On applying the matching law to between-subject data

Abstract

The matching law is a descriptive mathematical model thatconceptualizes an organism’s response rates as a function of rela-tive rates of reinforcement (Herrnstein 1961). This relation isexpressed by equation (1):B1=B2 ¼ R1=R2 (1)where Bs represent rates of behaviour and Rs represent reinforcerrates. However, extensive research has confirmed that organisms’behaviour systematically deviates from predictions of equation (1)(Baum 1974, 1979 , 1983; Wearden & Burgess 1982; Davison M McDowell 2012). The power function version,known as the generalized matching law (Baum 1974), providesbetter descriptions of animals’ and humans’ choices in concurrentschedules and is represented logarithmically by equation (2):log ðB1=B2Þ¼a log ðR1=R2Þþlog c (2)where Bs and Rs are the same as in equation (1), a is referred to assensitivity and represents the organism’s adjustment to the con-tingencies, and c is called the bias parameter and represents theorganism’s preference when choice is asymmetrical. Note thatequation (2) reduces to equation (1) when a ¼ c ¼ 1, and is usuallycalled strict matching. Furthermore, most studies have found thatthe sensitivity value varies around 0.8, which is referred to asundermatching.Herrnstein (1961) developed the matching law to describe in-dividual choices. A misunderstanding of this fundamentalassumption has led several authors (i.e. Schroeder & Holland 1969;Conger & Killeen 1974; Schmitt 1974; Vollmer & Bourret 2000;Borrero & Vollmer 2002; Alferink et al. 2009; McDowell & Caron2010a, b) to conduct their analyses incorrectly. For instance, theyused a small number of levels of the independent variable andpooled their subjects’ data, which combined idiosyncraticdependent variables into a single analysis and did not account forwithin-subject and within-level variances. At best, between-subject data will estimate an inaccurate model with erroneousparameter values and, at worst, will violate several statistical as-sumptions and lead to uninterpretable results. The purpose of thecurrent study is to prevent future misunderstanding by clarifyingthese errors.Pooling Different VariablesStatistically, the dependent variable must be the same throughall levels of the independent variable. Applied to the matching law,this means that the response ratio must remain the same at alllevels of the reinforcer ratios. Therefore, between-subject analysesmust rely on the same responses for every subject. However, if thesubjects had different target behaviours, then their log response