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

Source: R/makeScales.R
makeScales.Rd

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

Usage

makeScales(n, means, sds, lowerbound = 1, upperbound = 5, items = 1, 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). Default = 1.

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). Default = 5.

items

(positive, int) a vector of length k of number of items in each scale. Default = 1.

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

Details

makeScales() 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.

See also

lfast, lcor

Examples


## Example 1: four correlated items (questions)

### 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
)

scale_names <- c("Q1", "Q2", "Q3", "Q4")
rownames(corMat) <- scale_names
colnames(corMat) <- scale_names

### apply function

df1 <- makeScales(
  n = n, means = dfMeans, sds = dfSds,
  lowerbound = lowerbound, upperbound = upperbound, cormatrix = corMat
)
#> Variable  1 :  Q1  - 
#> reached maximum of 1024 iterations
#> Variable  2 :  Q2  - 
#> reached maximum of 1024 iterations
#> Variable  3 :  Q3  - 
#> reached maximum of 1024 iterations
#> Variable  4 :  Q4  - 
#> reached maximum of 1024 iterations
#> 
#> Arranging data to match correlations
#> 
#> Successfully generated correlated variables
#> 

### test function

str(df1)
#> 'data.frame':	16 obs. of  4 variables:
#>  $ Q1: num  1 3 2 3 1 4 2 2 2 4 ...
#>  $ Q2: num  3 1 2 5 2 4 2 3 3 3 ...
#>  $ Q3: num  3 1 4 4 1 5 1 4 1 5 ...
#>  $ Q4: num  3 3 3 4 2 5 3 4 3 4 ...

#### means
apply(df1, 2, mean) |> round(3)
#>  Q1  Q2  Q3  Q4 
#> 2.5 3.0 3.0 3.5 

#### standard deviations
apply(df1, 2, sd) |> round(3)
#>    Q1    Q2    Q3    Q4 
#> 1.033 1.033 1.506 0.730 

#### correlations
cor(df1) |> round(3)
#>       Q1    Q2    Q3    Q4
#> Q1 1.000 0.313 0.386 0.619
#> Q2 0.313 1.000 0.557 0.707
#> Q3 0.386 0.557 1.000 0.788
#> Q4 0.619 0.707 0.788 1.000


## Example 2: five correlated Likert scales

### a study on employee engagement and organizational climate:
# Job Satisfaction (JS)
# Organizational Commitment (OC)
# Perceived Supervisor Support (PSS)
# Work Engagement (WE)
# Turnover Intention (TI) (reverse-related to others).

### define parameters

n <- 128
dfMeans <- c(3.8, 3.6, 3.7, 3.9, 2.2)
dfSds <- c(0.7, 0.8, 0.7, 0.6, 0.9)
lowerbound <- rep(1, 5)
upperbound <- rep(5, 5)
items <- c(4, 4, 3, 3, 3)

corMat <- matrix(
  c(
    1.00, 0.72, 0.58, 0.65, -0.55,
    0.72, 1.00, 0.54, 0.60, -0.60,
    0.58, 0.54, 1.00, 0.57, -0.45,
    0.65, 0.60, 0.57, 1.00, -0.50,
    -0.55, -0.60, -0.45, -0.50, 1.00
  ),
  nrow = 5, ncol = 5
)

scale_names <- c("JS", "OC", "PSS", "WE", "TI")
rownames(corMat) <- scale_names
colnames(corMat) <- scale_names

### apply function

df2 <- makeScales(
  n = n, means = dfMeans, sds = dfSds,
  lowerbound = lowerbound, upperbound = upperbound,
  items = items, cormatrix = corMat
)
#> Variable  1 :  JS  - 
#> best solution in 429 iterations
#> Variable  2 :  OC  - 
#> best solution in 84 iterations
#> Variable  3 :  PSS  - 
#> best solution in 669 iterations
#> Variable  4 :  WE  - 
#> best solution in 407 iterations
#> Variable  5 :  TI  - 
#> best solution in 8873 iterations
#> 
#> Arranging data to match correlations
#> 
#> Successfully generated correlated variables
#> 

### test function

str(df2)
#> 'data.frame':	128 obs. of  5 variables:
#>  $ JS : num  4 2.75 4.75 4.25 4.25 3.5 4.75 4.5 3.25 4.5 ...
#>  $ OC : num  4.5 3 4.5 4.25 4.75 3.25 4.25 4.5 4.25 4.25 ...
#>  $ PSS: num  4 2 4.67 4 4.33 ...
#>  $ WE : num  4 3.67 4.67 3.67 4.33 ...
#>  $ TI : num  3 2.67 1 2.33 1.33 ...

#### means
apply(df2, 2, mean) |> round(3)
#>    JS    OC   PSS    WE    TI 
#> 3.801 3.602 3.701 3.901 2.201 

#### standard deviations
apply(df2, 2, sd) |> round(3)
#>    JS    OC   PSS    WE    TI 
#> 0.701 0.800 0.701 0.601 0.898 

#### correlations
cor(df2) |> round(3)
#>         JS    OC    PSS     WE     TI
#> JS   1.000  0.72  0.581  0.651 -0.550
#> OC   0.720  1.00  0.540  0.600 -0.600
#> PSS  0.581  0.54  1.000  0.571 -0.450
#> WE   0.651  0.60  0.571  1.000 -0.499
#> TI  -0.550 -0.60 -0.450 -0.499  1.000