calculate_pk <- function(x_in, y_in)

@title Compute the pk value to Measure the Performance of Anesthetic Depth Indicators.

@param x_in a vector, the indicator.
@param y_in a vector, the state.

@return a list containing all the matrices and variables during the calculation.
    The value list$type is "PK", which indicated the list is return-value of the function calculate_pk().
    The type of list$basic is also a list, which contains the most important results of the function.
    The type of list$matrices is also a list, which contains all the matrices during the calculation.
    The type of list$details is also a list, which contains all the intermediate variables during the calculation.

compare_pks <- function(pk1, pk2)

@title Compare two answers of the pk values.

@description Both of the two input have to be the output of the function calculate_pk().

@param pk1 a list, the output of the function calculate_pk().
@param pk2 a list, the output of the function calculate_pk().

@return a list containing all the variables during the calculation.
    The value list$type is "PKC", which indicated the list is return-value of the function compare_pk().
    The type of list$group is also a list, which contains the normal distribution test results for the group variables.
    The type of list$pair is also a list, which contains the t distribution test results for the pair variables.
    The type of list$details is also a list, which contains all the intermediate variables during the calculation.

x1 <- c(0, 0, 0, 0, 0, 0)
y1 <- c(1, 1, 1, 1, 1, 2)
ans1 <- calculate_pk(x1, y1)

## show the most important results.
print(ans1$basic)

x2 <- c(1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6)
y2 <- c(1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 3, 3, 3)
ans2 <- calculate_pk(x2, y2)

## show the full results.
print(ans2$basic)

$PK
[1] 0.5

$SE0
[1] 0

$SE1
[1] 0

$jack_ok
[1] FALSE

$PKj
[1] NaN

$SEj
[1] NaN

$PK
[1] 0.8670213

$SE0
[1] 0.06503734

$SE1
[1] 0.06587109

$jack_ok
[1] TRUE

$PKj
[1] 0.8664848

$SEj
[1] 0.07011821

x1 <- c(1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6)
y1 <- c(1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 3, 3, 3)

pk1 <- calculate_pk(x_in = x1, y_in = y1)

x2 <- c(1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6)
y2 <- c(1, 1, 2, 1, 1, 2, 1, 2, 3, 3, 2, 2, 1, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2)

pk2 <- calculate_pk(x_in = x2, y_in = y2)

ans <- compare_pks(pk1, pk2)
print(ans$group)
print(ans$pair)

$PKD
[1] 0.06757172

$SED
[1] 0.1010385

$ZD
[1] 0.6687717

$ZP
[1] 0.5036411

$ZJ
[1] "P > 0.05"

$DF
[1] 23

$PKDJ
[1] 0.02971846

$SEDJ
[1] 0.06558182

$TD
[1] 0.4531508

$TP
[1] 0.3273431

$TJ
[1] "P > 0.05"