Evolutionary change in New World Monkey (NWM) skulls occurred primarily along the line of least resistance defined by size (including allometric) variation (gmax). Although the direction of evolution was aligned with this axis, it was not clear whether this macroevolutionary pattern results from the conservation of within population genetic covariance patterns (long-term constraint) or long-term selection along a size dimension, or whether both, constraints and selection, were inextricably involved. Furthermore, G-matrix stability can also be a consequence of selection, which implies that both, constraints embodied in gmax and evolutionary changes observed on the trait averages, would be influenced by selection. Here, we describe a combination of approaches that allows one to test whether any particular instance of size evolution is a correlated by-product due to constraints (gmax) or is due to direct selection on size and apply it to NWM lineages as a case study. The approach is based on comparing the direction and amount of evolutionary change produced by two different simulated sets of net-selection gradients (β), a size (isometric and allometric size) and a nonsize set. Using this approach it is possible to distinguish between the two hypotheses (indirect size evolution due to constraints or direct selection on size), because although both may produce an evolutionary response aligned with gmax, the amount of change produced by random selection operating through the variance/covariance patterns (constraints hypothesis) will be much smaller than that produced by selection on size (selection hypothesis). Furthermore, the alignment of simulated evolutionary changes with gmax when selection is not on size is not as tight as when selection is actually on size, allowing a statistical test of whether a particular observed case of evolution along the line of least resistance is the result of selection along it or not. Also, with matrix diagonalization (principal components [PC]) it is possible to calculate directly the net-selection gradient on size alone (first PC [PC1]) by dividing the amount of phenotypic difference between any two populations by the amount of variation in PC1, which allows one to benchmark whether selection was on size or not.

Evolution, at all biological organizational levels (from DNA to complex morphologies or behavior), is contingent upon genetic variability. When dealing with the evolution of complex systems, like the mammalian skull, not only should genetic variation of each individual dimension be considered, but perhaps more importantly, the amount and pattern of covariation between traits (G-matrix) also needs to be taken into account. Genetic and phenotypic patterns of variation and covariation among traits are critical for understanding the evolution of complex systems, whatever the evolutionary processes (genetic drift or natural selection) impacting the population. The reason is straightforward: genetic covariance is produced when traits share some of their genetic basis and this covariance implies that traits cannot evolve independently (Lande 1979; Lande and Arnold 1983; Arnold et al. 2001). Therefore, G-matrices are the quantitative expression of constraints (Cheverud 1984; Arnold 1992), interacting with genetic drift and selection to produce evolutionary change and at same time also being shaped by those same process along with mutation (Wagner et al. 2007).

In a pathbreaking paper, Schluter (1996) examined the relationship between the G-matrix and evolutionary divergence. In particular, the focus was on what he called the “genetic line of least resistance” or gmax for simplicity. This line of least resistance is a single simple factor: the multivariate direction of greatest genetic (or its phenotypic surrogate) variation (gmax or pmax). This factor is a linear combination of a suite of morphological traits that displays the maximum within-population variance (first principal component [PC1]). It defines a line of least resistance to evolutionary change by either natural selection or genetic drift (Arnold et al. 2001) simply because it is the axis with the largest genetic variance. Schluter (1996) showed that this line of least resistance can affect both the direction and the magnitude of evolutionary change. It is also possible to extend the same rationale to other eigenvectors holding sequentially smaller portions of the total variation embodied in the G-matrix (Blows and Higgie 2003; Blows et al. 2004; McGuigan et al. 2005).

One way to appreciate the effects of genetic covariance upon the magnitude and direction of evolution is portrayed in Figure 1 (modified from Arnold et al. 2001). In this hypothetical adaptive landscape, three populations differ in their current position in relation to the adaptive peak. All the three populations share the same basic genetic covariance pattern. Selection will push all the three populations to the nearest adaptive peak, but the orientation of the selection gradient will differ due to their differences in current position on the landscape. Although selection in all the three populations specifies the shortest linear path to the peak, the realized evolutionary trajectory (Δz) from generation to generation may be quite different. In fact, if the two axes of major genetic variance are not aligned (population C) with the direction of selection, the evolutionary response to selection will be curvilinear (Fig. 1). Furthermore, this curvilinear trajectory would be biased by the line of least resistance embodied in the G-matrix (gray ellipses in the figure). Because in this simple example (with only two dimensions) the first line of least resistance holds almost twice as much variation as the second one, the initial response in population C would be strongly biased in the direction of the largest genetic variance (gmax). It is also important to note how the line of least resistance influences not only the direction but also the magnitude of the evolutionary response along the path of selection. This point is made it clear when comparing populations A and B in Figure 1. The magnitude of the response in population A is much larger than in population B. This reflects the fact that in population A the first line of least resistance is aligned with the selected dimension whereas in population B the second line of least resistance is the one aligned with the direction of selection (β). The difference in response magnitude between A and B due to the variance differences even overcomes differences in strength of selection due to the fact that the path between A and the peak has a shallower slope than the path between B and the peak that is steeper and therefore reflects stronger selection.

Figure 1.

A hypothetical adaptive landscape for two characters with the cross marking the adaptive optima (peak) and oval lines indicating isoclines of subsequent smaller height (fitness). Three populations (A, B, C) are show with their corresponding variance/covariance pattern (gray small ellipses) and averages for both traits (the ellipse center corresponding to the crossing of the two major axes of variation). The long and short axes within the gray ellipses represent the first (gmax) and second (g2) lines of least resistances or the two largest eigenvalues of the G-matrix. On the left the selection gradients (β) operating upon these three populations are show. On the right side the evolutionary responses (Δz) are shown, with the arrows indicating the direction and magnitude of those responses.

Another important question, and one that it is still underexplored in the evolutionary literature, is the relationship between the adaptive landscape and constraints embodied in the G-matrix (Cheverud 1984; Arnold et al. 2001; Arnold 2005). Although our comparisons of P- and G-matrices in mammals (Cheverud 1995, 1996; Marroig and Cheverud 2001; Oliveira et al. 2009; Porto et al. 2009) with an a priori hypotheses based on functional and developmental factors imply that stabilizing selection should be an important force keeping those patterns stable in mammals (at least in the skull), two fundamental processes that potentially have an impact on structuring the P- and G-matrices are not usually addressed: mutation and directional selection. The difficulties in estimating the M-matrix (the variance/covariance matrix of mutational effects) usually prevents it from being measured, even though it has been suggested for some time now (Cheverud 1984) that the interaction between the M-matrix and the W-matrix (the matrix describing the covariance among traits due to selection) is critical to the problem of stability or plasticity in G-matrices. In other words, in an adaptive radiation like the one observed in New World Monkeys (NWMs) where P- and G-matrices remain relatively stable (Cheverud 1995, 1996; Marroig and Cheverud 2001; see also below) are the long-term patterns of evolutionary change influenced by the constraints (G-matrix) irrespective of selection regimes, are they a result of selection and constraints operating together, or finally do both constraint and selection interact to produce evolutionary change whereas the constraints themselves are shaped by selection? If so, should we expect an alignment between them and the covariance in the net-selection gradients through time?

The present paper builds on our previous contributions (see also Material and methods below) that showed that the evolutionary divergence of cranial morphology in NWMs was not compatible with a null hypothesis of genetic drift (Marroig and Cheverud 2004). Additionally, Marroig and Cheverud (2005) showed that the line of least resistance affected both the direction and magnitude of evolutionary change in size during the adaptive radiation of NWMs. However, net-selection gradients (β's) reconstructed by Marroig and Cheverud (2005) were not especially similar to size (pmax) vectors (fig. 4 in Marroig and Cheverud 2005) suggesting that the bulk of the size evolution observed in NWM crania was due to the influence of the constraints. Yet, the possibility that those net-selection gradients were not representative of the past selection regimes due to either the Turelli's effect (Turelli 1988; see also Jones et al. 2004) or the error involved in matrix estimation, in particular in the inverted matrices (G. Marroig and D. Melo, unpubl. ms.), cannot be disregarded. The so-called Turelli's effect could happen if there is stochastic variation in the G-matrix and a covariance between the G-matrix and the net-selection gradient (Turelli 1988; Jones et al. 2004). However, Jones et al. (2004) present evidence that usually this effect will be small. Furthermore, although the observed macroevolutionary pattern of changes along gmax could be accounted for by the conservation and influence of the line of least resistance, there was also good evidence that dietary diversification might have been the ecological factor driving such diversification through natural selection on size due to metabolic and foraging restrictions (Marroig and Cheverud 2004, 2005). This correlation between size and diet suggests an alignment of adaptive peaks with size during NWM diversification and raises the question of whether it is possible to test whether or not size evolution is a direct consequence of selection on size (selection + constraints hypothesis) or is an indirect consequence of gmax redirecting evolutionary response (constraint hypothesis; Björklund and Merilä 1994). Here an approach based on extensive simulations applying the multivariate response to selection equation (Δz=) is used to evaluate these two hypotheses; specifically, looking at the direction and magnitude of evolutionary response under two distinct simulated selection scenarios, nonsize selection and size selection. We also introduce a benchmark to compare the observed amount of evolutionary change in size (or any other multivariate dimension) among groups and infer whether size was under direct selection or not. This benchmark is equal to the correlation of the direction of selection and the line of least resistance when both vectors are normalized to a length of one. Our results show that both constraints alone and selection aligned with constraints can produce evolutionary change aligned with gmax. However, there is substantial difference in the alignment between evolutionary responses and gmax when the two hypotheses are compared in terms of the variance of the two distributions. Also, the magnitude of the evolutionary change along the line of least resistance produced by these two processes is quite different allowing tests of whether selection was operating directly on size or size evolution was an indirect consequence of the attractor effect of the line of least resistance.

Materials and Methods

Thirty-nine skull measurements were obtained from 5222 crania of 16 genera of NWMs (Platyrrhini). This 39 trait-set was obtained from the distances between three-dimensional coordinates recorded for 36 landmarks using a Polhemus 3Draw digitizer. The sample was described elsewhere (Marroig and Cheverud 2001) and the general procedure for measuring specimens and the landmark definitions follow Cheverud (1995). Within-group phenotypic variance/covariance matrices (P-matrix for simplicity from now on) were obtained for each genus, removing all other sources of variation (sex, species, and species by sex interaction whenever appropriate) and details can be found in Marroig and Cheverud (2001, 2005). These variance/covariance matrices represent within-population constraints and were previously compared among NWMs by Cheverud (1996) and Marroig and Cheverud (2001). Throughout this paper we will use these P-matrices as the best representation of the underlying G-matrices. However, we highlight this point by presenting the random skewer (RS) correlation (see Marroig and Cheverud 2001; Cheverud and Marroig 2007) between the G-matrix obtained (Cheverud 1996) for a population of tamarins (Saguinus) and the P-matrices of each genus used here. We also present the G-matrix versus P-matrix comparison using a different technique (Krzanowski 1979; Blows et al. 2004) that allows the comparison of any two variance/covariance matrices by calculating the angles between the best-matched pairs of orthogonal axes (Principal components [PC]). The following relationship is used to find a projection matrix, S, based on a subspace of the first 19 PCs of the 39 PCs in the full dimensionality space


where A corresponds to the first 19 PCs (norm = 1) column arranged of the first variance–covariance matrix being compared, B stands for the first 19 PCs of the second matrix, and T for the transpose. Because matrix S represents the minimum angles between an arbitrary set of orthogonal vectors in the subspace of A and a set of orthogonal vectors closest to the same directions in the subspace of B, the eigenvalues of S can then be used to determine a similarity index between the two subspaces representing the two matrices (Blows et al. 2004). The sum of the eigenvalues of S equals the sum of squares of the cosines of the angles between the two sets of orthogonal axes (Blows et al. 2004) and will lie in the range of 0–19 (in our case), as all eigenvalues of S will have values between 0 and 1. The sum of the eigenvalues of S therefore represents a convenient measure of the similarity of the two subspaces because it is bounded within a range of values that have a straightforward interpretation (Krzanowski 1979; Blows et al. 2004). If the sum is close to 0, the two subspaces are dissimilar and are approaching orthogonality, whereas a sum equal to k= 19 would indicate that two original matrices share the same orientation (Blows et al. 2004). We report the sum of the eigenvalues divided by the maximum value (k= 19) and therefore this index potentially can range from zero (no structural similarity) to one (full structural similarity in the 19 PCs subspace).

All matrices used from now on were pooled for each node on the tree from the within-population matrices observed in the terminal taxa (genus) using the phylogenetic tree of Platyrrhini (Wildman et al. 2009) as described in Marroig and Cheverud (2005). Moreover, it is worth noting here that although our previous studies showed that G- and P-patterns are quite similar in NWM, this conservation has been recently extended by Oliveira et al. (2009) across all anthropoid primates. P-matrices are used here as a surrogate for G-matrices except where specifically stated, but we caution against application of this approach to other organisms or groups without, at least, performing an empirical comparison of P-and G-matrices. Also, we have shown previously that pmax and gmax are closely aligned and conserved among NWM (Marroig and Cheverud 2005) and therefore we will use pmax calculated for each P-matrix as a surrogate for the only gmax available for our dataset (Cheverud 1996). The direction of pmax was calculated from the P-matrix as the PC1 for each comparison.

Average values for the 16 genera were used along with the phylogeny (Wildman et al. 2009) to obtain estimates of the direction of evolution (Δz). This is not the same phylogeny used in Marroig and Cheverud (2005) because at that time the basal split between Pithecidae, Atelidae, and Cebidae (including all marmosets and tamarins) was not fully resolved. We presented the results back in 2005 based on the following topology: (Atelidae-Pithecidae, Cebidae) but also tested the two alternative topologies and reported that there was no difference in the results between these alternatives. After Wildman et al. (2009) it is clear that Pithecidae (CacajaoChiropotes, Pithecia, and Callicebus) are the sister group to a clade with all remaining NWM. Notice also that Wildman et al.'s (2009) phylogeny although well resolved does not include the genus Cebuella. We use here the previous estimates of branch length (see Marroig and Cheverud 2005) for the CebuellaCallithrix case and the phylogeny used here is presented in NEXUS format in the Supporting Information (MIPOD input file, see below). Accordingly, given that the best estimate of the NWM phylogeny topology is somewhat different from the one used in 2005 we have recomputed here all Δz's and P-matrices along the phylogeny. The conclusions and results from Marroig and Cheverud (2005) are essentially the same. These averages were obtained as least squares means from the general linear model module in SYSTAT with genera and species nested within genera as independent factors and the 39 skull trait-set as dependent variables (see Marroig and Cheverud 2005 for details). Estimates of ancestral states for this 39 continuous trait-set were obtained using a maximum likelihood (ML) approach (Schluter et al. 1997) implemented in the program ANCML available from D. Schluter's home page ( These are estimates of ancestral states and are equivalent to those reconstructed using maximum parsimony and minimizing the squared difference among taxa (Maddison and Maddison 2003). After the estimation of ancestral data, the direction of evolution (Δz) within each branch was obtained simply as the difference vector in the 39 averages of each taxon/node pair along the phylogeny. Only the 16 living genera were considered here ignoring the deepest nodes in tree.

Throughout the paper we use vector correlation to measure the similarity of any two vectors. The vector correlation (Blackith and Reyment 1971) is a measure of vector orientation similarity in a P-dimensional space (P being the number of traits). The correlation between two vectors is equal to the cosine of the angle θ formed between them. The expected range of correlations commonly occurring among 39 element vectors by chance alone is −0.4 < r < 0.4 with 99% of all values within this interval. For the calculation of a random expectation for vector correlations, only absolute values need to be taken into consideration because an angle of θ > 90° is equivalent to one of 180°−θ (Ackermann and Cheverud 2000). We use a broken stick model (Schluter 1996) to obtain 6000 random 39 element vectors from a normal distribution and then correlate each random vector to a fixed isometric vector (all elements equal to 0.160) to obtain the null distribution. Our sample showed a normal distribution of vector correlations with an average absolute value of 0.127 and standard deviation (SD) = 0.095, both values being approximately the same no matter which fixed vector was used for comparisons with the random vectors. The simulated distribution based on a random sample of 39 element vectors, allows us to test whether the correlation of any two observed vectors is significantly different from the correlation between two random vectors expected by chance. We use the average and empirically determined (nonparametric) 95% confidence intervals in the simulations tests below.

Having both Δz and pmax estimated, we previously obtained (Marroig and Cheverud 2005) the vector correlation between them as a measure of how closely morphological diversification follows the line of least resistance. When evolutionary change occurs primarily along the line of least resistance, the correlation between Δz-pmax will approach one and, conversely, as Δz deviates more from pmax, the vector correlation will decrease. Additionally, the magnitude of the evolutionary change is estimated as the square root of the sum of squared differences of the averages for the 39 traits for each genus-pair, or the length (norm) of the vector (mamount in Marroig and Cheverud 2005, which is a measure of the amount of morphological differentiation). This is equivalent to the respondability as defined by Hansen and Houle (2008), and their terminology will be employed here. Although, it should be noticed that our observed respondabilities are not normalized by the length of the net-selection gradient because that length is typically unknown (see below).


Figure 2 illustrates the basic principle of the simulation approach (see Pielou 1984). We define two types of selection vectors, size and nonsize. Nonsize vectors are composed of positive and negative elements whereas size vectors are composed of positive elements only (Figure 2). The first of the two simulated samples (nonsize) is composed of 6000 nonsize selection vectors (the two white quadrants in Fig. 2) obtained from a normal distribution random number generator. The numbers for each cell in the vector can vary from −1 to +1, which means that nonsize vectors are combinations of positive and negative elements simulating positive selection upon some traits and negative selection upon others. Each vector is then multiplied by the samples’P-matrices to obtain the response vectors (ΔzNONSIZE). Notice, however, that by chance alone some of the 6000 nonsize random vectors should bear some similarity to size vectors and accordingly any simulated selection vector with a correlation of 0.2 or higher with an isometric size vector was eliminated from the sample before being used in any subsequent analyses. The simulated sample of 6000 size selection vectors (upper right gray quadrant in Fig. 2) is obtained from the absolute value of the nonsize net-selection gradients. It is possible to have either selection to increase size (upper right gray quadrant) or to decrease size (lower left quadrant). The results are not different because this is a simple rotation operation and therefore we always use vectors with all elements positive. The rotation also does not have any effect upon the length of the response vector (Δz) because vector length is measured as the square root of the sum of squares of the elements. As described previously, all vectors were then multiplied by the samplesP-matrices to obtain the corresponding response vectors (ΔzSIZE). Of course, once obtained these two “populations” of simulated vectors (size and nonsize) remain constant and are applied to all P-matrices so that all comparisons obtained from this procedure are strictly comparable.

Figure 2.

A two-dimensional system showing the size (gray quadrants) and nonsize (white quadrants) spaces of the selection gradients simulations. The gray ellipses represent the variance–covariance patterns of three populations subjected to those simulated β's. At the right upper side the distribution of the simulated vector length are presented. On the right middle the distribution of the random vectors correlation with 39 elements is presented. On the right lower corner the distribution of individual elements for trait 1 as an example of the distribution of selection gradients elements.

Although this characterization of the size and nonsize vectors may appear somewhat arbitrary, this can be easily visualized and quantified by the vector correlation of each simulated vector against any fixed size-vector. We use an isometric size vector with all 39 elements equal to 0.160 for this purpose (see Fig. 2), but in fact any size vector can be used and the results would be basically the same. Also, it is important to note that each of these simulated selection vectors should be interpreted as the long-term net-selection gradient β vector (net-β sensu Lande 1979; see also Jones et al. 2004) which is equivalent to the accumulation over time of many generations of different β’s or, in other words, the resultant vector of selection over different directions through time (Lande 1979). Furthermore, although this approach could be used as an indirect metric to compare variance–covariance matrices (see Marroig and Cheverud 2001; Cheverud and Marroig 2007; Hansen and Houle 2008), here it is used in a different way, to explore the realm of evolutionary possibilities (meaning both the direction and magnitude of evolutionary change) embodied in the P-matrices in a large exploration of the directional selection spectrum (see also Marroig et al. 2009).

In addition to the direction, the length of the selection vectors is a very important piece of information for our purposes. The reason is straightforward: evolutionary change under selection (see equation above) is dependent on the amount and pattern of variance/covariance (G-matrix) and the strength of the net-selection gradient (β). Both selection and constraints impact the direction and magnitude of evolutionary responses. In principle, the length or norm of the net-selection vectors (β) (a measure of its strength) can have any value. Usually this information would not be easily available for macroevolutionary studies. However, an extensive review of selection strength in natural populations (Kingsolver et al. 2001; see also Endler 1986) showed that estimates of individual values of β for each trait are usually in the range of −1 to +1 with an average around 0.22 for the empirical exponential distribution of the inline image values found in a collection of 993 studies. These reported β's were standardized to estimate selection on a trait in terms of the effects on relative fitness in units of (phenotypic) SDs of the trait (Kingsolver et al. 2001). Therefore, by setting the lower and upper limits of −1 to +1 for each individual element in the random normal number generator to obtain the nonsize and size simulations of selection gradients, we have a population of a 6000 vectors whose lengths (norms) should be representative of naturally occurring vectors with 39 elements. In fact our distribution of random net-β values for any particular element of the 39 traits vectors fits well with the empirical distribution reported by Kingsolver et al. 2001 (Fig. 2). Although we are aware that the approach described below making use of the lengths of the response vectors under nonsize and size net-β can be criticized on the ground that we cannot be sure of the true magnitude of the net-selection gradient in the past and eventually some of them could have been stronger than the ones simulated here, the extensive review of published selection gradients by Endler (1986) and Kingsolver et al. (2001) suggest to us that this is a reasonable first approach. Any vector correlation comparison involving simulated β's were, of course, computed with their normalized versions to allow us to compute the correlation of simulated β's with simulated Δz’s.


Although pmax in NWMs are quite similar and all of them have significant similarities with an isometric size vector, with gmax (see Supporting Information, Marroig and Cheverud 2005) some differences are also observed. For example, the average correlation between the pmax and gmax is 0.89, with the lowest value observed being 0.81 (corresponding to an angle of 36°). The comparisons of all pmax's among NWMs present an average of 0.93 with the lowest value being 0.76 (an angle of around 40°, see Supporting Information). The vector correlation between the pmax and the isometric vector showed an average of 0.86 with the largest value being 0.90 and the lowest 0.83. Giving these small differences in orientation of pmax and alignment with the isometric vector we will focus on comparisons between simulated responses and the isometric pure size-vector.

After multiplying each P-matrix by both size and nonsize selection vectors, two sets of evolutionary responses (ΔzSIZE and ΔzNONSIZE) were obtained. The orientation of those vectors is compared with pmax and an isometric vector to check how similar the simulated responses are to the line of least resistance or to size. The comparison between the ΔzNONSIZE with an isometric size vector is quite informative because it addresses some fundamental questions if selection is not aligned with size would the population respond along a size dimension anyway? How often and to what extent? This approach allows us to test the strength of the constraint represented by the line of least resistance (Marroig et al. 2009). Moreover, it allows us to evaluate the ability to evolve along dimensions quite distinct from the line of least resistance when selection is not aligned with it. But perhaps, more importantly for developing an statistical test of the null hypothesis of size evolution due to the line of least resistance, the comparisons of the observed Δz× isometric vector with the distribution of simulated ones under nonsize selection allow us to test whether the observed association between the evolutionary changes and pmax or the isometric vector can be explained by the attractor effect of the line of least resistance alone. Furthermore, the magnitude of evolutionary change (respondability) can be calculated for each response vector (Marroig and Cheverud 2005) allowing us to compare respondabilities under size and nonsize selection with the observed respondability (called mamount in Marroig and Cheverud 2005) between any pair of interest in NWM phylogeny. We will present both, normalized respondabilities (divided by the norm of the selection gradient) and unnormalized respondabilities. The latter provides an indication of how much evolutionary change we would expect under size and nonsize selection regimes if the true net-β had magnitudes similar to those observed in nature (Endler 1986; Kingsolver et al. 2001) whereas the normalized version corresponds to strictly comparable respondabilities.

Considering that size and nonsize net-selection gradients simulated here have exactly the same magnitude distribution (Fig. 2), if selection is aligned with the line of least resistance (pmax), we would expect that evolutionary responses (Δz) are also aligned with “size variation.” Furthermore, given that both selection and constraints are acting together in the same direction we would expect large magnitudes for the evolutionary response vectors (respondabilities). Conversely, if selection is not aligned with the line of least resistance we expect that evolutionary responses will to some degree be biased by pmax, but would also present a much wider range of response directions, not necessarily aligned with size variation. Moreover, given that selection is acting in directions distinct from the line of least resistance, smaller magnitudes of evolutionary response are expected. In other words, constraints alone could cause evolutionary response vectors to be aligned with size variation regardless of the direction of selection, but the amount of evolutionary change along the line of least resistance should be smaller because selection is acting in other directions.

In the present context of testing for size evolution as a correlated change influenced by the line of least resistance or size evolution as a consequence of selection acting directly on size, these two measures (respondability and correlation between Δz with size) are the fundamental quantities to be evaluated in comparative retrospective analyses as pursued here.


Another approach to explore whether size evolution was a result of selection directly on size or a by-product of constraints is based on the diagonalization of the P-matrix. P-matrices are reduced to their PCs. The PCs are ordered by their level of variance (eigenvalues) and are uncorrelated with one another so that on the scale of the PCs, the P-matrix is a simple diagonal matrix with no covariances among components. PC scores for each PC are then calculated for each population by the sum of the inner products of trait means by the standardized within-population PC loadings (sum of square of the PC coefficients equal to one). The PC1 is always size/allometry in NWM and is quite similar among genera (Marroig and Cheverud 2005; see Table A1). Therefore, after computing the average value for each genus along the pmax dimension of the appropriate P-matrix for each comparison, we can directly estimate the Δz in size for each genus-ancestor in the phylogeny.

The pmax eigenvalue is an estimate of the within-population genetic (phenotypic) variation along the size dimension and can be used to benchmark observed size evolution between groups. This can be accomplished by dividing the observed Δz projected onto pmax by its corresponding eigenvalue. This is equivalent to computing the net-β on size/allometry (PC1) alone. If selection was directly on size we expect a β value either positive or negative but significantly different from zero. Conversely when size evolution is a result of the constraints embodied in the P-matrix (selection on something different from size) the estimated β should be close to zero. Accordingly, using this approach we are back to a univariate version of the multivariate breeder's equation, where Δz can be calculated by the difference in the two population's pmax scores, the G-matrix is represented by the pmax eigenvalue, and β can be calculated from the ratio Δz/G. Thus, a model of size evolution by size selection versus nonsize selection or drift can be distinguished by measuring the amount of size evolution relative to the magnitude of the first eigenvalue of the within-group variance for size. This can be tested by making use again of the simulation strategy outlined above and comparing the observed net-β on size/allometry with the simulated ones under nonsize selection and testing whether the observed ones are significantly different from the simulated values or not. Notice that the benchmark is actually equal to the correlation between the net-β and pmax if both vectors are compared with a norm equal to one. Because the historical information about the length of the net-β is not available in macroevolutionary studies like this, we will present the observed values both as raw values and also by dividing them by the average net-β length in our sample which corresponds with those observed by Kingsolver et al. (2001). Recently, Hansen and Houle (2008) came to a similar solution from a different perspective when discussing measures of univariate evolvability and selection strength.


The evolution of population trait means under natural selection is dependent upon both genetic constraints and the selection gradient as shown above with the multivariate response to selection equation. Independent populations experiencing their own selection regimes can display correlated trait changes due to both shared constraints and or similar selection regimes. Felsenstein (1988) and Zeng (1988) reached a similar equation to describe the variance and covariance of changes among lineages:


where V is the variance/covariance matrix of evolutionary changes among lineages (or in other words the variance–covariance matrix of the Δz's among independent lineages), G is the G-matrix, and W is the variance/covariance of the selection gradients, that is, the covariance matrix of the slopes of log inline image. Using this equation retrospectively, it is possible to calculate the matrix W from the covariance among changes on independent branches of a tree and an estimate of the G-matrix (or P-matrix), as long as the assumption of similar G-matrices along the tree holds. With the W-matrix estimated it is possible to compare it, in particular its major axis of variation (wmax), with the major axis of variation in the G-matrix (pmax or gmax), the major axis (dmax) in the interpopulation variance–covariance matrix of the population means (D-matrix), and to the V-matrix (vmax). Following the same strategy used by Marroig and Cheverud (2004), we transform the P-matrix to its PCs. Averages for each genus were obtained in this new space by projecting the original means of the 39 traits vector onto the PCs space. This is accomplished by computing the PC scores (averages) by multiplying each of the normalized PC vectors by the mean trait vector and obtaining the total sum of these cross-products (just like above for the benchmark approach). From this we then obtained the D-matrix as the variance/covariance matrix of those 39 PCs scores of the 16 genera. Any correlation between those scores is indicative of correlated selection as discussed in Marroig and Cheverud (2004). However, it is important to notice that the first four PCs of the D-matrix as well as of the V-matrix account for 99% of all the divergence among the NWM skulls and are dominated by the first four PCs of the P-matrix (see Table A3). The first four PCs of the NWM P-matrix (the best estimate of within-population variance–covariance patterns at the root of the tree) account for 57% of all variation and therefore we will present our results in this reduced subspace. This strategy has two additional benefits: One is that the whole morphometric system is more tractable and simpler; the second benefit is that by focusing on the first four PCs of the P-matrix, those with larger eigenvalues and therefore estimated with less error, we reduce the noise in our estimates. The problem of noise is even more pronounced in the D-, V-, and W-matrices, because they are based on only 16 (for the D-matrix) or 15 (V- and W-matrices) observations. This will be the subject of a future contribution dealing with noise in variance–covariance matrix estimation and how to remove that noise to find a better-estimated matrix. Here it is worth noting that noise is usually manifested in the last eigenvalues of any variance–covariance matrix and because 99% of all divergence in NWM is concentrated on the first four PCs of the P-matrix we have the additional benefit of simplifying the system and reducing the effect of noise on our estimates. The W-matrix is presented also in the Table A1 in this four PCs’ space.

After obtaining the PCs scores we use ANCML (Schluter et al. 1997) to reconstruct ancestral states along the NWM phylogeny in the same way as described above for the original traits. From these reconstructed ancestral states, it is easy to obtain the corresponding evolutionary changes (Δz's) along the phylogeny (within branches) as the difference between each of the PC scores at each node–terminal pair. Any covariance between those evolutionary changes would be indicative of coselected traits because the PCs are orthogonal within populations by definition. The V-matrix can then be calculated from the covariance of those Δz's along the phylogeny. With the P-matrix diagonalized, the W-matrix can be obtained by rearranging the terms in Zeng's (1988) and Felsenstein's (1988) equation above


After calculating all these matrices (P, D, V, and W) the major axis of orientation of each one of them can be easily obtained by a PC analysis. These vectors are named here as pmax, dmax, vmax, wmax. Notice that due to the change in base of the P-matrix to its PCs we reproject the major orientation axes of the matrices D, V, and W back to the original space of the P-matrix. The percentage of the total amount of variation captured by each one of these axes and its orientation relative to an isometric size vector (all 39 elements equal to 0.160) will be reported. We also repeat this analysis using the G-matrix of Saguinus.

An alternative approach to test for the alignment of the adaptive landscape with the G-matrix and the D-Matrix was developed recently by Hohenlohe and Arnold (2008). The MIPOD software implements a comparative method ML framework modeling neutral evolution by genetic drift that depends on the effective population size and the G-matrix. It allows one to predict a probability distribution for divergence of population trait means on a phylogeny. In this way it is possible to compare the divergence pattern corrected for historical relatedness to the ones expected under a null model of evolution where divergence should be proportional to the time elapsed (in generations) and the G-matrix. It is also possible to incorporate a test vector that describes the axis of major orientation of the adaptive landscape (wmax) to compare with the orientation of the G-matrix. Hohenlohe and Arnold's (2008) parametrization of the G-matrix includes different aspects of it such as size, shape, and orientation. This allows for a series of nested hypotheses in regard to whether the observed patterns of divergence corrected by the phylogeny can be explained by drift alone or not. For example, are taxa more (or less) divergent than expected by chance alone? Is the shape of the divergence pattern different from the shape of the G-matrix? If so, in what direction? Although we agree that the whole framework and this particular reparametrization of the G-matrix can be very useful, one problem in the current implementation is that the number of parameters will increase rapidly with the number of traits, in particular in regard to the orientation. With a simple two-trait system the approach is quite visual and easy to implement, but with 39 traits the task is difficult at the present time. While we have already shown that the null hypothesis of genetic drift can be rejected for NWM skull diversification (Marroig and Cheverud 2004), we apply the MIPOD framework here to test the series of hypotheses outlined above paying special attention to the orientation of the adaptive landscape. However, to avoid the problem of the complexity in the G-matrix and corresponding patterns of divergence we transform the G-matrix to its orthogonal form using PCs (see Supporting Information for the input file). Accordingly, the G-matrix it is now represented by eigenvalues on the diagonal and zeros in the off-diagonal elements (off-diagonal elements were actually set to 0.0001). The averages of the 16 genera were projected onto this space and accordingly we have four vectors corresponding to the 16 scores on those four PCs of the genera means. Another parameter necessary is the effective population size inline image, which is not available for any NWM population. We use an estimate available for baboons (Storz et al. 2002), which suggest that historical inline image was around 3500. Time in generations was obtained using the phylogeny and branches lengths in years from Schrago (2007), modified to fit the topology of Wildman et al. (2009), and the generation time estimates obtained from the literature (Hartwig 1996; Garber and Leigh 1997; Smith and Leigh 1998; Fleagle 1999; Lindenfors 2002; Porter and Garber 2004; see table 17.7 in Marroig and Cheverud 2009) for all NWM. These generation times correspond to the age of first reproduction and should be considered a conservative estimate for NWM because most likely generation lengths are longer than the values used here. Accordingly, the potential for genetic drift is maximized by using generation lengths with minimum estimates. The same rationale applies to the estimate of inline image in NWM, which are likely to be larger than those for baboons for most if not all taxa considered here. The same analysis was repeated for the P-matrix of all NWM. We use the first four PCs of the P-matrix of all NWMs (pooled within-groups P-matrix, which provides the best estimate of the variance covariance patterns at the root of the tree). We use two hypothetical test vectors for the wmax one involving coselection between size and all other dimensions (1, 1, 1, 1) and another with a coselection between PC1 and PC4 (1, 0, 0, 1) for the P-matrix and (1, 1, 0, 0) for the G-matrix. The PC4 of the P-matrix is equivalent to the PC2 in the G-matrix and previously we have shown that both are highly correlated among groups (see Marroig and Cheverud 2004). Finally, we also perform a sensitivity analysis of the results to variation in inline image using different values (from 500 to 50,000). The results are essentially the same except that the size parameter changes in magnitude. However, in all results the size parameter had a value well below the expectation under a pure drift model.


Table 1 presents the results of the similarity comparisons between the G-matrix of Saguinus and the P-matrices of all NWM. Both, RS average vector correlation and Krzanowski's (1979) (see also Blows et al. 2004) projection results are presented. Values presented in Table 1 can be envisaged as a “correlation” but the interpretation of the values for the two techniques applied is not the same. The Krzanowski's projection can be thought as simple metric of shared orientation in both matrices compared (at least of the subspace considered, 19 PCs for each matrix which usually explain from 91% to almost 99% of all variation in our samples). The RS correlation is a metric that compares the similarity in the evolutionary responses of two matrices to the same selection vector. Neither metric captures the magnitude of the variation in the matrices compared (see Marroig and Cheverud 2001; Hohenlohe and Arnold 2008; Porto et al. 2009). Notice that G- and P-matrices are not identical but are quite similar indicating that throughout the diversification of NWM constraints embodied in the G-matrices have remained quite stable.

Table 1.  Comparisons between the SaguinusG-matrix and all P-matrix for the 16 NWM genera. Results for both comparison methods are presented (the Krzanowski projection and the Random Skewers). Values were adjusted for matrix repeatability which are presented in Marroig and Cheverud (2001) and for the G-matrix (t=0.75) in Cheverud (1996).
Genus P-matrixKrzanowski G-matrix SaguinusRandom skewer G-matrix Saguinus

Table 2 presents the average and empirically determined 95% confidence interval of the vector correlations of both simulated responses (ΔzSIZE and ΔzNONSIZE) against pmax as well as against the isometric size vector. The observed values are also presented for the correlation of Δz with pmax and Δz with an isometric size vector. The 95% empirical limits draw from the simulation are higher for the correlation between Δz and pmax in comparison with those found for the correlation between Δz with the isometric size vector for both size and nonsize selection. This reflects the fact that pmax is the axis holding the larger portion of the within-population variance and accordingly is the direction with the strong potential for evolutionary change and the strong attractor effect. Notice that eight of the 16 cases show (in bold font) a correlation between the observed Δz and an isometric size vector higher than expected by nonsize selection alone. Furthermore, five of these eight cases have observed values within the 95% empirical confidence interval expected by selection on size if we compare the Δz with an isometric vector. If we consider the correlation between Δz and pmax only four cases show an observed correlation outside the empirically obtained 95% confidence interval based on nonsize selection gradients. These results suggest that a number between 50% and 25% of all the cases analyzed here most likely were under some sort of selection regime that involved a component of size selection.

Table 2.  Results of the size and nonsize net-β selection simulations. The 95% empirical confidence interval is show for the vector correlations between the Δz obtained from the simulation against pmax and the isometric size vector. Observed values are also presented. For the size selection empirical confidence interval the sign was reversed in those cases in which the observed correlation between Δz and pmax is negative. Significant values are presented in bold (P<0.05).
  Non-size Size
 Observed Δz×Iso. Vector Simulation Observed Δz×pmax Simulation Observed Δz×Iso. Vector Simulation Observed Δz×pmax Simulation
 P2.5% P97.5% P2.5% P97.5% P2.5% P97.5% P2.5% P97.5%
Ateles 0.672−0.7240.723 0.680−0.8600.861 0.672 0.863 0.930 0.680 0.888 0.994
Alouatta 0.648−0.8310.829 0.772−0.9350.938 0.648 0.895 0.919 0.772 0.985 0.999
Brachyteles0.810−0.7530.747 0.764−0.8630.860 0.810 0.880 0.928 0.764 0.951 0.998
Lagothrix 0.216−0.7530.747 0.388−0.8630.860 0.216 0.880 0.928 0.388 0.951 0.998
Cacajao0.847−0.7570.758 0.759−0.8970.903 0.847 0.861 0.917 0.759 0.961 0.997
Chiropotes−0.041−0.7570.758 0.093−0.8970.903−0.041−0.917−0.861 0.093 0.961 0.997
Aotus 0.040−0.7590.763−0.060−0.8930.899 0.040 0.867 0.930−0.060−0.997−0.941
Cebus0.860−0.7730.780 0.871−0.9180.9230.8600.8500.926 0.871 0.916 0.995

Figure 3 summarizes the simulation results, which are basically the same for all cases. First, when selection is along size all the responses are strongly aligned with size (Fig. 3 black distribution). Second, when selection is not on size the distribution of evolutionary responses is highly variable, but it is also skewed to extreme values (either positive or negative), towards correspondence with a size vector or pmax. The implication is clear: there is a trend for a larger proportion of the responses being deflected towards a size response even when selection is not on size (Fig. 3 gray distributions).

Figure 3.

Simulation results for size (black distribution) and nonsize selection (gray distribution). The distribution of the vector correlation between the evolutionary responses (Δz) obtained from the simulations against an isometric size vector is show for one example (Alouatta ancestral variance–covariance matrix). All P-matrices present basically the same patterns depicted here.

Table 3 presents the average and the 95% empirical confidence interval of the respondability for both size and nonsize simulations normalized by the length of the selection gradients. Observed values were also normalized by the larger length observed in our sample of net-β. Figure 4 summarizes the results for the respondability. It is interesting to notice that the two distributions are completely separate for all genera. In other words, the amount of evolutionary change expected under nonsize random selection (including those that were aligned with size) is always smaller than the amount expected under selection for size. In fact, on average, the magnitude of the responses for nonsize selection is around 20–30% of those obtained for selection on size. This result is clear-cut: when both selection and constraint are aligned, the magnitude of the responses is three to four times larger than those observed when selection is pushing populations in directions distinct from the line of least resistance. The comparison of the observed magnitudes to both nonsize and size respondability distributions is quite informative (Table 3). Seven of the 16 cases have observed values of respondability well within the expected distribution of evolutionary response magnitudes under nonsize selection. Conversely, eight cases have observed values of respondability above that expected under nonsize selection and either within or even above that expected under size-selection. Additionally, one other case (Alouatta) has an observed respondability above the expected distribution under nonsize selection but smaller than expected under size selection. One interesting pattern that emerges is that seven of the eight cases (except Callicebus) suggestive of positive selection on size are concentrated within the Callithrichinae subfamily.

Table 3.  Simulation results of the magnitude of evolutionary change (respondability) normalized by the net-β length. Results of the magnitude of the evolutionary change (respondability) obtained for the nonsize (left) and size (right) selection simulations. Observed values also normalized by the maximum value (3.15) observed in our sample of net-β are presented in the middle. In bold are those cases in which observed respondability is higher than expected by nonsize selection and within (or even above) the range expected by size selection. In italic are those observed values above the nonsize selection expectation but smaller that the size selection expectation.
GenusNon-size Observed (normalized)Size
Cacajao1.4430.5803.2323.221 6.8184.7339.864
Pithecia1.4240.5773.1952.645 7.4315.19910.685
Chiropotes1.4430.5803.2321.563 6.8184.7339.864
Figure 4.

Simulation results of the magnitude of evolutionary change (respondability) for size (black distribution) and nonsize selection (gray distribution) gradients. Raw respondabilities are presented on the left panel and normalized respondabilities (divided by the length of the net-selection gradient) on the right. The results presented are for the Alouatta ancestral matrix as an example but the same pattern is observed in all other matrices.

Table 4 shows the results of the selection benchmark approach. Recall that the values correspond to the observed Δz's between each genus and its immediate ancestral node projected on the size dimension (pmax) and divided by the corresponding pmax eigenvalue obtained from the P-matrix. The 95% empirical confidence interval is also presented based on the simulation of nonsize net-selection gradients. Considering the normalized benchmark, these results (Table 4) are similar to the ones obtained in Table 2 and indicate that at least in some cases size evolution was indeed the result of direct selection on size and not a correlated response due to variational constraints. In particular, genera Callicebus, Cebus, Saguinus, Callimico, Callithrix, and Cebuella seem likely to have had some sort of selection on size when one considers all the results together (Tables 2, 3 and 4).

Table 4.  Results of the benchmark approach (net-selection gradient (β) values on size alone). Observed benchmark values (raw) and observed benchmark divided by the selection gradient length (observed in our sample of net-β) are presented (normalized benchmark). The 95% empirical confidence intervals for the statistic based on nonsize selection simulations are presented. Cases in which size selection is observed are in bold.
Ateles 0.304  0.101 −0.262 0.267 
Alouatta0.416 0.138−0.2620.268
Brachyteles0.400 0.133−0.2630.267
Lagothrix 0.095 0.031−0.2630.267
Cacajao0.388 0.129−0.2620.269
Chiropotes 0.023 0.008−0.2620.269

Table 5 presents the MIPOD results. Notice that genetic drift can be ruled out because there are significant differences in size, shape, and orientation of the divergence among groups relative to the G-matrix (the same result is obtained using the P-matrix, the only difference is that the PC2 in the G-matrix corresponds to PC4 in the P-matrix). Notice that σ is significantly smaller than expected under drift indicating that stabilizing selection was preventing greater differentiation among groups. Also, the first shape parameter is significantly different and larger than the previous value indicating that the divergence among groups is more “cigar-shaped” than expected by the G-matrix alone (more divergence along gmax). Furthermore, there is no significant difference between orientation of the divergence and the first trait (gmax) as shown by the P-value of the first orientation. Moreover, there is a significant difference for the second orientation parameter indicating that the divergence among groups does not conform to the orientation of the G-matrix. Finally, there is no significant difference between the models for both hypothetical test vectors (adjusted in separate runs) describing the major orientation of the adaptive landscape (1, 1, 0, 0 and 1, 1, 1, 1), suggesting that divergence among NWMs comply with extensive coselection among the first four PCs of the G-matrix.

Table 5.  Results of the MIPOD approach based on the G-matrix of Saguinus simplified to the first four principal components. Notice the significant value for the hierarchical tests of size, shape (1), and orientation (2). Notice also that there is no significant difference using the two test vectors describing two hypothetical vectors representing the orientation of the adaptive landscape.
Additional maximum likelihood parameter estimateSigmaG-matrix parameters
 lnL(theta) LRdfP value
Size (sigma)0.2795−233.67408.3510.0000
Shape 1 (epsilon)0.3624−228.2510.8410.0010
Shape 2 (epsilon)0.3332−227.990.5310.4653
Shape 3 (epsilon)0.3668−231.17−6.3711.0000
Orientation 1 (phi)0.3685−230.591.1610.2814
Orientation 2 (phi)0.3762−227.376.4420.0400
Orientation 3 (phi)0.3677−226.491.7730.6212
Test Vector (1, 1, 0, 0)0.4744−220.61−11.7531.0000
Test Vector (1, 1, 1, 1)0.4701−220.41−12.1531.0000

Table 6 presents the major axes of orientation of the matrices D, V, G, and W along with the proportion of the total variation explained by each of them in their respective matrices. Notice that all vectors are significantly similar to a pure size vector with the following correlations observed with the isometric size vector: −0.67 (wmax based on the P-matrix), 0.89 (pmax), 0.92 (vmax), 0.92 (dmax), 0.78 (wmax based on the G-matrix with 4 PCs) and 0.84 (gmax). Using 9 PCs of the G-matrix (accounting for 81% of all variation in the G-matrix) nearly the same result is obtained with a wmax showing a 0.61 correlation with a pure size dimension. The major axis of variation in the adaptive landscape (wmax-G4) estimated using the four PCs of the G-matrix has a 0.85 correlation with gmax and a 0.42 correlation with the second PC of the G-matrix. Similarly, the major axis of variation in the adaptive landscape (wmax-P) estimated using the four PCs of the P-matrix has a −0.56 correlation with pmax and 0.74 correlation with the fourth PC of the P-matrix (PC4 of the P-matrix corresponds to the PC2 in the G-matrix). All of these results are in accord with the MIPOD results and suggest that the orientation of the adaptive landscape and the G- and V-matrices are similar.

Table 6.  Major orientation axes of the P-matrix (pmax), G-matrix (gmax), V-matrix (vmax), D-matrix (dmax), W-matrix based on the P-matrix (wmax-P), W-matrix based on the G-matrix (wmax-G4) represented by the first four principal components, W-matrix based on the G-matrix (wmax-G9) represented by the first nine principal components, and an hypothetical isometric vector. The % of the total variation explained by each axis is also presented.


What happens when selection is aligned with the line of least evolutionary resistance? Basically, all the evolutionary responses obtained for the genera closely followed a size vector (Fig. 3). In fact, pmax acts as an attractor for evolutionary responses and even an oblique alignment between the selection gradient and pmax will cause the response vector to be aligned with the line of least resistance. So, whenever selection and constraints are aligned, the evolutionary response will also be aligned with them. Furthermore, because the axis with the largest genetic/phenotypic variation is aligned with the direction of selection, the largest magnitudes of evolutionary responses are observed (Fig. 4) as predicted by theory (Fig. 1).

Yet, what happens when selection is not aligned with the line of least resistance? In this situation evolutionary responses of many kinds (directions) are observed (Fig. 3 gray distributions), including some size responses. But perhaps the most interesting result is the fact that when selection is distinct from the line of least resistance, even with a substantial portion of the evolutionary response still aligned with size, the magnitude of the evolutionary responses are three to four times lower than those obtained when both selection and constraint are acting in the same direction (Fig. 4). Furthermore, the comparisons between the respondability distributions for the size and nonsize selection regimes show a complete separation of values. All these results are consistent with the theory (Schluter 1996; Arnold et al. 2001) summarized in Figure 1 and raise the possibility of testing whether a particular case of evolutionary change along the line of least resistance can be explained by a simple hypothesis of correlated evolution due to constraint or if selection along the line of least resistance (size in our case) was involved.

We illustrate this possibility here by reexamining the results of Marroig and Cheverud (2005) in NWMs. Based on the comparisons of the observed Δz's and pmax we suggested that most of the living NWM genera (12 of 16) diversified along the line of least evolutionary resistance (see fig. 2 in Marroig and Cheverud 2005). However, was that a result of nonsize selection regimes acting on a common variance–covariance pattern (Marroig and Cheverud, 2001) in separate lineages producing a similar size-related response (pmax) because of the attractor effect exerted by the line of least resistance (the constraints hypothesis)? Or was the similarity between evolutionary response and the line of least resistance a consequence of long-term selection along the size dimension triggered by ecological factors related to the availability of diet-based adaptive zones (the selection hypothesis, see Marroig and Cheverud 2005)? These two hypotheses can also be envisaged in terms of two adaptive landscape models. In the first case (constraints), the location of the peaks is not aligned with the major orientation of the G-matrix and the stability in gmax among groups is responsible for the diversification of trait averages being aligned with gmax. Stability in variance–covariance patterns in this case results from internal stabilizing selection for developmental/functional combination of traits as discussed in Marroig and Cheverud (2001). The second hypothesis (constraints and selection) implies that adaptive peaks were aligned with gmax and so that the observed size evolution in NWMs results from selection on size along with the line of least resistance. Such an adaptive landscape with multiple peaks aligned with gmax would also help explain conservation of the G- and P-matrices in NWM.

The comparison of both the direction and the magnitude of evolution can be useful to test these two hypotheses (Tables 2 and 3, Figs. 3 and 4). For example, five of the 16 cases present an observed correlation between the observed Δz and a pure size vector larger (either increasing or decreasing size) than expected if selection was not aligned with size (nonsize selection vectors) and at same time display correlations between Δz and the size vector well within the interval for selection aligned with size. These cases of lineage diversification, marked in bold in Table 2, are most likely due to size selection. Complementarily, comparing the observed respondability values normalized by the average expected length for the net-β (Kingsolver et al. 2001) against the values obtained from the simulations under nonsize selection regimes there are eight cases (bold font in Table 3) suggestive of higher magnitudes of evolutionary change than expected under nonsize selection and well within (or even above) the expectation under size selection. There is a close agreement between the respondabilities in Table 3 with the alignment observed in Table 2. Most of the normalized respondabilities that are larger than expected under nonsize selection regimes are concentrated in the Callithrichinae clade that display an long-term trend for reduced body size (see Ford 1980; Marroig and Cheverud 2005), which in general agrees with the alignment observed in Table 2. So, considering both the alignment with size and the magnitude of evolutionary change there are at least five significant cases (Callicebus, Saimiri, Cebus, Saguinus, Cebuella, and Callithrix) that are more strongly aligned with size and have larger magnitudes of evolutionary change than expected under a model of nonsize selection. The differentiation of these genera was likely due to size selection (Table A1).

Are these results consistent with the benchmark approach? In general, yes. Cebuella, Callitrhix, Callimico, Saguinus, Cebus, and Callicebus all have observed values for the β on size larger than expected by the constraints hypothesis in which size evolution is a by-product of the attractor effect of the line of least resistance. One caveat of the benchmark approach, shared with the respondability comparisons, is that one cannot be certain of the length (magnitude) of the net-β vector that operated in the past for any lineage during its differentiation. We minimize this problem here by dividing the observed values by the maximum value of our simulated net-β vectors that are based on the estimates of net-β from nature summarized by Kingsolver et al. (2001). The distribution of net-β lengths can be observed in Figure 2 and results from the sum of the random normal distributions of each of the 39 traits with the distributional properties similar to those found by Kingsolver et al. (2001) in their survey of selection in nature. The empirical 95% confidence interval of the simulation results (Table 4) are based on normalized net-β vectors (norm = 1) and accordingly correspond to the correlation between simulated β and the line of least resistance.

Summarizing the results so far, based on the alignment and magnitude of evolutionary change at least five and perhaps as many as eight of the 16 cases studied here seem to involve size selection during their diversification. The respondability and benchmark approach include the caveat of being dependent on the magnitude of the net-β in the past, information usually not available in macroevolutionary studies, but the alignment between the Δz and the isometric vector does not suffer from this additional assumption. So, it seems likely that some size selection operated during the diversification of NWMs. This illustrates that the approach described here can be a useful tool in testing whether changes along the line of least resistance are a by-product of the attractor effect exerted by gmax or are a consequence of selection and constraints being aligned. However, one question should be posed at this point: why were the net-β's reconstructed by Marroig and Cheverud (2005) not aligned with size? Particularly informative is an inspection of figure 4 in Marroig and Cheverud (2005). The observed correlations between net-β reconstructed and the Δz are almost all below 0.4, a threshold too low if we compare those values with the simulated ones here that have a range between 0.4 and 0.8 (see also Marroig et al. 2009). The correlation between the net-β and the Δz was called “flexibility” by Marroig et al. (2009) and it is a useful metric to establish a link between evolutionary theory and modularity. Although this is not the subject of the present contribution it is worth noting that those observed values in figure 4 of Marroig and Cheverud (2005) are below the 99% empirical confidence interval for the values of flexibility in all NWM P-matrices in our simulations (results not shown). This suggests that those net-β vectors are probably poor estimates of the selection gradients acting in each of those lineages. The reason for this is related to error in matrix estimation and the fact that net-β vectors are estimated retrospectively by multiplying the inverted G-matrix by the Δz vector. Those dimensions in the G- or P-matrix that represent noise in the estimates (smaller eigenvalues) are actually the dimensions which will dominate the inverted matrix. Hence, estimates of net-β will be dominated by the minor, unreliable eigenvectors and be poor estimates of the “true” selection gradients. This will be the subject of another contribution (G. Marroig and D. Melo, unpubl. ms.) as well as a possible solution to this problem. For now it suffices to note that the most likely reason that estimates of net-β in Marroig and Cheverud (2005) are not similar to size selection in at least some genera is the error in G−1 estimates. Another possibility to explain why net-β vectors reconstructed by Marroig and Cheverud (2005) are inaccurate is due to stochastic fluctuations in the G-matrix resulting from a possible covariance between the G-matrix and the net-selection gradient in the same generation, also known as the Turelli's effect (Turelli 1988). Although Jones et al. (2004) found that usually this effect is not pronounced, in a situation where the selection gradients through time are highly correlated (autocorrelation), the variation caused in the G-matrix can potentially have a substantial effect.

The final question to be addressed is related to the interplay between selection and constraints. We have shown previously that P-matrices in NWM are quite stable in terms of the patterns of intertrait relationships and that this is related to common developmental and functional factors which we interpreted as exerting a stabilizing selection upon variance/covariance patterns (Marroig and Cheverud 2001). Here we also extend this result showing the overall similarity between the G-matrix of Saguinus and the P-matrices of all other NWM (Table 1). The average similarity between the G-matrix and the P-matrices is around 0.81 using the Krzanowski (1979) projection and 0.82 with the RS (Cheverud and Marroig 2007). Furthermore, we showed previously that gmax, the genetic line of least resistance obtained from SaguinusG-matrix, and pmax of all NWM genera are quite similar (see Marroig and Cheverud 2005). We here extend this result showing all pairwise comparisons between gmax and pmax calculated from all the P-matrices representing the ancestral patterns along the tree (Table A2). Overall we have good evidence for the conservation of constraints in NWM. The fact that pmax and gmax remain stable across such a long history of diversification (Table A1 and A2), especially given the evidence for the adaptive nature of the morphological evolution in NWM (Marroig and Cheverud 2004), might indicate that constraints themselves were also shaped by selection. We test the potential alignment between the adaptive landscape and the G- and P-matrices using two comparative approaches, one based on MIPOD (Hohenlohe and Arnold 2008) and the other based on the equation 2 (see above). The results of both approaches are in complete agreement. First, MIPOD results (Table 5) reject the hypothesis of genetic drift and point to significant differences in the size, shape and orientation of the divergence matrix among NWM relative to that expected given the within-population variance–covariance patterns (for both G- and P-matrices). Although in general NWMs are less divergent than expected by drift alone (size in Table 5) indicating stabilizing selection (a result robust to variation in the effective population size), the larger divergence along gmax is consistent with directional selection operating on this dimension whereas the others experienced stabilizing selection. These results agree with our previous findings (Marroig and Cheverud 2004). Furthermore, there is no significant difference between the orientation of the test vector (the hypothetical vector describing the major orientation in the adaptive landscape as a coselected dimension between PC1 and PC4 of the P-matrix, or PC1 and PC2 in the G-matrix) and the wmax computed from the data using MIPOD. This suggests that a ridge on the adaptive landscape of NWMs exists corresponding to a correlation between size (PC1) and PC4. The fourth PC contrasts zygomatic traits with cranial vault traits, zygomatic trait distances are negatively associated with high PC4 scores whereas cranial vault distances are positively associated with high PC4 scores (see Table 1 in Marroig and Cheverud 2004). These results are confirmed by the approach based on equation 2 (Tables 6 and A3). The PC1 of the W-matrix accounts for 93.1% of all variation in the adaptive landscape and corresponds to coselection of PC1 and PC4 of the P-matrix. This wmax has a correlation of −0.56 with pmax and 0.74 with PC4 and a vector correlation of −0.67 with a pure size isometric vector. The result based on the G-matrix suggests an even stronger alignment between the adaptive landscape and the line of least resistance with a 0.85 correlation between the wmax and gmax and 0.78 for wmax and isometric size vector. These results agree with the MIPOD results and reinforce two important conclusions. First, a substantial aspect of cranial morphological evolution in NWM was due to size selection, either for increased or decreased size. Second, constraints embodied in the P-matrices and G-matrix are aligned with the orientation of the adaptive landscape. This may help explain why P- and G-matrix patterns are conserved throughout the adaptive radiation of NWMs. This conclusion has two consequences: (1) If we take a narrow perspective the methods described here based on simulations of net-selection gradients allows one to test whether selection was closely aligned with size/gmax whenever evolution along pmax or gmax is observed as is the case for most NWM; and (2) Taking a broader perspective, gmax and selection are inextricably woven together because even in those cases in which evolution was along the line of least resistance and we cannot reject the null hypothesis that selection was not aligned with gmax (size), the constraints themselves being embodied in the line of least resistance as a historical product of stabilizing selection and the selective covariance of size with other aspects of morphology.


The approach described here allows us to go beyond the major focus of evolutionary quantitative genetics during the last two or three decades, that is, the comparison of variance/covariance patterns among populations or species (Marroig and Cheverud 2001; Steppan et al. 2002; Baker and Wilkinson 2003; Bégin and Roff 2003; Jones et al. 2004), and delve into another fundamental question: how variance/covariance patterns impact the evolutionary dynamics of complex systems. In other words, although our focus on estimating and comparing G- and P-matrices (the evolution of G-patterns) is sound and interesting, we should also exploit G- and P-patterns as evolutionary tools with which to investigate phenotypic evolution, particularly in complex systems with many traits (Arnold 1992; Roff 1996, 2003; Schluter 1996; Marroig and Cheverud 2005; Roff and Mousseau 2005; McGuigan et al. 2005; McGuigan 2006; Oliveira et al. 2009; Porto et al. 2009). We show here that combining information about the direction and magnitude of evolutionary change under simulated selection along size and nonsize dimensions and comparing those simulated responses to observed evolution, one can test whether selection and constraints were aligned in the past and whether, in particular instances, evolution along the line of least resistance cannot be explained by the attractor effect alone. This makes it possible to test whether any particular observed size evolution is likely to be due to direct selection on size (selection hypothesis) or due to a correlated response to selection or drift causing the observed size change (constraints hypothesis). This is particularly true for situations like those found in NWM in which G- and P-matrices are similar and remain quite stable through the time period under consideration (Marroig and Cheverud 2001) and especially if the size dimension (pmax and gmax) is conserved (Marroig and Cheverud 2005). Perhaps, the approach described here will also be useful in those situations in which changes in G-matrices are substantial, allowing one to test whether the evolutionary dynamics will empirically differ among populations by making use of simulations or not.

Associate Editor: J. Hermisson


We thank those people and institutions who provided generous help and access to the NWMs skeletal material: R. Voss (American Museum of Natural History); the staff at the British Museum of Natural History; B. Paterson, B. Stanley, and L. Heaney (Field Museum of Natural History); L. Salles, J. Oliveira, F. Barbosa, and S. Franco (Museu Nacional do Rio de Janeiro); S. Costa and J. de Queiroz (Museu Paraense Emílio Goeldi); Museo de la Universidad Nacional Mayor de San Marcos; M. de Vivo (Museu de Zoologia da Universidade de São Paulo); and R. Thorington and R. Chapman (National Museum of Natural History). Many thanks also to the following people. Leila Shirai made many corrections and suggestions on a previous version of this manuscript; Stevan Arnold, an anonymous reviewer, and Joachim Hermisson made numerous comments and suggestions on an earlier version of the article which helped us making it more focused and hopefully more strong on the test side of the article as well as on the subject of the alignment between the adaptive landscape and constraints; GM students which gave us important feedback during the review of the article (D. Rossoni, B. Costa, H. S. Silva, A. P. Assis, G. Garcia, and D. Melo). This research was supported by grants from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), an American Museum of Natural History Collections Study Grant, a visiting scholarship from the Field Museum of Natural History, and NSF grant BCS-0725068.