Despite the growing interest in incorporating response time data into item response models, there has been a lack of research investigating how the effect of speed on the probability of a correct response varies across different groups (e.g., experimental conditions) for various items (i.e., differential response time item analysis). Furthermore, previous research has shown a complex relationship between response time and accuracy, necessitating a functional analysis to understand the patterns that manifest from this relationship. In this study, response time data are incorporated into an item response model for two purposes: (a) to examine how individuals' speed within an experimental condition affects their response accuracy on an item, and (b) to detect the differences in individuals' speed between conditions in the presence of within-condition effects. For these two purposes, by-variable smooth functions are employed to model differential and functional response time effects by experimental condition for each item. This model is illustrated using an empirical data set to describe the effect of individuals' speed on their reading comprehension ability in two experimental conditions of reading medium (paper vs. digital) by item. A simulation study showed that the recovery of parameters and by-variable smooth functions of response time was satisfactory, and that the type I error rate and power of the test for the by-variable smooth function of response time were acceptable in conditions similar to the empirical data set. In addition, the proposed method correctly identified the range of response time where between-condition differences in the effect of response time on the probability of a correct response were accurate.