Normal ogive models

4 Normal ogive models

4.1 Derivation

In the Thurstonian tradition of scaling, the normal distribution has been used extensively to model unexplained variation in responses. When the observed behaviour is discrete or binary, an underlying, not observed continuous response variable is assumed to operate, together with a kind of boundary or threshold. The observed or overt response then is thought to come about by a comparison of the unobserved continuous response to the threshold. This is exemplified in Figure 8.5. When answering an item, a latent response, z, is drawn from a normal distribution with standard deviation equal to one. The mean of this distribution is the ability of the student. The item defines a threshold, ␤ i , and the overt response is correct if the latent response is greater than the threshold.

The probability that the answer is incorrect is given by

PX ( i = 0 |) θ =

∫ exp[ − ( z − )] 2 θ dz = Φ ( 2 βθ i − ),

2 π −∞ where ⌽(.) denotes the standard normal distribution function. The IRF for the

model is then readily found by using the symmetry of the normal distribution: P(X i = 1|␪) = 1 – ⌽(␤ i – ␪) = ⌽(␪ – ␤ i ).

(13) The graph of the function ⌽(.) is known as the normal ogive, and it looks very

similar to the curves in Figure 8.2.

z Figure 8.5 The normal ogive model using latent responses

Using Item Response Theory 169

4.2 The one- and two-parameter normal ogive models

When the above derivation applies equally to all items, the one-parameter normal ogive model results. The main feature of this is that it is assumed that for every item the standard deviation of the latent responses is equal. The two-parameter model is obtained when it is assumed that the standard deviations of the latent responses may vary across items. Representing these standard deviations by ␴ i , one finds readily that

The graphs of these functions are steeper the smaller the standard deviation is, meaning that the inverse of the standard deviation has the same interpretation as the discrimination parameter in the 2PLM. Defining ␣ i =␴ –1 i , one finds the standard expression for the two-parameter normal ogive model:

P(X i = 1|␪) = ⌽[␣ i (␪ – ␤ i )]. (15)

4.3 Relation between logistic and normal ogive models

The derivation shown for the normal ogive models can also be applied to the Rasch model and the 2PLM, although historically, this does not seem to have been done. The only difference is that the normal distribution is replaced by the logistic distribution. Using the function symbol ⌿(.) for the logistic distribu- tion function, we then find an expression for the 2PLM analogous to (15):

P(X i = 1|␪) = ⌿[␣ i (␪ – ␤ i )]. (16) Although the two distribution functions ⌽(.) and ⌿(.) have very similar graphs,

the graphs will be quite dissimilar if in (15) and (16) the same values are used for the parameters ␣ i and ␤ i . The reason for this is that the standard logistic distribution does not have a standard deviation equal to one. A close similarity is found if the discrimination parameter in the 2PLM is 1.7 times the discrimination parameter in the two-parameter normal ogive model; the difficulty parameters can be treated as equal. This gives

⌽[␣ i (␪ – ␤ i )] ⬇ ⌿[1.7␣ i (␪ – ␤ i )]. The scale factor 1.7 is often found in textbooks treating the 2PLM and the

3PLM. In Figure 8.6, a graph of both IRFs is given for the interval (0, 3) with ␤ i = 0 for both models; the discrimination parameter for the normal model

equals one, and for the logistic model it is 1.7.

170 Different methodological orientations

Normal Logistic

0 1 2 3 Figure 8.6 Similarity of the normal ogive and the logistic models

As the differences are very small, it might be concluded that the models can be used interchangeably, and for practical purposes this is undoubtedly the case. Nevertheless, models of the logistic family have been more popular than the normal ogive models in the history of modern psychometrics. At least two reasons can be given for this. First, the mathematics needed to study the characteristics of the models are easier and more elegant in the logistic family than with the normal models. A thorough study of the 2PLM and the 3PLM can be found in the influential book by Lord and Novick (1968), especially the contributions of A. Birnbaum. The great attention that IRT has got in Europe is certainly due to the important work of Rasch (1960) and the pioneering work of Fischer. Both these authors have spent a lot of effort demonstrating that the Rasch model follows necessarily from a few requirements that can be attributed to objective measurement. The interested reader is referred to Fischer (1974, 1995) for a detailed (and often quite difficult) discussion on these topics.

In the 1990s, renewed attention was given to the normal ogive models. Estimation techniques based on sampling procedures, such as the Gibbs sampler, became popular and feasible due to the increasing computing speed of modern computers. It appeared that these techniques were easier to implement with the normal ogive models than with models from the logistic family.