Lecture 5 Flashcards
Discrete latent variables are called ….
Continuous latent variables are called ….
Discrete latent variables -> latent classes
Continuous latent variables -> latent traits
Definition of item response function
The item response function is the function that relates the latent variables to the item response distribution.
What is the difference between constrained and unconstrained item response functions?
An unconstrained IRF can take any of its admissible values, whereas the admissible values of a constrained IRF are restricted
What assumes the independence model?
The independence model assumes that all test takers belong to the same latent class, which means that there are no individual differences in test takers’ item response behavior.
Independence model: 1 latent class, no latent traits
What assumes the nominal latent class model?
The nominal latent class model assumes two or more unordered latent classes. The IRF of a nominal latent class consists of the probabilities of giving the corrects answer to an item per latent class. The latent classes are unordered, and therefore, these probabilities are not further constrained.
What assumes an ordinal latent class model?
An ordinal latent class model assumes two or more latent classes that are unordered. As for a nominal latent class model, the IRF of an ordinal latent class model consists of the probabilities of giving the correct answer to the item per latent class. However, the latent classes are ordered, and, therefore, these probabilities are constrained.
What is a deterministic ordinal latent class model?
This is an ordinal latent class model that is further constrained by assuming that masters have a probability of 1 of giving the correct answer, and non-masters a probability of 0.
What is a probabilistic ordinal latent class model?
An probabilistic ordinal latent class model assumes only that the probabilities of the latent classes are ordered, but it does not assume that these probabilities are 1 and 0. A masters probability of giving the correct answer is higher than than non-masters probability.
What assumes an unidimensional latent trait model?
This model assumes one continuous latent variable. The IRF of an unidimensional latent trait model is a non-decreasing function.
What is the Guttman model?
A deterministic unidimensional latent trait model
Each item response model makes assumptions on …
- the distribution of the observed item response variable
- the latent variable(s)
- the IRF, which connects the latent and observed variables
Fourth assumption is common to all models - assumption of local independence
What is local independence ?
Local independence assumes that the responses to the N items of the test are independently distributed across (hypothetical) repeated test administrations for each of the test takers of a population of N persons
Local independence of an n-item test applies if two conditions are fulfilled:
- the prob of any response pattern of the n-item test is equal to the product of the probabilities of the corresponding test item probabilities
- the probability of a response pattern on every subtest of the n-item test is equal to the product of the probabilities of the corresponding subtest item probabilities
Which statistic can be used to examine the fit of an IRM? And how can it be computed?
Use the chi square statistic.
E
How many degrees of freedom should you use for the chi square statistic?
Number of free parameters - number of estimated parameters
What is the weakness of CTT?
The scoring of items and tests lack any theoretical justification
A well known statistical method for estimating parameters is the….
Maximum likelihood procedure
In contrast to latent class models, latent trait models always assume …
Latent trait models always assume that the IRF is an increasing or on-decreasing function of the latent trait(s)
Three types of IRF for latent trait models are distinguished…
- Deterministic latent trait models: assumes that the IRF can take only 2 values; 0 or 1
- Probabilistic parametric latent trait models: the IRF of this model is an increasing function of the latent trait that can take any value between 0 and 1 and the IRF is characterized by one or more parameters.
- Probabilistic nonparametric latent trait models: the IRF of this type of model is a non-decreasing function of the latent trait that can take any value between 0 and 1, but the IRF is not characterized by parameters
When a item response pattern is given, how can you determine to which latent class the respondent should be assigned?
Computing the probabilities that the item response pattern of the child belongs to each of the three latent classes, and assigning the child to the latent class that has the highest of these three probabilities.
1. Probabilities of belonging to each of the latent classes = proportion of children per latent class.
2. Compute the probability of the given response pattern per latent class
3. Use Bayes theorem:
Boven de streep: proportion children first latent class * probability of the given response pattern in the first latent class
Divide by:
Prop children first lat classprob response pattern in the first lat class + prop children second lat classprob response pattern in the second class etc etc
What is the Guttman model
The Guttman model assumes one continuous latent variable. Therefore, it is a unidimensional latent trait model because it assumes only one latent trait. It is a deterministic model because it assumes that the IRF can only take the values 0 and 1.
What is an example of a probabilistic version of the Guttman model?
The proctor model. The proctor model assumes that the IRF of the k item of a test is the following threshold function of the latent trait
(1-m) for theta bigger than b
M for theta smaller than b
What is a logistic IRF and give two examples of logistic IRFs
The logistic IRF is a continuous function that has no thresholds where the function jumps from one value to another. Two examples : birnbaum’s two-parameter logistic model and the rasch one-parameter logistic model
The birnbaum’s IRF has a number of properties …
- The IRF is bounded by 0 and 1
- The function is continuous, and has no discontinuities or jumps at thresholds
- The function is always increasing in the latent trait, and is never constant when the latent trait increases.
- If the latent trait value is equal to the b parameter, the IRF takes the value of 0.5
- The IRF has one inflexion point at the latent trait value b.
- The slope of the tangent line of the IRF at the inflexion point is 0.25a
What determines the b parameter (item difficulty / item attractiveness) in the IRF from the birnbaum’s two-parametric logistic model?
The b parameter determines the location of the IRF.
What indicated the a parameter (discrimination parameter) of the birnbaum’s two parameter logistic model?
The discrimination parameter indicated the steepness of the IRF
What is the difference between birnbaum’s model and the rasch model?
Birnbaum’s model has two parameters the a-parameter is the discrimination parameter and determines the steepness of the function. The b-parameter is the item difficulty parameter and determines the position of the IRF.
The rasch model only has 1 parameter, the item difficulty parameter (b-parameter).
The rasch model is a somewhat restrictive model, but it has some good measurement properties, name 4
- The b-parameter can unequivocally be interpreted as item difficulty.
- The rasch model does not need an extra assumption. (The birnbaum’s model was extended with an extra assumption: normal distribution of the latent trait)
- A number of statistical tests were developed to test the fit of the rash model using sample data.
- The rasch model has the property of specific objectivity
What is specific objectivity?
Test measurements are said to be specific objective if:
- Statements that quantitatively compare any two test takers only involve parameters of other test takers, and
- Statements that quantitatively compare any two items only involve parameters of the two items concerned
name two non parametric models
- the Mokken’s monotone homogeneity model
- the Mokken’s double monotonicity model
what assumes the Mokken’s monotone homogeneity model?
that the IRF is a monotonically non-decreasing function of the latent trait. Moreover, the function is always greater than 0 and smaller than 1, which makes it a probabilistic item response model. The IRF is not further restricted. Therefore, the IRFs of differrent items of a test may or may not intersect
What assumes the Mokken’s double monotonicity model?
It also assumes that the IRF is a monotonically non-decreasing function of the latent trait that is between 0 and 1. Moreover, it assumes that the IRFs of the different items of a test do not intersect.
What measures the Loevinger’s homogeneity coefficient?
- The homogeneity of pairs of items
- The homogeneity of one item with respect to the other test items
- The homogeneity of the total test
The Loevinger’s homogeneity coefficient will be derived by considering two benchmark models:
the guttman model and the independence model
Hkl (Loevinger’s homogeneity coefficient between items k and l) = 1 means ….
Hkl = 0 means
Hkl = 1 --> if the Guttman model perfectly applies to the data of a population of test takers. Hkl = 0 --> if the independence model perfectly applies to the data
What is a Guttman error?
An item response pattern that is inadmissible under the Guttman model, but occurs in empirical data
How is Hkl computed?
Hkl = 1 - (population propotion of Guttman errors / under independence model expected population proportion of Guttman errors)
What about the H coefficient if Mokken model applies?
If Mokken model applies, then all the H coefficients are >0
When is the H coefficient strong, moderate or weak?
> 0.5 = strong
Between 0.4 and 0.5 = moderate
