Choice behavior of rats in a concurrent-chains schedule: Amount and delay of reinforcement

Rats were exposed to concurrent-chains schedules in which the terminal links were equal fixed-interval schedules terminating in one or three food pellets. Choice proportions for large reward increased with increases in delay intervals programmed on fixed-interval schedules and supported the predictions derived from a general choice model originally formulated by Fantino and later developed by Navarick and Fantino. In addition, a functional equivalence of two alternatives was established by increasing delay intervals with large reward, whereas delay intervals for small reward were held constant. Functionally equivalent delay intervals with large reward, for each delay interval with small reward, can be described by a power function with exponent smaller than 1.0. A better prediction of choice proportions resulted when this function was used to derive predicted choice proportions.     

Naturalism Need Not Be "Made Safe": A Response to William Rottschaefer's Misunderstandings

In this article, I respond to William Rottschaefer's analysis of my writings on religion and science, especially my Religion, Science and Naturalism (1996). I show that I am not trying "to make naturalism safe," as Rottschaefer contends, but rather attempting to explore options available when one endorses naturalistic approaches. I also explain why I object to the label "supernaturalistic naturalism" used by Rottschaefer. Possible limitations to naturalistic projects are discussed, not as limitations imposed but rather as features uncovered.     

Discerning the Limits of Religious Naturalism

In response to my "How to Make Naturalism Safe for Supernaturalism: An Evaluation of Willem Drees's Supernaturalistic Naturalism" (Rottschaefer 2001), Willem Drees maintains that I have misunderstood his purpose and views and have failed to make the case against his view that naturalism is intrinsically limited. In this response, I comment on these concerns.     

Multilevel Modeling with Correlated Effects

When there exist omitted effects, measurement error, and/or simultaneity in multilevel models, explanatory variables may be correlated with random components, and standard estimation methods do not provide consistent estimates of model parameters. This paper introduces estimators that are consistent under such conditions. By employing generalized method of moments (GMM) estimation techniques in multilevel modeling, the authors present a series of estimators along a robust to efficient continuum. This continuum depends on the assumptions that the analyst makes regarding the extent of the correlated effects. It is shown that the GMM approach provides an overarching framework that encompasses well-known estimators such as fixed and random effects estimators and also provides more options. These GMM estimators can be expressed as instrumental variable (IV) estimators which enhances their interpretability. Moreover, by exploiting the hierarchical structure of the data, the current technique does not require additional variables unlike traditional IV methods. Further, statistical tests are developed to compare the different estimators. A simulation study examines the finite sample properties of the estimators and tests and confirms the theoretical order of the estimators with respect to their robustness and efficiency. It further shows that not only are regression coefficients biased, but variance components may be severely underestimated in the presence of correlated effects. Empirical standard errors are employed as they are less sensitive to correlated effects when compared to model-based standard errors. An example using student achievement data shows that GMM estimators can be effectively used in a search for the most efficient among unbiased estimators. This research was supported by the National Academy of Education/Spencer Foundation and the National Science Foundation, grant number SES-0436274. We thank the editor, associate editor, and referees for detailed feedback that helped improve the paper.