #2 cronbach's alpha
alpha(cor(Att_2)) |> round(3)

## 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

### Attitude #3 correlations
cor(Att_3) |> round(2)

### Attitude #2 cronbach's alpha
alpha(cor(Att_3)) |> round(3)


## 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
```



##### correlateScales parameters

```{r 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

```{r my_correlated_scales}
### apply correlateScales() function
my_correlated_scales <- correlateScales(
  dataframes = data_frames,
  scalecors = scale_cors
)
```

#### plot the new correlated scale items

```{r fig6, fig.height=9, fig.width=9, fig.align='center', echo=FALSE, warning=FALSE,  crop = TRUE}
# Correlation panel
panel.cor <- function(x, y) {
  usr <- par("usr")
  on.exit(par(usr))
  par(usr = c(0, 1, 0, 1))
  r <- round(cor(x, y), digits = 2)
  txt <- paste0(r)
  cex.cor <- 0.8 / strwidth(txt)
  text(0.5, 0.5, txt, cex = 1.25)
}
# Customize upper panel
upper.panel <- function(x, y) {
  points(x, y, pch = 19, col = "#0000ff11")
}
# diagonals
panel.hist <- function(x, ...) {
  usr <- par("usr")
  on.exit(par(usr))
  par(usr = c(usr[1:2], 0, 1.5))
  h <- hist(x, plot = FALSE)
  breaks <- h$breaks
  nB <- length(breaks)
  y <- h$counts
  y <- y / max(y)
  rect(breaks[-nB], 0, breaks[-1], y, col = "#0000ff50")
}
# Create the plots
pairs(my_correlated_scales,
  lower.panel = panel.cor,
  upper.panel = upper.panel,
  diag.panel = panel.hist
)
```


###### Check the properties of our derived dataframe

```{r newdata_check}
## data structure
str(my_correlated_scales)
```


```{r fig7, fig.height=4, fig.width=5, fig.align='center', echo = TRUE,  crop = TRUE}
## eigenvalues of dataframe correlations
Cor_Correlated_Scales <- cor(my_correlated_scales)
eigenvalues(cormatrix = Cor_Correlated_Scales, scree = TRUE) |> round(2)
```

```{r fig7a, fig.height=4, fig.width=5, fig.align='center', echo = TRUE,  crop = TRUE}
#### Eigenvalues of predictor variable items only
Cor_Attitude_items <- cor(my_correlated_scales[, -16])
eigenvalues(cormatrix = Cor_Attitude_items, scree = TRUE) |> round(2)
```



____

## 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

```{r alphaExample}
## 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)
alpha(data = df)
alpha(NULL, df)
alpha(corMat, df)
```



### eigenvalues()

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

### eigenvalues() examples


```{r eigenExample}
## 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)

print(evals)
```

##### eigenvalues() function with optional scree plot

```{r fig8, fig.height=4, fig.width=5, fig.align='center', echo = TRUE, crop = TRUE}
evals <- eigenvalues(correlationMatrix, 1)
print(evals)
```



### 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

```{r reliabilityExample, warning=FALSE, message=FALSE}

## 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
)

## run function
reliability(my_data)

reliability(
  my_data,
  include = c("lambda6", "polychoric")
)


## bootstrapped Confidence intervals can be slow!
reliability(
  my_data,
  include = "polychoric",
  ci = TRUE,
  n_boot = 64
)

```

____


# 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)](https://glaswasser.github.io/simulating-correlated-likert-scale-data/) 
for an excellent and short example using the following packages:

- [truncnorm](https://cran.r-project.org/package=truncnorm)
 
- [faux](https://cran.r-project.org/package=faux) 
 
- See also the _rLikert()_ function from the excellent
 [latent2likert](https://github.com/markolalovic/latent2likert) package, 
 [Lalovic (2024)](https://latent2likert.lalovic.io/), 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.
 
 
```{r, eval = FALSE}
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](https://CRAN.R-project.org/package=SimCorrMix) on CRAN.
 
- [SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types]( https://CRAN.R-project.org/package=SimMultiCorrData) on CRAN.

- [lsasim: Functions to Facilitate the Simulation of Large Scale Assessment Data](https://CRAN.R-project.org/package=lsasim) on CRAN. 
See [Matta et al. (2018)](https://doi.org/10.1186/s40536-018-0068-8)

- [GenOrd:Simulation of Discrete Random Variables with Given Correlation Matrix and Marginal Distributions](https://CRAN.R-project.org/package=GenOrd) on CRAN.

- [SimCorMultRes: Simulates Correlated Multinomial Responses](https://cran.r-project.org/package=SimCorMultRes) on CRAN. 
See [Touloumis (2016)](https://doi.org/10.32614/RJ-2016-034)
 
- [covsim: VITA, IG and PLSIM Simulation for Given Covariance and Marginals](https://cran.r-project.org/package=covsim) on CRAN. 
See [Grønneberg et al. (2022)](https://www.jstatsoft.org/article/view/v102i03/)

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

The [psych package](https://CRAN.R-project.org/package=psych) has several 
excellent functions for simulating rating-scale data based on factor loadings. <br /> 
These focus on factor and item correlations rather than item moments. <br />
Highly recommended.

- [**_psych::sim.item_**  Generate simulated data structures for circumplex, spherical, or simple structure](https://CRAN.R-project.org/package=psych)

- [**_psych::sim.congeneric_** Simulate a congeneric data set with or without minor factors](https://CRAN.R-project.org/package=psych)
See [Revelle (in prep)](https://personality-project.org/r/book/)

Also:

[**_simsem_**](https://CRAN.R-project.org/package=simsem) has many 
functions for simulating and testing data for application in 
Structural Equation modelling. 
See examples at [https://simsem.org/](https://simsem.org/) 

### General data simulation

[**_simpr_**](https://CRAN.R-project.org/package=simpr) provides a general, 
simple, and tidyverse-friendly framework for generating simulated data, 
fitting models on simulations, and tidying model results.


____


# References

D'Alessandro, S., H. Winzar, B. Lowe, B.J. Babin, W. Zikmund (2020). 
_Marketing Research_ 5<sup>ed</sup>, 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>

