In a situation where the population can be divided into different and exclusive categories, we can calculate the Bayes Linear Estimator for the proportion of individuals in each category with the BLE_Categorical() function, which receives the following parameters:
Letting \(\rho_{ii} \to 1\), that is, assuming prior ignorance, the resulting point estimate will be the same as the one seen in the design-based context for categorical data.
This can be achieved using the BLE_Categorical() function by omitting either the prior proportions and/or the parameter rho, that is:
If the calculation of matrices R and Vs results in non-positive definite matrices, a warning will be displayed. In general this does not produce incorrect/ inconsistent results for the proportion estimate but for its associated variance. It is suggested to review the prior correlation coefficients (parameter rho).
ys <- c(0.2614, 0.7386)
n <- 153
N <- 15288
m <- c(0.7, 0.3)
rho <- matrix(0.1, 1)
Estimator <- BLE_Categorical(ys,n,N,m,rho)
Estimator$est.prop
#> [1] 0.2855228 0.7144772
Estimator$Vest.prop
#> [,1] [,2]
#> [1,] 0.001155671 -0.001155671
#> [2,] -0.001155671 0.001155671
Bellow we can see that the greater the correlation coefficient, the closer our estimation will get to the sample proportions.
ys <- c(0.2614, 0.7386)
n <- 153
N <- 15288
m <- c(0.7, 0.3)
rho <- matrix(0.5, 1)
Estimator <- BLE_Categorical(ys,n,N,m,rho)
Estimator$est.prop
#> [1] 0.2642195 0.7357805
Estimator$Vest.prop
#> [,1] [,2]
#> [1,] 0.0006750388 -0.0006750388
#> [2,] -0.0006750388 0.0006750388
ys <- c(0.2, 0.5, 0.3)
n <- 100
N <- 10000
m <- c(0.4, 0.1, 0.5)
mat <- c(0.4, 0.1, 0.1, 0.1, 0.2, 0.1, 0.1, 0.1, 0.6)
rho <- matrix(mat, 3, 3)
Estimator <- BLE_Categorical(ys,n,N,m,rho)
Estimator$est.prop
#> [1] 0.2221967 0.4785131 0.2992902
Estimator$Vest.prop
#> [,1] [,2] [,3]
#> [1,] 0.0013711226 -0.0004980297 -0.0008730929
#> [2,] -0.0004980297 0.0006722052 -0.0001741755
#> [3,] -0.0008730929 -0.0001741755 0.0010472684
Same example, but with no prior correlation coefficients informed (non-informative prior)
ys <- c(0.2, 0.5, 0.3)
n <- 100
N <- 10000
m <- c(0.4, 0.1, 0.5)
Estimator <- BLE_Categorical(ys,n,N,m,rho=NULL)
#> parameter 'rho' not informed, non informative prior correlation coefficients used in estimations
#> Warning in BLE_Categorical(ys, n, N, m, rho = NULL): 'Vest.prop' should have
#> only positive diagonal values. Review prior specification and verify calculated
#> matrices 'R' and 'Vs'.
Estimator$est.prop
#> [1] 0.2017585 0.4996729 0.2985685