Package 'LikertMakeR'

Calculate Cronbach's Alpha from a correlation matrix or dataframe

Description

alpha() calculate Cronbach's Alpha from a given correlation matrix or a given dataframe.

Usage

alpha(cormatrix = NULL, data = NULL)

Arguments

cormatrix	(real) a square symmetrical matrix with values ranging from -1 to +1 and '1' in the diagonal
data	(real) a dataframe or matrix

Value

a single value

Examples

## Sample data frame
df <- data.frame(
  V1  =  c(4, 2, 4, 3, 2, 2, 2, 1),
  V2  =  c(4, 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)
)

## example correlation matrix
corMat <- matrix(
  c(
    1.00, 0.35, 0.45, 0.70,
    0.35, 1.00, 0.60, 0.55,
    0.45, 0.60, 1.00, 0.65,
    0.70, 0.55, 0.65, 1.00
  ),
  nrow = 4, ncol = 4
)

## apply function examples

alpha(cormatrix = corMat)

alpha(, df)

alpha(corMat, df)

---

Create a dataframe of correlated scales from different dataframes of scale items

Description

correlateScales() creates a dataframe of scale items representing correlated constructs, as one might find in a completed questionnaire.

Usage

correlateScales(dataframes, scalecors)

Arguments

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

Details

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.

Value

Returns a dataframe whose columns are taken from the starter dataframes and whose summated values are correlated according to a user-specified correlation matrix

Examples

## three attitudes and a behavioural intention
n <- 32
lower <- 1
upper <- 5

### attitude #1
cor_1 <- makeCorrAlpha(items = 4, alpha = 0.90)
means_1 <- c(2.5, 2.5, 3.0, 3.5)
sds_1 <- c(0.9, 1.0, 0.9, 1.0)

Att_1 <- makeItems(
  n = n, means = means_1, sds = sds_1,
  lowerbound = rep(lower, 4), upperbound = rep(upper, 4),
  cormatrix = cor_1
)



### attitude #2
cor_2 <- makeCorrAlpha(items = 5, alpha = 0.85)
means_2 <- c(2.5, 2.5, 3.0, 3.0, 3.5)
sds_2 <- c(1.0, 1.0, 0.9, 1.0, 1.5)

Att_2 <- makeItems(
  n = n, means = means_2, sds = sds_2,
  lowerbound = rep(lower, 5), upperbound = rep(upper, 5),
  cormatrix = cor_2
)



### attitude #3
cor_3 <- makeCorrAlpha(items = 6, alpha = 0.75)
means_3 <- c(2.5, 2.5, 3.0, 3.0, 3.5, 3.5)
sds_3 <- c(1.0, 1.5, 1.0, 1.5, 1.0, 1.5)

Att_3 <- makeItems(
  n = n, means = means_3, sds = sds_3,
  lowerbound = rep(lower, 6), upperbound = rep(upper, 6),
  cormatrix = cor_3
)



### behavioural intention
intent <- lfast(n, mean = 3.0, sd = 3, lowerbound = 0, upperbound = 10) |>
  data.frame()
names(intent) <- "int"


### target scale correlation matrix
scale_cors <- matrix(
  c(
    1.0, 0.6, 0.5, 0.3,
    0.6, 1.0, 0.4, 0.2,
    0.5, 0.4, 1.0, 0.1,
    0.3, 0.2, 0.1, 1.0
  ),
  nrow = 4
)

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


### apply the function
my_correlated_scales <- correlateScales(
  dataframes = data_frames,
  scalecors = scale_cors
)
head(my_correlated_scales)

---

calculate eigenvalues of a correlation matrix with optional scree plot

Description

eigenvalues() calculate eigenvalues of a correlation matrix and optionally produces a scree plot.

Usage

eigenvalues(cormatrix, scree = FALSE)

Arguments

cormatrix	(real, matrix) a correlation matrix
scree	(logical) default = FALSE. If TRUE (or 1), then eigenvalues() produces a scree plot to illustrate the eigenvalues

Value

a vector of eigenvalues

report on positive-definite status of cormatrix

Examples

## define parameters

correlationMatrix <- matrix(
  c(
    1.00, 0.25, 0.35, 0.40,
    0.25, 1.00, 0.70, 0.75,
    0.35, 0.70, 1.00, 0.80,
    0.40, 0.75, 0.80, 1.00
  ),
  nrow = 4, ncol = 4
)

## apply function

evals <- eigenvalues(cormatrix = correlationMatrix)
evals <- eigenvalues(correlationMatrix, 1)

---

Rearrange columns in a data-frame to fit a predefined correlation matrix

Description

lcor_C() rearranges values in each column of a data-frame so that columns are correlated to match a predefined correlation matrix.

Usage

lcor(data, target)

Arguments

data	data-frame that is to be rearranged
target	target correlation matrix - should be a symmetric k*k positive-semi-definite matrix

Details

Values in a column do not change, so univariate statistics remain the same.

Value

Returns a dataframe whose column-wise correlations approximate a user-specified correlation matrix

Examples

## parameters
n <- 32
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)
)

cor(mydat3) |> round(3)

tgt3 <- matrix(
  c(
    1.00, 0.50, 0.75,
    0.50, 1.00, 0.25,
    0.75, 0.25, 1.00
  ),
  nrow = 3, ncol = 3
)

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

## test output
cor(new3) |> round(3)

---

Deprecated. Use lfast() instead

Description

lexact is DEPRECATED. Replaced by new version of lfast.

lexact remains as a legacy for earlier package users. It is now just a wrapper for lfast

Previously, lexact used a Differential Evolution (DE) algorithm to find an optimum solution with desired mean and standard deviation, but we found that the updated lfast function is much faster and just as accurate.

Also the package is much less bulky.

Usage

lexact(n, mean, sd, lowerbound, upperbound, items = 1)

Arguments

n	(positive, int) number of observations to generate
mean	(real) target mean
sd	(real) target standard deviation
lowerbound	(positive, int) lower bound (e.g. '1' for a 1-5 rating scale)
upperbound	(positive, int) upper bound (e.g. '5' for a 1-5 rating scale)
items	(positive, int) number of items in the rating scale. Default = 1

Value

a vector of simulated data approximating user-specified conditions.

Examples

x <- lexact(
  n = 256,
  mean = 4.0,
  sd = 1.0,
  lowerbound = 1,
  upperbound = 7,
  items = 6
)

x <- lexact(256, 2, 1.8, 0, 10)

---

Synthesise rating-scale data with predefined mean and standard deviation

Description

lfast() applies a simple Evolutionary Algorithm to find a vector that best fits the desired moments.

lfast() generates random discrete values from a scaled Beta distribution so the data replicate a rating scale - for example, a 1-5 Likert scale made from 5 items (questions) or 0-10 likelihood-of-purchase scale.

Usage

lfast(n, mean, sd, lowerbound, upperbound, items = 1, precision = 0)

Arguments

n	(positive, int) number of observations to generate
mean	(real) target mean, between upper and lower bounds
sd	(positive, real) target standard deviation
lowerbound	(positive, int) lower bound (e.g. '1' for a 1-5 rating scale)
upperbound	(positive, int) upper bound (e.g. '5' for a 1-5 rating scale)
items	(positive, int) number of items in the rating scale. Default = 1
precision	(positive, real) can relax the level of accuracy required. (e.g. '1' generally generates a vector with moments correct within '0.025', '2' generally within '0.05') Default = 0

Value

a vector approximating user-specified conditions.

Examples

## six-item 1-7 rating scale
x <- lfast(
  n = 256,
  mean = 4.0,
  sd = 1.25,
  lowerbound = 1,
  upperbound = 7,
  items = 6
)

## four-item 1-5 rating scale with medium variation
x <- lfast(
  n = 128,
  mean = 3.0,
  sd = 1.00,
  lowerbound = 1,
  upperbound = 5,
  items = 4,
  precision = 5
)

## eleven-point 'likelihood of purchase' scale
x <- lfast(256, 3, 3.0, 0, 10)

---

Correlation matrix from Cronbach's Alpha

Description

makeCorrAlpha() generates a random correlation matrix of given dimensions and predefined Cronbach's Alpha

Usage

makeCorrAlpha(items, alpha, variance = 0.5, precision = 0)

Arguments

items	(positive, int) matrix dimensions: number of rows & columns to generate
alpha	(real) target Cronbach's Alpha (usually positive, must be between -1 and +1)
variance	(positive, real) Default = 0.5. User-provided standard deviation of values sampled from a normally-distributed log transformation.
precision	(positive, real) Default = 0. User-defined value ranging from '0' to '3' to add some random variation around the target Cronbach's Alpha. '0' gives an exact alpha (to two decimal places)

Value

a correlation matrix

Note

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 still be possible. The user is encouraged to try again, possibly several times, to find one.

Examples

# define parameters
items <- 4
alpha <- 0.85
variance <- 0.5

# apply function
set.seed(42)
cor_matrix <- makeCorrAlpha(items = items, alpha = alpha, variance = variance)

# test function output
print(cor_matrix)
alpha(cor_matrix)
eigenvalues(cor_matrix, 1)

# higher alpha, more items
cor_matrix2 <- makeCorrAlpha(items = 8, alpha = 0.95)

# test output
cor_matrix2 |> round(2)
alpha(cor_matrix2) |> round(3)
eigenvalues(cor_matrix2, 1) |> round(3)


# large random variation around alpha
set.seed(42)
cor_matrix3 <- makeCorrAlpha(items = 6, alpha = 0.85, precision = 2)

# test output
cor_matrix3 |> round(2)
alpha(cor_matrix3) |> round(3)
eigenvalues(cor_matrix3, 1) |> round(3)

---

Synthetic rating-scale data with given first and second moments and a predefined correlation matrix

Description

makeItems() generates a dataframe of random discrete values so the data replicate a rating scale, and are correlated close to a predefined correlation matrix.

makeItems() is wrapper function for:

• lfast(), generates a dataframe 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.

Usage

makeItems(n, means, sds, lowerbound, upperbound, cormatrix)

Arguments

n	(positive, int) sample-size - number of observations
means	(real) target means: a vector of length k of mean values for each scale item
sds	(positive, real) target standard deviations: a vector of length k of standard deviation values for each scale item
lowerbound	(positive, int) a vector of length k (same as rows & columns of correlation matrix) of values for lower bound of each scale item (e.g. '1' for a 1-5 rating scale)
upperbound	(positive, int) a vector of length k (same as rows & columns of correlation matrix) of values for upper bound of each scale item (e.g. '5' for a 1-5 rating scale)
cormatrix	(real, matrix) the target correlation matrix: a square symmetric positive-semi-definite matrix of values ranging between -1 and +1, and '1' in the diagonal.

Value

a dataframe of rating-scale values

Examples

## define parameters

n <- 16
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, 0.30, 0.40, 0.60,
    0.30, 1.00, 0.50, 0.70,
    0.40, 0.50, 1.00, 0.80,
    0.60, 0.70, 0.80, 1.00
  ),
  nrow = 4, ncol = 4
)

## apply function

df <- makeItems(
  n = n, means = dfMeans, sds = dfSds,
  lowerbound = lowerbound, upperbound = upperbound, cormatrix = corMat
)

## test function

str(df)

# means
apply(df, 2, mean) |> round(3)

# standard deviations
apply(df, 2, sd) |> round(3)

# correlations
cor(df) |> round(3)

---

scale items from a summated scale and desired Cronbach's Alpha

Description

makeItemsScale() generates a random dataframe of scale items based on a predefined summated scale (such as created by the lfast() function), and a desired Cronbach's Alpha.

scale, lowerbound, upperbound, items, alpha, variance

Usage

makeItemsScale(
  scale,
  lowerbound,
  upperbound,
  items,
  alpha = 0.8,
  variance = 0.5
)

Arguments

scale	(int) a vector or dataframe of the summated rating scale. Should range from ('lowerbound' * 'items') to ('upperbound' * 'items')
lowerbound	(int) lower bound of the scale item (example: '1' in a '1' to '5' rating)
upperbound	(int) upper bound of the scale item (example: '5' in a '1' to '5' rating)
items	(positive, int) k, or number of columns to generate
alpha	(posiitve, real) desired Cronbach's Alpha for the new dataframe of items. Default = '0.8'. See ⁠@details⁠ for further information on the alpha parameter
variance	(positive, real) the quantile from which to select items that give given summated scores. Must lie between '0' and '1'. Default = '0.5'. See ⁠@details⁠ for further information on the variance parameter

Details

alpha

makeItemsScale() rearranges the item values within each row, attempting to give a dataframe of Likert-scale items that produce a predefined Cronbach's Alpha.

Default value for target alpha is '0.8'.

More extreme values for the 'variance' parameter may reduce the chances of achieving the desired Alpha. So you may need to experiment a little.

variance

There may be many ways to find a combination of integers that sum to a specific value, and these combinations have different levels of variance:

• low-variance: '3 + 4 = 7'

• high-variance: '1 + 6 = 7'

The 'variance' parameter defines guidelines for the amount of variance among item values that your new dataframe should have.

For example, consider a summated value of '9' on which we apply the makeItemsScale() function to generate three items. With zero variance (variance parameter = '0'), then we see all items with the same value, the mean of '3'. With variance = '1', then we see all items with values that give the maximum variance among those items.

variance	v1	v2	v3	sum
0.0	3	3	3	9
0.2	3	3	3	9
0.4	2	3	4	9
0.6	1	4	4	9
0.8	2	2	5	9
1.0	1	3	5	9

Similarly, the same mean value applied to six items with makeItemsScale() gives the following combinations at different values of the 'variance' parameter.

variance	v1	v2	v3	v4	v5	v6	sum
0.0	3	3	3	3	3	3	18
0.2	1	3	3	3	4	4	18
0.4	1	2	3	4	4	4	18
0.6	1	1	4	4	4	4	18
0.8	1	1	3	4	4	5	18
1.0	1	1	1	5	5	5	18

And a mean value of '3.5' gives the following combinations.

variance	v1	v2	v3	v4	v5	v6	sum
0.0	3	3	3	4	4	4	21
0.2	3	3	3	3	4	5	21
0.4	2	2	4	4	4	5	21
0.6	1	3	4	4	4	5	21
0.8	1	2	4	4	5	5	21
1.0	1	1	4	5	5	5	21

The default value for 'variance' is '0.5' which gives a reasonable range of item values. But if you want 'responses' that are more consistent then choose a lower variance value.

Value

a dataframe with 'items' columns and 'length(scale)' rows

Examples

## define parameters
k <- 4
lower <- 1
upper <- 5

## scale properties
n <- 64
mean <- 3.0
sd <- 0.85

## create scale
set.seed(42)
meanScale <- lfast(
  n = n, mean = mean, sd = sd,
  lowerbound = lower, upperbound = upper,
  items = k
)
summatedScale <- meanScale * k

## create new items
newItems <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lower, upperbound = upper,
  items = k
)

### test new items
str(newItems)
alpha(data = newItems) |> round(2)


## create items with higher Alpha but same summated scale
newItems <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lower, upperbound = upper,
  items = k,
  alpha = 0.9,
  variance = 0.5
)

### test new items
str(newItems)
alpha(data = newItems) |> round(2)


## very low variance usually gives higher Cronbach's Alpha
mydat_20 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lower, upperbound = upper,
  items = k, alpha = 0.8, variance = 0.20
)

### test new data frame
str(mydat_20)

moments <- data.frame(
  means = apply(mydat_20, MARGIN = 2, FUN = mean) |> round(3),
  sds = apply(mydat_20, MARGIN = 2, FUN = sd) |> round(3)
) |> t()

moments

cor(mydat_20) |> round(2)
alpha(data = mydat_20) |> round(2)


## default alpha (0.8) and higher variance (0.8)
mydat_80 <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lower, upperbound = upper,
  items = k, variance = 0.80
)

### test new dataframe
str(mydat_80)

moments <- data.frame(
  means = apply(mydat_80, MARGIN = 2, FUN = mean) |> round(3),
  sds = apply(mydat_80, MARGIN = 2, FUN = sd) |> round(3)
) |> t()

moments

cor(mydat_80) |> round(2)
alpha(data = mydat_80) |> round(2)

---