A conceptual introduction to psychometrics by G, J., Mellenbergh - a summary
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A conceptual introduction to psychometrics
Chapter 6
Classical analysis of item scores
The conventional way of scoring items is by assigning ordinal numbers to the response categories.
Usually, these item scores are ordered with respect to the attribute that the item is assumed to measure. But, these assignment of these ordinal numbers lacks a theoretical justification.
Usually, the analysis of test scores is supplemented by an analysis of the item scores.
The scores of a given item have a distribution in a population of N persons.
Classical item difficulty and attractiveness
The location of the item score distribution is used to define the classical item difficulty (maximum performance tests) and classical item attractiveness (typical performance tests) concepts.
The two definitions are the same.
Classical item difficulty and attractiveness are defined in a population of persons.
Population-dependent and may differ between populations.
The mean in mainly used for this.
The mean of a dichotomously scored item is called the item p-value.
Item score variance and standard deviation
The most common parameters that are used in classical item score analysis are the variance and the standard deviation of the item scores.
Items that have a small item score variance, have little effect on the test score variance.
The variance of dichotomous item scores is a function of the item p-value.
For a given sample size, the variance has its maximum value at p=.5.
Location and dispersion parameters yield useful information on the items of a test.
But, these parameters do not indicate the extent to which an item contributes to the aim of a test to assess individual differences in the attribute that is measured by the test.
Classical item discrimination: a parameter that indicates the extent to which the item differentiates between the true test scores of a population of persons.
Defined in a population of persons, may vary between different populations.
The item-test and item-rest correlations
An appropriate index for discrimination between the true scores would be the product moment correlation between the item score and the true score in the population of persons.
Test taker j’s observed score is the estimator of his true score.
The population item-test correlation is estimated by the sample correlation.
Item-rest correlation: the product moment correlation between the item scores and the rest scores of the test, where the studied item is deleted.
The item reliability index
Item reliability index: the kth item reliability index is the product of the item-test correlation and item score standard deviation of the kth item in a population of persons.
A maximum performance item presents the test taker with a problem that has to be solved.
The response mode of this type of item is either the free(constructed) response or the choice (selected) response mode.
The most common response mode is multiple choice.
Item distractor popularity
Classical difficulty of a dichotomously scored multiple-choice item: the proportion of persons of the population who selected the correct answer to the item.
Item distractor popularity: the proportion of persons of a population who selected the distractor.
The item difficulty and the item distractor popularities are estimated in a sample of test takers by the proportions of test takes of the sample who selected the correct answer and the distractors, respectively.
The distractor popularities yield information on the appropriateness of the distractors. An unpopulair distractor is selected by only a small proportion of test takers.
Apparently, most test takers know that the distractor is an incorrect answer to the item, which means that the item can do without this distractor.
Item distractor-item correlations
The usual way of scoring multiple-choice items is by assigning a 1 to the choice of the correct option, and assigning a 0 to the choice of a distractor.
Item distractor-rest correlations: the product moment correlations of the separate dichotomous correct answer/distractor variables and the rest score.
The item distractor-rest correlations yield detailed information on item quality.
A positive distractor-rest correlation indicates that the distractor tends to attract test takers who have lower true scores than test takers who selected the correct answer.
A negative distractor-rest correlatoin indicates that the distractor discriminates in the wrong direction because the distractor tends to attract test takers who have higher true scores than test takers who selected the correct answers.
The observed test score is the unweighted or weighted sum of the item scores.
The basic idea is that in a population of test takers the items that measure the same attribute are more highly associated among each other than associated with items that measure another attribute.
The association between items can be assessed by different coefficients.
Analysis of inter-item product moment correlations
The conventional approach is to compute product moment correlations between the item scores, and to look for clusters of items that are highly correlated within the cluster and less correlated with items of other clusters.
The product moment correlation between the scores of two items in a population of persons is estimated by the sample product moment correlation.
Phi coefficient: the product moment correlation coefficient between two dichotomous variables.
The phi coefficient between two items can only reach a maximum of 1 if the p-values are equal.
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This is a summary of the book A conceptual introduction to psychometrics by G, J., Mellenbergh. The summary contains chapter 1 to 6, and focusus on developing psychological tests.
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