This version of the model is known as the one-parameter logistic model (1PLM), or Rasch model (Rasch 1960). To illustrate the model, consider an individual with a score θj of 1 on the latent trait, and a particular item with parameter β = 1. Then the probability of a positive response from this individual on this item equals exp(1 − 1)/(1 + exp(1 − 1)) = exp(0)/(1 + exp(0)) = 1/2 = 50%. An individual with a score higher than 1 has a higher probability of showing a positive response, whereas an individual scoring lower than 1 has a lower probability. Individuals with a latent score of −1 have a probability of exp(−2)/(1 + exp(−2)) = 12%. With a simple multiplicative transformation of the scale, the logistic and normal ogive curves are very similar and indistinguishable for all practical work (see, for instance, Lord 1980).
