makeCorrAlpha() generates a random correlation
matrix of given dimensions and predefined Cronbach's Alpha.
Such a correlation matrix can be applied to the makeScales()
function to generate synthetic data with the predefined alpha.
Usage
makeCorrAlpha(
items,
alpha,
variance = 0.5,
precision = 0,
sort_cors = FALSE,
diagnostics = FALSE
)Arguments
- items
(positive, int) matrix dimensions: number of rows & columns to generate
- alpha
(real) target Cronbach's Alpha (usually positive, must be between about -0.3 and +1)
- variance
(positive, real) Default = 0.5. User-provided standard deviation of values sampled from a normally-distributed log transformation. Caution: Larger values increase chance of a non-positive-definite matrix. 'TRUE' is faster, but produces less natural output. Default = FALSE
- 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)
- sort_cors
(logical) If 'TRUE', sorts the correlation coefficients in the final correlation matrix. Similar to an earlier version of this function.
- 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.
Value
If 'diagnostics = FALSE', a k x k correlation matrix. If 'diagnostics = TRUE', a list with components:
- R
k x k correlation matrix
- diagnostics
list of summary statistics
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
)
#> correlation values consistent with desired alpha in 59 iterations
# test function output
print(cor_matrix)
#> item01 item02 item03 item04
#> item01 1.0000000 0.7658611 0.6926037 0.4331446
#> item02 0.7658611 1.0000000 0.6936888 0.4251139
#> item03 0.6926037 0.6936888 1.0000000 0.5069007
#> item04 0.4331446 0.4251139 0.5069007 1.0000000
alpha(cor_matrix)
#> [1] 0.8500063
eigenvalues(cor_matrix, 1)
#> cor_matrix is positive-definite
#>
#> [1] 2.7831667 0.6670820 0.3157114 0.2340400
# higher alpha, more items
cor_matrix2 <- makeCorrAlpha(
items = 8,
alpha = 0.95
)
#> correlation values consistent with desired alpha in 731 iterations
#> Correlation matrix is not yet positive definite
#> Working on it
#>
#> improved at swap - 1 (min eigenvalue: -0.124095)
#> improved at swap - 2 (min eigenvalue: -0.122669)
#> improved at swap - 19 (min eigenvalue: -0.102987)
#> improved at swap - 20 (min eigenvalue: -0.073935)
#> improved at swap - 22 (min eigenvalue: -0.064356)
#> improved at swap - 26 (min eigenvalue: -0.057259)
#> improved at swap - 27 (min eigenvalue: -0.041386)
#> improved at swap - 31 (min eigenvalue: -0.031194)
#> improved at swap - 32 (min eigenvalue: -0.030663)
#> improved at swap - 33 (min eigenvalue: -0.010804)
#> improved at swap - 67 (min eigenvalue: -0.007359)
#> improved at swap - 71 (min eigenvalue: -0.005866)
#> improved at swap - 77 (min eigenvalue: -0.000102)
#> improved at swap - 102 (min eigenvalue: 0.002371)
#> positive definite at swap - 102
# test output
cor_matrix2 |> round(2)
#> item01 item02 item03 item04 item05 item06 item07 item08
#> item01 1.00 0.69 0.83 0.76 0.76 0.69 0.72 0.51
#> item02 0.69 1.00 0.86 0.79 0.75 0.67 0.45 0.89
#> item03 0.83 0.86 1.00 0.73 0.89 0.70 0.58 0.69
#> item04 0.76 0.79 0.73 1.00 0.78 0.73 0.87 0.62
#> item05 0.76 0.75 0.89 0.78 1.00 0.81 0.73 0.58
#> item06 0.69 0.67 0.70 0.73 0.81 1.00 0.68 0.71
#> item07 0.72 0.45 0.58 0.87 0.73 0.68 1.00 0.25
#> item08 0.51 0.89 0.69 0.62 0.58 0.71 0.25 1.00
alpha(cor_matrix2) |> round(3)
#> [1] 0.95
eigenvalues(cor_matrix2, 1) |> round(3)
#> cor_matrix2 is positive-definite
#>
#> [1] 5.954 0.983 0.418 0.340 0.223 0.047 0.032 0.002
# large random variation around alpha
set.seed(42)
cor_matrix3 <- makeCorrAlpha(
items = 6,
alpha = 0.85,
precision = 2
)
#> correlation values consistent with desired alpha in 2484 iterations
#> Correlation matrix is not yet positive definite
#> Working on it
#>
#> improved at swap - 3 (min eigenvalue: -0.034581)
#> improved at swap - 6 (min eigenvalue: -0.013659)
#> improved at swap - 7 (min eigenvalue: -0.003994)
#> improved at swap - 9 (min eigenvalue: 0.006886)
#> positive definite at swap - 9
# test output
cor_matrix3 |> round(2)
#> item01 item02 item03 item04 item05 item06
#> item01 1.00 0.74 0.47 0.68 0.77 0.78
#> item02 0.74 1.00 0.77 0.85 0.71 0.78
#> item03 0.47 0.77 1.00 0.76 0.71 0.48
#> item04 0.68 0.85 0.76 1.00 0.74 0.90
#> item05 0.77 0.71 0.71 0.74 1.00 0.72
#> item06 0.78 0.78 0.48 0.90 0.72 1.00
alpha(cor_matrix3) |> round(3)
#> [1] 0.94
eigenvalues(cor_matrix3, 1) |> round(3)
#> cor_matrix3 is positive-definite
#>
#> [1] 4.638 0.641 0.390 0.235 0.089 0.007
# with diagnostics
cor_matrix4 <- makeCorrAlpha(
items = 4,
alpha = 0.80,
diagnostics = TRUE
)
#> correlation values consistent with desired alpha in 1400 iterations
#> Correlation matrix is not yet positive definite
#> Working on it
#>
#> improved at swap - 1 (min eigenvalue: 0.060505)
#> positive definite at swap - 1
# test output
cor_matrix4
#> $R
#> item01 item02 item03 item04
#> item01 1.0000000 0.6754423 0.1099580 0.3831963
#> item02 0.6754423 1.0000000 0.5653570 0.4757467
#> item03 0.1099580 0.5653570 1.0000000 0.7902754
#> item04 0.3831963 0.4757467 0.7902754 1.0000000
#>
#> $diagnostics
#> $diagnostics$items
#> [1] 4
#>
#> $diagnostics$alpha_target
#> [1] 0.8
#>
#> $diagnostics$alpha_achieved
#> [1] 0.7999974
#>
#> $diagnostics$average_r
#> [1] 0.499996
#>
#> $diagnostics$eigenvalues
#> [1] 2.52156119 1.02842128 0.38951300 0.06050454
#>
#> $diagnostics$is_positive_definite
#> [1] TRUE
#>
#> $diagnostics$variance
#> [1] 0.5
#>
#> $diagnostics$precision
#> [1] 0
#>
#> $diagnostics$sort_cors
#> [1] FALSE
#>
#>