Arne Mooers
email: amooers@sfu.ca.

Olivier Gascuel

Tanja Stadler

Heyang Li

Mike Steel

BRANCH LENGTHS ON TREES Branch Lengths on Birth-Death Trees and the Expected Loss of Phylogenetic Diversity

HAL is

present both makes average branches one half what we might expect and makes pendant and interior branches roughly equal in length. Although dependent on whether we condition on the size of the tree, its age, or both, these results hold both for the Yule model and for birth-death models with moderate extinction. Importantly, the rough equivalency in interior and exterior branch lengths means the loss of evolutionary history with loss of species can be roughly linear. Under these models, the Tree of Life may offer limited redundancy in the face of ongoing species loss.

(Keywords: Phylogenetic tree, Yule process, extinction, phylogenetic diversity )

In a well-cited paper, [START_REF] Nee | Extinction and the loss of evolutionary history[END_REF] state that "80% of the underlying tree of life can survive even when approximately 95% of species are lost." This quote has percolated through the literature (see, e.g. [START_REF] Erwin | Extinction as the loss of evolutionary history[END_REF][START_REF] Purvis | Phylogenetic approaches to the study of extinction[END_REF][START_REF] Roy | Phylogenetic conservatism of extinctions in marine bivalves[END_REF][START_REF] Santos | Edge-related loss of tree phylogenetic diversity in the severely fragmented brazilian atlantic forest[END_REF][START_REF] Vamosi | Nonrandom extinction leads to elevated loss of angiosperm evolutionary history[END_REF]). This high level of phylogenetic redundancy is due to Nee and May using coalescent-type models of tree shape, where pendant edges are expected to be much shorter than interior edges. Here, we test the robustness of this result by building on recent algebraic results from [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF] to derive the expected branch lengths on phylogenies produced under alternative Yule and birth-death models of diversification. We highlight three findings: (i) the average length of branches in pure-birth (Yule) trees is roughly one half our naive expectation; (ii) the expected length of the interior branches and those leading to species are the same or nearly so, and this means that (iii) the relationship between the loss of species to extinction and the loss of phylogenetic diversity [START_REF] Faith | Conservation evaluation and phylogenetic diversity[END_REF]) can be much more precipitous than that quoted above [START_REF] Nee | Extinction and the loss of evolutionary history[END_REF]. All three findings hold for birth-death trees with low to moderate relative extinction rates.

For much of what follows, we will consider a pure-birth Yule tree with diversification rate λ. We note that inferred phylogenetic trees are often more imbalanced than Yule trees [START_REF] Mooers | Macroevolution and the shapes of phylogenetic trees[END_REF], but currently, no biological model captures this empirical distribution. More importantly for what follows, the Yule process produces a distribution of splitting events on the tree from past to present that is intermediate between that expected under an adaptive radiation [START_REF] Gavrilets | Dynamics of adaptive radiation[END_REF][START_REF] Rabosky | Explosive evolutionary radiations: Decreasing speciation or increasing extinction through time?[END_REF], where splits are concentrated nearer the root, that expected under long-term equilibrium models of diversification [START_REF] Hey | Using phylogenetic trees to study speciation and extinction[END_REF][START_REF] Hubbell | The Unified Neutral Theory of Biodiversity and Biogeography[END_REF], where splits are concentrated nearer the present. Our main motivation for focusing on this model is that trees sampled from the literature tend to have splitting times concentrated nearer the root [START_REF] Mcpeek | The ecological dynamics of clade diversification and community assembly[END_REF][START_REF] Morlon | Inferring the dynamics of diversification: a coalescent approach[END_REF], making the Yule model a conservative model when measuring phylogenetic redundancy.

We refer to branches that lead to the tips of a tree as pendant edges (with expected average length p n , where n is the number of tips) and branches found deeper within the tree as interior edges (with expected average length i n ). The term 'expected average length' clarifies that two random processes are at work -the production of a Yule tree and the selection of an edge from that tree. The expected phylogenetic diversity of such a tree is the sum of the expected pendant and interior edge lengths, i.e. L n = np n +(n-2)i n . We will assume throughout that the tree starts as an initial bifurcation, such that at some time t in the past it has two lineages each of length 0 (as in [START_REF] Nee | Inferring speciation rates from phylogenies[END_REF]). After time t from the initial bifurcation, we produce a binary tree with n tips (as in [START_REF] Nee | Inferring speciation rates from phylogenies[END_REF]; [START_REF] Yang | Bayesian phylogenetic inference using dna sequences: A markov chain monte carlo method[END_REF]), and several properties of this process have been well-studied by these and other authors. In particular, the expected number of tips in the tree is 2e λt . Given rate λ, the time that a given lineage persists until it splits on a Yule tree has an exponential distribution with a mean of 1 λ . This motivates our naive expectation that the expected average edge length on such a tree would also be 1 λ . We first present a simple proof that the expected average edge length in a Yule tree is actually 1 2λ . This provides an underlying intuition that is absent from the purely algebraic proof of [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF]. We then summarise and extend some results from [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF] to describe how the relative lengths of pendant and interior edges are affected by (i) conditioning on, (ii) estimating, or (iii) not knowing, three related quantities: n, the number of tips of the tree; t, the depth of the tree; and λ, the diversification rate. We then further extend our results to birth-death trees, and finally revisit the provocative question: at what rate do we lose phylogenetic diversity as we lose species on a tree?

Expected length of a branch on a Yule tree sampled at the present Let us assume that we observe a Yule tree at the moment that it has grown to n + 1 tips (n = 4 in Fig. 1). We do not condition on its depth (t). We can designate the edge that has just split as an interior edge, and disregard the two zero-length branches that have just arisen. Doing so designates an equal number (n -1) of interior and pendant edges on this tree. One might think of this Yule tree as one that has been 'cut at' (or conditional on) the observation of n + 1 tips. Intuitively, even though the expected length of an edge on an uncensored tree would be 1 λ , the designated pendant edges will be shorter due to this conditioning. However, interior branches are also affected by this censoring: particularly long interior branches would stretch to the present, and so would be pendant edges. This means that the expected lengths of interior edges are also shorter than 1 λ .

Theorem 1. In a Yule tree, at the latest speciation event, the expected length of a randomly drawn edge is 1 2λ .

Proof: Consider the late sampling scenario described in the preceding paragraph, and let the n -1 remaining pendant edges each grow under the Yule process until they also split, disregarding all the new infinitesimal edges that result. Each of these grown pendant edges has an expected length g n and is made up of two segments -its expected length before the tree had n + 1 edges (= p b ), and its expected length as it continued to grow after the tree had n + 1 tips (= p a ), such that g n = p b + p a . Importantly, given the memoryless nature of exponential processes, the length of any pendant edge segment observed from the time that n + 1 tips are produced (the dashed lines in Fig. 1) is drawn from one common exponential distribution, with the same parameter λ. Also, p n on the censored tree = p b on the uncensored tree.

Given an equal number of interior and pendant edges on this uncensored tree, we can write an expression for the expected length (call it E[L]) of any randomly drawn edge on this tree as:

E[L] = 1 2 • i n + 1 2 • (p b + p a ) = 1 2 • i n + 1 2 • (p b + 1 λ ). (1)
Any single lineage has E[L] = 1 λ , and so we can substitute this for E[L] to obtain:

1 2 • i n + 1 2 • p b = 1 2λ , (2)
because p n = p b . The left member in equation ( 2) is the expected length of a randomly drawn edge in the censored Yule tree, which completes the proof.

This proof does not say anything about the relative lengths of internal vs. pendant edges per se -it might be that internal edges are still much longer than pendant ones on Yule trees that we observe at a single time slice, and it may be that the result hinges on observing the tree at exactly the moment that a speciation event occurs. We turn to these issues now.

tree reaches n+1 = 5 tips

Figure 1: Growing a Yule tree, to illustrate the proof of Theorem 1. The horizontal line is the observation time, when n + 1 tips first appear. Below this line is the censored tree whose edge lengths we are modelling. The uncensored tree has each pendant edge continuing to lengthen till it speciates in turn. The thick lines denote interior branches, the thin lines are pendant edges on the censored tree, and the dashed lines are the segments that accrue to produce the uncensored tree.

Expected pendant vs. interior edge lengths as function (only) of n

In the above construction of the Yule tree we made the convention that the edge that has just split is an interior edge of the resulting tree. However, we could have alternatively classified it as a pendant edge. In that case we have n pendant edges and n -2 interior edges, and one can again consider the expected average pendant and interior branch lengths, which we will denote as i ′ n and p ′ n . [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF] used a recursive argument to establish the following exact result: For all n ≥ 3, we have:

i ′ n = p ′ n = 1 2λ . (3)
This result tells us exactly how 1 λ is shared out between the two terms in Theorem 1. Due to the memoryless nature of the exponential distribution, the pendant edge that was chosen to split at our observation time is random with respect to its length, and so we can express the lengths of the interior and pendant edges on the censored tree as:

i n = 1 n -1 ((n -2)i ′ n + p ′ n ), and p n = 1 n -1 (np ′ n -p ′ n ).
Eqn.

(3) implies that, for all n ≥ 3:

i n = i ′ n = 1 2λ and p n = p ′ n = 1 2λ .
In particular, the terms i n and p n in Theorem 1 are equal. We note that Theorem 1 is for a late sampling scenario, when we show up just when n + 1 tips first appear. However, if we only condition on n, but show up at a random time between the interval when n and n+1

tips exist (i.e. if we 'show up' at the present to sample our tree), any pendant edge has the same expected average length as in the late sampling scenario. This result is analogous to the bus-stop problem: if buses arrive at a certain rate b under an exponential process, if one shows up at a random time, the expected time since the last bus is b -1 rather than something less than that. This property was formally proven for model trees (Gernhard 2008a) and also used recently by [START_REF] Hartmann | Sampling trees from evolutionary models[END_REF] in the context of sampling trees from evolutionary models.

Expected pendant vs. interior edge lengths as functions of t (alone or with n)

The expected number of tips in a Yule tree at time t is given by N (t) = 2e λt , since each of the two initial lineages has a geometrically distributed distribution, with a mean of e λt (see e.g. [START_REF] Nee | The reconstructed evolutionary process[END_REF], or Beichett and Fatti (2002) (Example 6.10, pp. 193)). We now introduce P as the sum of all pendant edges, I as the sum of all interior edges, and, as in the introduction, L as the total tree length, L=P +I. These quantities, conditional on either n or t or both, should be noted, as they will be useful for many of the proofs that follow. If we let P (t) and I(t) denote, respectively, the expected sum of the lengths of the pendant and interior edges of a Yule tree grown for time t and let L(t) = P (t) + I(t), then, from [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF], we have the following equalities:

L(t) = 2 λ (e λt -1); P (t) = 1 λ (e λt -e -λt
) and

I(t) = 1 λ (e λt + e -λt -2). ( 4
)
Thus the ratio of the expected average lengths of the pendant and interior edges of a Yule tree of depth t converges to 1 exponentially fast with increasing t. P (t) is slightly larger than than I(t), but the difference becomes rapidly negligible. In particular, the ratio P (t)/L(t) converges quickly to 1/2; we will consider this ratio further when we allow for extinction.

Importantly, for most phylogenetic trees, both n and t will be known from the data.

Do the observations on edge lengths made above also hold when we condition on both n and t? The expected total length of a Yule tree conditional on it having grown for time t and having exactly n tips at time t is given by:

L n (t) = t • 2 + n -2 x (1 -y(x)) , (5)
where x = λt and y(x) := xe -x 1-e -x , which is a function that decreases from 1 towards 0 as x ≥ 0 grows (for details, see [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF] , sum of all edges (L n (t)) and their ratio ( Rn (t)) for various tree sizes n when both n and t are fixed and λ is set to its maximum likelihood value.

conditional on it having grown for time t and having exactly n tips at time t. Thus,

I n (t) + P n (t) = L n (t) (
given by Eqn. ( 5)).

A proof of the following result is provided in the Appendix.

Theorem 2. The expected length of a randomly picked pendant edge in a Yule tree on n extant species and of age t is,

1 n P n (t) = t• 2 n(n -1) + (n -2) [(n + 5) -4(1 + n + 2x)e -x + (3n -1 + 2(n + 1)x)e -2x ] 2xn(n -1)(1 -e -x ) 2 ,
where x = λt.

In particular, if we set λ to its maximum likelihood estimate, i.e. λ M L = log( n 2 )/t [START_REF] Magallon | Absolute diversification rates in Angiosperm clades[END_REF], then the ratio Rn := P n (t)/L n (t) of the expected total length of the pendant edges to the expected total length of all edges in a Yule tree on n extant species and age t is independent of t and is given by:

Rn = n 3 -3n 2 -4n log(n/2) + 4n -4 2(n -1)(n -2) 2 ,
which tends to 1/2 as n → ∞.

Table 1 presents P n (t), L n (t) and their ratio Rn (t) (ie, R n (t) conditioned on λ taking its maximum likelihood estimate) for a range of tree sizes.

Extension to birth-death models

Allowing for random extinction (as well as speciation) introduces additional complexity into the analyses presented above. We first consider what happens if we condition just on n (and adopt the assumption that the time of origin of the initial linage is a parameter of the birth-death model). To do this, we have to assume a prior distribution for the time of origin when conditioning the trees to have n extant species. We make the common assumption that the first species originated at any time in the past with uniform probability [START_REF] Aldous | A critical branching process model for biodiversity[END_REF]. This is also called an improper prior on (0, ∞).

Conditioning the resulting tree to have n extant species yields a proper distribution for the time of origin (Gernhard 2008a). Note that, under the Yule model where µ = 0, this scenario is equivalent to stopping the process just before the n + 1-th speciation event [START_REF] Hartmann | Sampling trees from evolutionary models[END_REF], which is the setting we considered in the first two sections of this paper. The following result generalizes those earlier findings to birth-death models (a proof is provided in the Appendix). As usual, λ is the per lineage speciation rate and µ is the per-lineage extinction rate.

Theorem 3. The expected length of a pendant edge on a birth-death tree conditioned on n is, for 0 < µ < λ,

E[p|n] = µ + (λ -µ) log(1 -µ/λ) µ 2 ; ( 6
)
for µ = λ, we have:

E[p|n] = 1 λ ;
and for µ = 0, we have:

E[p|n] = 1 2λ .
We can also obtain exact results for the lengths of the edges in a birth-death tree if

we condition (just) on time. In particular, we can provide extensions to equation (4)to allow for extinction. We begin, as usual, with two lineages of length 0. Let T R (t) denote the tree that is spanned by those taxa that are extant at time t; T R (t) is therefore referred to as the 'reconstructed' birth-death tree (the tree consisting of edges that survive to time t while extinct lineages are pruned away) [START_REF] Nee | The reconstructed evolutionary process[END_REF]Gernhard 2008a). If there are no taxa extant at time t, we say that T R (t) is empty. Let N R (t) denote the expected number of tips in the reconstructed birth-death tree, given by the well-known formula:

N R (t) = 2e (λ-µ)t , t ≥ 0.
Note that although N R (t) tends to infinity as t grows when λ > µ, it is quite possible that the actual number of lineages at time t is 0, in which case T R (t) is empty. Let L R (t) be the expected total length of the reconstructed birth-death tree, and let P R (t) be the expected sum of the pendant branch lengths of this tree. The proof of the following result is provided in the Appendix.

Theorem 4. Consider a birth-death tree with speciation rate λ > 0 and extinction rate µ that starts from two lineages of length 0. Let ρ = λ µ , r = λ -µ and let f ρ (s) = ρe s -1 (ρ-1)e s . Then, for t ≥ 0:

(i) L R (t) = 2e rt µ • (ln f ρ (rt)). (ii) P R (t) = 2e rt µ 1 -(ρ -1) • (ln f ρ (rt)) + 1 ρe rt -1 . (iii) For ρ > 1, the limiting ratio τ ρ := lim t→∞ P R (t) L R (t)
is given by:

τ ρ = 1 ln ρ ρ-1 -ρ + 1.
The function τ ρ from part (iii) is shown in Fig. 2. Note that the 0.5 asymptote agrees with the ratio of P R (t) and L R (t) as in the pure-birth model as calculated earlier (i.e. τ ρ → 1 2 as ρ → ∞). Interestingly, the asymptote is reached fairly quickly on large trees. For example, from Fig. 2, we see that when the extinction rate is one-third of the extinction rate (ρ = 3), then τ ρ = 0.47 and the expected pendant edge length is 87% the expected interior edge length. Mild extinction in a uniform birth-death model does not produce particularly short pendant edges. At the other extreme, as the extinction rate approaches the speciation rate (so r and ρ converge to 0 and 1 respectively) τ ρ can be easily shown to converge to 0, as suggested by Fig. 2. It is interesting to note that the expected sum of pendant edge lengths in the reconstructed tree at time t (i.e. P R (t)) divided by the expected number of extant taxa at time t (i.e. 2e (λ-µ)t ) converges to the same expression as given in Eqn. (6) as t → ∞.

Expected PD under simple Field-of-Bullets model for Yule trees

The expected lengths of edges in a tree are directly relevant for quantifying the expected loss of 'phylogenetic diversity' (PD) under simple models of extinction in which each tip is deleted with some fixed probability. In these models, edges that are 'deep' within the tree are more likely to contribute to the PD score of the surviving taxa than pendant edges of similar length, since they are more likely to have at least one non-extinct taxon in the clade they support. This redundancy leads to the nonlinear decrease of PD as more species are removed from a tree [START_REF] Nee | Extinction and the loss of evolutionary history[END_REF]. However, the ratio of the lengths of pendant to interior edges is also critical, as pendant edges will be the first to be deleted from the tree. In this section, we analyse the expected PD score of a Yule tree under random taxon deletion. Note that there are two random processes at play here: the Yule process that produces the tree, and then the extinction process that deletes taxa.

Consider then a Yule tree that starts with a split into two lineages at time 0 and is grown until time t > 0. At that time, each tip is selected independently with probability s, and the remaining tips are deleted (pruned). Thus s is the 'survival probability' of a taxon.

Let ψ t (s) be the P D of the resulted pruned tree, and let π t (s) = E[ψ t (s)], where E[.]

denotes expectation with respect to the random Yule tree and the random pruning operation. Thus, π t (1) is the expected P D of the (entire) Yule tree, namely

L(t) = 2
λ (e λt -1) (Eqn. 4). For s < 1, π t (s) is the expected P D one obtains by generating a Yule tree until time t and then applying a field-of-bullets pruning with survival probability s for each tip. The proof of the following result is provided in the Appendix.

Theorem 5.

π t (s) = 2s (1 -s)λ e λt • -log s + (1 -s)e -λt .
The ratio π t (s)/π t (1) of the expected PD in the pruned tree to the expected PD of the total tree therefore converges (quickly) with t to the limit:

π(s) := -s log(s) 1 -s
Theorem 5 implies that π t (s) ≥ s • π t (1) for all t > 0. Moreover, the limiting ratio π(s) is a continuous and concave, positive function that approaches 0 as s → 0 and approaches 1 as s → 1 (see Fig. 3). For s = 0.5, π(s) = log(2) = 0.69. The slope function π ′ (s) approaches infinity as s approaches 0 from above and π ′ (s) approaches 1 2 as s approaches 1 from below. This latter result can be seen by considering that pendant edges are the first to be lost from a tree undergoing extinction; under the Yule model, the sum of the pendant edges constitutes 0.5 of the total PD (Theorem 4).

The high level of redundancy reported by [START_REF] Nee | Extinction and the loss of evolutionary history[END_REF] is due to their use of coalescent-type models of tree shape with a constant population-size, where the pendant edges are expected to be much shorter than the interior edges. More precisely, the ratio of the expected total length of the pendant edges to the expected total length of the interior edges converges to 0 with increasing n, at a rate 1/ log(n), see e.g. [START_REF] Fu | Statistical tests of neutrality of mutations[END_REF] ). An example of the relationship between s and the proportion of the tree remaining under Nee and May's model (for n = 1000) is shown in Fig. 3. Under a Yule model, where interior and pendant edges have roughly the same expected length, the situation is quite different. If we take s=0.05, then π(s) = 0.157.

That is, in a large tree, if we lose 95% of species (randomly) then we would expect to lose more than 84% of the tree. This lower level of redundancy is also more in line with statistical [START_REF] Morlon | Spatial patterns of phylogenetic diversity[END_REF]) and empirical estimates of tree loss under extinction ). The dotted line shows the same quantity for the coalescent-style tree used by [START_REF] Nee | Extinction and the loss of evolutionary history[END_REF], for n=1000. The curve in between these two shows the same quantity but on a birth-death tree with µ = 0.5λ, as described by Eqn. (7).

regimes [START_REF] Von Euler | Selective extinction and rapid loss of evolutionary history in the bird fauna[END_REF][START_REF] Purvis | Nonrandom extinction and the loss of evolutionary history[END_REF][START_REF] Vamosi | Nonrandom extinction leads to elevated loss of angiosperm evolutionary history[END_REF], where tree shape and non-random extinction interact (see also [START_REF] Heard | Phylogenetically patterned speciation rates and extinction risks change the loss of evolutionary history during extinctions[END_REF][START_REF] Nee | Phylogenetic futures and the latest mass extinction[END_REF]).

Similar results hold with birth-death trees under mild extinction (see Fig. 2), where the sum of the pendant edges constitutes τ ρ of the total PD. In particular, for λ > µ > 0, the second formula presented in Theorem 5 can be modified as follows (see Appendix):

π(s) = s (a -s) • log s a • 1 -a log(a) , (7)
where a = 1 -µ/λ.

Fig. 3 exhibits an example curve π(s) on a birth-death tree constructed with µ = 0.5λ.

We note that this modified formula for µ > 0 should be used with care for larger values of µ for two reasons. Firstly, birth-death trees are increasingly likely to die out as µ approaches λ and so an asymptotic ratio of expected values such as π(s) may be a poor estimate of expected PD loss in such situations. Note in particular that in the limit as µ/λ → 1, we have π(s) = 1 for all s > 0. This of course does not mean that if 99.9% of the taxa are eliminated, then we would still expect to retain 100% of the phylogenetic diversity!

The second reason for caution is more empirically based. In the extreme (critical)

case where µ = λ then, as we have noted already, if we condition on a tree having n extant leaves (assuming a uniform prior distribution for the time of the origin of the tree, as in [START_REF] Aldous | A critical branching process model for biodiversity[END_REF]), then the expected distribution of branch lengths in this tree would be precisely that given by the coalescent process (Gernhard 2008b) that was used in the analysis by [START_REF] Nee | Extinction and the loss of evolutionary history[END_REF] . The problem now is that typical species-level phylogenetic trees look very different from such constant-size coalescent-shaped trees. Hey (1992), using a sample of only eight trees, was the first to point out that the coalescent model produced unreasonably short pendant edges (see also [START_REF] Morlon | Spatial patterns of phylogenetic diversity[END_REF], while

McPeek's recent compilation [START_REF] Mcpeek | The ecological dynamics of clade diversification and community assembly[END_REF] of 245 fairly-well sampled chordate, arthropod, mollusk, and magnoliophyte phylogenies, showed that these trees tended to have a branch length distribution in the opposite direction to the coalescent, with edges near the leaves tending to be, on average, slightly longer than expected under the Yule model. McPeek used the gamma statistic from [START_REF] Pybus | Testing macro-evolutionary models using incomplete molecular phylogenies[END_REF] to describe the distribution of branch lengths as one moves from the root of the tree to the tips, and found that the majority of trees had negative gamma values, rather than having them centered on 0 as expected under the Yule model [START_REF] Pybus | Testing macro-evolutionary models using incomplete molecular phylogenies[END_REF] and the positive values expected under the coalescent [START_REF] Pybus | New inferences from tree shape: numbers of missing taxa and population growth rates[END_REF]. Indeed, we show in the appendix that the expected value of gamma for a coalescent tree of increases indefinitely at a rate of √ 3n. [START_REF] Morlon | Inferring the dynamics of diversification: a coalescent approach[END_REF] applied a coalescent framework that allows for incomplete taxon sampling to an overlapping set of 289 trees and found that the majority of trees (> 80%) had splitting times that were either consistent with the Yule model or concentrated nearer the root. Though nonrandom sampling may be a concern [START_REF] Cusimano | Slowdowns in diversification rates from real phylogenies may not be real[END_REF], the observation that most nearly-complete phylogenetic trees have gamma values close to zero (or negative), as well as the explicit test of the Yule model by [START_REF] Morlon | Inferring the dynamics of diversification: a coalescent approach[END_REF] suggest that our use of this model in analyzing expected loss of PD may be conservative.

Conclusion

Although the Yule model of diversification is nearly 100 years old, it still holds some surprises. The fact that real trees are conditioned on t and that we show up at some random time after n tips have been produced leads to the observation that average pendant edge lengths (species ages) and internal edge lengths (those that anchor higher clades) are expected to be nearly equal under the Yule model. Although all edges are not expected to be the same length -for instance the two edges incident to the root are longer than others (results not shown) -this conditioning also makes randomly selected edge lengths one half of the naive expectation. These observations may be useful in informing prior distributions on edge lengths for tree inference.

Mild amounts of uniform extinction do not change these general observations. Indeed, the 'push of the past' [START_REF] Harvey | Phylogenies without fossils[END_REF][START_REF] Phillimore | Density dependent cladogenesis in birds[END_REF], which describes the expectation that those groups which diversified faster than expected early on are more likely to be sampled in the present, would lead to internal edges being even shorter relative to pendant edges. Non-uniform models, such as adaptive radiations where diversification actually slows down through time [START_REF] Rabosky | Explosive evolutionary radiations: Decreasing speciation or increasing extinction through time?[END_REF][START_REF] Morlon | Inferring the dynamics of diversification: a coalescent approach[END_REF], would do the same. All these processes work against the redundancy inherent in the Tree of Life. We predict that this redundancy may not be as great as hoped for. Of course, this prediction must await more complete, dated trees.

1 n -1 P n-1 . (8)
Notice that the term P n-1 = P n-1 (t) on the right-hand side of ( 8) can be replaced by L n-1 (t) -I n-1 (t) (with L n-1 (t) given by ( 5)). Moreover, when n = 2 we have the initial solution I 2 (t) = 0 (and P 2 (t) = 2t) for all t ≥ 0, and for each n we have the boundary condition I n (t) = 0 at t = 0.

It can now be verified that the expression given in Theorem 2 for P n (t) satisfies this system of linear differential equations subject to the boundary condition, and so is the unique solution.

For the second claim if we set λ to its maximum likelihood estimate, i.e.

λ M L = log( n 2 )/t, then,

P n (t) = 2 log(n/2) λ(n -1) + (n -2) (n + 5) -4(1 + n + 2 log(n/2))e -log(n/2) + (3n -1 + 2(n + 1) log(n/2))e -2 log(n/2) 2λ(n -1)(1 -e -log(n/2) ) 2 = 2 log(n/2) λ(n -1) + (n -2) (n + 5) -4(1 + n + 2 log(n/2))2n -1 + (3n -1 + 2(n + 1) log(n/2))(n/2) -2 2λ(n -1)(1 -2/n) 2 = t n 3 -3n 2 -4n log(n/2) + 4n -4 2 log(n/2)(n -1)(n -2)
The sum of all edge lengths is in expectation [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF],

L n (t) = t n-2 log(n/2) , and therefore the ratio R n is the expression given Theorem 2. From this expression it is easily seen that lim n→∞ Pn(t)

Ln(t) = 1/2. ✷

Proof of Theorem 3

The probability v(k) that a leaf is attached to the kth speciation event in a tree on n extant species under the Yule or birth-death model is, from [START_REF] Stadler | Lineages-through-time plots of neutral models for speciation[END_REF], given by:

v(k) = 2k n(n -1) . (9)
For 0 ≤ µ < λ, let:

p 0 (t) := (1 -e -(λ-µ)t ) λ -µe -(λ-µ)t , and p 1 (t) := (λ -µ) 2 e -(λ-µ)t (λ -µe -(λ-µ)t ) 2 ,
while for µ = λ, let:

p 0 (t) := t 1 + λt , and p 1 (t) := 1 (1 + λt) 2 .
The probability that a lineage produces 0 (resp. 1) offspring after time t is µp 0 (t) (resp. p 1 (t)) [START_REF] Kendall | Stochastic processes and population growth[END_REF]. We first establish the following result: Lemma 6. The length of a randomly picked pendant edge in a birth-death tree on n extant species has probability density function f p (t|n) = 2λp 1 (t)(1 -λp 0 (t)).

Proof. For proving the lemma, we will use the probability density of the time of the k-th speciation event in a birth-death tree with n extant species which is derived in Gernhard (2008a), and for µ < λ, we get,

f n,k (t) = (k + 1) n k + 1 λ n-k (λ -µ) k+2 e -(λ-µ)(k+1)t 1 -e -(λ-µ)t n-k-1 (λ -µe -(λ-µ)t ) n+1 . (10)
Using Equation ( 9) and ( 10), we can write,

f p (t|n) = n-1 k=1 v(k)f n,k (t) = 2 n-1 k=1 n -2 k -1 λ n-k (λ -µ) k+2 e -(λ-µ)(k+1)t 1 -e -(λ-µ)t n-k-1 (λ -µe -(λ-µ)t ) n+1 = 2λ n-1 (λ -µ) 3 e -2(λ-µ)t 1 -e -(λ-µ)t n-2 (λ -µe -(λ-µ)t ) n+1 n-1 k=1 n -2 k -1 (λ -µ)e -(λ-µ)t λ(1 -e -(λ-µ)t ) k-1 = 2λ(λ -µ) 3 e -2(λ-µ)t (λ -µe -(λ-µ)t ) 3 .
For µ = λ, we take the limit µ → λ (using the property e -ǫ ∼ 1 -ǫ), which establishes the lemma.

Note that the length of a pendant edge is independent of n. Theorem 3 now follows directly from Lemma 6 by evaluating ∞ 0 tf p (t|n)dt. ✷

Proof of Theorem 4

The quantity 1 fρ(rt) = r λ-µe -rt is the probability that a birth-death tree that starts with a single lineage at time 0 has at least one extant lineage at time t (Eqn. (2) of [START_REF] Nee | The reconstructed evolutionary process[END_REF]). Thus, by considering the first δ period of time in a birth-death tree that begins with a single lineage, the expected total sum S(t) of branch lengths spanning the leaves present at time t satisfies the differential expression:

S(t + δ) = 0 • µδ + 2S(t) • λδ + (S(t) + δ 1 f ρ (rt) ) • (1 -(µ + λ)δ) + O(δ 2 ),
(by considering whether or not the lineage becomes extinct, speciates, or persists unchanged within this initial δ period). Since L R (t) = 2S(t) this leads to the following

differential equation: dL R (t) dt = rL R (t) + 2/f ρ (rt). (11)
Solving Eqn. ( 11) subject to L R (0) = 0, gives part (i) of the Theorem. By considering the evolution of the tree from time t to t + δ a straightforward dynamical argument leads to a second differential equation that links L R (t) to P R (t) :

dL R (t) dt = N R (t) -µP R (t). (12)
Part (ii) follows by equating the right-hand sides of Eqns. ( 11) and ( 12) to express P R (t) in terms of quantities already determined. For Part (iii), observe that r > 0 and f ρ (rt) → (ρ -1)/ρ as t → ∞ and so, from parts (i), (ii), we have the asymptotic equivalences L R (t)/2e rt ∼ µ -1 ln[ρ/(ρ -1)], P R (t)/2e rt ∼ µ -1 (1 -(ρ -1) ln[ρ/(ρ -1)]).

Taking the ratio of these quantities gives the result claimed.

Proof of Theorem 5

Let φ t = φ t (s) be the analogue of ψ t (s) if we start the Yule tree with a single (rather than 2) lineages at time t = 0; thus,

π t (s) = E[ψ t (s)] = 2E[φ t (s)], (13)
(the behaviour of φ is slightly easier to analyse than ψ). Let X t denote the number of tips in the Yule tree (starting with a single lineage at time 0) at time t. Consider φ t+δ , for a small value δ > 0. In the first δ period of time the initial lineage can either (i) speciate (with probability λδ + O(δ 2 )) or (ii) fail to speciate (with probability 1 -λδ + O(δ 2 )) and so we have:

φ t+δ =        φ 1 t +
where

E[Y t |X t+δ = n] = δ • (1 -(1 -s) n ),
and φ 0 t , φ 1 t and φ 2 t are independent random variables having the same distribution as φ t

(the contribution of δ to the PD score of the tree applies precisely if at least one of the tips at time t + δ is sampled, and this event, conditional on X t+δ = n, has probability 1 -(1 -s) n ). Now, P(X t+δ = n|X δ = 1) = P(X t = n|X 0 = 1), and it is a classic result that this latter probability has a geometric distribution with mean e λt (see e.g. Beichett and Fatti (2002), Example 6.10, pp. 193) and so:

E[Y t ] = δ • (1 -E[(1 -s) Xt ]) = δ • 1 - n≥1
(1 -s) n e -λt (1 -e -λt ) n-1 = δ • s s + qe -λt (15) where q = 1 -s. Let π ′ t (s) := E[φ t (s)]. Taking expectation of ( 14) (with respect to both the Yule tree and the random sampling process) and applying (15) leads to the following differential relationship for π ′ t (s):

π ′ t+δ (s) = 2λδ • π ′ t (s) + (1 -λδ) • π ′ t (s) + δ • s s + qe -λt + O(δ 2 ).
This leads to the following first-order, linear differential equation for π ′ t (s):

dπ ′ t (s) dt -λπ ′ t (s) = s s + qe -λt .
Solving this equation gives π ′ t (s), and thereby the stated value for µ t (s) = 2π ′ t (s) (by ( 13)).

The modification of this result to give Eqn. ( 7) in the birth-death setting, with 0 < µ < λ following a similar case analysis (but allowing for the possibility of extinction) leads to the differential equation for M t (s) = E[φ t (s)]: dM t (s) dt = (λ -µ)M t (s) + P(φ t (s) = 0). ( 16)

Now, by Eqn. (1) of (Yang and Rannala 1997) (or see [START_REF] Nee | The reconstructed evolutionary process[END_REF])) we have:

P(φ t (s) = 0) = as s -(s -a)e -(λ-µ)t , (17)
where a = 1 -µ/λ. Now π t (s) lies between 2M t (s) and 2M t (s) -t (depending on whether we add the lengths of all the edges from the extant taxa to the root, or just the edges from the extant taxa to their most recent common ancestor), from which Eqn. (7) follows by evaluating the limit of the ratio π t (s)/π t (1) as t → ∞.

✷.

Figure 2 :

 2 Figure2: Graph of τ ρ , which is the limiting ratio (for large t) of the sum of pendant edge lengths to the sum of all edge lengths in a birth-death tree, in which the speciation rate is ρ > 1 times the extinction rate.

Figure 3 :

 3 Figure3: Lower solid line shows the proportion of PD remaining when random extinction occurs with probability 1-s on a Yule tree (from π(s) from Theorem 5). The dotted line shows the same quantity for the coalescent-style tree used by[START_REF] Nee | Extinction and the loss of evolutionary history[END_REF], for n=1000. The curve in between these two shows the same quantity but on a birth-death tree with µ = 0.5λ, as described by Eqn. (7).

Table 1 :

 1 ). Let I n (t) and P n (t) denote the expected sum of the interior and pendant edge lengths (respectively) of a Yule tree, Sum of pendant edges (P n (t))

	n =	P n (t)	L n (t)	Rn (t)
	4 16 64 256 1024	3.03296 • t 3.8697 • t 9.2373 • t 26.3815 • t 82.0735 • t 163.8260 • t 0.5010 2.8854 • t 1.0511 6.7326 • t 0.5748 17.8894 • t 0.5163 52.3492 • t 0.5040

  φ 2 t + O(δ), with probability λδ + O(δ 2 );

	φ 0 t + Y

t , with probability 1 -λδ + O(δ 2 );

log e (n)+C ) where C is Euler's constant (0.5772...), and simulations confirm this improved fit. ✷

Acknowledgements

We thank our funders for support, and two anonymous reviewers, various audiences and the Associate Editor Cécile Ané for very helpful comments. *

Funding

This work was funded by the Royal Society of New Zealand James Cook Fellowship and Marsden Funds (MS and LH), the PhyloSpace project (ANR -Programme la 6ème Extinction, OG), and the Natural Sciences and Engineering Research Council of Canada Discovery Grants programme (AOM).

Appendix: Proofs of Theorems

Proof of Theorem 2

We can modify the argument that leads to the differential equation dI(t) dt = λP (t) from [START_REF] Steel | The expected length of pendant and interior edges of a yule tree[END_REF]) so as to take into account conditioning on n as well as tthe analysis consists of calculating quantities such as P[X t = n -1|X t+δ = n], where X t denotes the number of species present at time t, for which Eqn. ( 4) of [START_REF] Nee | Inferring speciation rates from phylogenies[END_REF]) is helpful. In this way one can derive the following sequence of first-order linear differential equations for I n = I n (t):

The expected value of gamma under the coalescent process

Under a Yule (pure-birth) model, the gamma statistic has a standard normal distribution with mean 0, while under a coalescent model it is positive. Under the coalescent model, the original γ statistic grows at the asymptotic rate of √ n as the number of tips n grows.

Theorem 6.

For a coalescent tree with n leaves, γ/ √ n converges in probability to √ 3 with increasing n.

For a rooted binary tree with n ≥ 2 leaves, let g 2 , g 3 , . . . , g n be times between successive speciation events, measured from the root to the leaves, and let T n = n j=2 jg j . From [START_REF] Pybus | Testing macro-evolutionary models using incomplete molecular phylogenies[END_REF] we have γ = Xn Yn , where X n can be written in the form:

and

.

Now, under the coalescent, the random variables g 2 , . . . , g n are independently distributed, and with g j having an exponential distribution with mean 1 ( j 2 )

. It follows that Tn 2 log(n) and Xn log(n) have expected values that converge to 1, and variances that converge to 0 as n → ∞, and so Tn 2 log(n) and Xn log(n) each converge in probability to the constant 1 as n → ∞. Consequently, the ratio X n /T n converges in probability to 1/2 as n → ∞, and so

converges in probability to √ 3, as claimed.

Finally, a more careful asymptotic analysis provides a closer approximation to γ/ √ n by the formula √ 3 • (1 -


Frederique Gayraud

Hye-Ran Lee

Melissa Barkat-Defradas
email: melissa.barkat@univ-montp3.fr

Melissa Barkat- Defradas

Running head: Speech dysfluencies in Alzheimer's disease 1 'Title page (not for review)' Syntactic and lexical context of pauses and hesitations in the discourse of Alzheimer patients and healthy elderly subjects

Keywords: Alzheimer's disease, spontaneous speech, speech rate, pauses, lexical frequency, french language Alzheimer's disease, spontaneous speech, speech rate, pauses, lexical frequency, french language

Running head: Speech dysfluencies in Alzheimer's disease 2 Psycholinguistic studies dealing with Alzheimer's disease (AD) commonly consider verbal aspects of language. In this paper we investigated both verbal and non-verbal aspects of speech production in AD. We used pauses and hesitations as markers of planning difficulties and hypothesized that AD patients show different patterns in the process of discourse production. We compared the distribution, the duration and the frequency of speech dysfluencies in the spontaneous discourse of 20 AD patients with 20 age, gender and socioeconomic matched healthy peers. We found that patients and controls differ along several lines: patients' discourse displays more frequent silent pauses, which occur more often outside syntactic boundaries and are followed by more frequent words. Overall patients show more lexical retrieval and planning difficulties, but where controls signal their planning difficulties using filled pauses, AD patients do not.

Syntactic and lexical context of pauses and hesitations in the discourse of Alzheimer patients and healthy elderly subjectsLanguage production is a complex activity involving almost simultaneous planning of content, syntactic structure, lexical retrieval and articulation [START_REF] Levelt | Speaking: From intention to Articulation[END_REF] involving executive functions to control the execution of this complex, goaldirected activity. In addition, this activity is performed under strict temporal pressure since the average rate of production is around two or three syllables per second.

Temporal characteristics of speech

Speech production cannot be continuously exercised for several reasons including physiological reasons such as respiratory activity and cognitive planning [START_REF] Zellner | The temporal structure of speech[END_REF].

Psycholinguists investigated the location and duration of cognitive pauses in order to identify a valid psychological unit of planning. Syntactic units rapidly appeared as good candidates.

Early studies on pause duration and location demonstrated that hesitation phenomena tend to occur at major syntactic boundaries. The pioneering study by [START_REF] Maclay | Hesitation phenomena in spontaneous English speech[END_REF] hypothesized a non-random distribution of pauses. [START_REF] Hawkins | The syntactic location of hesitation pauses[END_REF] claimed that the clause is an important encoding unit of speech on the basis of the finding that two thirds of pauses were located at clause boundaries and accounted for three quarters of total pause time. The well-known survey by [START_REF] Goldman-Eisler | Pauses, clauses, sentences[END_REF] aiming at assessing the psychological validity of syntactic units also established the link between syntactic properties and temporal organization of speech production. It was shown that the higher the degree of integration between clauses, the fewer pauses were encountered at the junction. [START_REF] Grosjean | The patterns of silence: Performance structures in sentence production[END_REF] findings were similar in reading experiments. When asking participants to read sentences at a low rate in order to make pauses more salient, they found that pauses tended to be long and frequent between words that had little cohesion and were shorter and scarcer between words with a high degree of cohesion. [START_REF] Gee | Performance structures: A psycholinguistic and linguistic appraisal[END_REF] investigated pause patterns further, focusing on the lack of correspondence between syntactic structures and psycholinguistic structures called 'performance structures'. They found that the sentences are broken into basic units that are 'rather small' (3.46 syllables in a reading experiment in French, [START_REF] Monnin | Les structures de performance en français: caractérisation et prédiction[END_REF], independent of respiratory constraints, they show a rich hierarchical structure and they tend to be symmetrical, that is the main pause occurs around the middle of the sentence. The latter characteristic is crucial for explaining the mismatch between syntactic description and performance: while syntactic structures are insensitive to the length of their constituents, length of constituents matters for performance structures [START_REF] Zellner | The temporal structure of speech[END_REF].

Speech rate and pauses in normal and pathological aging

Since normal aging involves a slowing of neural processes, it may be expected that speech segments will take longer to produce, and that speech rate should decrease with age. [START_REF] Penny | Some aspects of speech and voice in healthy ageing people[END_REF] actually found no difference in speech rate between 65 and 90 years of age. With regards to pausing however, several studies on temporal characteristics of speech production in normal aging showed an increase in the frequency and duration of silent pauses, reflecting lexical retrieval difficulties and/or cognitive slowing down [START_REF] Nef | Les manifestations du vieillissement normal dans le langage spontané oral et écrit[END_REF][START_REF] Kemper | Adults' sentence fragments: Who, what, when, where, and why[END_REF][START_REF] Schmitter-Edgecombe | Aging and word finding: aA comparison of discourse and nondiscourse testsspontaneous and constrained naming tests[END_REF]Burke and Shafto, 1994). Indeed word finding difficulties are known to increase with age (Shwartz, 2005;[START_REF] Mortensen | Age-related effects on speech production: A review[END_REF] and they are reflected in pausing [START_REF] Zellner-Keller | Comment est-ce qu'on dit ?" Vieillissement et manque de mot en conversation[END_REF]. By contrast, [START_REF] Kynette | Aging and the loss of grammatical forms: A crosssectional study of language performance[END_REF] did not find significant changes in the production of fillers or incomplete sentences between 50 and 90 years of age.

Turning to Alzheimer's disease (henceforth AD), studies comparing temporal features of AD patients to controls found that the former produce more silent pauses than the latter [START_REF] Illes | Neurolinguistic features of spontaneous production dissociate three forms of neurodegenerative disease: Alzheimer's, Huntington's and Parkinson's[END_REF][START_REF] Bucks | Linguistic analysis of spontaneous, conversational speech in probable Alzheimer's disease: A comparison with normal older adults[END_REF]Singh, Burke and Cuerden 2001;Hoffmann, Nemeth, Dye, Pakaski, Iinyi and Kalman, 2010). In addition, [START_REF] Singh | Evaluation of an objective technique for analysing temporal variables in DAT spontaneous speech[END_REF] computed other temporal measures including verbal rate and phonation rate. They found that the proportion of speech time dedicated to pauses is not of critical importance to discriminate AD patients from controls, but these results need to be ascertained because of the small number of participants in the study. A recent work by [START_REF] Hoffmann | Temporal parameters of spontaneous speech in Alzheimer's disease[END_REF] tends to confirm those results: comparing four temporal parameters (articulation rate, speech tempo, hesitation ratio (i.e. pauses), and rate of grammatical errors) across AD patients and controls, they found significant differences between both groups, especially for hesitation ratio. In addition, temporal characteristics of speech seem to be of critical importance to discriminate AD from other neurodegenerative pathologies [START_REF] Illes | Neurolinguistic features of spontaneous production dissociate three forms of neurodegenerative disease: Alzheimer's, Huntington's and Parkinson's[END_REF], and to discriminate healthy elderly individuals from AD individuals, even at the earliest stages of the disease [START_REF] Hoffmann | Temporal parameters of spontaneous speech in Alzheimer's disease[END_REF]. Fewer studies have investigated the linguistic context of occurrence of pauses.

In an attempt to identify which linguistic forms are more resistant and which ones are more vulnerable to Alzheimer's disease, [START_REF] Tapir-Ladino | Analisis de los enunciados producidos despues de una pausa en la conversacion de enfermos de Alzheimer y de senescentes[END_REF] demonstrated that verbs are more often preceded by a pause than nouns, suggesting that the latter grammatical category is more robust.

One should be very careful however when interpreting these results because they vary along several lines relating to methodological issues. Concerning the participants, some studies include only a small sample (Illes, 1989: n=10;Singh et al., 2001: n=8). In addition, some studies include participants with mild to moderate stages symptoms of AD (e.g. [START_REF] Illes | Neurolinguistic features of spontaneous production dissociate three forms of neurodegenerative disease: Alzheimer's, Huntington's and Parkinson's[END_REF] while others also include participants at severe stages of the disease [START_REF] Singh | Evaluation of an objective technique for analysing temporal variables in DAT spontaneous speech[END_REF][START_REF] Bucks | Linguistic analysis of spontaneous, conversational speech in probable Alzheimer's disease: A comparison with normal older adults[END_REF]. Another source of variation lies in the control of the level of education of the participants, which we know is likely to have a strong impact on AD patients' cognitive functioning [START_REF] Turell | Socioeconomic Position Across the Lifecourse and Cognitive Function in Late Middle Age[END_REF][START_REF] Karp | Relation of education and occupation-based socioeconomic status to incident Alzheimer's disease[END_REF]Wilson, Scherr, Hoganbson, Cathy 2/4/10 17:48 Commentaire: Do you mean 'replicated' ? [START_REF] Wilson | Early Life Socioeconomic Status and late Life Risk of Alzheimer's Disease[END_REF]. In Illes' study, participants are matched for this variable, as they are in [START_REF] Hoffmann | Temporal parameters of spontaneous speech in Alzheimer's disease[END_REF] where the participants all have a relatively high level of education, with a mean around 12 years. The other studies do not mention whether this possibly important variable was taken into consideration. Another methodological issue relates to the identification, labelling and measurement of the different temporal variables.

First, the threshold for detecting silent pauses varies across studies: 200 ms in [START_REF] Illes | Neurolinguistic features of spontaneous production dissociate three forms of neurodegenerative disease: Alzheimer's, Huntington's and Parkinson's[END_REF], 300 ms in [START_REF] Hoffmann | Temporal parameters of spontaneous speech in Alzheimer's disease[END_REF] while in [START_REF] Tapir-Ladino | Analisis de los enunciados producidos despues de una pausa en la conversacion de enfermos de Alzheimer y de senescentes[END_REF], [START_REF] Bucks | Linguistic analysis of spontaneous, conversational speech in probable Alzheimer's disease: A comparison with normal older adults[END_REF] and [START_REF] Singh | Evaluation of an objective technique for analysing temporal variables in DAT spontaneous speech[END_REF] the threshold is not mentioned. In the latter studies however, one can guess that the threshold was very high since pauses were measured with a stop watch and they found durations around two seconds. With the exception of [START_REF] Illes | Neurolinguistic features of spontaneous production dissociate three forms of neurodegenerative disease: Alzheimer's, Huntington's and Parkinson's[END_REF], filled pauses are usually not taken into consideration. For speech rate, different units are used: number of words per minute [START_REF] Illes | Neurolinguistic features of spontaneous production dissociate three forms of neurodegenerative disease: Alzheimer's, Huntington's and Parkinson's[END_REF][START_REF] Singh | Evaluation of an objective technique for analysing temporal variables in DAT spontaneous speech[END_REF], number of phonemes per second [START_REF] Hoffmann | Temporal parameters of spontaneous speech in Alzheimer's disease[END_REF]. Another possible source of confusion lies in the labelling of the different temporal variables: some authors use the same label to refer to different phenomena.

For instance in [START_REF] Illes | Neurolinguistic features of spontaneous production dissociate three forms of neurodegenerative disease: Alzheimer's, Huntington's and Parkinson's[END_REF], 'Verbal rate' refers to the total number of words per speaking time (minus hesitation time) while in [START_REF] Hoffmann | Temporal parameters of spontaneous speech in Alzheimer's disease[END_REF], the same term is used to refer to the total number of phonemes per speaking time (including hesitation time). We claim that these methodological discrepancies should be kept in mind and that comparisons across studies should be very careful. In our view variables such as level of education or socioeconomic status should be carefully controlled and pause time data must be measured as precisely as possible.

Finally, most studies focus on silent pauses without paying attention to filled pauses which, even though less frequent, are nevertheless functionally important as they stand for reliable pragmatic indicators of 'turn protection' [START_REF] Maclay | Hesitation phenomena in spontaneous English speech[END_REF] as well as markers of 'preparedness problems' [START_REF] Clark | Using uh and um in spontaneous speech[END_REF] that can be observed during the process of lexical retrieval.

Word retrieval difficulties

The language of AD patients shows pervasive disturbances even at the earliest stages of the disease, especially in the lexico-semantic domain (see [START_REF] Barkat-Defradas | Les troubles de la parole dans la maladie d'Alzheimer[END_REF] for a recent review). Word retrieval difficulties are experienced by any speaker, but this phenomenon tends to increase with age [START_REF] Schwartz | The relation of tip-of-the-tongue states and aging: Contrasting psycholinguistic and metacognitive perspectives[END_REF][START_REF] Mortensen | Age-related effects on speech production: A review[END_REF][START_REF] Zellner-Keller | Comment est-ce qu'on dit ?" Vieillissement et manque de mot en conversation[END_REF], and is an early sign of AD as demonstrated by naming experiments.

But speech production analysis also documents word finding difficulties [START_REF] Croisile | Comparative study of oral and written description with Alzheimer's disease[END_REF].

Several factors are known to affect lexical retrieval in general as well as in AD patients, among which we will focus on word frequency and grammatical category. Lymperopoulou, [START_REF] Lymperopoulou | The effects of age of acquisition and word frequency on object naming accuracy in Alzheimer's disease[END_REF] investigated the effect of Age-of-Acquisition and frequency in a naming task performed by patients with Alzheimer's disease. They found that the naming accuracy was affected by both factors, which is confirmed in Ferreiro, Davies, Gonzalez-Nosti, [START_REF] Ferreiro | Name agreement, frequency and age of acquisition, but not grammatical class, affect object and action naming in Spanish speaking participants with Alzheimer's disease[END_REF]. In free speech, open class items are typically replaced by more general thus more frequent terms such as 'thing, stuff, do' [START_REF] Bird | The rise and fall of frequency and imageability: noun and verb production in semantic dementia[END_REF][START_REF] Meteyard | The relation between content and structure in language production: An analysis of speech errors in semantic dementia[END_REF]. Turning to the grammatical category of the lexical items prone to retrieval difficulties, naming experiments bring contradictory findings regarding AD individuals' performances in object versus action naming. While some studies find an advantage for verbs (for instanceRobinson, Grossman, White-Devine and D 'Esposito, 1996), others fail to find differences between verbs and nouns [START_REF] Cappa | Object and action naming in Alzheimer's disease and fronto-temporal dementia[END_REF][START_REF] Lee | Object and action naming in two types of Chinese brain damage patients: Alzheimer's disease and frontal lesion[END_REF] or find an advantage for nouns [START_REF] Robinson | Category specific difficulty naming with verbs in Alzheimer's disease[END_REF][START_REF] Kim | Verb deficits in Alzheimer's disease and agrammatism: Implications for lexical organization[END_REF] Running head: Speech dysfluencies in Alzheimer's disease 8 2004). [START_REF] Ferreiro | Name agreement, frequency and age of acquisition, but not grammatical class, affect object and action naming in Spanish speaking participants with Alzheimer's disease[END_REF] hypothesize that this lack of consensus is due to insufficient control over the stimuli used. When controlling for name agreement, visual complexity, frequency, orthographic neighbourhood, word length, imageability and age of acquisition, they did not find any difference for the grammatical category. In addition, the authors found that AD individuals tend to replace specific verbs by more generic ones in their discourse.

Beyond the inconsistency in the results of previous studies, [START_REF] Kim | Verb deficits in Alzheimer's disease and agrammatism: Implications for lexical organization[END_REF] work is of special relevance for our research since it includes, among other tasks, a narrative production, which is definitely more ecological than naming tasks. Another promising step lies in focusing on those moments -typical of spontaneous speech -that are not filled with linguistic material. Indeed, we claim that the search for the most vulnerable components of language in AD can benefit from the study of speech temporal characteristics as well as oral dysfluencies.

Pauses as a reliable cue for lexical retrieval difficulties

Hypothesis

Because of the impairment of executive functions and word finding difficulties, AD patients should (i) produce more and longer pauses as well as more hesitations than controls, whilst speech rate should not differ; (ii) break their discourse into smaller units than controls (performance structures); (iii) produce more pauses at minor syntactic boundaries. Besides, if verbs are more difficult to process they should be preceded more often by a dysfluency than nouns. Finally, if patients tend to replace specific words with more generic ones, words following a pause should be more frequent in AD patients.

Method

Participants.Twenty participants diagnosed with probable Alzheimer's disease and twenty healthy controls matched for age, sex, level of education and socioeconomic status (measured with Poitrenaud's scale, 1995) participated in the study. The AD participants were diagnosed using GDS Reisberg, Ferris, de Leon and Crook(1982) and NINCDS-ADRDA criteria [START_REF] Mckhann | Clinical diagnosis of Alzheimer's disease: report of the NINCDS-ADRDA Work Group under the auspices of Department of Health and Human Services Task Force on Alzheimer's Disease[END_REF] suffered from mild to moderate dementia as gauged by the Mini-Mental State Examination [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF]. All participants were native speakers of French. The participants were tested at home or in a nursing home. Informed consent was obtained either from the participant or from a close relative for some patients prior to testing. Demographic information of the participants is provided in Table 1.

Insert table 1 about here

Procedure. Spontaneous speech data were elicited by the experimenter asking the participants about biographical details such as the best/worst day in their life. The data were then carefully transcribed using Transcriber ® [START_REF] Barras | Transcriber: development and use of a tool for assisting speech corpora production[END_REF] a freeware tool for segmenting, labelling and transcribing speech.

Coding.

(i) Any silence exceeding 200 ms was coded as a silent pause and its duration was precisely measured. Filled pauses and vocalic lengthenings were also measured. Finally hesitation phenomena were coded. Example (1) shows an excerpt of the transcribed data. The mean number (per number of words) and the mean duration of each type of dysfluency were computed.

(1) PAT: ah ben de toutes façons <Silent pause: 0.67> pour le moment ce :<Lengthening: 0.45> même <Hesitation : c-ce > qui m'a: <Lengthening: 0.45> <Silent pause: 0.71> handicapée <Silent pause: 0.25> toute la vie ma belle-mère <Silent pause: 0.58> (Well, anyway / so far wha:t / handicaped me / my whole life my mother-in-law ) Running head: Speech dysfluencies in Alzheimer's disease 10 (ii) Speech rate was computed by dividing the number of syllables per the total amount of time, including pause time. Articulation rate was computed by dividing the number of syllables per the total amount of time, excluding pause time.

(iii) These dysfluencies were used as delimitators of speech chunks which were measured in terms of duration and number of syllables.

(iv) Filled and silent pauses were then coded depending on their location following Grosjean and Deschamp's (1972) procedure: at a major syntactic boundary 'external pause' as in (2) or inside a phrase 'internal pause' as in (3). In (2) pauses occur between clauses. In (3), silent or filled pauses break the verb phrase between the auxiliary and the past participle, or break the noun phrase between the determiner and the noun.

(2) PAT: alors je vais vous dire quand j'avais 5 ans <Silent pause: 1.21> j'étais vilaine <Silent pause: 1.08> je prenais le jour pour la nuit bébé déjà <Silent pause: 0.87> et pis j'étais très vilaine.

(So let me tell you when I was five years-old / I was naughty / already as a baby I was mistaking the day for the night / and I was very naughty)

(3) PAT: la même entreprise <Silent pause: 0.49> dont j'avais <Filled pause: 0.32> organisé le <Silent pause: 1.89>: le service (The same company / for which I had / organized the / service) (v) Each word reflecting difficulties of lexical retrieval (following Zellner-Keller's principles, 2007) was coded for its grammatical category (Noun, Verb, Adjective) and for its frequency determined in Lexique 3.01 database [START_REF] New | Une base de données lexicales du français contemporain sur internet: LEXIQUE[END_REF]. The dependant variables under examination are summarized in Table 2.

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Results

The first set of variables under consideration is related to pause time data. Table 3 provides the mean number of silent and filled pauses, lengthenings and hesitations.

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As can be seen from table 3, silent pauses and hesitations are more frequent in patients' than in controls' speech (t (38) =2.73, p< 0.01 for silent pauses and t (38) =3.64, p< 0.001 for hesitations). By contrast, there are no significant differences in the number of filled pauses and lengthenings.

Turning to the proportion of time dedicated to speech and dysfluencies over the total duration of the recording, Table 4 provides the main results.

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Patients dedicate significantly less time to speech and conversely more time to hesitation phenomena than controls (t (38) =2.42, p= 0.02). The proportion of silent pauses are marginally higher in patients (t (38) =1.85, p= 0.07) while the difference fails to reach statistical significance for filled pauses and lengthenings.

Turning to the duration of each type of dysfluency, Table 5 shows the main results.

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Even though pauses (filled and silent) and lengthenings appear slightly longer in patients than in controls, these differences are not statistically significant.

Table 6 shows the results for speech rate and articulation rate.

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Overall, patients produce significantly fewer syllables per minute than controls (t (38) =2.17, p=0.035). However, if pause time is removed in order to calculate articulation rate, this difference is not longer significant.

As for the duration of verbal sequences, as shown in Table 7, it is longer in controls than in patients (t (38) =2.52, p=0.01). Similarly, verbal sequences are longer in terms of number of syllables in controls (t (38) =2.58, p=0.01). =2.22, p=0.03). By contrast no difference is found for verbs.

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Insert table 8 about here

Table 9 shows that, as expected, patients experience significantly more trouble with lexical retrieval than healthy elderly subjects (t (38) =2.033, p=0.05). Regarding the mean frequency of words following a pause in patients' and controls' speech, our study reveals that these lexical items are significantly more frequent in patients' speech than in controls' (t (38) =4.008, p=0.0003).

Insert table 9 about here

Discussion

The results presented in this study were obtained using a strictly controlled methodology which included a reasonably large number of AD participants (n=20) matched for sex, age and sociocultural / socioeconomic levels with controls. Our group of patients was homogenous in terms of the degree of severity of the disease (i.e. no patient at a severe stage of AD was included). Natural spontaneous speech was collected and transcribed manually by experts who carefully measured pause duration using the standard auditory threshold for perception of pauses (i.e. 200-250 ms) as defined by [START_REF] Goldman-Eisler | Pauses, clauses, sentences[END_REF], Grosjean and[START_REF] Grosjean | Analyse des variables temporelles du français spontané[END_REF][START_REF] Zellner | The temporal structure of speech[END_REF].

Though previous studies have underlined that silent pause duration rather than pause frequency makes a difference between AD individuals and healthy elderly individual's speech [START_REF] Singh | Evaluation of an objective technique for analysing temporal variables in DAT spontaneous speech[END_REF][START_REF] Tapir-Ladino | Analisis de los enunciados producidos despues de una pausa en la conversacion de enfermos de Alzheimer y de senescentes[END_REF], the present work does not replicate these findings. We find no significant differences in the length of silent pauses but instead, we find that their frequency is significantly higher in AD patients' speech. We also find that AD patients produce more hesitations, which is in accordance with previous work. Contrary to our expectations, no statistical difference was noted for filled pauses and lengthenings. These phenomena, that are very frequent in spontaneous speech, indicate uncertainty and are used to maintain control of a conversation while thinking of what to say next [START_REF] Clark | Using uh and um in spontaneous speech[END_REF]. According to [START_REF] Clark | Running head: Speech dysfluencies in Alzheimer's disease 16[END_REF], in the framework of the theory of performance, speakers proceed along two tracks of communication simultaneously. They use signals in the primary track to refer to the theme of the discourse and they also use signals in the collateral track to refer to the performance itself, that is to timing, delays, rephrasings, mistakes, repairs, intentions to speak, and the like. A related proposal is that fillers are elements 'whereby the speaker, momentarily unable or unwilling to produce the required word or phrase, gives audible evidence that he is engaged in speech-productive labor' [START_REF] Goffman | Forms of Talk[END_REF]. The analysis we carried out reveals that AD patients do not follow this pattern: although they experience more planning difficulties than healthy elderly individuals, their proportion of filled pauses and lengthening does not increase accordingly. In other words, they do not use

  Table 8 below shows the grammatical context where dysfluencies are found. Patients produce more pauses at sites which do not correspond to syntactic boundaries (t (38) =2.24, p=0.03). Turning to the grammatical category our results show that controls tend to experience more word finding difficulties with nouns (t (38) =1.95, p=0.58) whilst patients have more word retrieval difficulties with adjectives (t (38)

the conventional signals in the collateral track and tend to remain silent. This explains why the total amount of speech is greater in controls' discourse than in patients'.

Regarding speech and articulatory rates, our findings reveal on the one hand that patients' speech rate is slower while articulatory rate does not differ across groups. This suggests that motor aspects of speech production are preserved in AD and underlines the fact that pauses are responsible for the decrease of speech rate in AD. Combining this result with the higher frequency of pauses in AD patients' speech, one can easily understand why verbal sequences are shorter in their discourse and why pauses occur more often outside major syntactic boundaries.

Further analysis enables us to characterize the words occurring next to speech dysfluencies and allows us to identify precisely which lexical category yields more often to dysfluency. We find no difference in the process of retrieval of nouns or verbs. This is consistent with [START_REF] Ferreiro | Name agreement, frequency and age of acquisition, but not grammatical class, affect object and action naming in Spanish speaking participants with Alzheimer's disease[END_REF]. Similarly, our study on lexical frequency confirms patients' recourse to generic and frequent words when facing word production failure. Finally, an original aspect of our study lies in the use of spontaneous data that enabled us to examine other categories than nouns and verbs that are commonly addressed in previous studies based on naming tasks. We show that the retrieval of adjectives is more problematic for patients. One possible explanation for this can be related to the optional feature of attributive adjectives. Further investigations should be conducted on this little studied but promising category so as to ascertain our preliminary findings.