Generate scale items from a summated scale, with desired Cronbach's Alpha

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

Value

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

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.

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
)
#> best solution in 2841 iterations
summatedScale <- meanScale * k

## create new items
newItems <- makeItemsScale(
  scale = summatedScale,
  lowerbound = lower, upperbound = upper,
  items = k
)
#> generate 64 rows
#> rearrange 4 values within each of 64 rows
#> Complete!
#> desired Cronbach's alpha = 0.8 (achieved alpha = 0.8002)

### 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
)
#> generate 64 rows
#> rearrange 4 values within each of 64 rows
#> Complete!
#> desired Cronbach's alpha = 0.8 (achieved alpha = 0.8011)

### 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
)
#> generate 64 rows
#> rearrange 4 values within each of 64 rows
#> Complete!
#> desired Cronbach's alpha = 0.8 (achieved alpha = 0.8003)

### 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)