Chakib Fettal

Lazhar Labiod

Mohamed Nadif

More Discriminative Sentence Embeddings via Semantic Graph Smoothing

This paper explores an empirical approach to learn more discriminantive sentence representations in an unsupervised fashion. Leveraging semantic graph smoothing, we enhance sentence embeddings obtained from pretrained models to improve results for the text clustering and classification tasks. Our method, validated on eight benchmarks, demonstrates consistent improvements, showcasing the potential of semantic graph smoothing in improving sentence embeddings for the supervised and unsupervised document categorization tasks.

Introduction

Text categorization, also known as document categorization, is a natural language processing (NLP) task that involves arranging texts into coherent groups based on their content. It has many applications such as spam detection [START_REF] Jindal | Review spam detection[END_REF], sentiment analysis [START_REF] Melville | Sentiment analysis of blogs by combining lexical knowledge with text classification[END_REF], content recommendation [START_REF] Michael | Contentbased recommendation systems[END_REF], etc. There are two main approaches to text categorization: classification (supervised learning) and clustering (unsupervised learning). In text classification, the process involves training a model using a labeled dataset, where each document is associated with a specific category. The model learns patterns and relationships between the text features and the corresponding categories during the training phase. Text clustering, however, aims to group similar documents together without prior knowledge of their categories. Unlike text classification, clustering does not require labeled data. Instead, it focuses on finding inherent patterns and similarities in the text data to create clusters.

In the field of NLP, pretrained models have attained state-of-the-art performances in a variety of tasks [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF][START_REF] Liu | Roberta: A robustly optimized bert pretraining approach[END_REF][START_REF] Reimers | Sentence-bert: Sentence embeddings using siamese bert-networks[END_REF], one of which is text classification. In spite of that, text clustering using such models did not garner significant attention.

To this day most text clustering techniques use the representations of texts generated by some pretrained model such as Sentence-BERT [START_REF] Reimers | Sentence-bert: Sentence embeddings using siamese bert-networks[END_REF] and often use classical clustering approaches such as k-means to obtain a partition of the texts. This is done without any finetuning due to the unsupervised nature of the clustering problem.

Recently, graph filtering has appeared as an efficient and effective technique for learning representations for attributed network nodes. The effectiveness of this technique has made it a backbone for popular deep learning architectures for graphs such as the graph convolutional network (GCN) [START_REF] Thomas | Semisupervised classification with graph convolutional networks[END_REF]. Simplified versions of this deep architecture have been proposed wherein the learning of large sets of weights has been deemed unnecessary. Their representation learning scheme works similar to Laplacian smoothing and, by extension, graph filtering. We can give as examples of these simplified techniques the simple graph convolution (SGC) [START_REF] Wu | Simplifying graph convolutional networks[END_REF], and the simple spectral graph convolution (S²GC) [START_REF] Zhu | Simple spectral graph convolution[END_REF]. Some researchers used GCNs for the task of text classification. [START_REF] Yao | Graph convolutional networks for text classification[END_REF] proposed TextGCN which is GCN with a custom adjacency matrix built from word PMI and the TF-IDF of the documents with the attributes being word count vectors. [START_REF] Lin | Bertgcn: Transductive text classification by combining gcn and bert[END_REF] proposed BertGCN which is similar to TextGCN with the difference that they use BERT representation for the GCN and combine their training losses. The issue is that these approaches are not suitable for learning unsupervised representations since labels are needed. This is a significant limitation towards their use in unsupervised tasks. Recently some graph-based unsupervised approaches were proposed to deal with text data represented using document-term matrices [START_REF] Fettal | Subspace co-clustering with two-way graph convolution[END_REF][START_REF] Fettal | Boosting subspace co-clustering via bilateral graph convolution[END_REF].

In this paper, we propose to use the concept of graph smoothing/filtering, which is the main component accredited with the success of GCNs [START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF][START_REF] Thomas | Semisupervised classification with graph convolutional networks[END_REF][START_REF] Li | Deeper insights into graph convolutional networks for semi-supervised learning[END_REF], to semantically "fine-tune" the representations obtained via sentence embedding models to help traditional clustering and classification algorithms better distinguish between semantically different texts and group together texts which have similar meanings, all in an unsupervised manner.

To do this, we build a graph with respect to the text which describes the semantic similarity between the different documents based on the popular cosine similarity measure. Our approach yields almost systematic improvement when using filtering on the textual representations as opposed to using them without filtering in both facets of document categorization: classification and clustering. Experiments on eight popular benchmark datasets support these observations.

The code for the experiments is available at1 .

Background: Graph Filtering and Smoothing

Graph Signal Processing [START_REF] David I Shuman | The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains[END_REF][START_REF] Ortega | Graph signal processing: Overview, challenges, and applications[END_REF]) provides a framework to analyze and process signals defined on graphs, by extending traditional signal processing concepts and tools to the graph domain. This allows for the representation and manipulation of signals in a way that is tailored to the specific structure of the graph.

In what follows we refer to matrices in boldface uppercase and vectors in boldface lowercase.

Graph Signals Graph signals are mappings from the set of vertices to the real numbers. A graph signal for a given graph G can be represented

using vector f = [f (v 1 ), . . . , f (v n )] ⊤ such that f : V → R is a real-valued function on the vertex set.
The smoothness of a signal f over graph G can be characterized using the Laplacian quadratic form associated with Laplacian L:

f ⊤ Lf = 1 2 i,j a ij (f i -f j ) 2 .
(1)

These signals can be high dimensional and can represent many kinds of data. In our case, signals will represent text embeddings.

Graph Filters Smoother graph signals can be obtained by minimizing the quantity described in Formula (1). That is the goal of graph filters and the filtering is generally done from a spectral perspective. A specific class of filters that additionally has an intuitive interpretation from a vertex perspective is that of the polynomial filters. When the filter is a P -th order polynomial of the form ĥ(L) = p m=0 θ m L m , the filtered signal at vertex i, is a linear combination of the components of the input signal at vertices within a P -hop local neighborhood of vertex i:

f out i = α ii f in i + j∈N (i,p) α ij f in j (2)
where N (i, p) is the P -th order neighborhood of vertex i. It is possible to then make the correspondence with a polynomial filter (from a spectral perspective) as follows:

α ij = p m=d G (i,j) θ m (L m ) ij (3)
where d G is the shortest distance between node i and j. Several polynomial filters have been proposed in the literature such as the ones associated with Simple Graph Convolution (SGC) [START_REF] Wu | Simplifying graph convolutional networks[END_REF], simple spectral Graph Convolution (S²GC) [START_REF] Zhu | Simple spectral graph convolution[END_REF], approximate personalized propagation of neural predictions (APPNP) [START_REF] Gasteiger | Predict then propagate: Graph neural networks meet personalized pagerank[END_REF] and Decoupled Graph Convolution (DGC) [START_REF] Wang | Dissecting the diffusion process in linear graph convolutional networks[END_REF].

Proposed Methodology: Smoothing Sentence Embeddings

In this paper, we theorize that smoothing sentence embeddings with a semantic similarity graph can help supervised and unsupervised categorization models better differentiate between the similar and dissimilar documents, leading to performance gains. A common choice for quantifying semantic similarity of text is the cosine similarity; given two sentence embedding vectors x i , x j ∈ R d we have

cos(x i , x j ) = x ⊤ i x j ∥x i ∥∥x j ∥ .
We build a k-nearest neighbors connectivity graph which we denote G based on this similarity measure i.e. a graph for which each node has exactly k neighbors and whose edge weights are all equal to one. We characterize the graph G using its adjacency matrix A, we denote its Laplacian as L.

Given the adjacency matrix, a standard trick to obtain better node representations consists in adding a self-loop

 = A + λI (4)
where λ is a hyperparameter controlling the number of self-loops. As such in what follows we consider the symmetrically normalized version of Â, that is

S = D-1/2 Â D-1/2 . ( 5
)
Now given a node embedding matrix X and the previous semantic similarity graph. We consider four polynomial graph filters whose propagation rules we describe in Table 1.

Table 1: The propagation rules associated with the different polynomial filters. H (0) is the X. P is the propagation order. α and T are filter-specific hyperparameters.

Filter Propagation Rule

F SGC H (p+1) ← SH (p) F S²GC H (p+1) ← H (p) + SH (p) F APPNP H (p+1) ← (1 -α)SH (p) + αH (0) F DGC H (p+1) ← (1 -T P )H (p) + T P SH (p)

Experiments

In this section we evaluate our semantically smoothed representations obtained through four filters on two tasks, clustering and classification, with respect to the original representations obtained from SentenceBERT [START_REF] Reimers | Sentence-bert: Sentence embeddings using siamese bert-networks[END_REF] as well as two large language models baselines: BERT and RoBERTa.

Datasets and Metrics

We use eight benchmark datasets of varying sizes and number of clusters, and we report their summary statistics in Table 2. For the metrics, in the supervised context, we use the F1 score as the quality metric while in the unsupervised context we use the adjusted rand index (ARI) [START_REF] Hubert | Comparing partitions[END_REF] and the adjusted mutual information (AMI) [START_REF] Nguyen | Information theoretic measures for clusterings comparison: is a correction for chance necessary[END_REF].

Experimental Settings

For the classification task, we use a random stratified 64%-16%-20% train-val-test split. We also tune the hyperparameters k of the k-nn graph, order of propagation P , the parameter λ and the filter specfic parameters α and T . For the clustering task, we use k = 10 for the k-nn graph, set P = 2 as the propagation order, λ = 1, α = 0.1 and T = 5. We report the averages of the metrics as well as their standard deviations over 10 runs (for the classification task, we omit standard deviation due to them being insignificant).

Experimental Results

Clustering Results

We compare the results of the k-means algorithm (kM) applied on Sentence-BERT (we refer to it as SBERT or SB) embeddings with and without the different filtering operations. Note that instead of using kM we can use any other clustering algorithms including variants of kM such as k-means++ [START_REF] Arthur | K-means++ the advantages of careful seeding[END_REF] and entropy kM [START_REF] Chakraborty | Entropy weighted power kmeans clustering[END_REF].

In addition to this, we add a baseline which uses an ensemble technique [START_REF] Ait-Saada | How to leverage a multi-layered transformer language model for text clustering: an ensemble approach[END_REF] on the layer outputs of the word embedding of BERT and RoBERTa, this method improves over considering a single layer or taking the mean. We report the clustering results in Table 3. The filtering operation systematically leads to better results on the benchmark with respect to the filterless clustering scheme on all datasets we have used. These increases are statistically significant in most cases. It also significantly beats the ensemble approach on most datasets.

Classification Results Similar to the clustering setting, we compare results from a Logistic Regression (LR) applied on the original sentence embeddings with and without the filtering operation we introduced. We also use fine-tuned BERT and RoBERTa (2 epochs) as baselines; we use the base versions due to computational restrictions. We report the results in Table 4. We see that this op- eration leads to better performances on the classification task on the majority of the datasets with respect to the filterless Sentence-BERT but this performance increase is not as pronounced as for the clustering task. We also see that the representations we learn lead to competitive results with respect to BERT and RoBERTa despite Sentence-BERT not being suited to classification.

Statistical Significance Testing Using the Bonferroni-Dunn post-hoc mean rank test [START_REF] Demšar | Statistical comparisons of classifiers over multiple data sets[END_REF], we analyze the average ranks of the clustering and classification over the Sentence-BERT representations with and without filtering in terms of AMI and ARI, for the clustering task, as well as the F1 score for the classification task on the eight datasets. Figure 1 shows that the clustering and classification results when using the pro- posed semantically smoothed representations are statistically similar and that they all outperform the Sentence-BERT variant with no filtering in a statistically significant manner at a confidence level of 95%.

Conclusion

We proposed a simple yet effective empirical approach that consists in using similarity graphs in an unsupervised manner to smooth sentence embeddings obtained from pretrained models in a semantically aware manner. The systematic improvements in performance on both clustering and classification tasks on several benchmark datasets of different scales and balance underscore the effectiveness of using semantic graph smoothing to improve sentence representations.

Limitations

The main limitation of our approach is the additional computational complexity entailed by creating the k-nn graph from the data, performing the smoothing. Add to that, the hyperparameter tuning that is necessary for the classification task. While this increase is in no way prohibitive even for large datasets, a performance-speed compromise is to be considered.

Figure 1 :

 1 Figure 1: Bonferroni-Dunn average rank test at a confidence level of 95%.

Table 2 :

 2 Summary statistics of the datasets. Balance refers to the ratio of the most frequent class over the least frequent class. Length refers to the average sentence length in the corpus.

	Dataset	Docs Classes Balance Length
	20News	18,846	20	1.6	221
	DBpedia	12,000	14	1.1	46
	AGNews	8,000	4	1.1	31
	BBCNews 2,225	5	1.3	384
	Classic3	3,891	3	1.4	152
	Classic4	7,095	4	3.9	107
	R8	7,674	8	76.9	65
	Ohsumed	7,400	23	61.8	135

Table 3 :

 3 Clustering results in terms of AMI and ARI on the eight datasets. The best results are highlighted in bold. If our best performing variant outperforms the best comparative method in a statistically significant matter (t-test at a confidence level of 95%), we highlight it in blue.

		20News	AGNews	BBCNews	Classic3
		AMI	ARI	AMI	ARI	AMI	ARI	AMI	ARI
	ENS BERT-base	37.5 ±2.5 15.3 ±1.7 54.1 ±3.6 51.4 ±5.8 81.0 ±5.5 80.0 ±8.5 98.6 ±0.1 99.4 ±0.0
	ENS BERT-large	46.1 ±0.7 21.4 ±0.6 58.5 ±2.8 58.2 ±5.9 86.0 ±3.5 86.5 ±6.3 98.4 ±0.2 99.3 ±0.1
	ENS RoBERTa-base 37.5 ±1.4 15.9 ±1.8 55.9 ±4.1 52.1 ±4.1 80.0 ±5.3 77.2 ±9.4 98.4 ±0.1 99.3 ±0.1
	ENS RoBERTa-large 48.0 ±0.8 23.2 ±1.2 56.7 ±4.6 52.8 ±5.1 85.8 ±3.8 85.1 ±7.2 98.7 ±0.1 99.4 ±0.1
	SBERT+kM	62.9 ±0.3 47.4 ±1.0 57.9 ±0.1 60.5 ±0.1 90.8 ±0.2 93.0 ±0.1 96.0 ±0.1 97.6 ±0.1
	SB+F SGC +kM	65.4 ±0.4 49.1 ±1.1 60.6 ±0.1 62.4 ±0.3 90.6 ±0.1 92.9 ±0.1 98.8 ±0.0 99.5 ±0.0
	SB+F S²GC +kM	64.9 ±0.4 49.0 ±1.1 60.1 ±0.2 62.2 ±0.2 90.9 ±0.1 93.1 ±0.1 98.3 ±0.0 99.2 ±0.0
	SB+F APPNP +kM 65.4 ±0.4 49.8 ±1.2 60.6 ±0.0 62.5 ±0.0 90.6 ±0.1 92.9 ±0.1 98.5 ±0.0 99.3 ±0.0
	SB+F DGC +kM	65.6 ±0.7 48.8 ±1.0 60.5 ±1.5 60.5 ±2.2 90.2 ±0.1 92.5 ±0.1 99.1 ±0.0 99.6 ±0.0
		Classic4	DBpedia	Ohsumed	R8
		AMI	ARI	AMI	ARI	AMI	ARI	AMI	ARI
	ENS BERT-base	71.4 ±3.5 49.0 ±4.0 73.4 ±2.5 51.0 ±4.0 15.2 ±1.0 9.1 ±1.2 35.3 ±2.0 22.7 ±2.4
	ENS BERT-large	73.0 ±1.8 51.1 ±3.2 72.4 ±2.1 47.2 ±4.2 16.1 ±0.9 9.3 ±0.7 35.7 ±3.5 22.8 ±3.1
	ENS RoBERTa-base 72.1 ±4.7 51.0 ±4.1 74.2 ±2.6 52.5 ±4.7 17.5 ±0.7 11.4 ±0.8 25.6 ±1.0 13.6 ±1.2
	ENS RoBERTa-large 74.1 ±3.5 52.5 ±3.9 72.5 ±2.5 49.0 ±4.4 19.4 ±0.7 12.7 ±0.7 42.4 ±5.6 32.9 ±9.2
	SBERT+kM	84.5 ±0.1 86.2 ±0.1 86.0 ±1.4 80.0 ±3.1 39.3 ±0.7 23.5 ±1.2 63.1 ±1.8 45.5 ±3.7
	SB+F SGC +kM	85.8 ±2.8 85.6 ±7.4 85.6 ±1.0 78.5 ±2.7 41.8 ±0.5 25.2 ±1.0 65.6 ±0.5 49.0 ±0.6
	SB+F S²GC +kM	86.0 ±0.0 86.9 ±0.0 86.6 ±1.2 80.4 ±2.8 41.0 ±0.8 24.5 ±1.5 64.8 ±1.1 47.8 ±0.7
	SB+F APPNP +kM 86.2 ±0.0 87.0 ±0.0 85.8 ±1.0 78.9 ±2.7 41.6 ±0.7 24.9 ±1.5 65.1 ±1.6 48.5 ±1.0
	SB+F DGC +kM	86.9 ±0.0 87.7 ±0.0 85.4±1.0 78.4 ±2.2 41.8 ±0.7 24.8±1.7 65.6 ±0.5 49.3 ±0.4

Table 4 :

 4 Classification results in terms of F1 score on the eight data sets.

		20News	R8	AGNews BBCNews Classic3 Classic4 DBpedia Ohsumed
	BERT base	80.7	89.94	89.78	95.51	100.0	98.58	97.84	56.48
	RoBERTa base	85.48	89.42	88.06	96.73	99.16	96.47	98.22	58.11
	SBERT+LR	83.35	90.22	86.25	98.62	99.61	98.19	97.33	62.87
	SB+F APPNP +LR	87.54	90.9	87.9	99.06	99.75	98.36	97.14	67.6
	SB+F DGC +LR	87.11	90.08	87.59	98.19	99.61	98.52	97.38	67.09
	SB+F S²GC +LR	87.36	91.19	88.33	98.62	99.75	98.19	97.26	67.42
	SB+F SGC +LR	87.26	89.22	88.05	99.06	99.61	98.32	97.01	67.05

https://github.com/chakib401/ smoothing_sentence_embeddings


Early survival is highly variable and strongly influences observed population growth rates in most vertebrate populations. One of the major potential drivers of survival variation among juveniles is body mass. Heavy juveniles are better fed and have greater body reserves, and are thus assumed to survive better than light individuals. In spite of this, some studies have failed to detect an influence of body mass on offspring survival, questioning whether offspring body mass does indeed consistently influence juvenile survival, or whether this occurs in particular species/environments. Furthermore, the causes for variation in offspring mass are poorly understood, although maternal mass has often been reported to play a crucial role. To understand why offspring differ in body mass, and how this influences juvenile survival, we performed phylogenetically corrected meta-analyses of both the relationship between offspring body mass and offspring survival in birds and mammals and the relationship between maternal mass and offspring mass in mammals. We found strong support for an overall positive effect of offspring body mass on survival, with a more pronounced influence V.R.

Causes and consequences of variation in offspring body mass: meta-analyses in birds and mammals

Victor Ronget 1* , Jean-Michel Gaillard 1 , Tim Coulson 2 , Michael Garratt 3 , François Gueyffier 1 , Jean-Christophe Lega 1 and Jean-François Lemaître 1 in mammals than in birds. An increase of one standard deviation of body mass increased the odds of offspring survival by 71% in mammals and by 44% in birds. A cost of being too fat in birds in terms of flight performance might explain why body mass is a less reliable predictor of offspring survival in birds. We then looked for moderators explaining the among-study differences reported in the intensity of this relationship. Surprisingly, sex did not influence the intensity of the offspring mass-survival relationship and phylogeny only accounted for a small proportion of observed variation in the intensity of that relationship. Among the potential factors that might affect the relationship between mass and survival in juveniles, only environmental conditions was influent in mammals. Offspring survival was most strongly influenced by body mass in captive populations and wild populations in the absence of predation. We also found support for the expected positive effect of maternal mass on offspring mass in mammals (rpearson = 0.387). As body mass is a strong predictor of early survival, we expected heavier mothers to allocate more to their offspring, leading them to be heavier and so to have a higher survival. However, none of the potential factors we tested for variation in the maternal mass-offspring mass relationship had a detectable influence. Further studies should focus on linking these two relationships to determine whether a strong effect of offspring size on early survival is associated with a high correlation coefficient between maternal mass and offspring mass.

I. INTRODUCTION

Getting reliable estimates of demographic parameters including survival and reproduction is a major step in assessing population dynamics (Caswell, 2001). Individuals vary greatly in terms of lifespan and reproductive success, which lead them to differ strongly in their contribution to population dynamics. Life-history theory is built on the premise that individual traits that determine reproduction and survival throughout life are shaped by natural selection to maximize individual fitness [START_REF] Gadgil | Life historical consequences of natural selection[END_REF][START_REF] Stearns | The evolution of life histories[END_REF]. It is therefore of crucial importance to assess the relationship among individual traits, demographic parameters, and individual fitness (Cam et al., 2002).

Juvenile survival is an important fitness component because it determines whether or not an individual will reach maturity and therefore reproduce [START_REF] Lindström | Early development and fitness in birds and mammals[END_REF]. In long-lived species of mammals and birds the juvenile period is a particularly critical life stage because mortality risks are much higher than after sexual maturity. Since offspring survival often drives population dynamics of long-lived species [START_REF] Gaillard | Temporal variation in fitness components and population dynamics of large herbivores[END_REF][START_REF] Ozgul | Coupled dynamics of body mass and population growth in response to environmental change[END_REF], it is particularly important to understand the ecological and biological factors that will modulate this fitness component. Numerous studies have investigated the influence of phenotypic traits on offspring survival, with a particular emphasis on body mass (Magrath, 1991, Maness & Anderson, 2013). Generally, these studies have reported that body mass is a reliable predictor of offspring survival (e.g. [START_REF] Hamel | Individual quality, early-life conditions, and reproductive success in contrasted populations of large herbivores[END_REF]Mackas et al., 2010).

Body mass is known to be positively correlated with body fat, which represents the main component of body reserves in birds and mammals [START_REF] Garnett | Body size, its heritability and influence on juvenile survival among great tits[END_REF][START_REF] Labocha | Morphometric indices of body condition in birds: A review[END_REF][START_REF] Monteith | Life-history characteristics of mule deer: effects of nutrition in a variable environment[END_REF]Bennett et al., 2015) and allows large individuals to survive over periods of food shortage. Furthermore, since body mass and body size are generally closely correlated across individuals within a given population, body size also has a positive effect on offspring survival (e.g. McMahon et al., 2015). For instance, in temperate ecosystems, individuals with greater body size survive better than those with low body reserves over the winter (Ringsby, Saether & Solberg, 1998). In addition, energy demands for growth are high during the juvenile stage [START_REF] Parker | Nutrition integrates environmental responses of ungulates[END_REF] and when food availability is low, body reserves allow growth to continue [START_REF] Lee | Growth, weaning and maternal investment from a comparative perspective[END_REF]. However, some studies have failed to detect a positive relationship between offspring body mass and juvenile survival (e.g. [START_REF] Williams | Chick growth and survival in gentoo penguins (Pygoscelis papua): effect of hatching asynchrony and variation in food supply[END_REF][START_REF] Ylönen | Effect of birth and weaning mass on growth, survival and reproduction in the bank vole[END_REF]Reading et al., 2009). The most common explanation for these results involves quite constant and abundant food resources during the critical juvenile stage that lead body reserves, and consequently mass, to have less impact on survival [START_REF] Van Vuren | Differential investment in twin offspring by female pronghorns (Antilocapra americana)[END_REF]. Likewise, in environments where most juvenile mortality is caused by predation, high individual body mass might not confer a particularly strong survival advantage [START_REF] Warren | Mortality of lambs in free-ranging domestic sheep (Ovis aries) in northern Norway[END_REF]. Based on such contrasting results, it remains difficult to infer a general pattern for the effect of body mass on juvenile survival.

Among the factors that influence offspring body mass, maternal condition has been one of the most studied. Maternal body mass is indeed expected to account for a substantial proportion of the variation observed in offspring body mass [START_REF] Pomeroy | Consequences of maternal size for reproductive expenditure and pupping success of grey seals at North Rona, Scotland[END_REF]Hamel, Craine & Towne, 2012a) because heavy females can typically allocate more resources to their offspring during both pre-and postnatal stages (e.g. gestation and lactation in mammals), which leads to increased offspring mass and thereby offspring survival. Such relationships between maternal body mass and offspring body mass have been repeatedly documented in the literature (Clutton Brock et al., 1996;[START_REF] Monclús | Yellow-bellied marmots do not compensate for a late start: the role of maternal allocation in shaping life-history trajectories[END_REF]. However, some case studies failed to detect such relationships (Campbell & Slade, 1995;[START_REF] Wheatley | Influence of maternal mass and condition on energy transfer in Weddell seals[END_REF]Foster & Taggart, 2008). Common explanations for this inconsistency involve the offspring number-size trade-off [START_REF] Michener | Reproductive effort during gestation and lactation by Richardson's ground squirrels[END_REF]), which appears to be the rule among shortlived species that produce multiple offspring per reproductive attempt [START_REF] Smith | The Optimal balance between size and number of offspring[END_REF]. Moreover, females of long-lived species often trade current allocation to reproduction for allocation to their own future survival [START_REF] Tavecchia | Predictors of reproductive cost in female Soay sheep[END_REF][START_REF] Hamel | Fitness costs of reproduction depend on life speed: empirical evidence from mammalian populations[END_REF]. In harsh years, females of long-lived species are expected to put the emphasis on their own survival, which may produce costs in terms of losing their offspring or of producing offspring of reduced size [START_REF] Skogland | The effects of food and maternal conditions in fetal growth and size in wild reindeer[END_REF], Festa-Bianchet & Jorgenson, 1998). Although [START_REF] Lim | Heterogeneity in individual quality and reproductive trade-offs within species[END_REF] performed a pioneering meta-analysis to assess the direction of the relationship between mother and offspring body size and found support for an overall positive relationship, they included only a limited number of bird and mammal species (22 birds and eight mammals) and did not identify the factors driving the observed variation in the strength of that relationship.

To fill this knowledge gap, we review empirical evidence of the strength of the relationships both between offspring body mass and offspring survival and between offspring mass and maternal body mass from published data. We restricted our analysis to birds and mammals because most detailed studies of free-ranging populations have been performed in these two vertebrate classes (Clutton-Brock & Sheldon, 2010). We first performed two phylogenetically corrected meta-analyses (i.e. one for each relationship) to assess the direction and magnitude of these relationships. In a second step, we looked for biological factors that drive observed variation in each of the two relationships and could explain the conflicting results reported in literature.

We particularly focused on biological moderators that have previously been suggested to influence the relationships between mother and offspring mass, and juvenile survival.

Offspring sex was included as one of these because male offspring of dimorphic and polygynous species are more susceptible to harsh conditions than females (Clutton-Brock, Albon & Guinness, 1985). We thus expected that reserves and also body mass will have more influence on male than on female survival, which should ultimately lead to between-sex differences in the relationship between maternal mass and offspring mass. In addition, in polytocous species, the trade-off between offspring mass and offspring number should influence the relationship between offspring mass and maternal mass (Charnov & Ernest, 2006). Thus we accounted for variation in litter size in the analysis of each relationship.

Finally, we also tested for an influence of the species mating system because different mating systems lead to different patterns of maternal allocation [START_REF] Zeveloff | Parental investment and mating systems in mammals[END_REF] and thereby to expected differences in the relationship between offspring mass and maternal mass.

II. METHODS (1) Literature survey

We collected published papers by using the database of ISI Web of Science following a strict search protocol. The key words ("mass" or "weight" or "size") and ("survival" or "mortality") were used to identify studies investigating relationships between offspring survival and mass and the key words ("mass" or "weight") and ("mother" or "maternal" or "adult") and ("newborn" or "offspring" or "neonate") were used to identify studies testing for a relationship between maternal mass and offspring mass. The search was conducted in December 2015. We restricted the results to the topics "Ecology", "Zoology", "Ornithology" and "Evolutionary Biology". We deliberately used broad key words because much of the required information can be hidden within papers on different topics (e.g. [START_REF] Serra | Seasonal decline of offspring quality in the European starling Sturnus vulgaris: an immune challenge experiment[END_REF].

We identified 20,240 papers related to offspring mass and survival and 1,414 papers related to maternal mass and offspring mass. We applied a first selection procedure to this list based on the title and only retained papers dealing with mammalian or avian populations. Then, we read all the abstracts to check whether the relationships were explicitly reported in these papers. Finally, we checked the references cited in these articles for any relevant studies missed. A total of 103 papers on mammals and 133 papers on birds were retrieved for the relationship between offspring mass and survival (Fig. 1A). For the relationship between maternal mass and offspring mass we recovered 85 papers on mammals but only three papers on birds (Fig. 1B). We thus did not have enough data on birds to perform a meta-analysis for the relationship between maternal mass and offspring mass. This low amount of published data in birds is discussed later in Section IV.

(2) Data reported (a) Information collected for each case study For the relationship between offspring mass and survival and for the relationship between maternal mass and offspring mass we retained any relationship including mass or any indicator of mass such as structural size or body condition. When different measurements of mass were used in one paper, we extracted the strict measurement of mass. We did not consider pre-birth measurements such as egg or fetus mass. When the relationship was analysed at different ages (i.e. survival-offspring mass relationship at birth and at weaning), the earliest relationship was retained to avoid pseudo-replication due to repeated measures of the same individuals (Hurlbert, 1984). When the relationship was assessed independently for both sexes, we included sex-specific relationships in the analysis.

All information required for the identification of the paper (i.e. title, first author, year of publication, journal, location and species studied) was recorded. We also reported the timing of offspring measurement, the type of measurement and the data quality (see Section II.4b), which could potentially influence the results of the meta-analysis. We included these factors as moderators in the meta-analysis. We also recorded whether the relationship was assessed for both sexes separately, or for pooled sexes. Lastly we reported whether the focal population was captive or not and if the individuals were subjected to predation.

(b) Information collected for each species

To assess the potential influence of biological factors on the relationships identified from our meta-analysis, we searched in the literature for information about mating system and litter size (mammal) or brood size (bird) for each species included in our data set (see online Supporting Information, Appendix S1, S2 and S3 for all data used for the analysis).

(3) Extraction of effect sizes (a) Relationship between offspring mass and offspring survival This relationship was generally reported as a logistic function because survival follows a binomial distribution. The slope of the logistic regression was reported with its standard error. When the slope was not provided but the raw data or the logistic curve were graphically displayed in the paper, we extracted the data from the figure using WebPlotDigitizer (http://arohatgi.info/WebPlotDigitizer/) and then ran a logistic regression with the package betareg in R (version R.3.3.0, R Development Core Team 2015). In cases where the standard error was missing but the Wald statistics was reported, we used the Wald statistics to obtain the standard error. We calculated the Wald statistics as (θ -θ0)² / var(θ), which is to be compared to a χ² distribution with θ0 equal to 0. When only the slope of the relationship was reported, the standard error could still be estimated when both the mean and the standard deviation of the offspring body mass were provided. We thus obtained the standard error by first simulating the survival data for each individual body mass using the published logistic relationship and then re-running a logistic regression (see R code in Appendix S4). The relationship was sometimes presented with a quadratic term (e.g. [START_REF] Verboven | Seasonal variation in local recruitment of great tits: the importance of being early[END_REF] and in such situations, when the raw data were available in the paper, we ran a new logistic regression without the quadratic term. This relationship was also sometimes presented as a linear relationship (e.g. [START_REF] Garnett | Body size, its heritability and influence on juvenile survival among great tits[END_REF]. In such cases, we converted the slope of the linear regression to a logistic slope following the procedure given in [START_REF] Hamel | Statistical evaluation of parameters estimating autocorrelation and individual heterogeneity in longitudinal studies[END_REF]. The linear relationship corresponds to a portion of a logistic that is quasi-linear, and multiplying the linear slope by a factor of 4 allows the slope of a logistic regression to be obtained. Occasionally, especially in old papers, the only results reported were the distributions of body mass with the mean and the standard deviation of the mass of dead and alive individuals. In such cases, we assumed that the masses of the dead individuals and of the live individuals were normally distributed and we simulated two normal distributions (one for each group) and ran a logistic regression. We replicated the procedure 10,000 times and retained the mean slope and standard error of this slope (see R code in Appendix S4).

When performing a meta-analysis, standardized coefficients are required to make results from the compiled studies comparable [START_REF] Nakagawa | Methodological issues and advances in biological meta-analysis[END_REF]. Only the measurement of mass was standardized in our data set because the slopes were obtained from very different species that have markedly different distributions of offspring mass. We did not standardize survival because survival is bounded between and 0 and 1 across all case studies and species. To standardize body mass, the standard deviation of mass was required. When not available, the range of mass was used to infer the standard deviation. We assumed that mass was normally distributed, so that the range corresponds to 4 standard deviations (because in a normal distribution 95% of the values belong to the interval encompassing approximatively two standard deviations). Semi-standard slopes were calculated by multiplying the slope with the standard error of the mass (Menard, 2011), the standard error being calculated in the same way. The effect sizes were reported in terms of odds ratios to facilitate interpretation [START_REF] Lipsey | Practical meta-analysis[END_REF]. The odds ratio is calculated as the exponential of the semi-standardized slope. When mass increases by one standard deviation, the odds of survival (i.e. the ratio between the probability to survive and the probability to die) is multiplied by one semi-standardized odds ratio. Thus, a coefficient greater than 1 corresponds to a positive effect of mass on survival. To assess a potential impact of the data extraction on the results for each effect size we scored the data quality as high when all the required information was reported in the paper and as low when we needed to report the data from the figure or to run simulations to obtain the information (see Tables 1, 2 and3

(b) Relationship between maternal mass and offspring mass

For this relationship, the coefficients extracted were Pearson correlation coefficients or partial correlation coefficients. Correlation coefficients can also be inferred from χ², t, and F statistics using the formulae provided in [START_REF] Lipsey | Practical meta-analysis[END_REF]. When only raw data were provided we extracted them with WebPlotDigitizer and ran the R function cor.test on the data.

For the meta-analysis, all these correlation coefficients were converted into a Fisher Zr, which is an unbounded measure of effect size for correlation coefficients [START_REF] Lipsey | Practical meta-analysis[END_REF]. This transformation allows the direct calculation of the standard error when the sample size is known. Following Cohen (1988), we considered that correlation coefficients of 0.1, 0.3, and 0.5 represent low, moderate, and strong effects, respectively. We reported the quality of the effect size as described in Section II.3a to check whether our transformation had any impact on the results of the meta-analysis.

(4) Statistical analysis (a) General model

A multi-level meta-analysis was performed because the effect sizes are not independent from each other. Correlation between the different effect sizes can arise when multiple effect sizes are recorded in one population or on the same species or in studies analysed by the same author. Moreover, the shared history among different species makes them non-independent [START_REF] Harvey | The Comparative Method in Evolutionary Biology[END_REF]. For such analyses, linear mixed models are recommended [START_REF] Nakagawa | Methodological issues and advances in biological meta-analysis[END_REF]. We used the function MCMCglmm of the package MCMCglmm [START_REF] Hadfield | MCMC methods for multi-respoinse generalized linear mixed models: the MCMCglmm R package[END_REF][START_REF] Hadfield | General quantitative genetic methods for comparative biology: phylogenies, taxonomies and multi-trait models for continuous and categorical characters[END_REF] to perform our analyses. Bayesian hierarchical models are especially recommended to handle phylogenetic meta-analyses in which several effect sizes are reported for the same species (e.g. [START_REF] Nakagawa | Methodological issues and advances in biological meta-analysis[END_REF].

In order to assess phylogenic relatedness among the different species, we used phylogenetic trees for avian (Jetz et al., 2012) and mammalian (Bininda-Edmonds et al., 2007) species. These phylogenies were used in the meta-analyses to correct for nonindependence between species-specific data points.

For each meta-analysis, linear mixed models were fitted with the effect size as the dependent variable and the error variance implemented for each effect size (with the mev argument in the function MCMCglmm). The covariance matrix among the species was extracted from the phylogeny. The phylogeny, species, population and first author were included in the model as random factors. We included another random effect as species independently of phylogeny because individuals from the same species can share characteristics that are independent of phylogeny (e.g. lifestyle). In the absence of clear a priori information, we used a non-informative prior (Inverse Wishart prior with ν = 0.02 and V = 1). To assess whether the prior impacted the results, we re-ran the analysis using a new parameter expanded prior (ν = 1, V = 1, alpha.mu = 0, alpha.V = 1000). This sensitivity analysis did not uncover any difference between the two models, meaning that the results we obtained were not dependent on the prior used. Each model was run with 2,000,000 iterations.

We ran several models and assessed convergence with the Gelmann diagnostic [START_REF] Gelmann | Inference from iterative simulation using multiple sequences[END_REF]) by using the Gelmann.diag function in R. This diagnostic detects statistically significant differences in the MCMC chains that could potentially occur between two models when these models do not converge. However, we did not detect any difference among models.

For each model, the mean of the posterior distribution was reported, which corresponds to the meta-analysis mean. We also reported the 95% credibility interval of the highest posterior density distribution (HPDI). The mean was considered as statistically significant when 0 (for Zr) or 1 (for the odds ratio) was not included in the credibility interval.

To quantify the importance of the different random effects, I² statistics were calculated for each random effect [START_REF] Nakagawa | Methodological issues and advances in biological meta-analysis[END_REF]. I² represents the percentage of the total variance that is accounted for by the random effect. Values of 25%, 50%, and 75% are classically interpreted as a low, moderate, and high percentage of variance explained, respectively [START_REF] Higgins | Measuring inconsistency in meta-analyses[END_REF]. The I² values are presented with their 95% highest posterior density credibility interval, this interval being bounded between 0 and 1.

(b) Models with moderators

To test the effect of moderators we included them with fixed effects in new models. In addition to the biological variables presented in the introduction we included other studyspecific variables that could influence the intensity of the relationships. We implemented the following moderators.

(1) The timing of the measurement, which is the life stage that includes the time elapsed between the offspring mass measurement and the record of offspring survivorship.

The timing of the measurement was included in models as a three-level factor (Early, Late and Total). 'Early' corresponds to pre-weaning (mammals) or pre-fledging (birds) survival. Mass is then recorded at or right after birth (mammals) or hatching (birds). 'Late' corresponds to post-weaning (mammals) or post-fledging (birds) survival before recruitment. Mass is then recorded at or close to weaning (mammals) or fledging (birds). 'Total' corresponds to a survival estimate encompassing both preand post-weaning (mammals) or fledging (birds). Mass is then recorded at or right after birth (mammals) or hatching (birds). We also used the same kind of metrics for the relationship between offspring mass and maternal mass. We distinguished between pre-and post-weaning (mammals) or fledging (birds) periods when possible because the weaning (mammals) and fledging (birds) periods are usually the most critical life stages (Clutton-Brock, 1991). In particular, at weaning, most mammals no longer rely on parental care for survival.

(2) The type of mass measurement was fitted as a two-level factor (Mass versus Condition index). We included this moderator to assess whether the use of different measures impacted our results. In some cases, condition index and mass can be related to body reserves with different intensities (e.g. [START_REF] Wilder | Moving beyond body condition indices as an estimate of fitness in ecological and evolutionary studies[END_REF].

(3) The fact that the data were obtained from wild or captive conditions was recorded as a two-level factor (Wild versus Captive). We considered a population as being captive when the individuals were kept in an enclosure and artificially fed. Captive animals do not display the same mortality patterns as free-ranging animals (e.g. [START_REF] Lemaître | Comparing free-ranging and captive populations reveals intra-specific variation in aging rates in large herbivores[END_REF]Tidière et al., 2016). In particular, captive individuals have access to veterinary care that can markedly influence the magnitude of the offspring mass-survival relationship. As all bird populations included in the meta-analysis were free-living, we only tested an effect of captive versus wild conditions in mammals.

(4) The occurrence of predation in the studied population was implemented as a two-level factor (Predation versus No predation) for the analysis of the relationship between offspring survival and body mass. We first considered the information provided in the paper about the occurrence of predation. When no information about the causes of mortality was reported, we searched for other papers about the same population to find out whether the focal population was subjected to predation. We expected that predation should decrease the effect of body mass on offspring survival because predators generally prey upon juveniles independently from their mass [START_REF] Griffin | Neonatal mortality of elk driven by climate, predator phenology and predator community composition[END_REF]Keech et al., 2011). This moderator was only tested for mammalian populations because all of the bird populations included in our data set were subjected to predation. We did not report any information about hunting in populations because juveniles are typically not hunted.

(5) Offspring sex was included as a three-level factor (Female, Male or Combined). 'Combined' corresponds to studies in which individuals from both sexes were pooled within the same relationship. 'Male' and 'Female' correspond to studies in which sexspecific relationships were provided. We looked for potential sex differences in the effect sizes of the relationships.

(6) The influence of species-specific mating system was tested differently in mammals and birds. As only two mating systems occurred in our set of mammalian species, we included this moderator as a two-level factor (Polygynous versus Promiscuous). Our bird species were principally socially monogamous, although a high rate of extra-pair paternities occurred in several species [START_REF] Garamszegi | Testosterone, testes size, and mating success in birds: a comparative study[END_REF]. As the degree of monogamy can impact the amount of parental care and thereby influence the offspring mass-survival relationship (Jašarević et al., 2013) we distinguished between strict monogamy and other mating systems. We defined species as being strictly monogamous when the rate of extra-pair paternities was lower than 5%. As the magnitude of between-sex differences might differ in relation to mating systems, we included a test of the interaction between mating system and sex in our analyses.

(7) Litter size was implemented as a two-level factor in mammals (Monotocous versus Polytocous). Monotocous species have a mean litter size of one, whereas polytocous species produce more than one offspring per litter. Brood size in birds was measured as the average clutch size for each species (i.e. a continuous variable). Litter (mammals) or clutch (birds) size could influence the offspring mass-survival relationship because of the expected offspring size-number trade-off [START_REF] Smith | The Optimal balance between size and number of offspring[END_REF]. We also tested for the interaction between litter size and mating system for mammals because siblings in species displaying a promiscuous mating system are expected to face higher sibling competition than siblings in species with other mating systems [START_REF] Forstmeier | Female extrapair mating: adaptation or genetic constraint?[END_REF][START_REF] Garratt | Female promiscuity and maternally dependent offspring growth rates in mammals[END_REF].

(8) Data quality was implemented as a two-level factor (High quality versus Low quality).

Data were considered as high quality when all data required for the analysis were explicitly reported. Low-quality data corresponded to case studies for which the required data were extracted from graphs or obtained from simulations. We thus tested whether the data-extraction procedure had any detectable impact on the results.

To assess the impact of these different moderators on the relationships of interest, we reported the mean difference between the groups with the 95% highest posterior density interval (the odds ratios were log-transformed to obtain a meaningful mean difference between groups). The mean difference was considered as statistically significant when 0 did not fall within the credibility interval.

(5) Publication bias

If studies with no detectable effects are less likely to be published (Rosenthal, 1979), the meta-analysis performed from published information would lead to an overestimate of the true effect. To test whether such a publication bias was present in our data, funnel plots were built. The standard diagram plots the precision of the study (measured as the inverse of the standard error) against the mean of the study (Egger et al., 1997). The closer the mean is to the meta-analysis mean, the greater the precision. In the absence of any bias the diagram should be perfectly symmetrical around the mean. To test the symmetry of the diagram a linear regression of the means of each study as a function of their precision is performed. This test is known as the Egger regression (Egger et al., 1997). However, the means are not independent from each other, leading a key assumption of linear regression to be violated.

The only values that were independent between the different effect sizes were the residuals of the meta-analysis [START_REF] Nakagawa | Methodological issues and advances in biological meta-analysis[END_REF], which correspond to the variance that is not explained by the different random factors. The residuals should be symmetrically distributed around 0. A linear regression of residuals on the precision of the study was performed. A publication bias occurs when the intercept of the regression is statistically different from 0. To assess the influence of publication bias, the trim and fill method of the package metafor [START_REF] Viechtbauer | Conducting meta-analyses in R with the metafor Package[END_REF] was used. This method provides an estimate of the number of studies that are absent on one side of the funnel plot and adjusts the meta-analysis mean accordingly. It should not be interpreted as an exact correcting factor of the publication bias but rather provides an assessment of the magnitude of the publication bias.

III. RESULTS

(1) Relationship between offspring mass and offspring survival (a) Data set

In mammals, we extracted 75 effect sizes from 60 published papers. These effect sizes came from 33 different species. Overall, Carnivora (nine species), Artiodactyla (15 species) and Rodentia (six species) were the most represented mammalian orders (Table 1; Fig. 2A).

In birds, we extracted 86 effect sizes from 58 published studies. These effect sizes corresponded to 56 different species, mostly Passeriformes (25 species), Charadriiformes (11 species) and Anseriformes (seven species) (Table 2; Fig. 2B).

(b) Results from general meta-analyses

In mammals, offspring mass positively influenced offspring survival with a metaanalysis mean of 1.82. This effect was statistically significant because the highest posterior density interval of the odds ratio did not overlap 1 [HPDI = (1.37; 2.41)] (Fig. 3A). In birds, the same positive effect of mass occurred for offspring survival (meta-analysis mean = 1.48).

This effect was also statistically significant [HPDI = (1.26; 1.72)] (Fig. 3B).

The heterogeneity analysis in mammals indicated that each random effect (the effect of phylogeny, of species independently of phylogeny, of population and of first author) included in our meta-analysis only accounted for a weak but similar proportion of heterogeneity among studies, with an I² near to 25% for each effect (Table 4). In birds, results

were similar with all I² near 25%, which indicates that each random effect included in our meta-analysis accounted for an equal and weak part of the heterogeneity among studies (Table 4). The credibility intervals were large for all the values, preventing us from relying on the exact I² value.

(c) Assessing the effects of moderators on the strength of the offspring mass-survival relationship

The age at which mass was measured, the type of mass measurement, data quality and sex did not have any detectable effect on the relationship between offspring mass and survival in either birds or mammals (Fig. 3, Table 5). Clutch/litter size did not influence the slope of the relationship either in birds [linear regression slope = -0.010, HPDI = (-0.040; 0.021)] or in mammals (Table 5). The meta-analysis mean was higher in captive than in wild mammals, and in mammal populations with no predation than in populations subjected to predation. We did not detect any influence of the mating system in birds but promiscuous mammals had a higher meta-analysis mean than polygynous ones, this difference being statistically significant (Table 5). The mating system was not independent from environmental conditions, since 97% of studies on polygynous species lived in the wild while 64% of studies on promiscuous species lived in captivity. This prevented us from reaching a firm conclusion on whether mating system influences the offspring mass-survival relationship, because polygynous species in this data set were virtually all from the wild, and our analysis indicated that living in the wild weakens the relationship between offspring mass and survival (see Section IV).

Including an interaction between mating system and sex did not reveal any detectable effect either in mammals [meanmalevsfemale promiscuous = 0.276, HPDI = (-0.394; 0.973); meanmalevsfemale polygynous = -0.142, HPDI = (-0.535; 0.222)] or in birds [meanmalevsfemale monogamous = 0.013, HPDI = (-0.232; 0.251); meanmalevsfemale othermating = -0.100, HPDI = (-0.264; 0.086)].

Likewise, we did not find any detectable interaction between litter size and mating system in mammals [meanmonotocousvspolytocous promiscuous = 0.181, HPDI = (-0.735; 1.067); meanmonotocousvspolytocous polygynous = 0.085, HPDI = (-0.307; 0.454)].

(d) Publication bias

The intercept of the Egger regression was statistically different from zero in mammals [intercept = 0.077, HPDI = (0.004, 0.152)]. The publication bias diagram was not symmetrical (Fig. 4A), indicating that a publication bias towards positive effects was likely. The trim and fill method indicated a lack of 18 studies on the left side of the funnel plot. The meta-analytic mean should thus be adjusted by -0.062, which results in a value of 1.71. In birds the intercept of the Egger regression also differed from 0 on statistical grounds [intercept = 0.156, HPDI = (0.065; 0.246)] (Fig. 4B). The trim and fill method indicated a lack of 15 studies on the left side of the funnel plot. The meta-analytic mean should thus be adjusted by -0.027, which results in a value of 1.44. Therefore, the slight publication bias we detected did not influence our conclusions.

(2) Relationship between maternal mass and offspring mass (a) Data set For this meta-analysis, we extracted 96 effect sizes from 60 published papers. We collected effect sizes for 38 different mammalian species with Carnivora (12 species), Rodentia (11 species) and Artiodactyla (10 species) as the most represented mammalian orders (Fig. 5; Table 3). This meta-analysis was performed in mammals only (see Section II.1).

(b) Results from the general meta-analysis

A positive relationship occurred between offspring and maternal mass (mean metaanalysis = 0.408, which is equivalent to a Pearson correlation coefficient of 0.387). This effect was statistically significant because the highest posterior density interval did not overlap 0 [HPDI = (0.223; 0.580)] (Fig. 6).

The heterogeneity analysis showed that all the random effects included in our metaanalysis contributed equally but weakly to the overall heterogeneity across studies, with an I² less than 25% (Table 6).

(c) Assessing the effects of moderators on the strength of the maternal mass-offspring mass relationship

The age at which offspring mass was measured, the type of mass measurement, sex, whether animals were captive or not, litter size, data quality, and mating system did not have any detectable effect on the magnitude of the relationship (Fig. 6; Table 7). Likewise, we did not detect any effect of interactions both between mating system and sex [meanmalevsfemale promiscuous = -0.064, HPDI = (-0.343; 0.247); meanmalevsfemale polygynous = 0.016, HPDI = (-0.010; 0.148)] and between litter size and mating system [meanmonotocousvspolytocous promiscuous = -0.050, HPDI = (-0.596; 0.523); meanmonotocousvspolytocous polygynous = 0.044, HPDI = (-0.151; 0.248)].

(d) Publication bias

The intercept of the Egger regression was almost statistically different from zero [intercept = 0.037, HPDI = (-0.001; 0.075)]. A direct inspection of the diagram suggests that some studies might be lacking on the left side since the funnel plot is not symmetrical (Fig. 7).

This indicates that a small publication bias might exist. However, the results of the Egger regression indicate that our results are robust to such a small bias.

IV. DISCUSSION

We assessed the sign and the magnitude of the relationships between offspring mass and offspring survival in mammals and birds and between maternal and offspring mass in mammals. The meta-analyses we performed provided strong support for positive relationships in all cases.

In mammals, on average, when offspring mass increases by 1 standard deviation of the offspring body mass distribution in the population, the odds of offspring survival increase by 71%. We also highlighted the existence of a positive relationship between offspring mass and survival in birds. On average, when offspring mass increases by one standard deviation of the early mass distribution the odds of offspring survival increase by 44 %. Overall, these positive relationships support our expectation that offspring mass is a reliable proxy of individual survival in birds and mammals (e.g. [START_REF] Hamel | Individual quality, early-life conditions, and reproductive success in contrasted populations of large herbivores[END_REF]. The magnitude of the relationship was slightly weaker in birds. This difference might be due to the fact that birds and mammals are not subject to the same constraints. As 92% of our effect sizes were measured on postfledging survival, flight constraints are likely involved. The advantages of a greater body mass in birds might be not so strong because a high body mass increases the wing loading (Chandler & Mulvihill, 1992) and affects birds in terms of flying performance [START_REF] Norberg | How a long tail and changes in mass and wing shape affect the cost for flight in animals[END_REF]. There is an extensive literature about the cost of being too fat, especially when individuals need high flight performance to escape predators (e.g. [START_REF] Gosler | Predation risk and the cost of being fat[END_REF]Bonter et al., 2013;Rogers, 2015). In birds, there is clearly a trade-off between the advantage of being fat to avoid starvation and its costs in terms of predation.

Alternatively, a methodological issue might account for the weaker influence of mass on juvenile survival in birds compared to mammals. In bird studies, it is especially difficult to distinguish between death and emigration from the study site [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: a unified approach with case studies[END_REF][START_REF] Lebreton | The statistical analysis of survival in animal populations[END_REF][START_REF] Gilroy | A new approach to the 'apparent survival' problem: estimating true survival rates from mark-recapture studies[END_REF]. When the probability of emigration increases with body mass, the relationship between resighting rate (often used as a proxy of survival) and mass is weaker than the relationship between true survival and mass [START_REF] Stoleson | Hatching asynchrony, brood reduction, and food limitation in a neotropical parrot[END_REF]Barbraud, Johnson & Bertault, 2003).

From the heterogeneity analysis, we found weak effects of phylogeny, of species independent of phylogeny, and of population. As neither among-species nor amongpopulation differences accounted for a substantial proportion of the variation observed in the strength of the relationship between offspring survival and body mass, we can generalize our results to all mammals and birds. The absence of any detectable random effect to explain part of the heterogeneity highlights the importance of environmental variation on shaping these relationships. Juveniles from the same species can die from different causes and, even within the same population, juveniles born in different cohorts do not face the same environment (e.g. Keech et al., 2011;[START_REF] Garratt | High juvenile mortality is associated with sex-specific adult survival and lifespan in wild roe deer[END_REF]. In both birds and mammals, data quality did not influence our finding because we did not detect any difference between the mean of lowquality data and that of high-quality data. While a publication bias was detected in birds and to a lesser extent in mammals, it only involved a negligible decrease of the overall effect size, which left our conclusions unchanged.

To find potential major drivers explaining the variation in slopes reported in the literature for the offspring mass-survival relationship, we tested potential effects of the timing of the measurement. We examined three periods, including the period with high parental care from birth to weaning/fledging, the period of juvenile independence from weaning/fledging to adult stage, and the overall juvenile survival from birth to adult stage. A general objective behind this analysis was to assess in which period juvenile survival is most dependent on body mass. We did not identify a critical period likely because such effects could be masked by dominant mortality causes like predation, which is often less body-mass dependent than other causes of mortality such as starvation [START_REF] Monteith | Life-history characteristics of mule deer: effects of nutrition in a variable environment[END_REF]. A negative effect of predation on the strength of the offspring mass-survival relationship is confirmed by our findings in mammals that offspring survival in populations subjected to predation is less closely associated with body mass. However, the effective predation rate might strongly influence the strength of condition-dependence, which is expected to peak at some intermediate value of predation rate. Unfortunately, predation rates for the mammalian populations considered in our meta-analysis were not provided and it was thus impossible to assess accurately how predation affects the mass-survival relationship. It is also noteworthy that absolute body mass as analysed here might not reflect condition-dependent mortality through predation. Indeed, if we assume the existence of a limited mass range over which predators are able to prey upon juveniles, all juveniles in a population will be susceptible to predation initially, but the duration of the vulnerability period will be much lower for fastgrowing juveniles. In such cases, which encompass most ungulates (Byers, 1997), conditiondependent mortality is weak when using absolute body mass but could be much stronger when using individual growth rate instead of mass. In birds, several studies have reported that the critical period in terms of survival occurs just after fledging because the newly independent juveniles have little experience in foraging and so have to rely on their body reserves, which could be expected to strengthen the relationship between mass and survival (e.g. [START_REF] Sullivan | Predation and starvation: age-specific mortality in juvenile juncos (Junco phaenotus)[END_REF][START_REF] Stienen | Variation in growth in Sandwich tern chicks Sterna sandvicensis and the consequences for pre-and post-fledging mortality[END_REF]. However, fledging in birds also corresponds to a period when other causes of mortality occur, such as predation, likely explaining why late survival is not strongly associated with condition (Davies & Restani, 2006). The relationship between offspring survival and offspring body mass is driven by two parameters: the proportion of total mortality that is condition-independent or weakly condition-dependent and the strength of the relationship for each type of condition-dependent mortality. Condition-dependent mortality is mainly caused by starvation in relation to the depletion of body reserves of the juveniles [START_REF] Williams | Chick growth and survival in gentoo penguins (Pygoscelis papua): effect of hatching asynchrony and variation in food supply[END_REF]. As we compiled studies over a large range of environmental conditions and mortality causes, the absence of any influence of the juvenile period studied is not so surprising. Surprisingly, survival of captive mammals was more dependent on body mass than that of wild mammals. Wild animals have to face a much larger range of mortality factors, such as predation, which is often a major cause of offspring mortality (e.g. Linnell, Aanes & Andersen, 1995 in ungulates). Although accurate estimates of predation rates in the wild are generally lacking, it seems likely that predation, which is likely to be the highest during a limited time window of the juvenile stage, is only weakly related to absolute juvenile body mass. On the contrary, juveniles in captive populations are not subjected to predation and mostly die from infectious diseases or starvation, which can occur over the entire juvenile stage and are highly associated with absolute body mass [START_REF] Yapi | Factors associated with causes of preweaning lamb mortality[END_REF]Mandal et al., 2007). To assess the offspring mass-survival relationship in multiple case studies, different measures of mass were included. The most commonly used metric other than mass was body condition (i.e. mass corrected for size; Schultte-Hostedde et al., 2005).

Such heterogeneity in mass measurements could have led to an increase in variance across studies. However, a relatively low number of studies based on body condition were included in our analyses (two out of 75 for mammals and eight out of 86 for birds). Using other phenotypic traits to assess condition, such as growth rate, would improve our understanding of condition-dependent juvenile mortality.

We did not find any effect of sex on the magnitude of the relationship in mammals or birds. In particular, we did not find any evidence for disproportionately larger survival or mass advantage of increasing offspring mass in males than in females during early stages of life even when we accounted for the potential confounding effect of mating systems.

However, these results do not necessarily contradict the Trivers-Willard Hypothesis [START_REF] Trivers | Natural selection of parental ability to vary the sex ratio of offspring[END_REF] because we only looked at the early stages of life, whereas, as recently demonstrated, sex-specific reproductive value across the whole life course has to be considered to predict reliably a selective pressure for sex-biased allocation, even in the most sexually dimorphic and polygynous species (Schindler et al., 2015). Among the speciesspecific reproductive life-history traits, we considered only the mating system in mammals, which had a detectable influence on the offspring mass advantage. Offspring survival was more strongly mass-dependent in promiscuous species than in polygynous species. However, as the mating system had a confounding effect with environmental conditions, we cannot firmly conclude which of these factors generated the observed relationship. Furthermore, we did not find a higher effect of offspring body mass in polytocous and promiscuous mammal species for which we expected high sibling competition due to the existence of multi-paternity within litters.

In mammals, maternal mass was positively correlated to offspring mass with a mean correlation coefficient of 0.387, which corresponds to a moderate effect (sensu Cohen, 1988).

This finding matches the expectation that heavier mothers in a given population allocate more to their offspring than lighter ones, by allowing offspring to reach higher body mass and thereby higher survival. Interestingly, this finding supports recent results reported by [START_REF] Lim | Heterogeneity in individual quality and reproductive trade-offs within species[END_REF] who found a correlation coefficient of 0.414 between maternal size and offspring size for a wider set of taxonomic groups. The generally strong size-mass relationship explains the consistency of the results across studies (e.g. Dahle, Zedrosser & Swenson, 2006).

Both the species and the population random effects only accounted for a weak proportion of observed heterogeneity in our meta-analysis, which indicates that the positive effect we highlighted is consistent across mammalian species. As we included a large diversity of mammals, we can safely generalize our findings to the entire class of mammals.

The type of data used did not influence the results and the publication bias we detected had only a very weak effect on the final result. We were not able to perform this analysis for birds because of insufficient data. In birds more effort has been allocated to studying the relationship between maternal mass and egg mass, which is likely to be positive [START_REF] Wiggins | Sources of variation in egg mass of tree swallows Tachycineta bicolor[END_REF]Budden & Beissinger, 2005). Egg mass also relates to neonatal mass in birds (Krist, 2011), which leads us to expect that the relationship we identified in mammals also holds in birds. The few studies that assessed the maternal mass-offspring mass relationship in birds supported the existence of a positive relationship between maternal mass and offspring mass (Blums, Clark & Mednis, 2002;[START_REF] Parker | Maternal condition, reproductive investment, and offspring sex ratio in captive red junglefowl (Gallus gallus)[END_REF][START_REF] Newbrey | Growth of yellow-headed blackbird Xanthocephalus xanthocephalus nestlings in relation to maternal body condition, egg mass, and yolk carotenoids concentrations[END_REF].

We checked whether the timing of the offspring measurement could impact the magnitude of the mother-offspring mass relationship. Measuring offspring before weaning or after weaning led to similar results. Maternal mass thus provides a reliable predictor of both offspring birth mass and weaning mass in mammals. This result is not surprising because weaning mass is highly related to birth mass in mammals, with weaning mass being about four times the birth mass in pinnipeds, primates, and ungulates [START_REF] Lee | Growth, weaning and maternal investment from a comparative perspective[END_REF].

As in the analysis of the offspring mass-survival relationship, the use of different types of measurement did not have any impact on this meta-analysis. Likewise, wild and captive mammalian females allocate to their offspring with the same intensity at a given size. This is quite surprising when considering that body mass is more closely related to offspring survival in captive than in wild populations. However, we expect that females should increase their offspring body mass relative to their own mass only if an increase of the offspring body mass can give a sufficient increase in offspring survival compared to lighter ones. In captivity offspring body mass is more closely related to survival than in the wild but average offspring survival is typically higher in captivity than in the wild [START_REF] Littleton | Fifty years of chimpanzee demography at Taronga Park Zoo[END_REF]. Because offspring survival is already high in captivity, any increase in offspring body mass might not provide additional survival benefits.

The absence of any sex difference on the maternal-offspring mass relationship was an unexpected result, which indicates that mothers allocate the same relative amount of energy to male and female offspring irrespective of their body mass. Similar results were found in birds with no sex-biased allocation to egg size (Rutkowska, Dubiec & Nakagawa, 2014). In polygynous species the disproportionate mass or size advantage of offspring males is expected to be higher than in promiscuous species (Clutton-Brock, 1991) but we did not find any interaction between offspring sex and mating system. However, a similar correlation coefficient does not imply that there is no differential allocation between sexes. For a given mass, mothers can produce heavier male than female offspring. We did not detect any difference between the correlation coefficients but the intercept of the relationship was generally higher in males, indicating that mothers allocate more to male than to female offspring (e.g. Foster & Taggart, 2008). Furthermore, the mother can also allocate more toward males by biasing offspring sex ratio instead of increasing male mass, explaining why in some cases maternal mass can be related to offspring sex ratio (Arnbom, Fedak & Boyd, 1997). In addition, as recently pointed out by Schindler et al. (2015), the full sex-specific reproductive value has to be considered before stating that there are adaptive sex differences in maternal care. In 11 of our 17 studies that tested such differences, offspring mass was measured at birth, meaning that all the maternal allocation after birth was not accounted for.

Interestingly we did not find any difference in female allocation to offspring mass between monotocous and polytocous mammals. Mammals that produce multiple offspring can modify maternal allocation via two pathways: the offspring mass or the offspring number. We thus expected females of polytocous mammals to allocate less to offspring mass than females of monotocous species. However, in most cases, the expected offspring size-number trade-off does not show up among females within a population and both the mean mass of offspring and litter size increase with maternal mass (reviewed by [START_REF] Lim | Heterogeneity in individual quality and reproductive trade-offs within species[END_REF]. The mating system does not seem to impact this relationship. This is not so surprising because the difference between promiscuous and polygynous mating systems is only expected to impact paternal allocation (Adrian et al., 2005). As the degree of paternity certainty is lower in promiscuous species than in polygynous species, promiscuous fathers should allocate less than polygynous fathers [START_REF] Wright | Experimentally induced sex differences in parental care: an effect of certainty of paternity?[END_REF], whereas such differences are not expected for maternal allocation.

V. CONCLUSIONS

(1) Using meta-analyses we provide strong evidence of a positive relationship between offspring mass and offspring survival in birds and mammals. Our main finding shows the importance of considering body mass when analysing variation in early survival. Offspring mass offers a reliable indicator of offspring survival in both birds and mammals. However, the magnitude of the relationship was weaker for birds, likely because of flight constraints.

(2) We did not identify biological drivers that explained the differences we observed in the magnitude of the offspring mass-survival relationship across studies. We propose that this is because the offspring mass-survival relationship is highly dependent on the mortality causes in the focal populations. When most individuals die from weakly condition-dependent factors such as predation, a low magnitude of the relationship is expected, whereas when conditiondependent factors such as starvation mostly cause mortality, a higher magnitude of the relationship is expected.

(3) Offspring body mass, which drives individual differences in survival among offspring, is positively correlated with maternal body mass in mammals. This correlation was not quantitatively tested in birds due to a lack of data. However, from the limited information collected so far, there is support for a positive relationship. Further work, when sufficient data are available, should assess the correlation coefficient in birds for comparison with the coefficient obtained here for mammals. Because offspring survival is less related to offspring mass in birds, we expect a smaller coefficient of correlation in birds than in mammals.

(4) We did not identify any major driver that could explain the observed variability in the relationship between maternal mass and offspring mass. As we found large variation in condition-dependent survival in mammals in relation to variation in environmental conditions, we expected also to find large variation in the relationship between offspring and maternal mass. The link between the two relationships studied here is not clear and is worth further investigation.

Figure legends

  Key words: body size, individual heterogeneity, early survival, maternal size, Relationship between offspring mass and offspring survival……………………..20 (a) Data set (b) Results from the general meta-analysis (c) Assessing the effects of moderators on the strength of the offspring masssurvival relationship (d) Publication bias (2) Relationship between maternal mass and offspring mass………………………...22 (a) Data set (b) Results from the general meta-analysis (c) Assessing the effects of moderators on the strength of the maternal mass-

  for detailed information on the extraction procedure of data and on the quality assessment of each paper).

Fig. 1 .

 1 Fig. 1. PRISMA flow diagram [search procedure according to the PRISMA statement Liberati

Fig. 2 .

 2 Fig. 2. Phylogenies of (A) mammal (from Bininda-Edmonds et al., 2007) and (B) bird (from

Fig. 3 .

 3 Fig. 3. Meta-analysis means of each moderator (see Section II.4b) for (A) mammals and (B)

Fig. 4 .

 4 Fig. 4. Funnel plots of the different effect sizes in (A) mammals and (B) birds for the

Fig. 5 .

 5 Fig. 5. Phylogeny of mammal species included in the analysis of the relationship between

Fig. 6 .

 6 Fig. 6. Meta-analysis means of each moderator (see Section II.4b) in mammals for the

Fig. 7 .

 7 Fig. 7. Funnel plots of the different effect sizes in mammals for the relationship between

Figure 7

 7 Figure 7

Table 3 .

 3 Summary of the statistics and potential driving factors compiled in the meta-analysis for the relationship between offspring mass and maternal mass in mammals. Pearson correlation coefficient and sample size are reported. The extraction procedure is reported in parentheses as 1 if the statistic is directly calculated in the study, and 2 if data are reported from a figure. See Section II.3b for more information on the extraction procedure and Section II.4b for explanation of Mass or other relationships and Offspring age categories.

		Cleasby et al. (2010)	0.002 (1) 0.019 (1)	1 (1)	0.002	0.019	Mass	Total	Combined Wild
	Ficedula albicollis	Lindén et al. (1992)	0.0231 (1) 0.003 (1)	1 (1)	0.023	0.003	Mass	Late	Combined Wild
	Ficedula hypoleuca	Potti et al. (2002)	0.43(1)	0.17 (1)	1 (1)	0.430	0.170	Mass	Total	Female	Wild
			0.28 (1)	0.25 (1)	1 (1)	0.280	0.250	Mass	Total	Male	Wild
		Lobato et al. (2005)	-0.31 (3) 0.36 (2)	0.68 (3)	-0.211	0.245	Mass	Total	Combined Wild
	Erythropygia coryphaeus	Lloyd et al. (2009)	0.39 (1)	0.13 (1)	2.41 (2)	0.940	0.313	Mass	Total	Combined Wild
	Turdus merula	Snow (1958)	0.011 (2) 0.018 (2)	8.01 (2)	0.088	0.144	Mass	Total	Combined Wild
		Magrath (1991)	0.076 (2) 0.011 (2)	7.38 (2)	0.561	0.081	Mass	Early	Combined Wild
	Hylocichla mustelina	Brown & Roth (2004) 0.12 (2)	0.077 (2)	1 (1)	0.120	0.077	Mass	Total	Combined Wild
	Cinclus mexicanus	Mackas et al. (2010)	0.208 (2) 0.008 (2)	3.447 (2)	0.717	0.028	Condition	Total	Combined Wild
	Sturnus vulgaris	Serra et al. (2012)	0.056 (3) 0.044 (2)	10.8 (3)	0.605	0.475	Mass	Early	Female	Wild
	Parus major	Garnett (1981)	0.165 (1) 0.0468 (1)	1.25 (2)	0.206	0.059	Mass	Total	Combined Wild
		Smith et al. (1989)	0.347 (2) 0.247 (2)	1.192 (2)	0.414	0.294	Mass	Early	Combined Wild
		Lindén et al. (1992)	0.0047 (1) 0.003 (1)	1 (1)	0.005	0.003	Mass	Late	Combined Wild
		Verboven & Visser	0.135 (2) 0.035 (2)	1.44 (2)	0.194	0.050	Mass	Late	Female	Wild
		(1998)
			0.161 (2) 0.069 (2)	1.444 (2)	0.232	0.100	Mass	Late	Female	Wild
			0.244 (2) 0.035 (2)	1.518 (2)	0.370	0.053	Mass	Late	Male	Wild
			0.231 (2) 0.067 (2)	1.494 (2)	0.345	0.100	Mass	Late	Male	Wild
		Naef-Daenzer et al.	0.045 (1) 0.012 (1)	15.75 (2)	0.709	0.189	Mass	Late	Combined Wild
		(2001)
		Monrós et al. (2002)	0.14 (1)	0.05 (1)	1 (1)	0.140	0.050	Mass	Late	Combined Wild
		Greño et al. (2008)	0.26 (1)	0.07 (1)	1 (1)	0.260	0.070	Mass	Late	Combined Wild
	Parus ater	Naef-Daenzer et al.	0.045 (1) 0.012 (1)	15.75 (2)	0.709	0.189	Mass	Late	Combined Wild
		(2001)
	Parus caeruleus	Nur (1984)	0.297 (2) 0.13 (2)	1.02 (2)	0.303	0.133	Mass	Total	Combined Wild
		Raberg et al. (2005)	0.295 (1) 0.105 (1)	1 (1)	0.295	0.105	Mass	Total	Male	Wild
			0.29 (1)	0.153 (1)	1 (1)	0.290	0.153	Mass	Total	Female	Wild
	Corvus frugilegus	Patterson et al. (1988) 0.0084 (3) 0.0013 (2)	46.8 (3)	0.393	0.061	Mass	Late	Combined Wild
	Pica pica	Husby & Slagsvold	0.03 (3)	0.015 (2)	33.17 (3)	0.995	0.498	Mass	Total	Combined Wild
		(1992)

Table 4 .

 4 I² value associated with random effect of phylogeny, species, population, and author included when modelling the relationship between offspring mass and offspring survival in mammals and birds. For each value, the lower and upper high posterior density intervals (HPDI) limits of the credibility interval are reported.

	Mammals	Mean	Lower HPDI	Upper HPDI
	I² Phylogeny	18.64	0.44	47.44
	I² Species	18.43	0.73	44.92
	I² Population	10.29	0.54	30.66
	I² Author	39.74	4.02	72.24
	I² Residuals	10.95	0.47	34.11
	Birds	Mean	Lower HPDI	Upper HPDI
	I² Phylogeny	26.11	2.97	54.45
	I² Species	20.26	2.24	46.39
	I² Population	16.39	1.92	40.00
	I² Author	26.40	2.98	56.16
	I² Residuals	10.13	1.63	24.08

Table 6 .

 6 I² value associated with random effect of phylogeny, species, population, and author when modelling the relationship between offspring mass and maternal mass in mammals. For each value, the lower and upper high posterior density interval (HPDI) limits of the credibility interval are reported.

	Mammals	Mean	Lower HPDI	High HPDI
	I² Phylogeny	24.15	3.39	47.72
	I² Species	16.10	2.92	34.79
	I² Population	14.19	2.98	31.28
	I² Author	19.58	3.20	39.71
	I² Residuals	16.73	3.08	34.21

VI. ACKNOWLEDGMENTS

is funded by the French ministry of Education and Research. We thank Heiko Rödel,

Appendix S1. Data used for the analysis of the relationship between offspring body mass and offspring survival in mammals. See section II 4b for explanation of the different categories for the columns 'Age', 'Quality', 'Mating_system' and 'Litter_size'. Appendix S2. Data used for the analysis of the relationship between offspring body mass and offspring survival in birds. See section II 4b for explanation of the different categories for the columns 'Age', 'Quality', 'Mating_system' and 'Clutch_size'. Appendix S3. Data used for the analysis of the relationship between maternal mass and offspring mass in mammals. See section II 4b for explanation of the different categories for the columns 'Age', 'Quality', 'Mating_system' and 'Litter_size'. Appendix S4. R code for the extraction of the effect size for the relationship between offspring mass and offspring survival.

Table 1. Summary of the statistics and of the potential driving factors compiled in the meta-analysis for the relationship between offspring mass and offspring survival in mammals. For each study the logistic slope (Beta), its standard error (S.E.) and the standard deviation of the mass distribution (S.D. Mass), the standardized logistic slope (standardized Beta) and the standardized standard error (standardized S.E.) are reported. The extraction procedure is reported in parentheses as 1 if the statistic is directly calculated in the study, 2 if data are reported from a figure, and 3 if data are reported from our own simulation. See Section II.3a for further information on the extraction procedure and Section II.4b for explanation of Timing of measurement categories.