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The law of brevity in macaque vocal communication is not an artifact of analyzing mean call durations

Words follow the law of brevity, i.e. more frequent words tend to be shorter. From a statistical point of view, this qualitative definition of the law states that word length and word frequency are negatively correlated. Here the recent finding of patterning consistent with the law of brevity in Formosan macaque vocal communication (Semple et al., 2010) is revisited. It is shown that the negative correlation between mean duration and frequency of use in the vocalizations of Formosan macaques is not an artifact of the use of a mean duration for each call type instead of the customary 'word' length of studies of the law in human language. The key point demonstrated is that the total duration of calls of a particular type increases with the number of calls of that type. The finding of the law of brevity in the vocalizations of these macaques therefore defies a trivial explanation.


Introduction
In his pioneering work, G. K. Zipf observed that more frequent words tend to be shorter, and attributed this phenomenon to a general principle of least effort (Zipf, 1949). At the level of the dependency between length and frequency, that principle can be considered an informal precursor of a compression principle, i.e. assigning smaller lengths to more frequently used words, that has been studied with mathematical rigor in information theory (Cover & Thomas, 2006). The law of brevity has been reported for many languages (e.g., Zipf, 1949;Straus et al., 2007;Jayaram & Vidya, 2009). Recently, parallels of the law of brevity have been investigated in the behavior of other species. A negative correlation between frequency and size (in behavioural units) of surface behavioral patterns has been reported for dolphins (Ferrer-i-Cancho & Lusseau, 2009), and a negative correlation has been found between frequency of use and duration of vocalizations of Formosan macaques (Semple et al., 2010). However, support for this law has not been found in analyses of the vocalizations of two New World primates, common marmosets and golden-backed uakaris (Bezerra et al., 2011). The universality of the law in the behavior of other species is a matter of current discussion (Ferrer-i-Cancho & Hernández-Fernández 2012, Bezerra et al., 2011 In the study by Semple et al. (2010), the law of brevity was studied by means of a correlation analysis of the relationship between mean call duration and frequency of use (the latter being quantified as the number of calls of each type recorded). It has been argued that a negative correlation between two variables, X and another Z=Y/X (e.g., Z is the mean duration, X is the frequency of use and Y is the total duration) is an unavoidable consequence of the definition of Z as quotient involving X because then Z  1/X. (Solé, 2010). The argument is mathematically flawed, because Z  1/X requires that X and Y are uncorrelated (Hernández-Fernández et al., 2011;Li, 2012). However, the important message for quantitative linguistics researchers is that a negative correlation between Z=Y/X and X could potentially be a trivial consequence of the independence between X and Y.
With respect to the data analyzed by Semple et al. (2010), let D be the total duration of all calls of a particular type (D is the sum of all the durations of the calls of a given type). The main objective of this article is to demonstrate that the negative correlation between frequency of use f and mean call duration d = D/f in Formosan macaques is not a straightforward consequence of the definition of d as a mean. More formally, we want to reject d=c/f (where c is a constant) by showing that D and f are correlated. This potential problem in the study of the law of brevity by Semple et al. (2010) does not concern the analysis performed by Ferrer-i-Cancho & Lusseau (2009), where the correlation analysis focused on the dependency between size of a behavioral pattern in elementary behavioral units (not a mean size) and frequency.
For some quantitative linguistics researchers, another important problem, namely the fit of a particular function, e.g., d  f b where b is a constant, would need to be addressed as is customary in quantitative linguistics research (e.g., Jayaram & Vidya, 2009). Firstly, it should be noted that we want to stay neutral on the issue of the most appropriate function for human language or animal behavior in general in the present article. For instance, standard information theory suggests that d  log f could be a more appropriate function based upon optimal coding considerations (Cover & Thomas, 2006) but as far as we know, this alternative functional dependency has not been considered. Secondly, it is our aim here to contribute to defining a statistically rigorous methodology for studying via correlation analysis the law of brevity as a tendency for more frequent elements to be shorter.

Methods
The same dataset used by Semple et al. (2010) was reanalyzed here. Pearson's and Spearman's correlations, carried out in SPSS v. 17.0, were used to explore the relationship between D and f. As in Semple et al. (2010), two levels of analysis were considered, i.e. the whole repertoire of vocalizations (n=35 call types) and also just those vocalizations produced by all age classes (n=17 call types), as some vocalization are not produced by all age groups (Hsu et al., 2009). Figure 1 shows the dependency between D and f in the vocalizations of Formosan macaques at the two levels of analysis. Indeed, D and f are highly correlated in the whole data set (Pearson's correlation test: n=35, r=0.597, p<0.001; Spearman's correlation test: n=35, r s =0.727, p<0.001), and in the reduced data set of calls given by all age classes (Pearson's correlation test: n=17, r=0.671, p=0.003; Spearman's correlation test: n=17, r s =0.758, p<0.001).

Discussion
It has been shown that the law of brevity documented in the vocal communication of Formosan macaques defies a trivial explanation. A priori, uncorrelation between D and f would have been surprising from a mathematical perspective. Notice that D is defined as a sum of the durations of each vocalization type that is bounded below by a number greater than zero. To see this, introducing some notation is necessary.  i is defined as the duration of the i-th occurrence of vocalization type of frequency f, and  min is defined as the minimum duration of a vocalization type.
Then it follows that ∑ . (1) Notice that  min > 0 for a vocalization to be perceived. Eq. 1 and the fact that  min > 0 where a and b are constants. Notice that b = 0 is needed by the definition of D (Eq. 1) so that D = 0 when f = 0. Thus, if D followed Eq. 2 then d = D/f =a but d and f are significantly correlated (Semple et al. 2010). Here a very important problem for quantitative linguistics research and linguistic theory has been addressed: the statistical significance of statistical regularities of language. Contrary to what many have researchers have claimed (e.g., Miller & Chomsky, 1963;Suzuki et al., 2005;Solé, 2010), various statistical patterns of language are hard to explain in terms of artifacts or simplistic random processes such as the famous random typing experiment (Miller & Chomsky 1963); this applies not only to Zipf's law of brevity but also Zipf's law for word frequencies in both human language (Zipf, 1949)