# NORTA

library(faux)
library(ggplot2)
library(ggExtra)
library(patchwork)

NORTA stands for Normal to Anything. This procedure simulates correlated data from a normal distribution and converts it to any other distribution using quantile functions. The challenge is determining what correlation between normally distributed variables is equivalent to a specified correlation between non-normally distributed variables.

## rmulti

The current implementation of rmulti() is experimental. It has six arguments:

• n the number of samples required
• dist A named vector of the distributions of each variable
• params A list of lists of the arguments to pass to each distribution function
• r the correlations among the variables (can be a single number, vars*vars matrix, vars*vars vector, or a vars*(vars-1)/2 vector)
• empirical logical. If true, params specify the sample parameters, not the population parameters
• as.matrix logical. If true, returns a matrix

By default, it returns a data frame with 100 rows of two normally distributed values with means of 0, standard deviations of 1, and a correlation of 0.

dat <- rmulti()

get_params(dat)
#>     n var     A     B mean   sd
#> 1 100   A  1.00 -0.06 0.05 0.99
#> 2 100   B -0.06  1.00 0.05 0.93

The n, r, empirical and as.matrix arguments work the same as the do for rnorm_multi().

dat <- rmulti(n = 200, r = 0.5,
empirical = TRUE,
as.matrix = FALSE)

get_params(dat)
#>     n var   A   B mean sd
#> 1 200   A 1.0 0.5    0  1
#> 2 200   B 0.5 1.0    0  1

### Distributions and parameters

You can set the distribution for each variable with the dist argument. The options are any distribution function in the {stats} package, such as “norm”, “pois”, “binom”, and “unif”, plus the “truncnorm” distribution from the {truncnorm} package and the “likert” distribution from {faux}.

Set the params argument to a named list with a vector of named arguments for the random generation function for each distribution. For example, use ?rpois to find out what arguments the rpois() function needs to simulate values from a poisson distribution. Don’t set n in params; it will be set by the nin the rmulti() function.

dat <- rmulti(n = 1000,
dist = c(uniform_var = "unif",
poisson_var = "pois"),
params = list(uniform_var = c(min = 0, max = 100),
poisson_var = c(lambda = 3)),
r = 0.5)

get_params(dat)
#>      n         var uniform_var poisson_var  mean    sd
#> 1 1000 uniform_var        1.00        0.49 50.62 28.67
#> 2 1000 poisson_var        0.49        1.00  2.98  1.75 You can also simulate more than two variables. Set the correlations using the upper right triangle or a correlation matrix (e.g., from a pilot dataset you’re trying to simulate).

dat <- rmulti(
n = 1000,
dist = c(N = "norm",
T = "truncnorm",
L = "likert"),
params = list(
N = list(mean = 100, sd = 10),
T = list(a = 1, b = 7, mean = 3.5, sd = 2.1),
L = list(prob = c(much less = .10,
less      = .20,
equal     = .35,
more      = .25,
much more = .10))
),
r = c(-0.5, -0.6, 0.7)
)

# convert likert-scale variable to integer
dat$L <- as.integer(dat$L)
get_params(dat)
#>      n var     N     T     L   mean   sd
#> 1 1000   N  1.00 -0.47 -0.57 100.32 9.70
#> 2 1000   T -0.47  1.00  0.67   3.80 1.43
#> 3 1000   L -0.57  0.67  1.00   3.08 1.08 ### Impossible correlations

Not all correlations are possible for a given pair of distributions. At the extreme, it’s obvious that a normal distribution and a uniform distribution can’t be correlated 1.0, because they wouldn’t be different distributions then.

If you ask rmulti() for a correlation that is too high or low, you will get a message telling you the maximum and minimum correlations that can be generated.

dat <- rmulti(
dist = c(A = "binom", B = "pois", C = "norm"),
params = list(A = list(size = 1, prob = 0.5),
B = list(lambda = 3),
C = list(mean = 100, sd = 10)),
r = c(0.8, 0.9, 0.5)
)
#> Error in rmulti(dist = c(A = "binom", B = "pois", C = "norm"), params = list(A = list(size = 1, : Some of the correlations are not possible:
#>   *  A&B (-0.776 to 0.776)
#>   *  A&C (-0.798 to 0.798)

## Helper functions

I made a few helper functions for rmulti(). I’m not sure if they’ll be useful to anyone else, but they’re available.

### fh_bounds

Fréchet-Hoefding bounds are the limits to a correlation for a pair of distributions.

fh_bounds(dist1 = "truncnorm",
dist2 = "beta",
params1 = list(a = 0, b = 10, mean = 5, sd = 3),
params2 = list(shape1 = 1, shape2 = 5))
#> $min #>  -0.944 #> #>$max
#>  0.944

### convert_r

This gives the r-value you’d need to simulate for a pair of normally-distributed variables to achieve the target r-value after converting to the target distributions.

adjusted_r <- convert_r(
target_r = 0.75,
dist1 = "truncnorm",
dist2 = "binom",
params1 = list(a = 0, b = 10, mean = 5, sd = 3),
params2 = list(size = 1, prob = 0.5)
)

#>  0.911

What the rmulti() function does is use this adjusted r to generate a multivariate normal distribution, then convert each variable to the target distribution.

# simulate multivariate normal
dat <- rnorm_multi(n = 1000,
varnames = c("N1", "N2"),
empirical = TRUE)

# convert to target distributions
dat$T1 <- norm2trunc(dat$N1,
min = 0, max = 10,
mu = 5, sd = 3,
x_mu = 0, x_sd = 1)
dat$B2 = norm2binom(dat$N2,
size = 1, prob = 0.5,
mu = 0, sd = 1)

# check
get_params(dat)
#>      n var   N1   N2   T1   B2 mean   sd
#> 1 1000  N1 1.00 0.91 0.99 0.73 0.00 1.00
#> 2 1000  N2 0.91 1.00 0.90 0.79 0.00 1.00
#> 3 1000  T1 0.99 0.90 1.00 0.76 5.02 2.38
#> 4 1000  B2 0.73 0.79 0.76 1.00 0.52 0.50

Note that the correlation between T1 and B2 is unlikely to be exactly 0.75 unless the n is very large, especially for distributions that have very few unique values. 