G Ovarlez

Q Barral

P Coussot

Three-dimensional jamming and flows of soft glassy materials

Various disordered dense systems such as foams, gels, emulsions and colloidal suspensions, exhibit a jamming transition from a liquid state (they flow) to a solid state below a yield stress 1 . Their structure, thoroughly studied with powerful means of 3D characterization 2,3,4,5,6 , exhibits some analogy with that of glasses 1,7,8 which led to call them soft glassy materials 9 . However, despite its importance for geophysical and industrial applications 10,11 , their rheological behavior 2,12 , and its microscopic origin 1,13 , is still poorly known, in particular because of its nonlinear nature. Here we show from two original experiments that a simple 3D continuum description of the behaviour of soft glassy materials can be built. We first show that when a flow is imposed in some direction there is no yield resistance to a secondary flow: these systems are always unjammed simultaneously in all directions of space. The 3D jamming criterion appears to be the plasticity criterion encountered in most solids 14 . We also find that they behave as simple liquids in the direction orthogonal to that of the main flow; their viscosity is inversely proportional to the main flow shear rate, as a signature of shear-induced structural relaxation, in close similarity with the structural relaxations driven by temperature and density in other glassy systems.

 and elastic coupling between rearrangement zones 18 .

Most models, simulations and experiments consider the case of simple shear, which involves only scalars variables -the shear stress and the shear rate -. Generally 3D formulations of constitutive equations, essential for the description of complex flows in practical situations, cannot directly be inferred from these results. Besides, a better understanding of the structure of soft glassy materials and its link with flow properties might be gained by focusing on their 3D rheological properties. E.g., the 3D yield criterion is closely related to the shape of the energy landscape in which their elements are trapped, while the 3D resistance to shear at the neighbourhood of the jamming transition might provide an insight in their structural relaxation.

In this work, in the aim of investigating their 3D behaviour, we unjam soft glassy materials in a given direction and observe their behaviour in an orthogonal direction. We focus on three systems of different structures: an emulsion, a physical gel, and a colloidal gel (see Methods).

With these systems, we have different physical origins of unjamming: droplet deformation (emulsion), blob squeezing (physical gel), contact breaking (colloidal gel), these different effects leading to collective rearrangements and flow.

We first use spheres of radius R (much larger than the size of fluid elements) embedded in the sheared soft glassy materials to probe their structure in the direction orthogonal to shear. More precisely, our experiments (see Methods) consist in shearing the material horizontally in a Couette geometry and observing the possible vertical motion of solid particles under the action of gravity, thanks to a density mismatch ρ Δ between the particles and the material (Fig. 1a). For small ρ Δ , when the fluid is at rest the beads do not move, the gravity minus buoyancy force is not sufficient to break the jammed structure [START_REF] Tabuteau | Drag force on a sphere in steady motion through a yield-stress fluid[END_REF] . In contrast, when we shear the fluid and unjam it in the horizontal plane, the beads start moving perpendicularly at a constant velocity V (Fig. 2, Supplementary Fig. 1).

The first important result is that the sedimentation of particles that were stable at rest is induced by shear: the orthoradial shear unjams the system in both the orthoradial and the vertical direction whatever the shear intensity. Moreover, we observe that the sedimentation velocity strongly increases, i.e. the drag force decreases, when the shear rate in the main direction increases (Fig. 2); it also increases with the particle diameter and when the yield stress is decreased (Fig. 2). In other words, the material seems to behave like a liquid with no yield resistance to flow in the vertical direction, and that is less and less viscous as it moves away from the jamming transition.

To understand these features, we performed experiments in more controlled conditions. We developed a new test (see Methods) in which we impose at the same time a shear flow and a squeeze flow (Fig. 1b), and measure the material's resistance to both flows. These two flows involve independent components of the strain rate tensor ij d , so that we can characterize in detail the response to a secondary flow. The material is set up between two parallel disks. The Our experiments start by imposing a decreasing rotational shear rate ramp without squeezing motion ( 0 Γ = ). In that case we obtain the typical flow curve of soft glassy materials in simple shear, namely a shear thinning behaviour tending to a plateau at low shear rates (Fig. 3).

The plateau level corresponds to the yield stress below which the material remains jammed.

Then we start again the test, but now we impose an additional squeeze flow at constant squeeze shear rate 0 Γ ≠ . In that case the flow curve is superimposed to the previous one (without squeezing) at high shear rates only i.e. when the rotational shear flow is dominant ( Γ << γ , Regime 1) (Fig. 3). At low shear rates, when the squeeze flow becomes dominant ( Γ >> γ , Regime 2), the new flow curve starts to depart from the previous one and the shear stress decreases strongly below the yield stress. A remarkable result is that in this regime the flow curve tends to follow a line of slope 1 in logarithmic scale at any observable rate, typical of a Newtonian behaviour (Fig. 3). Moreover the viscous regime occurs for lower γ and is characterized by higher apparent viscosity when Γ is decreased. Similar features (not shown) were observed when studying the resistance to the squeeze flow: when the shear flow is dominant, the resistance to the squeeze flow is that of viscous fluid. This observation is critical as it further supports the generality of the subsequent theoretical analysis, independently of some specificity of the flows. In addition, we checked that these effects were not due to an experimental artifact such as wall slip. This experiment, in which both superimposed flows are well controlled and characterized, now allows us to address the question of the quantitative resistance to flow. Our starting point is that the viscous resistance to the secondary flow decreases when the shear rate Γ of the main flow increases. This leads us to propose a structural interpretation, in analogy with the behaviour of liquids and glasses. The main flow introduces a structural relaxation timescale 1 Γ . Any secondary flow at a shear rate γ < Γ then probes longer timescales, allowing for full relaxation of the material during this secondary flow which thus has a viscous behaviour.

For γ > Γ relaxation cannot take place and the nonlinear glassy behaviour is recovered. By analogy with simple liquids [START_REF] Jones | Soft Condensed Matter[END_REF] , we suggest that the apparent viscosity η in the viscous regime is proportional to the relaxation timescale 1 Γ and to a reference stress; a natural reference stress is that associated with the liquid/solid transition ( c τ ). This yields the viscosity of the Remarkably, these data are well fitted to the 1 Γ curve. Note that due to the complex flow involved (see Methods) the factor close to 1 here observed is only due to our specific choice of the (constant) factors in the characteristic shear rate expressions. Anyway, this shows that, close to the jamming transition, the viscous resistance to the secondary flow is simply controlled by the apparent viscosity experienced in the main flow, i.e. when Γ >> γ we have

( ) ∝ Γ τ η γ .
These results now allow analyzing quantitatively the sedimentation experiments: they suggest that the vertical drag force is that for a sphere moving through a Newtonian fluid of viscosity η equal to that experienced by the fluid in the (dominant) horizontal simple shear. To test this prediction, we scale the measured sedimentation velocity by

η ρ 9 2 2 . gR V Newt Δ =
, the velocity of a single sphere steadily falling through a Newtonian medium under the action of gravity, and we take ( ) = η τ γ γ . Close to the jamming transition, all the . Newt V V data fall around the same value (Fig. 2 inset), in excellent agreement with the theory. The obtained ratio, 1.4, is higher than expected for a Newtonian fluid (around 0.7 for a 5% suspension), but this likely reflects some additional complexity of the flow field around several spheres moving through a complex fluid [START_REF] Chhabra | Drops And Particles in Non-Newtonian Fluids[END_REF] . The collapse of the data obtained for various values of ρ Δ , R and η , proves the general validity of the suggested scaling, which confirms that the spheres basically "see" a simple viscous material in the direction orthogonal to shear.

These results make it possible to deduce the form of the 3D constitutive equation of soft glassy systems. In the liquid regime, the material has a similar apparent viscosity in the different directions when the flow in a given direction is dominant. This suggests that the stress tensor under such conditions takes a form analogous to that for a Newtonian fluid:

2 = ij ij d τ η , in which
( )

1 2 = + ij i j j i d
du dx du dx is the strain rate tensor ( i u is the material velocity in the direction i x ). In contrast, with Newtonian fluids η is not constant but depends on the flow intensity in the main direction. Under simple shear, it reads

( ) γ γ τ η n c k + =
. We can extrapolate this equation to find the 3D constitutive equation of soft glassy systems in their liquid regime, by using in the apparent viscosity, instead of the shear rate, a generalized

shear intensity 2 , 2 ij i j d d = ∑
which is an invariant of the strain rate tensor to ensure frame invariance:

( )

2 + = n c ij ij kd d d τ τ (1)
This expression is in good agreement with our observations: it predicts that for a dominant squeeze flow, near the jamming transition, the resistance to the rotational shear is characterized by a purely viscous stress that reads

c z = = Γ θ τ τ τ γ (with 2 z d = θ γ ).
The constitutive equation of soft glassy systems also includes a yielding criterion associated with unjamming, which marks the transition from the solid to the liquid regime. In a pure simple shear this transition occurs when the shear stress τ becomes larger than a single scalar c τ (the yield stress), but the above data show that as soon as the material is unjammed in one direction it is unjammed in another direction. This suggests that the 3D criterion for unjamming involves some distance of the stress tensor from a critical value. In this context, the most natural approach consists in using for this distance the sum of the squares of the shear stress components 2. It is remarkable that the same plasticity criterion is found in soft glassy materials as in most usual crystalline solids [START_REF] Hill | The Mathematical Theory of Plasticity[END_REF] , while the microscopic plasticity mechanisms seem radically different [START_REF] Schall | Structural rearrangements that govern flow in colloidal glasses[END_REF] . However, a yield criterion implying 2

2 rz z + θ τ τ
is not unexpected from a pure rational mechanics [START_REF] Truesdell | A First Course in Rational Continuum Mechanics[END_REF] point of view under the (unobvious) hypothesis that the material is isotropic at yield (which implies that the yielding criterion is a function of the three invariants of the stress tensor [START_REF] Coleman | Viscometric Flows of Non-Newtonian Fluids; Theory and Experiment[END_REF] ) and in the absence of normal stress differences.

Finally, Eqs. (1) and ( 2) constitute a 3D modelling of the jamming and flows of soft glassy systems deduced from experiments. Beyond their obvious interest for complex flow modelling these results provide generic quantitative explanations for shear-induced heterogeneities in industrial flows of suspensions in yield stress fluid, and for geophysical problems as the liquefaction of quicksand under shear [START_REF] Khaldoun | Rheology: Liquefaction of quicksand under stress[END_REF] . The generality of this constitutive equation is supported by the fact that we observed the same trends in sedimentation and squeeze-shear tests for our soft glassy materials (emulsions, bentonite suspensions, Carbopol gels) of different structures. This suggests that the main trends here observed may find a same phenomenological origin in all materials, namely the shear-induced structural relaxation of their structural units.

Our macroscopic observations actually shed light on the microscopic behaviour of such materials. It has been observed on several soft glassy systems [START_REF] Besseling | Three-dimensional imaging of colloidal glasses under steady shear[END_REF][START_REF] Ono | Effective temperatures of a driven system near jamming[END_REF][START_REF] Song | Experimental measurement of an effective temperature for jammed granular materials[END_REF][START_REF] Möbius | Relaxation and flow in linearly sheared two-dimensional foams[END_REF] that, under shear, in addition to cage diffusion observed at rest [START_REF] Kegel | Direct observation of dynamical heterogeneities in colloidal hard-sphere suspensions[END_REF][START_REF] Weeks | Three-dimensional direct imaging of structural relaxation near the colloidal glass transition[END_REF][START_REF] Oppong | Gelation on the microscopic scale[END_REF] , the elements undergo a simple diffusion process as in simple liquids in the three directions of space; other, consistent, observations are that the sheared colloidal glasses experience full exponential alpha relaxation in the directions orthogonal to shear [START_REF] Besseling | Three-dimensional imaging of colloidal glasses under steady shear[END_REF] as in a liquid state. Our results provide the missing macroscopic counterpart to these microscopic observations: if the Stokes-Einstein relation applies [START_REF] Ono | Effective temperatures of a driven system near jamming[END_REF] , the particles should indeed diffuse as in a simple liquid of viscosity η equal to that we observe in the direction orthogonal to shear, and the diffusion coefficient D should be proportional to 1 η . Then, in the low shear rate limit, our approach predicts that ∝ D γ . This is in quantitative agreement with some observations [START_REF] Ono | Effective temperatures of a driven system near jamming[END_REF][START_REF] Song | Experimental measurement of an effective temperature for jammed granular materials[END_REF] ; the nonlinear scaling of D with γ observed by others [START_REF] Besseling | Three-dimensional imaging of colloidal glasses under steady shear[END_REF][START_REF] Möbius | Relaxation and flow in linearly sheared two-dimensional foams[END_REF] , consistent with the alpha relaxation time measurements [START_REF] Besseling | Three-dimensional imaging of colloidal glasses under steady shear[END_REF] and with theoretical predictions [START_REF] Saltzman | Anomalous diffusion, structural relaxation and shearthinning in glassy hard sphere fluids[END_REF] , may be due to the fact that the diffusion coefficient considered in these studies is that of the materials constitutive elements, which do not probe the macroscopic behaviour.

Our results finally show with new quantitative results that there are striking similarities between the structural relaxation driven by shear, temperature, and density at the approach of the jamming transition. The flow curves of Fig. 3 are actually remarkably similar to those observed in colloidal suspensions [START_REF] Petekidis | Yielding and flow of sheared colloidal glasses[END_REF][START_REF] Fuchs | Flow curves of dense colloidal dispersions: Schematic model analysis of the shear-dependent viscosity near the colloidal glass transition[END_REF] and predicted in mode coupling theories [START_REF] Fuchs | Theory of nonlinear rheology and yielding of dense, colloidal suspensions[END_REF][START_REF] Fuchs | Flow curves of dense colloidal dispersions: Schematic model analysis of the shear-dependent viscosity near the colloidal glass transition[END_REF] (MCT) when varying the volume fraction φ of colloids near that of the colloidal glass transition g φ .

When < g φ φ , colloidal suspensions have a Newtonian behaviour below a low critical shear rate ( ) c γ φ , whereas they have the same shear-thinning behaviour as the colloidal glass at higher shear rates [START_REF] Petekidis | Yielding and flow of sheared colloidal glasses[END_REF][START_REF] Fuchs | Theory of nonlinear rheology and yielding of dense, colloidal suspensions[END_REF][START_REF] Fuchs | Flow curves of dense colloidal dispersions: Schematic model analysis of the shear-dependent viscosity near the colloidal glass transition[END_REF] . This Newtonian behaviour is observed when the timescale for the diffusion of the particles out of their cages is smaller than the typical flow timescale [START_REF] Fuchs | Theory of nonlinear rheology and yielding of dense, colloidal suspensions[END_REF] 1 γ . As a consequence, c γ decreases strongly as φ approaches g φ , similarly to our observations when varying the squeeze shear rate Γ . Similarly, a high viscosity plateau is found at lower and lower shear rates when the temperature of metallic glasses is decreased [START_REF] Lu | Deformation behavior of the Zr41.2Ti13.8CU12.5Ni10Be22.5 bulk metallic glass over a wide range of strain-rates and temperatures[END_REF] . Overall, MCT thus seems to be a very good candidate for general modelling of the jamming transition in all glassy systems, and to account properly for the effect of shear. In particular, it is remarkable that our finding of a Von Mises jamming criterion is consistent with very recent MCT findings [START_REF] Brader | Glass rheology: From modecoupling theory to a dynamical yield criterion[END_REF] . However, one still has to test MCT predictions in view of our observations and of the 3D constitutive law we propose.

METHODS

MATERIALS PREPARATION

The The material is loaded in a Couette geometry, the dimensions of which are: inner cylinder radius 4.1 cm, outer cylinder radius 6cm and inner cylinder height 11 cm. Sandpaper of roughness equivalent to the particles size is glued on the walls to avoid wall slip. The suspension is sheared at constant macroscopic shear rate, and the particles vertical velocity is obtained thanks to the evolution in time of the spatial distribution of the particle concentration measured through Magnetic Resonance Imaging (MRI) techniques. The MRI set up was described in detail elsewhere [START_REF] Ovarlez | Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging[END_REF] . The particle volume fraction can be obtained both in the radial and vertical directions from density imaging [START_REF] Ovarlez | Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging[END_REF] with an accuracy of 0.3%. Since the shear distribution within the gap of a Couette geometry is heterogeneous it is crucial to impose a large rotation velocity to avoid shear localization. Moreover, we measure the vertical concentration profiles in a 0.8mm thick layer in the middle of the gap to obtain information on particle motion in a zone of approximately homogeneous shear rate. This last shear rate is obtained locally through MRI thanks to the velocity profile measurement [START_REF] Ovarlez | Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging[END_REF] .

At rest, the particles appear to remain indefinitely in their initial position (Supplementary Fig. 1 inset): there is no observable difference between the vertical concentration profiles measured after loading and after a 24 h rest.

Then we start shearing the materials by rotating the inner cylinder at a given rotation velocity Ω between 20 and 130 rpm (corresponding to a shear rate ranging from 4 to 25 s -1 ). While the particles are stable at rest, we observe that there is sedimentation when the material is sheared. The sedimentation profiles show classical features of sedimentation in Newtonian fluids (Supplementary Fig. 1): the upper part is at a 0% concentration; the bottom part remains at the initial 5% concentration (note that the particles tend to accumulate out of the measurement window, below the inner cylinder, which explains that the concentration in the lower part of our profiles does not change); the transition zone between these two parts is rather narrow (with a typical thickness of 6mm), and defines a sedimentation front that moves regularly towards the bottom as the flow duration increases. We observe that this front moves linearly in time (Supplementary Fig. 1), defining a constant front velocity V that is identified with the individual particles sedimentation velocity.

COMBINED ROTATIONAL SHEAR AND SQUEEZE FLOWS EXPERIMENTS

A constant volume of material (typically 0.7 ml) is inserted within the gap of a parallel disk geometry of a Kinexus rheometer (Malvern Instruments) so as to partially fill the gap (initial height 0 h =1.1 mm). The material is then sheared by rotating the upper disk around its axis at a rotational velocity Ω , and at the same time it is squeezed by moving the upper disk towards the other at controlled velocity V (Fig. 1). We record the torque T exerted onto the axis of the upper plate and the normal force F exerted on the upper plate. Due to the squeeze flow, the material geometric characteristics change in time. In the following analysis, the relationships between the shear stresses and shear rates and the macroscopic quantities thus involve the instantaneous material outer radius R(t) and thickness h(t).

During this test, the flow characteristics within the gap are somewhat heterogeneous but as usual in viscometry it is possible to follow average variables which correctly reflect the relative variations of the local variables [START_REF] Coleman | Viscometric Flows of Non-Newtonian Fluids; Theory and Experiment[END_REF] . The applied torque induces an orthoradial motion associated with the velocity v θ (in the cylindrical frame shown in . Moreover, it may be shown that there is a relationship between τ and γ via a function depending only on the intrinsic material behavior [START_REF] Coleman | Viscometric Flows of Non-Newtonian Fluids; Theory and Experiment[END_REF] : these variables thus play the role of reference shear stress and shear rate the relative variations of which provide the characteristics of the material behavior.

In a similar way, the applied normal force is at the origin of a radial pressure gradient which induces mainly a radial shear flow [START_REF] Coussot | Rheophysics of pastes: a review of microscopic modelling approaches[END_REF] (as long as 1 << h R

) associated with the velocity r v , linked to a stress component rz τ . From mass conservation (which provides the relationship between the vertical velocity V and the average radial velocity) it is found that the resulting shear rate due to the vertical gradient of r v scales with . Again, it may be shown that there is a relationship between Σ and Γ via a function depending only on the intrinsic material behavior [START_REF] Coleman | Viscometric Flows of Non-Newtonian Fluids; Theory and Experiment[END_REF] . Thus, in this paper, we simply follow these quantities, namely τ and γ for the rotational shear flow and Σ and Γ for the squeeze flow. Note finally that the critical stresses for solid-liquid transition are here associated with the stress plateaus observed beyond a critical deformation for very low flow rates [START_REF] Coussot | Rheometry of Pastes, Suspensions and Granular Materials[END_REF] . Under these conditions, there are no radial stress heterogeneity issues since at the stress plateau the whole material is at yield. For the consistency of the mechanical characterization of our materials, we thus chose to use η τ γ γ . Note that in some systems sedimentation velocities at the highest shear rates tested are larger than the plateau value evidenced here, an effect likely due to non-Newtonian effects (such as normal stresses) on sedimentation [START_REF] Chhabra | Drops And Particles in Non-Newtonian Fluids[END_REF] , which is out of the scope of our study.

  rotation of the upper disk around its axis induces a simple orthoradial shear flow of A squeeze flow is superimposed by moving the upper disk towards the other at controlled velocity; it induces mainly a simple radial shear flow of characteristic shear rate 2

  secondary flow in the form Γ ∝ c τ η . In the inset of Fig.3, we plot the experimentally measured effective viscosity of the rotational flow η τ γ = in Regime 2, scaled by c τ , vs. Γ .

  be written more consistently as a function of an invariant of the stress tensor to include possible normal force differences: Von Mises criterion commonly used for solid materials[START_REF] Hill | The Mathematical Theory of Plasticity[END_REF] . In simple shear and simple squeeze, this criterion gives back the usual yielding criterion ( , . We have tested further the validity of this 3D yield criterion by combining squeeze and rotational shear flows and measuring simultaneously the ( rz τ , zθ τ )values at the onset of flow. The corresponding data are presented in Fig.4. The jammed region effectively lies within the disk defined by good agreement with Eq.

  emulsions used in the shear-induced sedimentation experiments are prepared by dispersing a 100 g/l water solution of CaCl 2 in a solution of Span 80 emulsifier (7%) in dodecane oil at 6000 rpm with a Silverson L4RT mixer. The emulsions used in the combined squeeze and rotational shear flows experiments are prepared by dispersing dodecane oil at 5000 rpm in a 2 g/L water solution of tetradecyltrimethylammonium Bromide (TTAB). The droplet size is 1 micron. Seven emulsions are used in this study; their droplet concentration is between 72 and 85% and their yield stresses range between 8.5 and 52 Pa. The physical gel is prepared by dispersing Carbopol 980 (Noveon) in water at a 0.4% concentration and neutralized with NaOH at pH=7. Carbopol gels may then be seen as polydisperse glasses made of individual swollen hydrophilic elastic sponges. The Carbopol gel used in this study has a 70 Pa yield stress. The colloidal gel is a suspension of 3.5% Na-Bentonite particles in water. Bentonite is a natural swelling clay with slightly flexible, large aspect ratio particles (typical length: 1 μm, typical thickness: 0.01 μm) which can aggregate via edge-to-face links, so that the suspension is a colloidal gel with a thixotropic yield stress fluid behaviour. The suspension was prepared by a strong mixing of the solid phase with water, then left at rest three months before any test, which avoids further irreversible (chemical) aging over the duration of the experiments. The bentonite suspension used in this study has a 4 Pa dynamic yield stress. While bentonite suspensions are thixotropic yield stress materials, time effects during flows are negligible for the suspensions we study and for the time scales of our experiments. The flowing suspension can be then considered as a simple yield stress fluid with a (dynamic) yield stress associated with stoppage for a progressive stress decrease. On the other hand, aging effects are important at rest and would imply additional complexity for determining a yield criterion with the (static) yield stress associated with flow start.SHEAR-INDUCED SEDIMENTATION EXPERIMENTMonodisperse glass beads of density 2.5, of diameter between 140 and 420 microns (with 10% polydispersity) are suspended in yield stress fluids at a 5% volume fraction to avoid important collective effects. For all materials at rest, the external force exerted on the particles, i.e. the gravity force minus the buoyancy force it is of order of 1% of c F in all cases).

  Fig.1b) linked to the stress component zθ τ . The resulting shear rate due to the vertical gradient of v θ scales with h R Ω = γ and from the momentum balance it is found that the shear stress scales with 3

  can be shown to provide the exact yield stress of a plastic material respectively in simple shear and simple squeeze 11,, A. J. & Nagel, S. R. (eds) Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales. Taylor & Francis, New York (2001).

Figure 1 |Figure 2 |

 12 Figure 1 | Sketches of the experiments. a) Sketch of the sedimentation experiment perpendicular to a shear flow: (left) bead suspended in the material at rest; (right) bead falling under the action of gravity in the material sheared in the horizontal plane. b) Sketch of the combined squeeze and rotational shear flow experiments: the material is loaded between two parallel plates, and is simultaneously sheared by rotating the upper disk around its axis at a rotational velocity Ω, and squeezed by moving the upper disk towards the other at controlled velocity V .

  single sphere that would fall in a viscous medium of viscosity ( ) =

Figure 3 |Figure 4 |

 34 Figure 3 | 3D flow curve. Rotational shear stress vs. rotational shear rate measured in a 28 Pa yield stress emulsion by rotating the upper plate of a parallel plate geometry, for a constant gap (squares) and while simultaneously squeezing the material (circles) at various squeeze shear rates Γ (from left to right: -1 0.027 s Γ = (red),

Supplementary Information

Position of the sedimentation front as a function of the time of shear, for a 5% suspension of 275 microns glass beads in an emulsion of 8.5 Pa yield stress, for various shear rates: 4 s -1 (squares), 8.8 s -1 (circles) 14 s -1 (up triangles) 18.6 s -1 (down triangles) 25 s -1 (diamonds). Inset: vertical volume fraction profiles observed in the gap of the Couette geometry in the same material as in Fig. 2, after a 24 h rest (squares) and after 15 min (black line) and 25 min (red line) of shear at 4 s -1 .


Professor Simon G Thompson
email: simon.thompson@mrc-bsu.cam.ac.uk

Stephen Kaptoge

I White

Angela M Wood

Statistical methods for the time-to-event analysis of individual participant data from multiple epidemiological studies

Keywords: Meta-analysis, Epidemiological studies, Individual participant data, Statistical methods, Survival analysis Meta-analysis, epidemiological studies, individual participant data, statistical methods, survival analysis

Methods: This paper describes statistical approaches adopted in the Emerging Risk Factors

Collaboration, in which primary data from over 1 million participants in over 100 prospective studies have been collated to enable detailed analyses of a variety of risk markers in relation to incident cardiovascular disease outcomes.

Results:

Analyses have been principally based on Cox proportional hazards regression models stratified by sex, undertaken in each study separately. Estimates of exposure-risk relationships, initially unadjusted and then adjusted for several confounders, have been combined over studies using meta-analysis. Methods for assessing the shape of exposure-risk associations and the proportional hazards assumption have been developed. Estimates of interactions have also been combined using meta-analysis, keeping separate within-and between-study information.

Regression dilution bias caused by measurement error and within-person variation in exposures and confounders has been addressed through the analysis of repeat measurements to estimate corrected regression coefficients. These methods are exemplified by analysis of plasma fibrinogen and risk of coronary heart disease, and Stata code is made available.

Conclusion:

Increasing numbers of meta-analyses of individual participant data from observational data are being conducted to enhance the statistical power and detail of epidemiological studies. The statistical methods developed here can be used to address the needs of such analyses.

INTRODUCTION

Combining information across several studies using meta-analysis can enhance precision for quantitative summaries of evidence. [START_REF] Egger | Systematic reviews in health care: metaanalysis in context[END_REF] Re-analysis of individual participant data (IPD) from multiple epidemiological studies has several advantages compared to meta-analysis of aggregated published data, including: harmonisation of definitions for risk markers as well as disease outcomes; ability to update follow-up information; consistent approaches to adjustment for confounding; characterisation of the shape of exposure-risk relationships; greater ability to correct for regression dilution bias; and determination of how exposure-risk relationships depend on age, sex and other potential effect modifiers. [START_REF] Stewart | Practical methodology of meta-analyses (overviews) using updated individual patient data[END_REF][START_REF]Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF][START_REF] Thompson | Can meta-analysis help target interventions at individuals most likely to benefit?[END_REF] This paper describes and illustrates statistical methods that are being used in the Emerging Risk Factors Collaboration (ERFC), an analysis of individual records from over 1.2 million participants in 116 prospective studies in predominantly Western populations of major cardiovascular disease outcomes. [START_REF]Collaborative meta-analysis of individual data on over 1 million participants in 96 prospective cohorts of lipid and inflammatory markers in cardiovascular diseases[END_REF][START_REF]Emerging Risk Factors Collaboration. Major lipids, apolipoproteins and risk of vascular disease[END_REF][START_REF]Lipoprotein(a) concentration and the risk of coronary heart disease, stroke and nonvascular mortality: individual data analysis of 121,944 participants from 34 prospective studies[END_REF][START_REF]Emerging Risk Factors Collaboration. C-reactive protein concentration and the risk of coronary heart disease, stroke and mortality: an individual participant meta-analysis[END_REF] The ERFC includes mostly prospective cohort studies (a few based in randomised trials), as well as some nested case-control and case-cohort studies. For each participant in the ERFC, the coordinating centre has collated, verified and harmonised individual records on baseline risk markers, confounders, other characteristics, major cardiovascular morbidity and cause-specific mortality. [START_REF]Collaborative meta-analysis of individual data on over 1 million participants in 96 prospective cohorts of lipid and inflammatory markers in cardiovascular diseases[END_REF] Available repeat survey data, which provide serial measurements, have also been collected to help address measurement error and within-person variability. [START_REF]Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF][START_REF] Macmahon | Blood pressure, stroke, and coronary heart disease. Part 1, Prolonged differences in blood pressure: prospective observational studies corrected for the regression dilution bias[END_REF] As the ERFC subsumes the Fibrinogen Studies Collaboration, [START_REF]Plasma fibrinogen and the risk of major cardiovascular diseases and nonvascular mortality: an individual participant meta-analysis[END_REF]

( ) β = + + ski si si sk s si s si h t E h t E X γ X (1)
The evolution of risk over time is thus modelled independently for each stratum in each study, as represented by the non-parametric baseline hazards h 0sk (t). The β s are the parameters of interest, being the log hazard ratios (HRs) per unit increase in the exposure in study s, adjusted for the confounding effects of the covariates X si .

These estimated log HRs can be combined over studies using random-effects meta-analysis, [START_REF] Higgins | A re-evaluation of random-effects metaanalysis[END_REF] which incorporates heterogeneity between studies as described below. Fixed-effects metaanalysis can also be used, [START_REF]Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF][START_REF]Collaborative overview ('meta-analysis') of prospective observational studies of the associations of usual blood pressure and usual cholesterol levels with common causes of death: protocol for the second cycle of the Prospective Studies Collaboration[END_REF][START_REF]Blood cholesterol and vascular mortality by age, sex, and blood pressure: a meta-analysis of individual data from 61 prospective studies with 55,000 vascular deaths[END_REF] and has been employed in parallel analyses in the ERFC. Writing the variance of the estimated β s as v s , the random-effects meta-analysis model is:

2 ˆ; where ~(0, )

; where ~(0, )

s s s s s s s s N v N β β ε ε β β η η τ = + = + (2)
Here β is the average log HR, the estimate of which combines within-study information on the relationship between exposure and risk, while allowing for heterogeneity in the true log HRs between studies as represented by the variance τ 2 . A standard moment estimator of τ 2 is used, [START_REF] Dersimonian | Meta-analysis in clinical trials[END_REF] although other estimation methods are available. [START_REF] Dersimonian | Random-effects model for meta-analysis of clinical trials: an update[END_REF] The statistical significance of the standard test for heterogeneity [START_REF] Cochran | The combination of estimates from different experiments[END_REF] reflects the strength of evidence for heterogeneity. The impact of heterogeneity on the imprecision of the overall log hazard ratio is expressed in terms of I 2 , the percentage of variance in the point estimates of the study-specific log HRs that is attributable to between-study variation as opposed to sampling variation, [START_REF] Higgins | Measuring inconsistency in meta-analysis[END_REF] for which a confidence interval is

F o r R e v i e w O n l y
6 also available. [START_REF] Higgins | Quantifying heterogeneity in a meta-analysis[END_REF] Values of I 2 close to 0% correspond to lack of heterogeneity. In addition, specific sources of heterogeneity are explored by investigating the impact of various factors (eg, age, sex and other potential effect modifiers) on the strength of the association between exposure and risk, as described in later sections.

The above procedure is a 2-step method: first each study is analysed separately in (1) and then the log HRs are combined in (2). A 1-step method would be preferable in principle, writing a combined model as: ; where ~(0, )

β β β η η τ = + + = + ski si si sk s si s si s s s h t E X h t E N γ X ( 3
)
Computational problems are, however, formidable in a dataset the size of the ERFC. [START_REF] Tudur-Smith | Investigating heterogeneity in an individual patient data meta-analysis of time to event outcomes[END_REF] A 2-step analysis has only the slight disadvantage that the first-step variances v s in a 2-step analysis are not in general exactly those implied in a 1-step method, although 1-step and 2-step methods usually produce very similar results. [START_REF] Olkin | Comparison of meta-analysis versus analysis of variance of individual patient data[END_REF][START_REF] Riley | Meta-analysis of continuous outcomes combining individual patient data and aggregate data[END_REF] For the case of fibrinogen and the risk of CHD, adjusting only for the linear effect of age at baseline in each study, these analyses are summarised in the upper part of

Choice of exposure scale

An assumption of the above model is that the log hazard increases linearly with the exposure. It might be more appropriate, however, to choose a log scale for some exposures to improve linearity. Alternatively, the use of a study-specific standard deviation (SD) score might reduce heterogeneity of the risk association between studies. In the case of fibrinogen, for example, the distributions were slightly positively skewed and the SDs varied considerably between studies. [START_REF]Associations of plasma fibrinogen levels with established cardiovascular risk factors, inflammatory markers and other characteristics: individual participant meta-analysis of 154 211 adults in 31 prospective studies[END_REF] It is also important to assess the possibility of non-linear risk relationships which could indicate a threshold or a plateau for risk.

To assess linearity, as in previous studies, [START_REF]Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF] the distributions of the exposure are divided into quantile groups such as fifths; such quantile groups can be defined within each study or across all studies. HRs in each quantile group, compared to the bottom group, are estimated using Cox PH regression in each study separately. These log HRs within each study are not independent (their correlations are available from standard regression software) because they are all relative to the same reference group. So the set of log HRs are pooled across studies using a multivariate version of random-effects meta-analysis [START_REF] Riley | An evaluation of bivariate random-effects meta-analysis for the joint synthesis of two correlated outcomes[END_REF][START_REF] White | Multivariate random-effects meta-analysis[END_REF] to allow for their inter-correlations both within and between studies. These pooled log HRs are plotted against the mean exposure level in each quantile group. Assessing linearity is easier using confidence intervals derived by floating absolute risk methods [START_REF] Easton | Floating absolute risk: an alternative to relative risk in survival and case-control analysis avoiding an arbitrary reference group[END_REF][START_REF] Plummer | Improved estimates of floating absolute risk[END_REF] so that each estimate (including that for the reference category) has a measure of uncertainty and is less correlated with the others. Judging linearity visually from strongly correlated estimates can be misleading: for example, if the reference group is small then all the standard confidence intervals will be wide and non-linearity cannot be ruled out.

Sensitivity analyses, employing different scales for the exposure (eg, log, SD-score) or assessing curvature using polynomial terms, are also used to investigate whether heterogeneity between studies is reduced or the substantive conclusions affected.

F o r R e v i e w O n l y 8
Figure 2 shows the results of an analysis by study-specific fifths of fibrinogen in relation to CHD risk, which suggests that a log-linear model for risk is satisfactory. [START_REF]Plasma fibrinogen and the risk of major cardiovascular diseases and nonvascular mortality: an individual participant meta-analysis[END_REF] Examples of sensitivity analyses are shown in the lower part of Table 1. These compare untransformed fibrinogen, log fibrinogen, study-specific SD fibrinogen score, and study-specific SD log fibrinogen score. For comparability, results are expressed as the HR per 1-SD higher baseline fibrinogen; in the first two analyses this refers to the pooled within-study SD (0.65g/L for untransformed fibrinogen). The results from all analyses are quantitatively similar, including the extent of heterogeneity. Including a quadratic term for untransformed fibrinogen in the first analysis provides little evidence of curvature in the risk relationship (P=0.09). In the case of fibrinogen, therefore, the heterogeneity between studies is not due to the choice of exposure scale.

A few technical issues in such analyses merit consideration. First, the visual assessment of linearity and the comparison between different exposure scales are informal. Secondly, while it might be preferable to use fractional polynomials [START_REF] Royston | The use of fractional polynomials to model continuous risk variables in epidemiology[END_REF] or splines [START_REF] Govindarajulu | Comparing smoothing techniques in Cox models for exposure-response relationships[END_REF] to investigate curvature, this is not straightforward in a 2-step random-effects meta-analysis because different functional forms might be appropriate for different studies. These problems would be reduced if a 1-step metaanalysis method were computationally feasible. Thirdly, in Figure 2, the choice of fifths is rather arbitrary, and it is not entirely clear what levels of fibrinogen the log HRs should be plotted against; for example, this could be the mean fibrinogen in each fifth weighted by the number of events rather than by the number of participants. Finally, the effect of measurement error and within-person variation in fibrinogen may distort the shape of the exposure-disease relationship, as discussed later.

Covariate adjustment

Age is the most important confounder in many epidemiological applications, and so adjustment for age demands particular attention. 2), and the extent of heterogeneity decreases. Thus some of the original heterogeneity between studies seems to be due to differing impacts of these confounders in different studies. The age-adjusted HR per 1g/L higher baseline fibrinogen falls from 1.57 to 1.38 on adjusting for these covariates, so that 29% (calculated as

Joint effects

An important advantage of IPD is that it provides the opportunity for systematic investigation of the exposure-risk relationship at different levels of other variables. This evaluation of factors that modify the overall log HRs estimated above involves assessing their interactions on this scale with the exposure of interest. When effect modifiers are variables measured in individuals, such as age or other risk markers, these interactions are most effectively assessed using withinstudy information. [START_REF] Thompson | Can meta-analysis help target interventions at individuals most likely to benefit?[END_REF][START_REF] Lambert | A comparison of summary patient-level covariates in meta-regression with individual patient data meta-analysis[END_REF] Here a 2-step procedure has again been adopted, first estimating the interaction in each study separately. For example, for a single potential effect modifier X si , the model in study s is:

0 log( ( | , )) log ( ) β δ = + + + ski si si sk s si s si s si si h t E X h t E γ X E X (4) F o r R e v i e w O n l y 10
The estimates of the interaction terms δ s are combined using random-effects meta-analysis, as in (2). The overall interaction, δ W say, is then based on only within-study information. Model (4) can be extended by including adjustments for other confounders, and indeed their interactions with the exposure of interest; this enables investigation of whether, as is possible, a particular interaction is confounded by other main effects or interactions.

Some potential effect modifiers are assessed only at the study level, for example the type of population recruited or the laboratory methods used for measuring the exposure. For such variables, any information on interactions relies entirely on between-study comparisons, which are assessed using random-effects meta-regression. [START_REF] Thompson | Explaining heterogeneity in meta-analysis: a comparison of methods[END_REF] Using the estimates of β s from (1), model ( 2) is extended to include a study level covariate X s by writing: ; where ~(0, )

β β ε ε β β δ η η τ = + = + + s s s s s s B s s s N s X N ( 5
)
δ B is the between-study interaction term, with statistical significance assessed allowing for the residual between-study heterogeneity τ 2 .

A few variables, notably sex and ethnic group, have potential interactions for which both withinstudy and between-study information may be important. For example, studies involving both men and women provide within-study information on sex interactions, while studies which comprise members of one sex alone can only be used to assess interactions across studies. In this case, the within-study interaction δ W is estimated as in model ( 4) based on studies of both sexes, and the between-study interaction δ B using model ( 5) in which X s is the proportion of women in each study. Provided they are similar, these two asymptotically independent estimates of interaction can themselves be combined. As between-study information on interactions is prone to numerous potential sources of between-study confounding, [START_REF] Thompson | How should meta-regression analyses be undertaken and interpreted?[END_REF] there is a trade-off between increased precision and possible bias in choosing whether to use between-study information in addition to within-study information. [START_REF] Riley | Meta-analysis of continuous outcomes combining individual patient data and aggregate data[END_REF][START_REF] Simmonds | Covariate heterogeneity in meta-analysis: criteria for deciding between meta-regression and individual patient data[END_REF][START_REF] Jackson | Improving ecological inference using individual-level data[END_REF] Presenting interactions in a way that is intelligible to readers is not easy. For a binary variable identifying two subgroups, the exponent of the interaction term is a ratio of HRs, but it is simpler

F o r R e v i e w O n l y 11
to present two separate meta-analyses, one in each subgroup. However, because the betweenstudy heterogeneity, τ 2 in (2), now affects each of these estimates, the (multivariate) metaanalytic weighting of study-specific subgroup estimates is different from the weighting of studyspecific interactions. So neither the estimates nor the confidence intervals of the subgroupspecific estimates are necessarily compatible with the estimate and confidence interval of the interaction term. In practice, this problem is not usually severe. For continuous variables, the exponent of the interaction term is a ratio of HRs per unit increase in the effect modifier.

Similarly, for presentation, it is easier to present the HR estimates according to study-specific quantile groups (for example thirds or fifths) of the effect modifier distribution.

Examples of interaction analyses for fibrinogen are shown in Table 3. The interactions with body mass index and age at baseline are clear, but the interactions with other variables are less marked. Including the body mass index and age interactions simultaneously hardly affects their respective estimates. There is more consistency in the interaction terms across studies than for the main effect of fibrinogen, as indicated by the lower values of I 2 . For investigating a possible sex interaction, δ B is estimated from a meta-regression of the study-specific log HRs on the proportion of women in each study. The SE of the interaction term is smaller for δ W than δ B , so the majority (73%) of the information comes from within-study information. It is sensible to rely on the within-study pooled interaction estimate, especially when it contributes the majority of the information, because of the potential for bias in the between-study estimate. [START_REF] Riley | Meta-analysis of continuous outcomes combining individual patient data and aggregate data[END_REF] The sex-specific combined log HRs (not shown) and the combined sex interaction term are similar but not identical. The sex interaction term represents the correct analysis, while the sex-specific HRs are probably the preferable method of presentation in applied publications, especially when given in a diagram. As noted above, effect modification is being assessed on the HR scale. Thus, while the HRs per unit higher fibrinogen decrease with increasing age, the absolute risk gradients increase (Figure 3).

Proportional hazards

An assumption of all the models considered so far is of proportional hazards (PH), meaning that the regression coefficients in model (1) do not change with time since baseline measurement.

F o r R e v i e w O n l y 12
While the effect of any covariate measured at baseline may plausibly decrease over time, the prime interest is whether the proportional hazards assumption is appropriate for the exposure of interest. This can be evaluated in each study separately by including an interaction between the exposure and time, or by the commonly used diagnostic based on Schoenfeld residuals. [START_REF] Collett | Modelling survival data in medical research[END_REF] These independent [START_REF] Stewart | Practical methodology of meta-analyses (overviews) using updated individual patient data[END_REF] 1 χ statistics can be summed across the S studies, yielding a 2 S χ statistic testing the hypothesis that PH holds in each study.

This approach is, however, not a powerful test against the plausible alternative hypothesis that HRs tend to decline over time in all studies. A better method is to combine the interaction terms between the exposure and time over studies. Using random-effects meta-analysis, and assuming linear time-dependence, the model is: ; where ~(0, )

β ξ ξ ξ η η τ = + + = + ski si si sk s si s si s s s h t E X h t E tE N ( 6
)
where β s are separate fixed effects, and the focus is on the estimate of ξ which can be tested using a 2 1 χ statistic.

The results of these analyses for fibrinogen are shown in Table 4. The summed 2 31 χ statistics are less than expectation, as is the more powerful 2 1 χ statistic. So, in this case (and perhaps surprisingly given the extent of data), there is no evidence of departures from proportional hazards for fibrinogen and no evidence of heterogeneity between studies in this regard. The final method provides an estimate of the non-PH parameter ξ, which indicates that over a 20-year period the estimated change in the exposure log HR is small. In ERFC, this random-effects pooling of the interaction terms between exposure and time is used to assess the PH assumption.

It provides extra power against a plausible alternative hypothesis, and is consistent with the approach described above for quantifying other interactions. If there was substantial evidence against the PH assumption, it would be necessary to summarise the exposure-risk relationship either in discrete intervals of time, or as a trend over time.

Other topics

When the focus is on estimating underlying aetiological associations, it is necessary to adjust for the effect of measurement error and within-person variation. For the exposure of interest, this addresses the often serious underestimation caused by regression dilution bias, [START_REF] Macmahon | Blood pressure, stroke, and coronary heart disease. Part 1, Prolonged differences in blood pressure: prospective observational studies corrected for the regression dilution bias[END_REF][START_REF] Clarke | Underestimation of risk associations due to regression dilution in long-term follow-up of prospective studies[END_REF][START_REF] Rosner | Correction of logistic regression relative risk estimates and confidence intervals for systematic within-person measurement error[END_REF] and for covariates it reduces residual confounding. [START_REF] Rosner | Correction of logistic regression relative risk estimates and confidence intervals for measurement error: the case of multiple covariates measured with error[END_REF] Methods exist to correct for regression dilution bias in exposure variables in IPD meta-analysis. [START_REF]Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF][START_REF]Collaborative overview ('meta-analysis') of prospective observational studies of the associations of usual blood pressure and usual cholesterol levels with common causes of death: protocol for the second cycle of the Prospective Studies Collaboration[END_REF][START_REF]Blood cholesterol and vascular mortality by age, sex, and blood pressure: a meta-analysis of individual data from 61 prospective studies with 55,000 vascular deaths[END_REF] Novel methods that enable concurrent adjustment for measurement error both in the exposure of interest and in covariates have been developed for use in the multiple study context of the ERFC, but they are technically demanding and have been described in full elsewhere. [START_REF]Regression dilution methods for meta-analysis: assessing long-term variability in plasma fibrinogen among 27,247 adults in 15 prospective studies[END_REF][START_REF]Correcting for multivariate measurement error by regression calibration in meta-analyses of epidemiological studies[END_REF] Examples of these analyses for fibrinogen are shown in Table 5. Since the within-person correlation of fibrinogen measurements on different occasions is about 0.5, [START_REF]Regression dilution methods for meta-analysis: assessing long-term variability in plasma fibrinogen among 27,247 adults in 15 prospective studies[END_REF] the overall log HR corrected for measurement error in fibrinogen alone is about twice the uncorrected estimate (leading to a corrected HR of 1.96 vs 1.38 uncorrected).

Multivariate correction for measurement error in four confounders makes the log HR for fibrinogen slightly less extreme, as expected because residual confounding is reduced, with an estimated HR of 1.85 per 1g/L higher "usual" (ie, long-term average) fibrinogen level. As methods that correct for regression dilution cannot correct for unmeasured covariates, residual confounding may persist after their use.

Some cohorts in the ERFC have analysed particular risk markers in a nested case-control or casecohort design. Nested case-control studies are analysed with similar methods to those described for cohort studies, but they involve logistic regression. [START_REF] Breslow | Statistical methods in cancer research[END_REF] For individually-matched studies, conditional logistic regression is appropriate. For frequency-matched studies, ordinary logistic regression is used with the matching factors as covariates. Such analyses either provide estimates of HRs (if matched controls were selected to be disease-free at the time the case had an event), or odds ratios (if the selected controls were disease-free at the end of the study). [START_REF] Rodrigues | Case-control designs in the study of common diseases: updates on the demise of the rare disease assumption and the choice of sampling scheme for controls[END_REF] Provided the disease is relatively rare (say fewer than 10% of the study's participants), odds ratios approximate HRs and it is reasonable to combine them in a meta-analysis. For nested case-cohort studies, the analysis should allow for the fact that some members of the randomly selected cohort also become cases. [START_REF] Prentice | A case-cohort design for epidemiologic cohort studies and disease prevention trials[END_REF] A modified PH regression model then provides estimates of Although some participants may have multiple events (eg, two CHD events at separate time points, or a CHD event followed by another type of event, such as a stroke or death from cancer), analyses in the ERFC focus on first events by censoring participants after their first CHD event, after another non-fatal event such as stroke (when cohorts have recorded them), and after death from any cause. The rationale for this approach is that major cardiovascular disease events may disrupt the association between baseline risk factors and subsequent disease risk. The ERFC does not, however, censor individuals at the time of cardiovascular investigations or interventions (such as angiography or coronary bypass operations) or at the diagnosis of angina because the incidence of such occurrences is not recorded reliably enough in sufficient studies.

Sensitivity analyses that implement alternative censoring criteria can assess potential biases that might arise through these decisions on censoring.

The constraints on comprehensiveness in IPD meta-analyses mainly relate to the identification of relevant studies and provision of data. In the ERFC, studies have been identified from publications, extensive literature searches, and correspondence with authors of relevant reports.

The ERFC has included the large majority of Western prospective studies with any relevant exposure markers and >20,000 person-years of follow-up. Hence, although publication and reporting biases are potential concerns in all meta-analyses, they may be less so in the ERFC.

DISCUSSION

The statistical methods used in the ERFC have been explicitly described and illustrated in this report to facilitate their adoption by others; example programs in Stata [START_REF]Stata: data analysis and statistical software[END_REF] A common practical problem in IPD meta-analyses is how to adjust for confounders which are measured only in a subset of the studies. For the fibrinogen example, age and four other confounders (Table 2) were measured in all participants in all studies. However, additional confounders, such as lipid fractions (HDL-C, LDL-C and triglycerides), were available in only about half of the studies. [START_REF]Plasma fibrinogen and the risk of major cardiovascular diseases and nonvascular mortality: an individual participant meta-analysis[END_REF] More comprehensive adjustment for confounding can only be easily achieved by restricting the dataset to the latter studies, but such restriction omits information on partial adjustment from the other studies. We have previously described an approach that uses the partially adjusted HRs, which can be estimated in all studies, and the more comprehensively adjusted log HRs, which can be estimated only in a subset of studies, in a bivariate metaanalysis. [START_REF]Systematically missing confounders in individual participant data meta-analysis of observational cohort studies[END_REF] This approach acknowledges the correlations between the partially and the more comprehensively adjusted log HRs within studies where both can be estimated, but uses the full dataset to contribute to the estimation of a combined more comprehensively adjusted log HR.

An unresolved issue concerns the estimation of a possibly non-linear exposure-risk relationship when the exposure is measured with error. Homogeneous measurement error, with a variance that does not depend on level of the exposure, will tend to make a non-linear association appear more linear. Conversely, measurement error that, for example, increases with level of the exposure will make a linear association appear non-linear. Characterising the shape of the underlying exposure-disease relationship, while taking into account possibly heterogeneous measurement error, is not well-studied, especially in the context of IPD meta-analysis. [START_REF] Carroll | Measurement error, instrumental variables and corrections for attenuation with applications to meta-analyses[END_REF] One approach may be to model the underlying association using fractional polynomials or splines, while carefully estimating measurement error variance as a function of exposure level.

As distinct from characterising the shape, magnitude and independence of associations between risk factors and disease (which may be relevant to judgements about an exposure's potential aetiological relevance), IPD meta-analyses of multiple studies can provide additional useful information. For example, we have previously described the ERFC's approach to characterising the cross-sectional correlates (and, hence, potential determinants) of risk markers. [START_REF]Associations of plasma fibrinogen levels with established cardiovascular risk factors, inflammatory markers and other characteristics: individual participant meta-analysis of 154 211 adults in 31 prospective studies[END_REF] Although

F o r R e v i e w O n l y
16 this paper has not addressed issues related to risk prediction (ie, the extent to which measuring an additional exposure could better identify the risk of disease outcomes for individuals), there is considerable interest in the use of information from multiple prospective studies to help inform risk stratification and/or screening strategies. A separate literature exists which involves discussion of how the "area under an ROC curve" can be adapted for time-to-event data [START_REF] Harrell | Multivariable prognostic models: issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors[END_REF] and the extent to which individuals are re-classified into risk groups that would affect the subsequent intervention offered. [START_REF] Cook | Use and misuse of the receiver operating characteristic curve in risk prediction[END_REF][START_REF] Pencina | Evaluating the added predictive ability of a new marker: from area under the ROC curve to reclassification and beyond[END_REF] We have adapted and illustrated some of these predictive metrics for use in the multiple study situation, [START_REF]Measures to assess the prognostic ability of the stratified Cox proportional hazards model[END_REF] and further such work comprises a future methodological research agenda.

Increasing numbers of IPD meta-analyses of observational data are being conducted in order to enhance the statistical power and detail of epidemiological studies. The scientific value of such approaches has now been demonstrated in relation to a variety of exposures and disease outcomes in many different consortia, exemplified by the Prospective Studies Collaboration, [START_REF]Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF] the Asia Pacific Cohort Studies Collaboration, [START_REF]Determinants of cardiovascular disease in the Asia Pacific region: protocol for a collaborative overview of cohort studies[END_REF] the Breast Cancer Genetics Linkage Consortium, [START_REF] Thompson | Cancer Incidence in BRCA1 mutation carriers[END_REF] the Collaborative Group on Hormonal Factors in Breast Cancer, [START_REF] Beral | Breast cancer and abortion: collaborative reanalysis of data from 53 epidemiological studies, including 83,000 women with breast cancer from 16 countries[END_REF] the US Pooling Project of Prospective Studies of Diet and Cancer [START_REF] Sa | Methods for pooling results of epidemiologic studies[END_REF] and the GENOMOS Genetic Markers for Osteoporosis Consortium. [START_REF] Uitterlinden | The association between common vitamin D receptor gene variations and osteoporosis: a participant-level meta-analysis[END_REF] The statistical methods developed here can be used to address the needs of such analyses. Appropriate meta-analytical methods may also have applications to analyses of large purpose-designed multi-centre prospective observational studies, such as the pan-European EPIC study, [START_REF] Bingham | Diet and cancer--the European Prospective Investigation into Cancer and Nutrition[END_REF] UK Biobank [START_REF] Uk Biobank | The UK Biobank sample handling and storage protocol for the collection, processing and archiving of human blood and urine[END_REF][START_REF]UK Biobank protocol[END_REF] and the subsequent planned meta-analysis of such studies. Interactions between baseline fibrinogen (g/L) and potential effect modifiers for risk of CHD: differences in log hazard ratios adjusted for the main effects of baseline age, smoking, total cholesterol, systolic blood pressure and body mass index. Hazard ratio for CHD per 1 g/L higher baseline fibrinogen Combined log hazard ratios with 95% CIs based on floating absolute risks for the relationship between baseline fibrinogen (g/L) and CHD risk, plotted against mean baseline fibrinogen in fifths. From multivariate random-effects meta-analysis, adjusted for a linear effect of age at baseline in each study separately.

Potential effect modifier

  Statistical methods for the time-to-event analysis of individual participant data from multiple epidemiological studies Meta-analysis of individual participant time-to-event data from multiple prospective epidemiological studies enables detailed investigation of exposure-risk relationships, but involves a number of analytical challenges.

  exposure-risk relationships based on individual time-to-event data from multiple studies enhances the detail and power of epidemiological analyses A 2-step meta-analysis method is proposed to combine study-specific associations estimated using Cox regression These methods allow investigation of the appropriate exposure scale, adjustment for confounders, and checking the proportional hazards assumption Within-study and between-study information for interactions need to be distinguished More technically demanding issues include adjustment for measurement error and within-person variation, and handling confounders which are not measured in all studies

[ 1 χ

 1 log1.57-log1.38]/log1.57) of the effect is 'explained' by the observed values of these confounders. The change in the respective Wald 2 statistics reflects a slight decrease in the strength of evidence for an association.[START_REF]Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies[END_REF]

2 2ˆ

 2 ; where ~(0, )

14 log

 14 HRs with robust standard errors,[START_REF] Barlow | Analysis of case-cohort designs[END_REF] although this modification generally has only a small effect.

Figure 1 Study

 1 Figure 1

32 Figure 3

 323 Figure 3

  This exposition initially assumes that all data derive from prospective cohorts; other designs are addressed towards the end of the paper. The main analyses are based on Cox proportional hazards (PH) models,[START_REF] Cox | Regression models and life tables[END_REF] estimated for each study separately. The PH models are stratified by sex and, if applicable, randomised group. So separately for each study s=1…S, with strata k=1…K s (for most studies K s =2 just for the two sexes) and individuals i=1…n s with exposure of interest E si and other covariates X si , the hazard at time t after baseline is modelled as:

	METHODS AND ILLUSTRATIVE ANALYSES
	Principal meta-analysis methods
	F o
	r
	R e v i e w
	O n
	l y
	of follow-up. Across the 31 studies, the number of CHD events ranged from 17 to 1474 and
	follow-up from 4 to 33 years; the crude mean fibrinogen was 3.02g/L (pooled within-study SD
	0.65g/L).
	5

we have illustrated the statistical methods used in the ERFC by analysis of plasma fibrinogen concentration and the risk of coronary heart disease (CHD) in the Fibrinogen Studies Collaboration dataset involving individual data on 154,211 participants from 31 prospective studies. CHD is defined as first non-fatal myocardial infarction or coronary death in those without known cardiovascular disease at the initial examination. 10 A total of 7118 CHD events occurred during an average of 9 years 0 log( ( | , )) log

Table 1 .

 1 The study-

	specific HRs are shown in Figure 1. The random-effects combined HR exp(β) is estimated as
	1.57 (95% CI 1.47 to 1.67) per 1g/L higher baseline fibrinogen concentration, and an I 2 of 64%
	(95% CI 48% to 76%) indicates substantial heterogeneity across studies (test for heterogeneity,
	P<0.0001). By comparison, a fixed-effects meta-analysis estimate gives a lower point estimate
	of 1.52 with a narrower 95% CI of 1.47 to 1.57.
	The above estimates and CIs relate to the overall mean HR across all studies. Also of interest is
	the range of true HRs across studies, representing those in different contexts or populations. It
	can be expressed by the 95% prediction interval for the true HR in a new study, and estimated
	from the random-effects meta-analysis by	ˆˆêxp( β	±	t	S	-	2	Var	( ) β τ +	2	)	where t is the 2.5-
	percentile of a t-distribution and S the number of studies. 12 In the case of the fibrinogen data,
	this 95% prediction interval is 1.18 to 2.08. Because of the presence of heterogeneity, this

Table 2 .

 2 Empirical comparison of alternatives as sensitivity analyses is useful to check for adequate age adjustment. In principle similar considerations apply to other covariates, but in practice the use of linear terms is usually sufficient unless the covariates are both highly prognostic and substantially correlated with the exposure of interest. One important practical problem often encountered is that not all studies measure all the desired confounders; an approach to this situation is described in the Discussion. The ERFC's 2-step approach allows the confounding effects (γ γ γ γ s in model 1) to be different in each study. Examples of age and other confounder adjustments for fibrinogen dataset are given in In this case, linear adjustment for age at baseline appears to be adequate, because

	covariates (especially sex).
	F o
	r
	R e v i e w
	O n
	l y
	9

For linear terms, age-at-baseline in a PH model is equivalent to including current age as a time-dependent variable, but the latter is computationally more difficult to fit. Assuming a simple linear term for age at baseline may, however, be inadequate, resulting in residual confounding. Alternatives include adjustment or stratification by age categories at baseline, and inclusion of polynomial terms and interactions with other more complex forms of adjustment hardly change the results. No precision is lost by stratification using narrow age bands. The overall HR for fibrinogen is reduced towards unity on adjusting for four additional covariates (last row of Table

Table 1

 1 ZUTE: EJM Feskens MD, Division of Human Nutrition, Wageningen University, Wageningen, Netherlands; D Kromhout PhD, Division of Human Nutrition, Wageningen University, Netherlands. Coordinating Centre: R Collins FMedSci, University of Oxford, UK; E Di Angelantonio MD, University of Cambridge, UK; S Erqou MD, University of Cambridge, UK; S Kaptoge PhD, University of Cambridge, UK; S Lewington DPhil, University of Oxford, UK; L Orfei MSc, University of Cambridge, UK; L Pennells MSc, University of Cambridge, UK; PL Perry MBChB, University of Cambridge, UK; KK Ray MD, University of Cambridge, UK; N Sarwar PhD, University of Cambridge, UK; M Alexander MPhil, University of Cambridge, UK; A Thompson PhD, University of Cambridge, UK; SG Thompson DSc, MRC Biostatistics Unit, UK; M Walker PhD, University of Cambridge, UK; S Watson MMath, University of Cambridge, UK; F Wensley MSc, University of Cambridge, UK; IR White MSc, MRC Biostatistics Unit, UK; AM Wood PhD, University of Cambridge, UK; J Danesh FRCP, University of Cambridge, UK (principal investigator). Methodological work in the ERFC has been supported by specific grants from the UK Medical ResearchCouncil. The ERFC Coordinating Centre is underpinned by a programme grant from the British Heart Foundation and supported by the BUPA Foundation and unrestricted educational grants from GlaxoSmithKline. A variety of sources have supported recruitment, follow-up and laboratory measurements in the cohorts contributing to the ERFC. Investigators of several of these studies have contributed to a list naming some of these funding sources, which can be found at http://www.phpc.cam.ac.uk/MEU/.Combined hazard ratios for the relationship between baseline fibrinogen (g/L) and CHD risk, adjusted for a linear effect of age at baseline in each study separately.

	62

Log hazard ratios per SD increase Random-effects meta-analysis

	Untransformed fibrinogen	1.34 (1.29 to 1.40)	0.294 (0.022)	0.008	<0.0001	64% (48, 76)
	Log fibrinogen	1.38 (1.32 to 1.45)	0.325 (0.025)	0.010	<0.0001	65% (48, 76)
	Study-specific SD score fibrinogen	1.34 (1.29 to 1.40)	0.292 (0.021)	0.007	<0.0001	63% (45, 75)
	Study-specific SD score log fibrinogen	1.37 (1.31 to 1.44)	0.316 (0.024)	0.009	<0.0001	64% (47, 76)
	Untransformed fibrinogen:
	Quadratic term for fibrinogen	0.96 (0.91 to 1.01)	-0.045 (0.027)	0.007	0.013	40% (7, 61)
	NA not applicable

Table 3

 3

Table 4

 4 Non-proportional hazards (PH) for CHD risk assessed by the interaction of baseline fibrinogen (g/L) and time (years). The models include adjustment for age at baseline as a linear term.

	Method		Estimated non-	χ 2 test	Heterogeneity
			PH parameter
			ξ (SE)	χ 2 (df)	p-value	I 2 (95% CI)
	Summed	2 1 χ statistics of non-PH parameter from	NA	24 (31)	0.80	NA
	each study
	Summed	2 1 χ statistics from tests of Schoenfeld	NA	21 (31)	0.90	NA
	residuals in each study
	Random-effects meta-analysis of study-specific	0.0016 (0.0045)	0.12 (1)	0.73	0% (0, 40)
	non-PH parameters
	NA not applicable

Table 5

 5 Combined hazard ratios for CHD per 1g/L increase in fibrinogen, corrected for measurement error. All analyses are adjusted for age at baseline, sex, smoking, total cholesterol, systolic blood pressure and body mass index.

	Measurement error correction	Hazard ratio	Log hazard ratio (SE)
		(95% CI)
	No measurement error correction	1.38 (1.31 to 1.45)	0.320 (0.026)
	Measurement error in fibrinogen	1.96 (1.76 to 2.17)	0.672 (0.053)
	Measurement error in fibrinogen, smoking, total	1.85 (1.66 to 2.06)	0.617 (0.055)
	cholesterol, systolic BP, and body mass index

18 National Institute for Health and Welfare, Finland; P Jousilahti MD, National Institute for Health and Welfare, Finland; E Vartiainen MD, National Institute for Health and Welfare, Finland; KIHD: JT Salonen MD, University of Kuopio, Finland; T-P Tuomainen MD, University of Kuopio, Finland; LASA: DJH Deeg MD, VU University Medical Centre, The Netherlands; JL Poppelaars, VU University Medical Centre, The Netherlands; LEADER: TW Meade FMedSci, London School of Hygiene and Tropical Medicine, UK; JA Cooper MSc, University College London, UK; MALMO: B Hedblad MD, Lund University, Sweden; G Berglund MD, Lund University, Sweden; G Engstrom; MD, Lund University, Sweden; MCVDRFP: WMM Verschuren PhD, National Institute of Public Health and the Environment, The Netherlands; A Blokstra MSc, National Institute for Public Health and the Environment, The Netherlands; MESA: M Cushman MD, University of Vermont, USA; S Shea MD, Columbia University, USA; MOGERAUG1, MOGERAUG2, MOGERAUG3: A Döring MD, German Research Center for Environmental Health, Germany; W Koenig MD, University of Ulm Medical Center, Germany; C Meisinger MD, German Research Center for Environmental Health, Germany; W Mraz MD, Klinikum der Universität München, Institute of Clinical Chemistry, Munich, Germany; MORGEN: WMM Verschuren PhD, National Institute of Public Health and the Environment, The Netherlands; A Blokstra MSc, National Institute for Public Health and the Environment, The Netherlands; H Bas Bueno-de-Mesquita PhD, National Institute for Public Health and the Environment, The Netherlands; MOSWEGOT: A Rosengren MD, Göteborg University, Sweden; G Lappas MD, Göteborg University, Sweden; MRFIT: LH Kuller MD, University of Pittsburgh, USA; G Grandits MS, University of Minnesota, USA; NCS: R Selmer PhD, Norwegian Institute of Public Health, Norway; A Tverdal PhD, Norwegian Institute of Public Health, Norway; W Nystad PhD, Norwegian Institute of Public Health, Norway; NHANES I, NHANES II, NHANES III: R Gillum MD, Centers for Disease Control and Prevention, USA; M Mussolino PhD, National Institutes of Health, USA; NHS: E Rimm ScD, Harvard University, USA; S Hankinson ScD, Harvard School of Public Health, USA; JE Manson MD, Harvard Medical School, USA; JK Pai ScD, Brigham and Women's Hospital, USA; NPHS I: TW Meade FMedSci, London School of Hygiene and Tropical Medicine, UK; JA Cooper MSc, University College London, UK; C Knottenbelt, MD; NPHS II: JA Cooper MSc, University College London, UK; KA Bauer MD, Harvard Medical School, USA; OSAKA: S Sato MD, Osaka Medical Center for Health Science and Promotion, Japan; A Kitamura MD, Osaka Medical Center for Health Science and Promotion, Japan; Y Naito MD, Mukogawa Women's University, Japan; H Iso MD, Osaka University, Japan; OSLO: I Holme PhD, Oslo University Hospital, Norway; R Selmer PhD, Norwegian Institute of Public Health, Norway; A Tverdal PhD, Norwegian Institute of Public Health, Norway; W Nystad PhD, Norwegian Institute of Public Health, Norway; OYABE: H Nakagawa MD, Kanazawa Medical University, Japan; K Miura MD, Shiga University of Medical Science, Japan; PARIS1: P Ducimetiere PhD, INSERM, France; X Jouven MD, INSERM, France; PRHHP: CJ Crespo DrPH, Portland State University, USA; MR Garcia Palmieri MD, University of Puerto Rico, USA; PRIME: P Amouyel MD, Institut Pasteur de Lille, France; D Arveiler MD, Université de Strasbourg, France; A Evans MD, The Queens University of Belfast, UK; J Ferrieres MD, University of Toulouse, France; PROCAM: H Schulte PhD, Assmann-Stiftung für Prävention, Germany; G Assmann FRCP, Assmann-Stiftung für Prävention, Germany; PROSPER: J Shepherd MD, Glasgow Royal Infirmary, UK; CJ Packard DSc, University of Glasgow, UK; N Sattar FRCPath, University of Glasgow, UK; I Ford PhD, University of Glasgow, UK; QUEBEC: B Cantin MD, Institut de Cardiologie de Québec, Hôpital Laval, Canada; J-P Després PhD, Centre de recherche de l'Institut universitaire de cardiologie et de pneumologie de Québec, Canada; GR Dagenais MD, Institut universitaire de cardiologie et pneumologie de Québec, Canada; RANCHO: E Barrett-Connor MD, University of California, USA; DL Wingard PhD, University of California, USA; R Bettencourt MS, University of California, USA; REYK: V Gudnason MD, University of Iceland, Iceland; T Aspelund PhD, University of Iceland, Iceland; G Sigurdsson, MD, University of Iceland, Iceland; B Thorsson MD, Icelandic Heart Association, Iceland; RIFLE: M Trevisan MD, Nevada System of Higher Education, USA; ROTT: J Witteman PhD, Erasmus MC, The Netherlands; I Kardys MD, Erasmus MC, the Netherlands; M Breteler MD, Erasmus MC, The Netherlands; A Hofman MD, Erasmus MC, The Netherlands; SHHEC: H Tunstall-Pedoe MD, University of Dundee, UK; R Tavendale PhD, University of Dundee, UK; GDO Lowe DSc, University of Glasgow, UK; M Woodward PhD, Mount Sinai School of Medicine, USA; SPEED: Y Ben-Shlomo PhD, University of Bristol, UK; G Davey-Smith MD, University of Bristol, UK; SHS: BV Howard PhD, Medstar Research Institute, USA; Y Zhang MD, University of Oklahoma Health Sciences Center, USA; J Umans MD, Georgetown University Medical Centre, USA; TARFS: A Onat MD, Istanbul University, Turkey; TPT: TW Meade FMedSci, London School of Hygiene and Tropical Medicine, UK; TROMSØ: T Wilsgaard PhD, University of Tromsø, Norway; ULSAM: E Ingelsson MD, Karolinska Institutet, Sweden; L Lind PhD, Department of Medical Sciences, Uppsala University, Uppsala, Sweden; V Giedraitis PhD, Department of Public Health and Geriatrics, Uppsala University Hospital, Uppsala, Sweden; L Lannfelt MD, Department of Public Health and Geriatrics, Uppsala University Hospital, Uppsala, Sweden; USPHS: JM Gaziano MD, Brigham and Women's Hospital, USA; P Ridker MD, Brigham and Women's Hospital, USA; USPHS2: JM Gaziano MD, Brigham and Women's Hospital, USA; P Ridker MD, Brigham and Women's Hospital, USA; VHMPP: H Ulmer PhD, Innsbruck Medical University, Austria; G Diem MD, Agency for Preventive and Social Medicine, Austria; H Concin MD, Agency for Preventive and Social Medicine, Austria; VITA: A Tosetto MD, San Bortolo Hospital, Italy; F Rodeghiero MD, San Bartolo Hospital, Italy; WHI: S Wassertheil-Smoller PhD, Albert Einstein College of Medicine, New York, USA; JE Manson MD, Harvard Medical School, USA; WHITE I: M Marmot FMedSci, University College London, UK; R Clarke MD, University of Oxford, UK; R Collins FMedSci, University of Oxford, UK; WHITE II: E Brunner PhD, University College London, UK; M Shipley MSc, University College London, UK; WHS: P Ridker MD, Brigham and Women's Hospital, USA; J Buring ScD, Brigham and Women's Hospital, USA; WOSCOPS: J Shepherd MD, Glasgow Royal Infirmary, UK; SM Cobbe FMedSci, BHF Glasgow Cardiovascular Research Centre, UK; I Ford PhD, University of Glasgow, UK; M Robertson BSc, University of Glasgow, UK; XIAN: Y He MD, Chinese PLA General Hospital, China; ZARAGOZA: A Marín Ibañez MD, San Jose Norte Health Centre,

List of Authors (Appendix 1 in reference 5 lists the study acronyms):

Writing Committee: SG Thompson DSc, MRC Biostatistics Unit, UK; S Kaptoge PhD, University of Cambridge, UK; IR White MSc, MRC Biostatistics Unit, UK; AM Wood PhD, University of Cambridge, UK; PL Perry MBChB, University of Cambridge, UK; J Danesh FRCP, University of Cambridge, UK.

Author contributions: All authors contributed to the conception and design of this work; SGT and IRW proposed the methods; SK and PLP analysed the data; SGT drafted the paper; all authors critically revised the paper and approved the final version. SGT is the guarantor. The authors declare no conflicts of interest.

Investigators