Simple Item Statistics

This function calculates statistics for each item, with different metrics available depending on the data type (binary, ordinal, or rated).

Usage

ItemStatistics(U, na = NULL, Z = NULL, w = NULL)

# Default S3 method
ItemStatistics(U, na = NULL, Z = NULL, w = NULL)

# S3 method for class 'binary'
ItemStatistics(U, na = NULL, Z = NULL, w = NULL)

# S3 method for class 'ordinal'
ItemStatistics(U, na = NULL, Z = NULL, w = NULL)

Arguments

U

Either an object of class "exametrika" or raw data. When raw data is given, it is converted to the exametrika class with the dataFormat function.

na

Values to be treated as missing values.

Z

Missing indicator matrix of type matrix or data.frame. Values of 1 indicate observed responses, while 0 indicates missing data.

w

Item weight vector specifying the relative importance of each item.

Value

For binary data:

ItemLabel

Label identifying each item

NR

Number of Respondents for each item

CRR

Correct Response Rate denoted as $p_j$.

ODDs

Item Odds is the ratio of the correct response rate to the incorrect response rate. Defined as \(o_j = \frac{p_j}{1-p_j}\)

Threshold

Item Threshold is a measure of difficulty based on a standard normal distribution.

Entropy

Item Entropy is an indicator of the variability or randomness of the responses. Defined as \(e_j=-p_j \log_2 p_j - (1-p_j)\log_2(1-p_j)\)

ITCrr

Item-total Correlation is a Pearson's correlation of an item with the Number-Right score.

For ordinal polytomous data:

ItemLabel

Label identifying each item

NR

Number of Respondents for each item

Threshold

Matrix of threshold values for each item's category boundaries, based on a standard normal distribution. For an item with K categories, there are K-1 thresholds.

Entropy

Item Entropy calculated using the category probabilities. Unlike binary data, this is calculated using the formula \(e_j = -\sum_{k=1}^{K_j} p_{jk} \log_{K_j} p_{jk}\), where \(K_j\) is the number of categories for item j.

ITCrr

Item-total Correlation calculated using polyserial correlation, which accounts for the ordinal nature of the item responses and the continuous total score.

Note

For rated data, the function processes the data as binary, with each response being compared to the correct answer to determine correctness.

Examples

# using sample dataset(binary)
ItemStatistics(J15S500)
#> Item Statistics
#>    ItemLabel  NR   CRR  ODDs Threshold Entropy ITCrr
#> 1     Item01 500 0.746 2.937    -0.662   0.818 0.375
#> 2     Item02 500 0.754 3.065    -0.687   0.805 0.393
#> 3     Item03 500 0.726 2.650    -0.601   0.847 0.321
#> 4     Item04 500 0.776 3.464    -0.759   0.767 0.503
#> 5     Item05 500 0.804 4.102    -0.856   0.714 0.329
#> 6     Item06 500 0.864 6.353    -1.098   0.574 0.377
#> 7     Item07 500 0.716 2.521    -0.571   0.861 0.483
#> 8     Item08 500 0.588 1.427    -0.222   0.978 0.405
#> 9     Item09 500 0.364 0.572     0.348   0.946 0.225
#> 10    Item10 500 0.662 1.959    -0.418   0.923 0.314
#> 11    Item11 500 0.286 0.401     0.565   0.863 0.455
#> 12    Item12 500 0.274 0.377     0.601   0.847 0.468
#> 13    Item13 500 0.634 1.732    -0.342   0.948 0.471
#> 14    Item14 500 0.764 3.237    -0.719   0.788 0.485
#> 15    Item15 500 0.706 2.401    -0.542   0.874 0.413