Predict Knowledge States from Basic Local Independence Models (BLIMs)

Predict knowledge state or state probabilities based on a fitted blim object given a response pattern.

# S3 method for class 'blim'
predict(object, newdata = NULL, type = c("state", "probs"),
        method = c("ML", "MD", "MDML"), quiet = FALSE,
        ties.method = c("min", "max", "random"), i.RK = NULL,
        incradius = object$incradius, as.pattern = TRUE, ...)

Arguments

object

an object of class blim, typically the result of a call to blim.

newdata

a character vector of response patterns with which to predict.

type

for each pattern, predict a knowledge state or the state probabilities.

method

how to compute the posterior state probabilities. (See Details.)

quiet

silence message when estimation and prediction methods differ.

ties.method

how to deal with a posteriori equally probable states.

i.RK

optional indicator matrix of states at minimum distance from response patterns.

incradius

see blim.

as.pattern

return a character vector via as.pattern or an indicator matrix.

...

further arguments passed to as.pattern.

Details

Predicted is the modal posterior state (type = "state") or the posterior distribution (type = "probs") of knowledge states given a response pattern. Depending on the method argument, the posterior distribution is defined as:

• Maximum likelihood (method = "ML") $$P(K | R)_{ML} = \frac{P(R | K) P(K)}{\sum_K P(R | K) P(K)} = \frac{P(R | K) P(K)}{P(R)}$$

• Minimum discrepancy (method = "MD") $$P(K | R)_{MD} = \frac{i_{RK}}{\sum_K i_{RK}}$$

• Minimum discrepancy ML (method = "MDML") $$P(K | R)_{MDML} = \frac{i_{RK} \cdot P(K | R)_{ML}} {\sum_K i_{RK} \cdot P(K | R)_{ML}}$$

where \(i_{RK}\) is a pattern-by-state indicator matrix that is one for each state \(K\) that is at minimum distance from pattern \(R\).

Value

If type = "state", a character vector of knowledge states (if as.pattern = TRUE) or a state-by-problem indicator matrix (if as.pattern = FALSE).

If type = "probs", a matrix of posterior state probabilities.

See also

blim, slm.

Examples

data(DoignonFalmagne7)
m <- blim(DoignonFalmagne7$K, DoignonFalmagne7$N.R)
predict(m)
#> estimation method is MD, prediction method is ML
#>  [1] "00000" "10000" "01000" "00000" "00000" "00000" "11000" "11100" "11010"
#> [10] "10000" "11100" "11010" "01000" "11110" "11101" "00000" "11100" "11010"
#> [19] "11101" "11110" "11101" "11010" "11110" "11101" "11010" "11111" "11110"
#> [28] "11101" "11010" "11111" "11111" "11111"
predict(m, newdata = c("00100", "10100", "01111"), as.pattern = FALSE)
#> estimation method is MD, prediction method is ML
#>       a b c d e
#> 00000 0 0 0 0 0
#> 11100 1 1 1 0 0
#> 11111 1 1 1 1 1

m <- slm(DoignonFalmagne7$K, DoignonFalmagne7$N.R)
predict(m, newdata = "00100")
#> estimation method is MD, prediction method is ML
#> [1] "00000"

data(endm)
m <- blim(endm$K, endm$N.R)
predict(m, type = "probs", method = "MD")
#>         0000      0110      0101 1110 1101 1011      1111
#> 1  1.0000000 0.0000000 0.0000000  0.0  0.0  0.0 0.0000000
#> 2  1.0000000 0.0000000 0.0000000  0.0  0.0  0.0 0.0000000
#> 3  0.3333333 0.3333333 0.3333333  0.0  0.0  0.0 0.0000000
#> 4  0.5000000 0.5000000 0.0000000  0.0  0.0  0.0 0.0000000
#> 5  0.5000000 0.0000000 0.5000000  0.0  0.0  0.0 0.0000000
#> 6  0.0000000 0.0000000 0.0000000  0.5  0.5  0.0 0.0000000
#> 7  0.0000000 0.0000000 0.0000000  0.5  0.0  0.5 0.0000000
#> 8  0.0000000 0.0000000 0.0000000  0.0  0.5  0.5 0.0000000
#> 9  0.0000000 1.0000000 0.0000000  0.0  0.0  0.0 0.0000000
#> 10 0.0000000 0.0000000 1.0000000  0.0  0.0  0.0 0.0000000
#> 11 0.0000000 0.0000000 0.0000000  0.0  0.0  1.0 0.0000000
#> 12 0.0000000 0.0000000 0.0000000  1.0  0.0  0.0 0.0000000
#> 13 0.0000000 0.0000000 0.0000000  0.0  1.0  0.0 0.0000000
#> 14 0.0000000 0.0000000 0.0000000  0.0  0.0  1.0 0.0000000
#> 15 0.0000000 0.3333333 0.3333333  0.0  0.0  0.0 0.3333333
#> 16 0.0000000 0.0000000 0.0000000  0.0  0.0  0.0 1.0000000
cbind(
  observed = names(m$N.R),
  min = predict(m, method = "MD", ties.method = "min"),
  max = predict(m, method = "MD", ties.method = "max"),
  rnd = predict(m, method = "MD", ties.method = "random")
) |> print(quote = FALSE)
#>       observed min  max  rnd 
#>  [1,] 0000     0000 0000 0000
#>  [2,] 1000     0000 0000 0000
#>  [3,] 0100     0000 0110 0101
#>  [4,] 0010     0000 0110 0000
#>  [5,] 0001     0000 0101 0101
#>  [6,] 1100     1110 1110 1110
#>  [7,] 1010     1110 1110 1011
#>  [8,] 1001     1101 1101 1101
#>  [9,] 0110     0110 0110 0110
#> [10,] 0101     0101 0101 0101
#> [11,] 0011     1011 1011 1011
#> [12,] 1110     1110 1110 1110
#> [13,] 1101     1101 1101 1101
#> [14,] 1011     1011 1011 1011
#> [15,] 0111     0110 1111 1111
#> [16,] 1111     1111 1111 1111