Technical Note Conjunction Analysis and Propositional Logic in fMRI Data Analysis Using Bayesian Statistics Thomas Rudert, * and Gabriele Lohmann, PhD Purpose: To evaluate logical expressions over different ef- fects in data analyses using the general linear model (GLM) and to evaluate logical expressions over different posterior probability maps (PPMs). Materials and Methods: In functional magnetic resonance imaging (fMRI) data analysis, the GLM was applied to esti- mate unknown regression parameters. Based on the GLM, Bayesian statistics can be used to determine the probability of conjunction, disjunction, implication, or any other arbi- trary logical expression over different effects or contrast. For second-level inferences, PPMs from individual sessions or subjects are utilized. These PPMs can be combined to a logical expression and its probability can be computed. The methods proposed in this article are applied to data from a STROOP experiment and the methods are compared to conjunction analysis approaches for test-statistics. Results: The combination of Bayesian statistics with prop- ositional logic provides a new approach for data analyses in fMRI. Two different methods are introduced for proposi- tional logic: the first for analyses using the GLM and the second for common inferences about different probability maps. Conclusion: The methods introduced extend the idea of conjunction analysis to a full propositional logic and adapt it from test-statistics to Bayesian statistics. The new ap- proaches allow inferences that are not possible with known standard methods in fMRI. Key Words: conjunction analysis; propositional logic; Bayesian statistics J. Magn. Reson. Imaging 2008;28:1533–1539. © 2008 Wiley-Liss, Inc. THE PURPOSE of this article is to make propositional logic available for data analysis in functional magnetic resonance imaging (fMRI). The methods we introduce can be used for first-, second-, or higher-level analysis, ie, single-subject, multisession or multisubject infer- ences. The methods are based on Bayesian statistics. In fMRI, an effect or an activation describes usually the difference in brain activity during different experi- mental conditions (eg, between an activation task and a baseline task). For fMRI data analyses using the general linear model (GLM) (1), different methods are available depending on the question to be answered: cognitive subtraction is used to test for a single effect. Factorial designs can be used to determine the interaction be- tween different experimental conditions. The aim of conjunction analysis is the conjoint testing for multiple effects in one subject or the conjoint testing for the same effect in different subjects. With propositional logic in combination with Bayesian statistics the prob- ability of a logical expression over different effects or over effects in different subjects can be determined. A very first implementation of conjunction analysis is the concept of cognitive conjunction introduced in Ref. (2) and discussed in Ref. (3). The method is also referred to as interaction masking (4). The idea of cognitive con- junction is to detect areas where there is a significant main effect (sum of all effects) and no significant differ- ences (interactions) between the single effects. These voxels are assumed to be significant activated for all effects. The cognitive conjunction approach uses clas- sical test-statistics. Two later approaches of conjunction analysis define conjunction as the joint refutation of multiple null hy- potheses. Both approaches are based on minimum sta- tistics but assume different distributions for the null hypothesis. First, the minimum statistics compared to the global null (MS/GN) has the null-hypothesis that there is no effect in any subject (5). Second, in the minimum statistics compared to the conjunction null (MS/CN), the null-hypothesis is that there is no effect in at least one subject (4). For further information see also Refs. (6,7). In this article we introduce two new methods that extend conjunction analysis to a full propositional logic. With our new methods it becomes possible to compute not only conjunction but also disjunction, implication, and any other arbitrary logical expression. The first method we propose is for data analysis using the GLM. With it, Bayesian statistics are utilized to determine the probability of logical expressions over different effects in the GLM. The second method requires independent probability maps obtained from previous analyses. With the method the probability of logical expressions Max-Planck-Institute for Human Cognitive and Brian Sciences, Depart- ment of Cognitive Neurology, Leipzig, Germany. *Address reprint requests to: T.R., Max-Planck-Institute for Human Cognitive and Brian Sciences, Stephanstrasse 1a, 04103 Leipzig, Ger- many. E-mail: rudert@cbs.mpg.de Received December 18, 2007; Accepted June 11, 2008. DOI 10.1002/jmri.21518 Published online in Wiley InterScience (www.interscience.wiley.com). JOURNAL OF MAGNETIC RESONANCE IMAGING 28:1533–1539 (2008) © 2008 Wiley-Liss, Inc. 1533