The study of biological systems and its component parts, whether molecules, cells, tissues, organisms and its forming parts, and even species and their interactions, is rapidly converging to the central theme of modularity. This refers to the connections among some of the component parts of a biological system (genes or morphological traits, for example) and the lack of such associations among other parts of the same system. (Olson and Miller 1958; Berg 1960; Wagner et al. 2007). The notion that interacting parts are not independent is intuitive and appears early in the history of Biology (see Mayr 1982). Therefore, modularity is quickly becoming one of the central questions in modern biology (Wagner et al. 2007; Klingenberg 2008) and a point of convergence of various specialties and areas (Mathematics, Statistics, Genetics and Genomics, Evolutionary Biology, Ecology, Biochemistry, and Physiology).

In biology, several types of modules have been recognized, including (1) functional, consisting of characters or features that act together on performing a task or function and are quasi-autonomous in relation to other functional sets; (2) developmental, which corresponds to parts of an embryo that are relatively autonomous with respect to pattern formation and differentiation, or an autonomous signaling cascade; (3) variational, composed of characters that vary together and are relatively independent of other such sets (Wagner et al. 2007).

The study of modularity is centered on statistical estimation of association among traits (Olson and Miller 1958; Berg 1960). Whether such association is measured by correlation, covariance, or distance/similarity measures, it is usually represented by matrices. Even if a particular system or network does not present a modular structure or is not being interpreted under this theory, associations among traits, parts, genes, or lineages will still be quantified by statistical association or dissociation matrices among these elements. We will focus here on correlation or covariance matrices (from now on **C**-matrix) among variables, although the same problem appears in any statistics of association/dissociation among the component parts of any system. As biologists, what we usually do is to sample nature and infer properties from natural systems using measures and statistics obtained from such samples. We should be aware of the fact that by sampling a population we do not have the true population parameter values but only estimates of these quantities. These estimates will be approximations that should converge to the true population value depending on a number of things, such as sample size, number of parameters considered, precision of the measuring device, and the quality of parameter estimators themselves, measured by their precision and accuracy (Sokal and Rohlf 1995). The general trend is that, as the ratio between number of parameters and sample size decreases, signal-to-noise ratio will increase. For **C**-matrices, this effect is summarized by their sampling distribution (the Wishart distribution) and can be expressed in terms of their eigenvalues (see next section and Meyer and Kirkpatrick 2008).

In this article, we illustrate the problem of noise in matrix estimation using modularity as our framework and addressing effects of matrix estimates in the context of natural selection. The evolutionary response of a set of quantitative traits is described by Δz = GβΔz is the vector of differences in means between generations, β is the selection gradient vector, and **G** is the additive genetic covariance matrix (Lande 1979). Rearranging the evolutionary response equation, the pattern of selection responsible for populations’ divergence can be reconstructed from observed mean differences using the relationship:

where ** β** is the cumulative selection gradient summed over generations (net-

**sensu Lande 1979),**

*β*

*G*^{−1}is the inverse of

**G**, and is the difference in means between populations

*i*and

*j*(Lande 1979; Lofsvold 1988; Cheverud 1996). Selection reconstruction can be extremely useful in understanding patterns of multivariate selection within a microevolutionary context (Boag 1983; Lande and Arnold 1983; Grant and Grant 1995), and if certain assumptions hold (see Marroig and Cheverud 2001, 2004, 2005, 2010) it can be extended to a macroevolutionary context.

We illustrate here how noise associated with sampling in matrix estimation can lead to error in ** β** reconstruction using a simulation approach. We also illustrate how changing both number of modules and magnitude of association among elements in a system will affect matrix estimation and

**reconstruction. We present possible solutions that could help ameliorate the noise problem in matrix estimation and test the performance of such solutions. Finally, we illustrate the analyses of selection reconstruction in a well-studied case, New World Monkeys (NWMs) skull evolution, and show how previously estimated β's (Marroig and Cheverud 2005) are most likely dominated by noise and how estimates of these gradients, after controlling for noise, lend an interpretation in full agreement with results obtained using a different approach on the same dataset (Marroig and Cheverud 2010).**

*β*