Covariance matrix, correlation matrix, variance and standard deviation for potential impact fractions

Computes the covariance (covariance) or correlation (correlation) for multiple potential impact fractions and the variance variance and standard deviation standard_deviationfor a potential impact fractions.

Usage

covariance(
  x,
  ...,
  var_p = NULL,
  var_beta = NULL,
  var_weights = NULL,
  var_pif_weights = NULL,
  var_pifs = NULL
)

variance(x, ...)

standard_deviation(x, ...)

correlation(x, ..., var_p = NULL, var_beta = NULL)

Arguments

x

A potential impact fraction

...

Multiple additional potential impact fraction objects separated by commas.

var_p

covariance matrix for the prevalences in both pif1 and pif2. Default NULL (no correlation).

var_beta

covariance matrix for the parameter beta in both pif1 and pif2. Default NULL (no correlation).

var_weights

Covariance between the weights of all the fractions contained in pif1 and all the fractions contained in pif2.

var_pif_weights

Covariance structure between the potential impact fractions in pif1 and the weights in pif2. Entry var_pif_weights[i,j] is the covariance between the ith fraction of pif1 and the jth weight of pif2. This refers to the term \(\operatorname{Cov}\Big(\widehat{\textrm{PIF}}_{A,i}, \hat{w}_j\Big)\) in the equation before.

var_pifs

Covariance vector between the potential impact fractions in pif_ensemble and pif_atomic. This refers to the term \(\operatorname{Cov}\Big( \textrm{PIF}_{A,i}, \widehat{\textrm{PIF}}_{B,j}\Big)\) in the equation below. If set to NULL its automatically calculated.

Examples

# Get the approximate link_variance of a pif object
my_pif <- pif(p = 0.5, p_cft = 0.25, beta = 1.3,
              var_p = 0.1, var_beta = 0.2)
variance(my_pif)
#> [1] 0.07233754

# This is the same as link_covariance with just 1 pIF
covariance(my_pif)
#>            [,1]
#> [1,] 0.07233754

# Calculate the link_covariance between 3 fractions with shared relative risk
beta <- 0.3
var_beta <- 0.1
pif1 <- pif(0.5, 0.2, beta, var_p = 0.5 * (1 - 0.5) / 100, var_beta = var_beta)
pif2 <- pif(0.3, 0.1, beta, var_p = 0.3 * (1 - 0.3) / 100, var_beta = var_beta)
pif3 <- pif(0.7, 0.3, beta, var_p = 0.7 * (1 - 0.7) / 100, var_beta = var_beta)
covariance(pif1, pif2, pif3)
#>             [,1]        [,2]        [,3]
#> [1,] 0.008789254 0.006486541 0.010220324
#> [2,] 0.006486541 0.005074095 0.007703812
#> [3,] 0.010220324 0.007703812 0.012268945

# The link_covariance between a pif and itself only has the link_variance as entries
covariance(pif1, pif1)
#>             [,1]        [,2]
#> [1,] 0.008789254 0.008789254
#> [2,] 0.008789254 0.008789254

# Or if there is a link_covariance structure between different betas you can specify with
# var_beta in the link_covariance
betas <- c(1.3, 1.2, 1.27)

# link_covariance among all betas
var_beta <- matrix(c(
  1.0000000, -0.12123053, 0.35429369,
  -0.1212305, 1.00000000, -0.04266409,
  0.3542937, -0.04266409, 1.00000000
), byrow = TRUE, ncol = 3)
pif1 <- pif(0.5, 0.2, betas[1], var_p = 0.5 * (1 - 0.5) / 100, var_beta = var_beta[1, 1])
pif2 <- pif(0.3, 0.1, betas[2], var_p = 0.3 * (1 - 0.3) / 100, var_beta = var_beta[2, 2])
pif3 <- pif(0.3, 0.1, betas[3], var_p = 0.3 * (1 - 0.3) / 100, var_beta = var_beta[3, 3])
covariance(pif1, pif2, pif3, var_beta = var_beta)
#>              [,1]          [,2]          [,3]
#> [1,]  0.042197697 -5.651801e-03  1.629740e-02
#> [2,] -0.005651801  5.536138e-02 -9.642887e-05
#> [3,]  0.016297402 -9.642887e-05  5.410105e-02

# Compute the correlation
correlation(pif1, pif2, pif3, var_beta = var_beta)
#>            [,1]         [,2]         [,3]
#> [1,]  1.0000000 -0.116933525  0.341091700
#> [2,] -0.1169335  1.000000000 -0.001761979
#> [3,]  0.3410917 -0.001761979  1.000000000