L Galtchouk
email: leonid.galtchouk@math.unistra.fr

S Pergamenshchikov
email: serge.pergamenchtchikov@univ-rouen.fr

Uniform concentration inequality for ergodic diffusion processes observed at discrete times. *

Keywords: Ergodic diffusion processes, Markov chains, Tail distribution, Upper exponential bound, Concentration inequality. AMS 2000 Subject Classifications: 60F10

In this paper a concentration inequality is proved for the deviation in the ergodic theorem in the case of discrete time observations of diffusion processes. The proof is based on the geometric ergodicity property for diffusion processes. As an application we consider the nonparametric pointwise estimation problem for the drift coefficient under discrete time observations.

Introduction

We consider the process (y t ) t≥0 governed by the stochastic differential equation dy t = S(y t ) dt + σ(y t )dW t , 0 ≤ t ≤ T , (1.1) where (W t , F t ) t≥0 is a standard Wiener process, y 0 is a initial condition and ϑ = (S, σ) are unknown functions. For this model we consider the pointwise estimation problem for the function S at a fixed point x 0 ∈ R (i.e. S(x 0 )), on the basis of the discrete time observations of the process (1.1), i.e.

(y

t j ) 1≤j≤N , (1.2)
where t j = jδ, N = [T /δ] and δ is some positive fixed observation frequency which will be specified later. Usually, for this problem one uses kernel estimators S N (x 0 ) defined as

S N (x 0 ) = N k=1 ψ h,x 0 (y t k ) ∆y t k N k=1 ψ h,x 0 (y t k ) ∆t k , ψ h,x 0 (y) = 1 h Ψ y -x 0 h , (1.3)
where Ψ(y) is a kernel function which equals to zero for |y| ≥ 2 and will be specified later, 0 < h < 1 is a bandwidth, ∆y t k = y t k -y t k-1 and ∆t k = δ. Main difficulty in this estimator is that the denominator is random. Therefore, to obtain the convergence rate for this estimator we have to study the behavior of the denominator, more precisely, one needs to show that N k=1 ψ h,x 0 (y t k )∆t k ≈ π ϑ (ψ h,x 0 )hT as T → ∞ , where π ϑ (ψ h,x 0 ) = R ψ h,x 0 (y) q ϑ (y) dy (1.4) and q ϑ is the ergodic density defined in (2.2). Unfortunately, the ergodic theorem does not permit to obtain this kind of result because the times t k and the bandwidth h depend on T . Usually one obtains such properties through concentration inequalities for the deviation in the ergodic theorem, i.e. one needs to study the limit behavior of the deviation

D T (φ) = N k=1 φ(y t k ) -π ϑ (φ) ∆t k (1.5)
for some functions φ which can be dependent on T , for example, φ(•) = ψ h,x 0 (•).

More precisely, we need to show, that for any ε > 0 and for any m > 0, uniformly over ϑ, lim

T →∞ T m P ϑ |D T (ψ h,x 0 )| > εT = 0 , (1.6)
where P ϑ is the law of the process (y t ) t≥0 under the coefficients ϑ = (S, σ). Usually, to get properties of type (1.6) one needs to establish an exponential inequality for the deviations (1.5).

There are a number of papers devoted to concentration inequalities for functions of independent random variables (we refer the reader to [START_REF] Boucheron | Concentration inequalities using the entropy method[END_REF] and references therein), for functions of dependent random variables (see [START_REF] Dedecker | A new covariance inequality and applications[END_REF], [START_REF] Dedecker | New dependence coefficients. Examples and applications to statistics[END_REF], [START_REF] Maume-Deschamps | Concentration inequalities and estimation of conditional probabilities[END_REF]). For Markov chains such inequalities were obtained in [START_REF] Bertail | Sharp bounds for the tails of functionals of Markov chains[END_REF]. For continuous time Markov processes an exponential concentration inequality was obtained in [START_REF] Cattiaux | Deviation bounds for additive functionals of Markov process[END_REF] (see also references therein). Some applications of concentration inequalities to statistics are presented in [START_REF] Massart | Some applications of concentration inequalities to statistics[END_REF]. Concentration inequalities for diffusion processes are given in [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF], [START_REF] Pergamenshchikov | On large deviation probabilities in ergodic theorem for singularly perturbed stochastic systems[END_REF], [START_REF] Yu | On large deviations for diffusion processes with measurable coefficients[END_REF].

For statistical applications, we need uniform upper bounds for the tail distribution over functions φ like to the exponential bounds in [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF]. We can not apply directly the method from [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF], since there it is based on the continuous times version of the Ito formula. In this paper we apply this approach through uniform (over the functions S) geometric ergodicity. We recall (see [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF]), that the geometric ergodicity yields a geometric rate in the convergence

lim t→∞ E ϑ (g(y t )|y 0 = x) = π ϑ (g)
for any integrable functions g and any initial value x ∈ R. Here E ϑ denotes the expectation with respect to the distribution P ϑ . In [START_REF] Galtchouk | Geometric ergodicity for families of homogeneous Markov chains[END_REF] through the Lyapunov functions method it is shown that the process (1.1) is geometrically ergodic uniformly over functions ϑ = (S, σ) from the functional class Θ defined in (2.1).

The paper is organized as follows. In the next section we formulate the main results. In Section 3 we introduce all the necessary parameters. In Section 4 we show a concentration inequality in ergodic theorem for the continuous observations of the process (1.1). In Section 5 we announce the uniform geometric ergodic property for the process (1.1). In Section 6 we give the Burkhölder inequality for dependent random variables. In Section 7 we prove all main results. The Appendix contains the proofs of some auxiliary results.

Main results

First we describe the functional class Θ for functions ϑ = (S, σ) defined in [START_REF] Galtchouk | Geometric ergodicity for families of homogeneous Markov chains[END_REF]. We start with some real numbers x * ≥ 1, M > 0 and L > 1 for which we denote by Σ L,M the class of functions S from C 1 (R) such that sup

|x|≤x * |S(x)| + | Ṡ(x)| ≤ M and -L ≤ inf |x|≥x * Ṡ(x) ≤ sup |x|≥x * Ṡ(x) ≤ -L -1 .
Furthermore, for some fixed numbers 0 < σ min ≤ σ max < ∞, we denote by V the class of the functions σ from C 2 (R) such that

σ min ≤ inf x∈R min (|σ(x)| , | σ(x)| , |σ(x)|) ≤ sup x∈R max (|σ(x)| , | σ(x)| , |σ(x)|) ≤ σ max . Finally, we set Θ = Σ L,M × V . (2.1)
It should be noted (see, for example, [START_REF] Gihman | Stochastic differential equations[END_REF]), that for any ϑ = (S, σ) ∈ Θ, the equation (1.1) has a unique strong solution which is a ergodic process with the invariant density q ϑ defined as

q ϑ (x) = R σ -2 (z) e S(z) dz -1 σ -2 (x) e S(x) , (2.2)
where S(x) = 2

x 0 S 1 (v)dv and S 1 (x) = S(x)/σ 2 (x). Now we describe the functional classes for the functions φ. First, for any parameters ν 0 > 0 and ν 1 > 0 we set

V ν 0 ,ν 1 = {φ ∈ C(R) : |φ| 1 ≤ ν 0 , |φ| * ≤ ν 1 } , (2.3)
where |φ| 1 = R |φ(y)| dy and |φ| * = sup y∈R |φ(y)|.

For any function φ from C 2 (R) we denote by L ϑ (φ) the generator operator for the process (1.1), i.e.

L ϑ (φ)(y) = S(y) φ(y) + σ 2 (y) 2 φ(y) .
Using this notation, we set

µ(φ) = sup ϑ∈Θ L ϑ (φ) * and µ(φ) = sup ϑ∈Θ | π ϑ (φ)| , (2.4)
where π ϑ (φ) = π ϑ (L ϑ (φ)). Now for any vector ν = (ν 0 , ν 1 , ν 2 , ν 3 , ν 4 ) from R 5 + we set

K ν = φ ∈ V ν 0 ,ν 1 : φ * ≤ ν 2 , µ(φ) ≤ ν 3 , µ(φ) ≤ ν 4 .
(2.5)

Theorem 2.1. For any vector ν = (ν 0 , ν 1 , ν 2 , ν 3 , ν 4 ) from R 5 + and any 0 < δ ≤ 1 there exist positive parameters z 0 = z 0 (δ, ν), γ = γ(δ, ν) and κ = κ(δ, ν) such that sup

T ≥1 sup z≥z 0 sup φ∈K ν sup ϑ∈Θ e z min(κz , γ) P ϑ |D T (φ)| ≥ z √ N ≤ 4 , (2.6)
where the parameters z 0 , γ and κ are defined in (3.5)-(3.6).

Now we apply this theorem to the pointwise estimation problem, i.e. for the functions ψ h,x 0 defined in (1.3). To this end we assume that the frequency δ in the observations (1.2) is of the following form

δ = δ T = 1 T l T , (2.7)
where the function l T is such that for any m > 0 lim

T →∞ l T T m = 0 and lim T →∞ l T ln T = +∞ . (2.8)
Further, let ǫ = ǫ T be a positive function satisfying the following properties lim

T →∞ ǫ T = 0, lim T →∞ l T T ǫ T = 0 and lim T →∞ ǫ 5 T l T ln T = +∞ . (2.9)
We can take, for example, for some ι > 0 l T = ln 1+6ι (T + 1) and ǫ T = 1 ln ι (T + 1) .

Theorem 2.2. Assume that the kernel function Ψ in (1.3) is two continuously differentiable. Moreover, assume that the functions δ T and l T satisfy the properties (2.7) and (2.9). Then there exist coefficients

z * 0 = z * 0 (Ψ) > 0 and γ * = γ * (Ψ) > 0 such that lim sup T →∞ e aγ * l T sup a≥a * sup h≥T -1/2 sup ϑ∈Θ P ϑ |D T (ψ h,x 0 )| ≥ a T ≤ 4 , (2.10)
where a * = z * 0 /l T , the parameters z * 0 and γ * are given in Section 3. This theorem implies immediately the following Corollary 2.1. Assume, that all conditions of Theorem 2.2 hold. Then, for any m > 0, lim sup

T →∞ T m sup a≥a * sup h≥T -1/2 sup ϑ∈Θ P ϑ |D T (ψ h,x 0 )| ≥ a T = 0 .

Now we study the deviation (1.5) for the function

χ h,x 0 (y) = 1 h χ y -x 0 h , (2.11)
where χ(y) = 1 {|y|≤1} .

Theorem 2.3. Assume that the parameter δ has the form (2.7). Then, for any m > 0, and for any function ǫ T , satisfying the condtions (2.8) and (2.9) lim

T →∞ T m sup h≥T -1/2 sup ϑ∈Θ P ϑ |D T (χ h,x 0 )| ≥ ǫ T T = 0 .
(2.12)

Remark 2.1. It is well known that to obtain the optimal rate in the estimation problem for a differentiable function S in the process (1.1) one needs to choose the bandwidth h as

h = T -1/(2α+1)
with the regularity parameter α ≥ 1. This means that, really for the pointwise estimation problem, h ≥ T -1/3 . But in the quadratic risk one needs to choose the parameter h as h = T -1/2 (see [START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF]- [START_REF] Galtchouk | Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes[END_REF], [START_REF] Galtchouk | Adaptive sequential estimation for ergodic diffusion processes in quadratic metric[END_REF]).

Parameters

In this section we introduce all necessary constants and parameters. First, we set

υ 1 = e β 2 1 /(4β 2 )
and

υ 2 = π/β 2 e β 2 1 /(4β 2 ) , (3.1)
where β 1 = 2M/σ 2 min and β 2 = 1/Lσ 2 max . Moreover, as we will see in Appendix, the ergodic density (2.2) is uniformly bounded by q * , where

q * = σ 2 max σ 2 min e β 1 x * +β 2 1 /(4β 2 ) . (3.2)
Now we set

r = r(ν 0 ) = 2ν 0 σ 2 min (1 + υ 1 + q * (x * + υ 2 )) e x * β 1 , (3.3)
where the parameter ν 0 is defined in (2.3). Now using this function we set

κ 0 = κ 0 (ν 0 ) = 1 108 r 2 (3ρ 2 + y 2 0 + 2σ 2 max ) (3.4) where ρ = max |y 0 | , σ max √ L , 2(x * + ML) .
Now for any δ > 0 and any parameter vector ν = (ν 0 , ν 1 , ν 2 , ν 3 , ν 4 ) from R 5

+ we set

z 0 = z 0 (δ, ν) = δ 3/2 max 2c * 1 ν 3 , 2c * 2 ν 2 , ν 4 T 1/2 , ν 1 T -1/2 , τ = τ (δ, ν) = δ 3/2 max c * 1 ν 3 , c * 2 ν 2 , (3.5)
where

c * 1 = 2e κ+1 R(1 + ρ) κ and c * 2 = √ 2eσ max .
The parameters R and κ are defined in Theorem 5.1. Finally we set

γ = 1 4τ and κ = κ(δ, ν) = 9κ 0 (1 -δ) 64δ . (3.6)
Now we set

M 1 = M + L (x * + |x 0 | + 2) . (3.7)
Now for any inegrated two times continuously differentiable R → R function Ψ we define

k * (Ψ) = max | Ψ| 1 , | Ψ| 1 , Ψ * , Ψ * , Ψ * . (3.8)
Using this operator we define the parameters

z * 0 = λ 1 k * (Ψ) and τ * = λ 2 k * (Ψ) , (3.9)
where

λ 1 = max 2c * 1 M 1 , 2c * 2 , M 1 q * , 1 and λ 2 = max c * 1 M 1 , c * 2 .
Finally, we set

γ * = 1 4τ * .
(3.10)

Continuous observations

In this section we study the deviation in the ergodic theorem for the continuous observation case, which in this case is defined as

∆ T (φ) = 1 √ T T 0 (φ(y t ) -π ϑ (φ)) dt , (4.1)
where φ is any integrated function, i.e. |φ| 1 < ∞. sup

φ∈V ν 0 ,ν 1 sup ϑ∈Θ P ϑ (|∆ T (φ)| ≥ z) ≤ 2 , (4.2)
where the parameter κ 0 is given in (3.4).

Proof. Similarly to [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF] firstly we show that the deviation (4.1) has an exponential moment, i.e. we show that for the parameter κ 0 sup

T ≥1 sup ϑ∈Θ E ϑ e κ 0 ∆ 2 T (φ) ≤ 2 . (4.3)
Indeed, to show this inequality we need to estimate the expectation of any even power for the deviation ∆ T (φ). To this end we have to represent this deviation as the sum of a continuous martingale and a negligible term. For this one needs to find a bounded solution for the following differential equation

vϑ (u) + 2 S(u) σ 2 (u) v ϑ (u) = 2 φ(u) σ 2 (u) , φ(u) = φ(u) -π ϑ (φ) . (4.4)
One can check directly that the function

v ϑ (u) = -2 ∞ u φ(y) σ 2 (y) exp{2 y u S 1 (z)dz} dy (4.5)
yields such a solution. We recall that the function S 1 is defined in (2.2). Moreover, due to Lemma A.2 from Appendix implies this function is uniform bounded. By applying the Ito formula to the function V (y) = y 0 v ϑ (u)du we following representation

T 0 φ(y s )ds = V (y T ) -V (y 0 ) -ζ T , (4.6)
where ζ T = T 0 v ϑ (y s )σ(y s )dw s . Therefore, for any T ≥ 1 through Lemma A.2 we can estimate ∆ T (φ) from above as

|∆ T (φ)| ≤ r|y T | + r|y 0 | + 1 √ T |ζ T | .
Moreover, taking into account (see [START_REF] Sh | Statistics of random processes[END_REF], Lemma 4.11), that for any m ≥ 1,

E ϑ (ζ T ) 2m ≤ (2m -1)!! r 2m σ 2m max T m
, we obtain by Proposition A.1 , that for any m ≥ 1

E ϑ |∆ T (φ)| 2m ≤ 3 2m-1 r 2m (E ϑ |y T | 2m + |y 0 | 2m ) + E ϑ (ζ T ) 2m T m ≤ (3 r) 2m 4(m + 1)(2m -1)!! ρ 2m + y 2m 0 + (2m -1)!! σ 2m max .
Therefore, taking into account the definition of κ 0 , we obtain

E ϑ e κ 0 ∆ 2 T (φ) = 1 + ∞ m=1 κ m 0 m! (3 r) 2m 4(2m + 1)!!ρ 2m + y 2m 0 + (2m -1)!!σ 2m max ≤ 1 + ∞ m=1 κ m 0 (3 r) 2m 4(3ρ 2 ) m + y 2m 0 + 2 m σ 2m max ≤ 1 + ∞ m=1 (1/2) m = 2 .
From here we obtain the inequality (4.3) and by the Chebychev inequality we come to the upper bound (4.2). Hence Proposition 4.1.

Remark 4.1. It should be noted that the inequality (4.2) is shown in [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF] for the process (1.1) with σ = 1. Thus Proposition 4.1 extends teh result from [START_REF] Galtchouk | Uniform concentration inequality for ergodic diffusion processes[END_REF] for any diffusion function σ.

Uniform geometric ergodicity

Here we announce a result on geometric ergodicity obtained in [START_REF] Galtchouk | Geometric ergodicity for families of homogeneous Markov chains[END_REF]. where the parameters R and κ are given in [START_REF] Galtchouk | Geometric ergodicity for families of homogeneous Markov chains[END_REF].

Burkhölder's inequality

In this section we give the following inequality from [START_REF] Dedecker | A new covariance inequality and applications[END_REF], [START_REF] Rio | Théorie asymptotique des processus faiblement dépendants[END_REF].

Proposition 6.1. Let (Ω, F , (F j ) 1≤j≤n , P) be a filtered probability space and (X j , F j ) 1≤j≤n be sequence of random variables such that for some p ≥ 2 Now we represent the deviation D T (φ) as

max 1≤j≤n E |X j | p < ∞ . Define b j,n (p) =   E (|X j | n k=j |E (X k |F j )|) p/2   2/p . Then E | n j=1 X j | p ≤ (2p) p/2   n j=1 b j,n (p)   p/2 . ( 6
D T (φ) = T 0 (φ(y t ) -π ϑ (φ))dt + A 1,T -A 2,T = √ T ∆ T (φ) + A 1,T -A 2,T , (7.2)
where

A 1,T = N j=1 t j t j-1
φ(y t j ) -φ(y t ) dt and A 2,T = T δN (φ(y t ) -π ϑ (φ))dt .

To estimate the term A 1,T we represent through the Ito formula the difference φ(y t j ) -φ(y t ) as

φ(y t j ) -φ(y t ) = t j t L ϑ (φ)(y s ) ds + t j t φ(y s )σ(y s )dW s = π ϑ (φ)(t j -t) + Ψ j (t) + t j t φ(y s )σ(y s )dW s ,
where

Ψ j (t) = t j t ψ(y s ) ds , ω j (t) = t j t φ(y s )σ(y s ) dW s
and ψ(y) = L ϑ (φ)(y) -π ϑ (φ). Now setting

X j = t j t j-1
Ψ j (t) dt and η j =

t j t j-1 ω j (t)dt , we obtain A 1,T = π ϑ (φ) Nδ 2 2 + N j=1 X j + N j=1 η j . (7.3)
To estimate the second term in the right-hand part of (7.3), we make use of the Proposition 6.1.We start with verifying its conditions. Putting F s = σ{y u , 0 ≤ u ≤ s}, we obtain by Theorem 5.1, that for any t ≥ s and for any φ from the functional class (2.5)

|E ϑ (ψ(y t )|F s ) | ≤ µ(φ) R (1 + |y s |) e -κ(t-s) ≤ ν 3 R (1 + |y s |) e -κ(t-s) .
Therefore, for any k > j,

|E ϑ (X k |F t j )| ≤ Re κ (1 + |y t j |) ν 3 δ 2 e -κδ(k-j) . (7.4)
It should be noted also, that the random variables X j are bounded, i.e.

|X j | ≤ ν 3 δ 2 .
To estimatie the probability tail for the sum n j=1 X j we will use the inequality (6.1). For this we need to estmate the coefficients b j,N (p) for any p ≥ 1. From here, taking into account that 1 -e -κδ ≥ κδe -κ and that for p ≥ 2

E ϑ (1 + |y t j |) p/2 2/p ≤ 1 + E ϑ |y t j | p/2 2/p , we can estimate the coefficient b j,N (p) as b j,N (p) ≤ 1 κ R e 2κ ς 2 1 + (E|y t j | p/2 ) 2/p ,
where ς 2 = ν 2 3 δ 3 . Now the inequality (7.1) yields

b j,N (p) ≤ R 1 ς 2 2 + p ≤ R 1 ς 2 2p , where R 1 = 1 κ R e 2κ (1 + ρ) .
Using this in (6.1) we obtain, that for any p > 2,

E ϑ | N k=1 X k | p ≤ (2p) p/2 N p/2 R p/2 1 ς p (2p) p/4 ≤ (2 R 1 ς) p N p/2 p p .
Therefore, by Chebyshev's inequality

P ϑ | N k=1 X k | ≥ z √ N ≤ e p ln(a)+p ln p
with a = 2 R 1 ς/z. Minimizing now the right-hand part over p ≥ 2, we obtain

for z ≥ 4e R 1 ν 3 δ 3/2 P ϑ | N k=1 X k | ≥ z √ N ≤ e -z/ς 1 , (7.5)
where

ς 1 = 2e R 1 ς.
Moreover, note that by the Burkholder-Davis-Gundy inequality, for any

α ≥ 1, E ϑ |ω j (t)| α ≤ (α) α/2 ν α 2 σ α max (t j -t) α/2
. Using this and the the Hölder inequality, we get

E ϑ |η j | α ≤ δ α-1 t j t j-1 E ϑ |ω j (t)| α dt ≤ δ 3α/2 α α/2 ν α 2 σ α max .
Note, that in this case in the right hand of the inequality (6.1)

b j,N = E ϑ |η j | p 2/p .
Therefore, similarly to the inequality (4.5) we find, that for all z ≥ 2ς 2 ,

P ϑ | N k=1 η k | ≥ z √ N ≤ e -z/ς 2 , (7.6)
where ς 2 = √ 2eδ 3/2 ν 2 σ max . Now from (7.3), (4.5)-(4.6) it follows that for z ≥ z 0

P ϑ |A 1,T | ≥ z √ N ≤ P ϑ | N k=1 X k | ≥ z √ N/4 + P ϑ | N k=1 η k | ≥ z √ N/4 ≤ 2 e -z/4τ , (7.7)
when the parameters z 0 and τ are given in (3.5). Moreover, note that due to (2.5) the last term in (7.2) is bounded, i.e.

|A 2,T | ≤ 2δ φ * ≤ 2δν 2 ≤ z 0 √ N /4 .
Finally, from (7.2) for z ≥ z 0 one has

P ϑ (|D T (φ)| ≥ z √ N ) ≤ P ϑ √ T |∆ T (φ)| + |A 1,T | ≥ 3z √ N /4 ≤ P ϑ √ T |∆ T (φ)| ≥ 3z √ N/8 + P ϑ |A 1,T | ≥ 3z √ N /8 .
Taking into account here, that N/T ≥ (1 -δ)/δ for any 0 < δ < 1 and T ≥ 1, we obtain, that

P ϑ (|D T (φ)| ≥ z √ N) ≤ P ϑ |∆ T (φ)| ≥ 3z (1 -δ) 8 √ δ + P ϑ |A 1,T | ≥ 3 8 z √ N .
Therefore, applying here the inequalities (4.2) and (7.7) we come to the upper bound (2.6) with the parameter κ given in (3.6). Hence Theorem 2.1.

Proof of Theorem 2.2

Firstly, note that in this case

|ψ h,x 0 | 1 = |Ψ| 1 , ψ h,x 0 * = 1 h Ψ * and ψh,x 0 * = 1 h 2 Ψ * .
Moreover, taking into account that |S(y)| ≤ M + Lx * + L|y|, we find that sup

|y|≤|x 0 |+2 |S(y)| ≤ M 1 , (7.8)
where M 1 is given in (3.7). Therefore, in view of the fact that 0 < h < 1, we can estimate from above the parametrs (2.4) as

µ(ψ h,x 0 ) ≤ µ * h -3 and µ(ψ h,x 0 ) ≤ µ * h -2 , (7.9)
where

µ * = max Ψ * , Ψ * M 1 and µ * = max | Ψ| 1 , | Ψ| 1 M 1 q * .
Therefore, the function ψ h,x 0 belongs to the class (2.5) with the following parameters

ν 0 = |Ψ| 1 , ν 1 = Ψ * h , ν 2 = Ψ * h 2 , ν 3 = µ * h 3 , ν 4 = µ * h 2 .
Therefore, in this case the coefficient (3.4) equals to κ 0 (|Ψ| 1 ) and the parameters (3.5) can be represented as

z 0 = δ 3/2 h 3 max 2c * 1 µ * , 2c * 2 Ψ * h , µ * hT 1/2 , Ψ * h 2 T -1/2 τ = δ 3/2 h 3 max c * 1 µ * , c * 2 Ψ * h . (7.10)
Therefore, thanks to the condition (2.8) for any

T -1/2 ≤ h ≤ 1 z 0 ≤ l -3/2 T z * 0 and τ ≤ l -3/2 T τ * , (7.11)
where the parameters z * 0 and τ * are given in (3.9). Note now that, by the condition (2.7)

P ϑ |D T (ψ h,x 0 )| ≥ a T ≤ P ϑ |D T (ψ h,x 0 )| ≥ z 1 √ N
where z 1 = a/ l T . The first inequality in (7.11) implies that z 1 ≥ z 0 for all a ≥ a * = z * 0 /l T . Moreover, from the last inequality in (7.11) it follows, that for a ≥ a * min (κz

1 , γ) = min κz 1 , 1 4τ ≥ min κ z * 0 l T l T , l T l T 4τ * .
Taking into account here the definition of κ in (3.6) and the form for δ given by (2.7) we obtain that for sufficiently large T

min κ z * 0 l T l T , l T l T 4τ * = l T l T 4τ * .
Thus, through Theorem 2.1 we come to the inequality (2.10). Hence Theorem 2.2

Proof of Theorem 2.3

First we represent the tail probability as

P ϑ |D T (χ h,x 0 )| ≥ ǫ T T = I 1 + I 2 ,
where

I 1 = P ϑ   N j=1 χ h,x 0 (y t j )∆t j ≤ (π ϑ (χ h,x 0 ) -ǫ T ) T  
and

I 2 = P ϑ   N j=1 χ h,x 0 (y t j )∆t j ≥ (π ϑ (χ h,x 0 ) + ǫ T ) T   .
Let us define now the following smoothing indicator functions

Ψ 1,η (u) = 1 η +∞ -∞ 1 {|z|≤1-η} V z -u η dz and Ψ 2,η (u) = 1 η +∞ -∞ 1 {|z|≤1+η} V z -u η dz ,
where η is a smoothing positive parameter which will be specified later, V is a two times continuously differentiable even R → R function such that V (z) = 0 for |z| ≥ 1 and

1 -1 V (z)dz = 1 .
It is easy to see that, for any y ∈ R and 0

< η ≤ 1/2, Ψ 1,η (u)(y) ≤ χ(y) ≤ Ψ 2,η (y)
and Ψ 2,η (y) = 0 for |y| ≥ 2. Moreover, for the functions

ψ i,h (y) = 1 h Ψ i,η y -z 0 h
using the inequality (A.4), we can estimate the difference between the cooresponding ergodic intergals (1.4) as

|π ϑ (χ h,x 0 ) -π ϑ (ψ i,h )| ≤ 4ηq * .
Therefore, choosing here η = ǫ 2 T we obtain, for sufficiently large T ,

I i ≤ P ϑ |D T (φ i,h )| ≥ ǫ T T /2 .
One can check directly that in this case the operator (3.8) has the following asymptotic (T → ∞) form

k * (Ψ i,η ) = O η -2 .
Therefore, from (3.9) and (7.11) it follows that for T → ∞ and

h ≥ T -1/2 z 0 (φ i,h ) = O η -2 l -3/2 T and τ (φ i,h ) = O η -2 l -3/2 T , with Υ(j, n) = X j n k=j E(X k |F j ) .
Therefore,

h n (t) = p(p -1) n-1 j=1 1 0 E|S j-1 + vX j | p-2 (-vX2 j + G(i, n, t))dv + p(p -1) 1 0 E|S n-1 + vtX n | p-2 t 2 (1 -v)X 2 n dv ,
where G(j, n, t) = Υ(j, n -1) + tX j E(X n |F j ) .

Moreover, we can estimate h n (t) as

h n (t) p2 ≤ n-1 j=1 1 0 E|S j-1 + vX j | p-2 |G(i, n, t)|dv + t 0 E|S n-1 + sX n | p-2 X 2 n ds
Now taking into account that for 0 ≤ t ≤ 1 E|G(j, n, t)| p/2 2/p ≤ b j,n (p) , we obtain by the Hölder inequality

1 0 E|S j-1 + vX j | p-2 |G(i, n, t)| dv ≤ 1 0 h α j (v) b j,n (p)dv ,
where α = 1 -2/p. Therefore,

h n (t) p2 ≤ n-1 j=1 b j,n (p) 1 0 h α j (v)dv + b n,n (p) t 0 h α n (s)ds
Now by the induction assumption for any 1

≤ j ≤ n -1 b j,n (p) 1 0 h α j (v)dv ≤ (2p) (p-2)/2 1 0 B (p-2)/2 j (v) dv b j,n (p) .
Moreover, taking into account that

B j (v) ≤ j-1 i=1 b i,n + vb j,n (p) ,

A.2 Uniform bound for the invariant density

Lemma A.1. The invariant density (2.2) is uniformly bounded:

sup x∈R sup ϑ∈Θ q ϑ (x) ≤ q * < ∞ , (A.4)
where the upper bound q * is given in (3.2).

Proof. First, note that through the definition of Θ we can check directly that for any |x| ≥ x *

2 x 0 S 1 (v) d v ≤ β 1 |x| -β 2 (|x| -x * ) 2 , (A.5)
where the coefficients β 1 and β 2 are given in (3.1). Therefore, taking into account, that for |x| ≥ x *

2 x 0 S 1 (v) d v ≤ β 1 x * ,
we obtain that 2 sup

x∈R x 0 S 1 (v) d v ≤ β 1 x * + β 1 4β 2 .
Estimating now the denominator in (2.2) from below as Taking into account here that sup x∈R σ 2 (x) ≤ σ 2 max we obtain, that for any m ≥ 1 and t ≥ 0 żt (m) ≤ 2mE ϑ y 2m-1 t S(y t ) + m(2m -1)σ 2 max z t (m -1) .

Now we need to estimate from above the function x 2m-1 S(x). Obviously, that for any

K > x * x 2m-1 S(x) ≤ K 2m-1 sup |x|≤K |S(x)|1 {|x|≤K} + x 2m S(x) x 1 {|x|>K} . Taking into account that sup |x|>x * | Ṡ(x)| ≤ L, we obtain, for any x ∈ [x * , K], |S(x)| ≤ |S(x * )| + L|x -x * | ≤ M + L(K -x * ) .
Similarly, we obtain the same upper bound for

x ∈ [-K, -x * ]. Therefore, sup |x|≤K |S(x)| ≤ M + L (K -x * ).
Consider now the case |x| > K. We recall, that sup |x|≥x * Ṡ(x) ≤ -L -1 .

Therefore,

S(x) x ≤ M K - K -x * LK . Choosing K = 2(x * + ML) yields S(x) x ≤ - 1 2L .
Therefore,

x 2m-1 S(x) ≤ K 2m-1 (M + L(K -x * )) - 1 2L x 2m 1 {|x|>K} = K 2m-1 (M + L(K -x * )) + β 2 x 2m 1 {|x|≤K} - 1 2L x 2m ≤ A m - β 2 x 2m , where A m = (2(x * + ML)) 2m-1 2M + x * L + L -1 + 2L 2 M From here it follows, that żt (m) ≤ 2m A m -L -1 m z t (m) + m(2m -1)σ 2 max z t (m -1) .
We can rewrite this inequality as follows

żt (m) = -L -1 mz t (m) + m(2m -1)σ 2 1 z t (m -1) + ψ t ,
where sup t≥0 ψ t ≤ 2m A m . This equality provides x * S 1 (z)dz dy .

z t (m) = z 0 (m)e -mL -1 t + m(2m -1)σ 2 max t 0 e -mL -1 (t-s) z s (m -1)ds + t 0 e -mL -1 (t-s) ψ s ds ≤ m(2m -1)σ 2 max t 0 e -mL -1 (t-s) z s (m -
To estimate this term note that similarly to (A.5) we can obtain that for any y ≥ a ≥ x *

Proposition 4 . 1 .

 41 For any ν 0 > 0 and ν 1 > 0 sup z≥0 e κ 0 z 2 sup T ≥1

Theorem 5 . 1 .

 51 There exist some constants R ≥ 1 and κ > 0 such that sup t≥0 e κt sup g * ≤1 sup x∈R sup ϑ∈Θ |E ϑ (g(y t )|y 0 = x) -π ϑ (g)| 1 + |x| ≤ R , (5.1)

R σ - 2 1 0σ - 2 (z) dz ≥ 1 σ 2 max, 0 E ϑ y 2m- 1 s

 212201 (z) e S(z) dz ≥ and taking into account the definition of q * we come to the upper uniform bound (A.4). Hence Proposition A.1.A.3 Moment bound for the process yt . Proposition A.1. For any m ≥ 1 sup t≥0 sup ϑ∈Θ E ϑ |y t | 2m ≤ 4(m + 1)(2m -1)!! ρ 2m ≤ 4(2m) m ρ 2m ,where ρ is given in (3.4).Proof. First note, that through the Ito formula we can write for the function z t (m) = E ϑ y 2m t the following intergal equalityz t (m) = z 0 (m) + 2m t S(y s )ds + m(2m -1) t 0 E ϑ y 2m-2 s σ 2 (y s )ds ,which can be rewritten as the differential equality żt (m) = 2mE ϑ y 2m-1 s S(y t ) + m(2m -1)E ϑ y 2m-2 t σ 2 (y t ) .

A. 4 5 )Lemma A. 2 .* β 1 σ 2 min(( 1 +

 45221 1)ds + B m , where B m = y 2m 0 + 2A m L. Setting B 0 = 1 and resolving this inequality by recurrence yieldsz t (m) ≤ 4 (2m -1)!! m j=0 σ 2 max L m-j B j .It is easy to see, thatB m ≤ 4 max |y 0 | 2 , 4(x * + ML) 2 m . Therefore sup t≥0 z t (m) ≤ 4(m + 1)(2m -1)!! ρ 2m ≤ 4(2m) m ρ 2m ,where ρ is defined in (3.4). Hence Proposition A.1. Properties of the function (4.For any integrated function φ the solution (4.5) is uniform bounded, i.e. sup ϑ∈Θ sup y∈R |v ϑ (y)| ≤ r , where the upper bound r is introduced in (3.3).Proof. Firstly we note, that for any ϑ from Θ and any intergatedR → R function φ |π ϑ (φ)| ≤ q * |φ| 1 .Moreover, by the definition of the parameter β 1 we get2 sup |u|≤x * |S 1 (u)| ≤ β 1 .Therefore, for 0 ≤ u ≤ x * we can estimate the function v ϑ as|v ϑ (u)| ≤ 2e x q * x * ) |φ| 1 + I(φ)) ,where β 1 is given in (3.1) andI(φ) = ∞ x * (|φ(y)| + q * |φ| 1 ) e 2 y

S 1 1 ∞ 0 e β 1 z-β 2 z 2 dz ≤ |φ| 1 sup z≥0 e β 1 z-β 2 z 2 + q * |φ| 1 ∞ 0 e β 1 (υ 1 + 0 S 1 S 1 (S 1 (

 11011210110111 (z)dz ≤ β 1 (y -a) -β 2 (y -a) 2 . (A.6)Using this inequlity for a = x * , we getI(φ) ≤ ∞ x * |φ(y)|e β 1 (y-x * )-β 2 (y-x * ) 2 dy + q * |φ| z-β 2 z 2 dz ≤ |φ| 1 (υ 1 + q * υ 2 ) ,where the parameters υ 1 and υ 2 are introduced in (3.1). Therefore, taking into account the definition (3.3), the last inequality implies sup ϑ∈Θ sup 0≤u≤x * |v ϑ (u)| ≤ r . (A.7) If u ≥ x * , then through the inequality (A.6) we estimate the function v ϑ (u) q * υ 2 ) ≤ r . Let now u ≤ 0. Taking into account that R φ(y) σ 2 (y) exp{2 y (z) dz} dy = 0 , we can represent the function v ϑ as v ϑ (u) = 2 -z) dz dy . Similarly to (A.6), one can check directly, that for any y ≥ a ≥ x * -z) dz ≤ β 1 (y -a) -β 2 (y -a) 2 . Therefore, by the same way as in the proof of (A.7) we can estimate the function v ϑ (u) as sup ϑ∈Θ sup u≤0 |v ϑ (u)| ≤ r . Hence Lemma A.2.

The second author is partially supported by the RFFI-Grant 09-01-00172-a.

i.e. z 0 (φ i,h ) = O 1

and τ (φ i,h ) = O 1

.

Now we have

where z 1 = ǫ T / l T . The last equality in (2.9) implies z 1 ≥ z 0 for sufficiently large T . Moreover, taking into account, that there exists a constant c * > 0 such that for sufficiently large T

T . Therefore, by Theorem 2.1 for sufficiently large T

Now the last condition in (2.9) yields the equality (2.12). Hence Theorem 2.3.

A Appendix

A.1 Proof of Proposition 6.1

We set

By the induction method we assume that for any 1

where

Note now that as is shown in [START_REF] Rio | Théorie asymptotique des processus faiblement dépendants[END_REF] (Theorem 2.3)

we obtain that

This implies for any 0 ≤ t ≤ 1

with

.

Now by setting

, we obtain the differential equation Ż(t) = k α n Z α (t) + g(t) with g(t) ≤ 0. From here we obtain


Rey Juan

MD Rafael Dal-Ré
email: rafael.dalre@urjc.es

PhD, MPH (* Alejandro Pedromingo

MD Manuel García- Losa

MD Juan Lahuerta

PhD, MHA Rafael Ortega

Are results from a pharmaceutical company-sponsored studies available to the public?

Keywords: clinical trials, results, disclosure, sponsor, pharmaceutical company

Purpose Only 53% and 63% of studies and clinical trials results presented at congresses are published.Company-sponsored trials' results are being posted on publicly accessible websites.We analyze the public availability (publication or posting on a website) rate, time to publication and factors predicting public availability of results of studies sponsored by a pharmaceutical company Methods Retrospective cohort study analyzing all studies conducted by GlaxoSmithKline in Spain between 2001-2006.Initiation and completion were defined as first subject/first visit and last subject/last visit (or their equivalents).Papers published up to March 31st,2009 were considered. Logistic regression models were used to identify factors predicting public availability of results.

Results

The cohort comprised 143 studies (94 clinical trials, of these,87 were included in international products' clinical development plans). Public availability rate was 80%(114/143) for all studies and 78%(73/94) for clinical trials; publication rates were 68% and 61%, respectively. The median time to publication for all studies and trials was 27.3 and 28.4 months, respectively. 'Study associated to a cancelled project' was the only significant factor associated to lower publication rate for all studies (OR:0.069; 95%CI 0.02-024;p<0.001) and trials (OR:0.075;95%CI 0.016-0.343;p=0.001) and a lower public availability rate (OR:0.052;95%CI 0.007-0.382;p=0.004) for trials' results.

Therapy area,sample size,positive trial results,duration of experimental phase and being a clinical trial,did not predict publication or public availability.

Conclusions

80% of studies included in this analysis are publicly available. Website posting increases clinical trial results' public availability rate from 61% to 78%.Cancellation of projects is the single factor negatively influencing publication and public availability rates.

Publication bias is well recognized [1] with important implications. One of them is that researchers have no access to all the data regarding the intervention of interest. The conclusions of their analyses are therefore bound to be biased but may nevertheless still influence treatment guidelines and decisions. Since it has been shown that positive studies are more likely to be published than negative or inconclusive ones [1][2][3][4][5][6][7][8][9][10][11][12][13],

reviews tend to overestimate the effects of the intervention. Publication of results could be influenced by investigators, sponsors, journal editors and regulations [14]. Many papers have addressed the publication rate and time to publication of studies and factors influencing these. Most of them are based on abstracts presented at scientific congresses [6,7,12,[15][16][17][18]24], studies approved by Research Ethics Committees (REC; or Institutional Review Boards in the US) [2][3][4]8,13,[19][20][21][22], studies funded by public agencies [5], or clinical trials submitted to Regulatory Agencies [10,11,23]. To our knowledge, none of them, however, has so far used internal data from a pharmaceutical company.

Indeed, controversy persists about how the industry, as compared to other sponsors, can affect the publication rate of study results. Some authors have found that industrysponsored studies tend to be less likely published than those funded or sponsored by non-commercial organisations [2,3,22]. This however has not been confirmed by others [15,20,24] Several other factors seem to influence this, with the type of studies under review (e.g. Phase 1 and 2 trials are less likely to be published [2,13,25]) being a critical one. Finally, its worth mentioning a recent report comprising phase 2-4 trials registered in Clinicaltrials.gov showed that industry-sponsored trials (44%) were lees likely to be published than non-industry/non-government-sponsored ones (56%), but there was no difference when compared with government-sponsored trials (40%) [26] Following a public debate on the publication of trial results, GlaxoSmithKline (GSK) launched in September 2004 a publicly available, internet-based clinical trial register (CSR; www.gsk-clinicalstudyregister.com/) in order to provide results from all GSKsponsored clinical trials of marketed medicines and vaccines completed since the formation of GSK in 2001 [27]. The aim is to assist physicians in their clinical practice and research, an initiative taken also by other companies and their US trade association.

The GSK register contains more than 3000 summaries of published and un-published trials conducted on 52 marketed products. The objective of this study was to describe the public availability rate and time to publication of studies managed by GSK in Spain, as well as to identify factors that could predict such public availability. The journal impact factor of papers was also determined.

METHODS

Studies

This is a retrospective cohort study based on all scientific studies managed by GSK's Medical Department in Spain. All studies initiated (i.e. first visit of the first subject -or its equivalent, e.g. first in vitro test performed; first clinical history reviewed) in 2001 or later, and completed (i.e. last visit from the last subject -or its equivalent) up to December 31 st , 2006, were included. Studies managed by international contract research organisations and all follow-up (extension) safety trials were excluded.

Data collection and definitions

A specific database was designed to contain all data gathered from the review of GSK files. Data were collected after a training session on abstraction of study characteristics; four authors (JL, MGL, RDR and RO) reviewed the data for consistency before entering it to the database. Discrepancies were resolved by consensus of all authors. Time to publication was defined as the period between study completion (last visit of last patient or its equivalent) and time when first paper on the study's primary endpoint, was published. Reason for not publication was captured. "Project" is defined as the group of studies comprising the product development plan for a given indication. "Cancelled projects" (i.e. those prematurely terminated) and the reason for such decision were also recorded.

Results of trials were classified as 'positive' if the protocol-defined hypothesis (primary end-point) was confirmed (i.e. statistically significant difference in favour of the experimental arm), or 'negative' if the hypothesis did not reach statistical significance, (i.e. not significant or significant in favour of the control arm). For non-inferiority trials, results meeting the protocol definition (below the pre-specified significance level, or a Confidence Interval (CI) excluding the pre-specified difference) were considered 'positive'. When no statistical test was performed, the results were considered as 'positive' if classified by investigators as "important" or "striking", and as 'negative', if classified as of "moderate or "little importance" or "not striking" [28].

Publication was defined only as an original article in a peer-reviewed journal, issued up to March 31 st , 2009 (cut-off date). For 'time to publication', only the month/year of publication were considered; an on-line article was included only if no paper publication was available. Journals' impact factors were obtained through ISI Web of Science (http://admin-apps.isiknowledge.com/JCR/JCR) for the year of each publication, between April 1st and 15 th, 2009.

Data management and statistical analyses

Quality control was conducted to ensure 100% accuracy and completeness for primary outcomes and main explanatory variables, certifying <1% error in secondary data.

Studies

Only 3 studies (2 follow-up safety trials and 1 trial not managed by GSK-Spain but by international Contract Research Organizations -this latter was eventually published in an international peer-reviewed journal) were excluded from the analysis. The total sample included 143 studies, two thirds (n=94) being clinical trials. Tables 1 and2 show the distribution by type of study and therapy area. All study protocols comprising human data, were approved by the relevant REC; clinical trials protocols were also approved by the Spanish Medicines Agency and the relevant competent authorities of the participating countries where appropriate. Six studies (4 clinical trials and 2 prospective, drug-related longitudinal observational studies) were terminated early due to safety (n=3) or recruitment issues (n=3). Twenty projects, accounting for 22 clinical trials (Table 3), were prematurely cancelled, most frequently because of lack of efficacy 1 and2, 68% (97/143) of all studies were published in peer-reviewed journals, and 1 was 'in-press', whereas 4 (3%) had already been submitted by the time of study cut-off date. Moreover, there were 17 (Tables 1 and2) not yet published but posted on CSR. Total public availability (published or posted on GSK CSR) thus reached 80% (114/143). Three out of the 6 studies prematurely terminated were published (2 trials due to safety reasons, and 1 prospective, longitudinal, observational drug-related study due to slow recruitment). GSK was b) Clinical trials (n=94). Sixty one percent (57/94) of trial results were published (Table 1). In addition, 2 manuscripts had been submitted for publication. Results of 43% (16/37) of the non-published trials were posted on CSR, thus reaching 78% (73/94) of public availability rate (Table 1). Reasons for no publication and publication and publicly available rates stratified by positive or negative results are presented in Table 4.

Results were publicly available for 87% (55/63) positive versus 58% (18/31) negative trials (Table 4). Results of trials associated with cancelled projects were only publicly available in 41% (9/22) of cases, as compared to 89% (64/72) of trials from noncancelled projects (Table 3). Results of 21 trials were neither published nor posted on CSR, most of them (13; 62%) belonging to cancelled projects Time to publication a) Total sample (n=97/143): Median (range) time from study completion (last visit of the last subject) to publication was 27.3 (6.0-61.9) months. Time (median) to publication by therapy area ranged between 21.3 (Anti-infectives) and 38.5 months (Vaccines). The median (range) window time available for publication (from first study reaching last subject/last visit -or its equivalent-up to cut-off date) was 5.5 (2.5-8.24) years. b) Clinical trials (n=57/94): Median (range) time from trial completion to publication was 28.4 (6.0-61.9) months. Phase 1 studies were published 16.7 months (median) after study completion. Remarkably, phase 2 (26.4 months) took about 10 or 6 months more (median) for publication than phase 3 (36.0 months) or 4 (30.4 months) trials, respectively. Time (median) to publication by therapy area ranged between 18.2 (Neurosciences) and 39.1 months (Vaccines). Trials with positive and negative results were published 26.9 (6.0-61.9) and 36.5 (8.2-55.8) months after study completion, respectively.

Impact factor

The results of 97 studies were published in 56 different peer-reviewed journals (89 papers in 51 international journals). All journals, except two, are indexed in PubMed; 91% (51/56) indexed by ISI Web of Science. The median impact factors of all published studies and of clinical trials were 3.6 and 3.9, respectively; negative trials had higher median impact factor (4.5) than positive ones (3.8) (tables 2 and 4).

Predictors for public availability and time to publication

One hundred and nineteen studies were included in the regressions models, i.e. the whole study sample (n=143) except Microbiology (n=20), Systematic Review (n=1), Mathematical Model (n=1) and Pharmacoeconomic (n=2) ones. Among the six factors selected as candidate predictors in the multivariate logistic regression models, 'Study associated to a cancelled project' was the only significant one predicting a lower publication rate for all studies (OR: 0.069; 95% CI 0.02-024; p<0.001) and trials (OR: 0.075; 95% CI 0.016-0.343; p=0.001) and a lower public availability rate (OR: 0.052; 95% CI 0.007-0.382; p=0.004) for trials' results. From the linear models, 'Impact factor' was the only significant contributor reducing the time to publication: papers submitted to journals with higher impact factors resulted in an earlier publication when total sample (B: -5.7; p=0.010) or only clinical trials (B: -6.9; p=0.007) are considered.

The results of the multivariate logistic and linear regression models are shown in tables 5 and 6.

DISCUSSION

Despite selective publication being frequently investigated [2][3][4][6][7][8][10][11][12][13][15][16][17][18][19][20][21][22], this study is the first analysis produced by a pharmaceutical company. Additionally, and as a novelty, it reports not only publication rate, but also availability of non-published study results posted on a website (GSK CSR) as a source of reliable information.

A recent Cochrane review [9] reported that after 9 years 53% of congress abstracts are published (with a median lag time of 17.9 months), this figure increasing to 63% for clinical trials. Our series shows a publication rate of 68% for all studies with a median lag time of 27.3 months; the corresponding figures being 61% and 28.4 months for clinical trials. Thus, GSK-sponsored studies in Spain have similar publication rates but with a shorter time to publication than those included in the Cochrane report (9 vs 5.5 years) [9]. When non-published trial results posted on CSR are added, the 'public availability' rate reaches 78%, (80% when considering the total sample). To put these results in context, only 80% of the Cochrane protocols were published as full reviews after more than 8 years of completion, with a median time to publication of 2.4 years [29], despite being the most reported systematic reviews [30].

When comparing publication lag time between different studies, a critical element is the time point considered as "start" for each study. We believe that last subject/last visit date is the best milestone for clinical trials, or its equivalent for other types of studies, given its objectivity. Thus, other factors that may influence publication lag time (e.g. database freeze delay) are avoided. Unfortunately, this time point is seldom considered by others, since it is not usually available. The few studies performed using this milestone indicate a median time of 2.4 years to publication for 36 National Institutes of Health-funded Human Immunodeficiency Virus trials [5], or 23 months from dataset finalization to full report publications [31]. These lag times are comparable to the 28.4 months for our 57 published trials. Of note is the fact that most authors when assessing publication rates used different 'start' time points, sometimes not taking into consideration many months or years after study completion; thus, times such as 3-5 years after abstract presentation at congresses [6,7,9,12,15,16,18], or 5-8.5 years after Food and Drug Administration drug approval [10,11] are common.

Although the pharmaceutical industry has been reportedly involved in selective publication [10,11,23], this is not an industry-specific issue. Thus, Chan [32] recently stated that "accumulating empiric evidence has shown that selective reporting of results is a systemic problem afflicting all types of trials, including those with no commercial input". It is well documented that positive trials are more likely [1-10, 12, 28, 33] and earlier [4,5,7] published, than negative ones. Hopewell et al [33] reported that positive trials are published in 4 to 5 years whereas negative or null results take 6 to 8 years. In our series, positive trials had higher publication (71%) and public availability (87%) rates; negative trials, were, on the other hand, less frequently published (39%) and publicly available (58%); for non-published studies, the proportion of positive/negative results was 49%/51%. This apparent positive publication bias is however rejected by the logistic regression model showing that the factor 'study associated to a cancelled project' but not 'positive trial result', was the only significant predictor for publication and public availability of a trial result. Hence, positive studies influencing the decision of publication become a confounding factor. Additionally, positive trials were not published significantly earlier. Although controversial, it is widely accepted that impact factor reflects to some extent the quality and scientific interest of the publication. Median and mean impact factors in our series were 3.9 and 7.5, respectively. Median impact factor ranged from 1.96 to 4.14 [12,16,17], and a mean of around 3 [18] in the few studies reporting this variable. In two studies, 24% and 37% of publications were in journals with an impact factor >4 [17] or >5 [13], respectively. As a comparison, 37% and 20% of the studies results of this study were published in journals with an impact factor > 4 and >5, respectively. On the other hand, we could speculate that the association of higher impact factor with a reduced time to publication may result from higher interest -and hence quicker submission of the paper-on the side of the authors and/or more agile review process by the more important journals.

In this analysis, data from 17 unpublished studies posted on CSR were added to publications. These summary results, as those on other websites, lack the context and interpretation that published papers provide [34] and therefore they should be a complement to publication rather than its substitute. Currently, publishing trial results involving only commercially available medicines -or even approved ones, as mandated by the US regulation [14] -is perceived as ethically insufficient. No longer is the aim only to provide information to health care professionals and researchers, but also to honour the implied contracts with study participants that expect their altruistic contribution to render useful information to science, and to prevent repetitive or risky trials with the same or similar compounds: hence, all trials results should be publicly available [35]. This is also a request included in the Declaration of Helsinki that states that positive, negative or inconclusive trial results, "should be published or otherwise made publicly available" [36] . Accordingly, GSK has recently committed to seek publication of results of all clinical trials, observational studies and meta-analyses, including those on prematurely terminated compounds; furthermore, when study results are not published, the CSR summaries will provide context and interpretation of the same [37].

Strengths and limitations

The availability of source documents for all published and unpublished studies is a plus for this internal analysis. Most reviews addressing publication fate of studies have been performed by external researchers not involved in them, usually without access to protocols and full reports. This uncommon circumstance [23] enabled all conducted studies to be accounted for, from their start until their final fate, with virtually no missing data. In particular, the availability of the study completion date, critical for calculating time lag to publication, is missing in most reviews [10,21]. This milestone coincides for almost all studies with the Food and Drug Administration requirement as the time point to count the 12-month period to disclose clinical trial results [14]. On the other hand, it could be argued that since the authors participated in the management of the studies and/or their publication, this might compromise their objectivity. Although this bias cannot be completely ruled out, it is tempered by the nature of the information collected, the quality control measures and the fact that most (93%) trials were part of international development projects, and therefore most decisions were made by others.

The external validity of this analysis could be questioned considering the limited number of studies and the even lower number of clinical trials, although greater than those followed until publication reported by others [4,5,13,19,21,23]. Another question is how well this trial sample actually represents the worldwide clinical development of new compounds. Data from the Clinicaltrials.gov database [START_REF] Karlberg | Industry clinical testing of new medicinal products requires 95000 study sites and 1300000 subjects annually[END_REF] show that in industrysponsored phase 2-4 trials the activity in Spain ranks parallel to the United Kingdom and Italy. Spain has participated in approximately 25% of all GSK-sponsored international phase 2-4 drug trials during the study period. Despite the limitations of the present analysis, the results could be considered nevertheless as a likely representation of the publication and public availability rates of GSK worldwide. Conversely, no data is provided for other pharmaceutical companies in Spain and, therefore, there is no justification for extrapolating these results to other organisations.

The definitions used in this study of 'positive' and 'negative' trials' results differ from those used by other researchers. In our analysis, the definition of "positive" result correspond to that stated in the protocol for each study, as done by only few other authors [8,9], instead of meeting some specific criteria defined a posteriori to be applied to studies [2][3][4][5][6][7][10][11][12]28]. By respecting the criteria set by the authors of each protocol, we rated as positive not only the judgement of the investigators but also those accepted by the RECs and regulatory agencies when approving the clinical trials' protocols. The time lag to publication analysis was applied to a completed set of studies (those with last subject/last visit date <1 Jan,2007). No additional censored data from studies finalizing beyond this date were considered and, therefore, Cox regression models simultaneously considering public availability and time lag to this endpoint were not employed; it should be noted that this model does not allow invariant calculation of publication and public availability rates. Although Cox regression analysis is used when assessing publication rate because of the nature of the datasets considered, we believe, as others did before [26], that our approach for a complete dataset over a defined time period is more informative.

Conclusion

Eighty percent of studies (80%) managed by GSK in Spain are publicly available. When clinical trials are considered, this figure is 78%, comprising a 61% journal publication rate plus an additional 17% of not published trial results posted on CSR -hence, substantially increasing public availability rate. As 93% of these clinical trials are multinational, it seems could be regarded as reasonably representative of what GSK activity is worldwide. Cancellation of projects is the single factor influencing a lower publication and public availability rates. There is, however, room for improvement for attaining a complete public availability of study results conducted by pharmaceutical companies.

F

Results

The Indeed, controversy persists about how the industry, as compared to other sponsors, can affect the publication rate of study results. Some authors have found that industrysponsored studies tend to be less likely published than those funded or sponsored by non-commercial organisations [2,3,22]. This however has not been confirmed by others [15,20,24] Several other factors seem to influence this, with the type of studies under review (e.g. Phase 1 and 2 trials are less likely to be published [2,13,25]) being a critical one. Finally, its worth mentioning a recent report comprising phase 2-4 trials registered in Clinicaltrials.gov showed that industry-sponsored trials (44%) were lees likely to be published than non-industry/non-government-sponsored ones (56%), but there was no difference when compared with government-sponsored trials (40%) [26] Following a public debate on the publication of trial results, GlaxoSmithKline (GSK) launched in September 2004 a publicly available, internet-based clinical trial register (CSR; www.gsk-clinicalstudyregister.com/) in order to provide results from all GSKsponsored clinical trials of marketed medicines and vaccines completed since the formation of GSK in 2001 [27]. The aim is to assist physicians in their clinical practice and research, an initiative taken also by other companies and their US trade association.

The GSK register contains more than 3000 summaries of published and un-published trials conducted on 52 marketed products. The objective of this study was to describe the public availability rate and time to publication of studies managed by GSK in Spain, as well as to identify factors that could predict such public availability. The journal impact factor of papers was also determined. organisations and all follow-up (extension) safety trials were excluded.

METHODS

Studies

Data collection and definitions

A specific database was designed to contain all data gathered from the review of GSK files. Data were collected after a training session on abstraction of study characteristics;

four authors (JL, MGL, RDR and RO) reviewed the data for consistency before entering it to the database. Discrepancies were resolved by consensus of all authors.

Time to publication was defined as the period between study completion (last visit of last patient or its equivalent) and time when first paper on the study's primary end-

Studies

Only 3 studies (2 follow-up safety trials and 1 trial not managed by GSK-Spain but by international Contract Research Organizations -this latter was eventually published in an international peer-reviewed journal) were excluded from the analysis. The total sample included 143 studies, two thirds (n=94) being clinical trials. Tables 1 and2 show the distribution by type of study and therapy area. 1 and2, 68% (97/143) of all studies were published in peer-reviewed journals, and 1 was 'in-press', whereas 4 (3%) had already been submitted by the time of study cut-off date. Moreover, there were 17 (Tables 1 and2) not yet published but posted on CSR. Total public availability (published or posted on GSK CSR) thus reached 80% (114/143). Three out of the 6 studies prematurely terminated were published (2 trials due to safety reasons, and 1 prospective, longitudinal, observational drug-related study due to slow recruitment). GSK was b) Clinical trials (n=94). Sixty one percent (57/94) of trial results were published (Table 1). In addition, 2 manuscripts had been submitted for publication. Results of 43% (16/37) of the non-published trials were posted on CSR, thus reaching 78% (73/94) of public availability rate (Table 1). Reasons for no publication and publication and publicly available rates stratified by positive or negative results are presented in Table 4.

Results were publicly available for 87% (55/63) positive versus 58% (18/31) negative trials (Table 4). Results of trials associated with cancelled projects were only publicly available in 41% (9/22) of cases, as compared to 89% (64/72) of trials from noncancelled projects (Table 3). 8 studies and of clinical trials were 3.6 and 3.9, respectively; negative trials had higher median impact factor (4.5) than positive ones (3.8) (tables 2 and 4).

Predictors for public availability and time to publication

One hundred and nineteen studies were included in the regressions models, i.e. the whole study sample (n=143) except Microbiology (n=20), Systematic Review (n=1), Mathematical Model (n=1) and Pharmacoeconomic (n=2) ones. Among the six factors selected as candidate predictors in the multivariate logistic regression models, 'Study associated to a cancelled project' was the only significant one predicting a lower publication rate for all studies (OR: 0.069; 95% CI 0.02-024; p<0.001) and trials (OR: 0.075; 95% CI 0.016-0.343; p=0.001) and a lower public availability rate (OR: 0.052; 95% CI 0.007-0.382; p=0.004) for trials' results. From the linear models, 'Impact factor' was the only significant contributor reducing the time to publication: papers submitted to journals with higher impact factors resulted in an earlier publication when total sample (B: -5.7; p=0.010) or only clinical trials (B: -6.9; p=0.007) are considered.

The results of the multivariate logistic and linear regression models are shown in tables 5 and 6.

DISCUSSION

Despite selective publication being frequently investigated [2][3][4][6][7][8][10][11][12][13][15][16][17][18][19][20][21][22], this study is the first analysis produced by a pharmaceutical company. Additionally, and as a novelty, it reports not only publication rate, but also availability of non-published study results posted on a website (GSK CSR) as a source of reliable information.

A recent Cochrane review [9] reported that after 9 years 53% of congress abstracts are published (with a median lag time of 17.9 months), this figure increasing to 63% for clinical trials. Our series shows a publication rate of 68% for all studies with a median lag time of 27.3 months; the corresponding figures being 61% and 28.4 months for clinical trials. Thus, GSK-sponsored studies in Spain have similar publication rates but with a shorter time to publication than those included in the Cochrane report (9 vs 5.5 years) [9]. When non-published trial results posted on CSR are added, the 'public availability' rate reaches 78%, (80% when considering the total sample). To put these results in context, only 80% of the Cochrane protocols were published as full reviews after more than 8 years of completion, with a median time to publication of 2.4 years [29], despite being the most reported systematic reviews [30].

When comparing publication lag time between different studies, a critical element is the time point considered as "start" for each study. We believe that last subject/last visit date is the best milestone for clinical trials, or its equivalent for other types of studies, given its objectivity. Thus, other factors that may influence publication lag time (e.g.

database freeze delay) are avoided. Unfortunately, this time point is seldom considered by others, since it is not usually available. The few studies performed using this milestone indicate a median time of 2.4 years to publication for 36 National Institutes of Health-funded Human Immunodeficiency Virus trials [5], or 23 months from dataset finalization to full report publications [31]. These lag times are comparable to the 28.4 months for our 57 published trials. Of note is the fact that most authors when assessing publication rates used different 'start' time points, sometimes not taking into consideration many months or years after study completion; thus, times such as 3-5 years after abstract presentation at congresses [6,7,9,12,15,16,18], or 5-8.5 years after Food and Drug Administration drug approval [10,11] are common.

Although the pharmaceutical industry has been reportedly involved in selective publication [10,11,23], this is not an industry-specific issue. Thus, Chan [32] recently stated that "accumulating empiric evidence has shown that selective reporting of results is a systemic problem afflicting all types of trials, including those with no commercial input". It is well documented that positive trials are more likely [1-10, 12, 28, 33] and earlier [4,5,7] published, than negative ones. Hopewell et al [33] reported that positive trials are published in 4 to 5 years whereas negative or null results take 6 to 8 years. In our series, positive trials had higher publication (71%) and public availability (87%) rates; negative trials, were, on the other hand, less frequently published (39%) and publicly available (58%); for non-published studies, the proportion of positive/negative results was 49%/51%. This apparent positive publication bias is however rejected by the logistic regression model showing that the factor 'study associated to a cancelled project' but not 'positive trial result', was the only significant predictor for publication and public availability of a trial result. Hence, positive studies influencing the decision of publication become a confounding factor. Additionally, positive trials were not published significantly earlier. Although controversial, it is widely accepted that impact factor reflects to some extent the quality and scientific interest of the publication. Median and mean impact factors in our series were 3.9 and 7.5, respectively. Median impact factor ranged from 1.96 to 4.14 [12,16,17], and a mean of around 3 [18] in the few studies reporting this variable. In two studies, 24% and 37% of publications were in journals with an impact factor >4 [17] or >5 [13], respectively. As a comparison, 37% and 20% of the studies results of this study were published in journals with an impact factor > 4 and >5, respectively. On the other hand, we could speculate that the association of higher impact factor with a reduced time to publication may result from higher interest -and hence quicker submission of the paper-on the side of the authors and/or more agile review process by the more important journals.

In this analysis, data from 17 unpublished studies posted on CSR were added to publications. These summary results, as those on other websites, lack the context and interpretation that published papers provide [34] and therefore they should be a complement to publication rather than its substitute. Currently, publishing trial results involving only commercially available medicines -or even approved ones, as mandated by the US regulation [14] -is perceived as ethically insufficient. No longer is the aim only to provide information to health care professionals and researchers, but also to honour the implied contracts with study participants that expect their altruistic contribution to render useful information to science, and to prevent repetitive or risky trials with the same or similar compounds: hence, all trials results should be publicly available [35]. This is also a request included in the Declaration of Helsinki that states that positive, negative or inconclusive trial results, "should be published or otherwise made publicly available" [36] . Accordingly, GSK has recently committed to seek publication of results of all clinical trials, observational studies and meta-analyses, including those on prematurely terminated compounds; furthermore, when study results are not published, the CSR summaries will provide context and interpretation of the same [37].

Strengths and limitations

The availability of source documents for all published and unpublished studies is a plus for this internal analysis. Most reviews addressing publication fate of studies have been performed by external researchers not involved in them, usually without access to protocols and full reports. This uncommon circumstance [23] enabled all conducted studies to be accounted for, from their start until their final fate, with virtually no missing data. In particular, the availability of the study completion date, critical for calculating time lag to publication, is missing in most reviews [10,21]. This milestone coincides for almost all studies with the Food and Drug Administration requirement as the time point to count the 12-month period to disclose clinical trial results [14]. On the other hand, it could be argued that since the authors participated in the management of the studies and/or their publication, this might compromise their objectivity. Although this bias cannot be completely ruled out, it is tempered by the nature of the information collected, the quality control measures and the fact that most (93%) trials were part of international development projects, and therefore most decisions were made by others.

The external validity of this analysis could be questioned considering the limited number of studies and the even lower number of clinical trials, although greater than those followed until publication reported by others [4,5,13,19,21,23]. Another question is how well this trial sample actually represents the worldwide clinical development of new compounds. Data from the Clinicaltrials.gov database [START_REF] Karlberg | Industry clinical testing of new medicinal products requires 95000 study sites and 1300000 subjects annually[END_REF] show that in industrysponsored phase 2-4 trials the activity in Spain ranks parallel to the United Kingdom and Italy. Spain has participated in approximately 25% of all GSK-sponsored international phase 2-4 drug trials during the study period. Despite the limitations of the present analysis, the results could be considered nevertheless as a likely representation of the publication and public availability rates of GSK worldwide. Conversely, no data is provided for other pharmaceutical companies in Spain and, therefore, there is no justification for extrapolating these results to other organisations.

The definitions used in this study of 'positive' and 'negative' trials' results differ from those used by other researchers. In our analysis, the definition of "positive" result correspond to that stated in the protocol for each study, as done by only few other authors [8,9], instead of meeting some specific criteria defined a posteriori to be applied to studies [2][3][4][5][6][7][10][11][12]28]. By respecting the criteria set by the authors of each protocol, we rated as positive not only the judgement of the investigators but also those accepted by the RECs and regulatory agencies when approving the clinical trials' protocols.

F

  Standard descriptive statistics were used for discrete and continuous data. A multivariate logistic regression model was performed to identify factors predicting publication. Two additional models with publication and public availability as outcomes were used only for clinical trials. Factors considered were 'Therapy Area', 'Study being associated to a cancelled project', 'Clinical trial', 'Sample size', 'Positive trial result' (only for clinical trials models) and 'Duration of experimental phase'. Odds Ratio and their 95% CI were calculated. Two multivariate linear regression models were used to identify potential factors explaining time to publication for studies and for clinical trials, respectively. Candidate factors were the same considered above plus the inclusion of 'Impact factor' which was Ln transformed. Non-standardized B coefficients were obtained. Only studies with data in all predicting factors were included in the multivariate models. All candidate factors were maintained in the final models; no stepwise procedures were used for selection. Factors were considered significant if P value <0.05. SPSS statistical software version 15.0 was used.

( 16 /

 16 20; 80%), and usually in phase 2 (15/20, 75%). An additional project was cancelled for lack of efficacy on an animal model, and after conducting an epidemiological longitudinal prospective study. Most trials (87/94, 93%) were part of the clinical development plans of the investigational medicines and vaccines. Median (range) sample size of clinical trials included in the analysis was 452 (12-5052) subjects. By clinical development phase, these figures were: 13 (12-56), 290 (13-1415), 569 (127-5052) and 458 (120-1395) for phase 1, 2, 3 and 4, respectively Public availability and Publication rates a) Total sample (n=143). As shown in Tables

  study sponsor and/or one, or more, GSK employees were included among study authors in all published manuscripts.

1 2 Purpose

 12 (a) All studies (n= 119) are included, except those belonging to Microbiology (n=20), Systematic Review, (n=1); Mathematical Model (n=1); Pharmacoeconomy (n=2) Are results from a pharmaceutical company-sponsored studies available to the public? Rafael Dal-Ré MD, PhD, MPH (*), Alejandro Pedromingo L Sc (a), Manuel García-Losa MD, Juan Lahuerta MD, PhD, MHA, Rafael Ortega MD Medical Department, GlaxoSmithKline SA, Tres Cantos, Madrid, Spain (a) Biometry Department, GlaxoSmithKline SA, Tres Cantos, Madrid, Spain (*) Corresponding author: Dr R Dal-Ré Current address: Dr R Dal-Ré; Department of Preventive Medicine, Public Health and Medical Immunology and Microbiology, School of Health Sciences, Rey Juan Carlos University, Avda Atenas s/n; 28922 Alcorcón, Madrid, Spain Phone: +34-91-488 8804; Fax: +34-91-488 8955 ; Email: rafael.dalre@urjc.es Only 53% and 63% of studies and clinical trials results presented at congresses are published.Company-sponsored trials' results are being posted on publicly accessible websites.We analyze the public availability (publication or posting on a website) rate, time to publication and factors predicting public availability of results of studies sponsored by a pharmaceutical company Methods Retrospective cohort study analyzing all studies conducted by GlaxoSmithKline in Spain between 2001-2006.Initiation and completion were defined as first subject/first visit and last subject/last visit (or their equivalents).Papers published up to March 31st,2009 were considered. Logistic regression models were used to identify factors predicting public availability of results.

  Most trials (87/94, 93%) were part of the clinical development plans of the investigational medicines and vaccines. Median (range) sample size of clinical trials included in the analysis was 452 (12-5052) subjects. By clinical development phase, these figures were: 13 (12-56), 290 (13-1415), 569 (127-5052) and 458 (120-1395) for phase 1, 2, 3 and 4, respectively Public availability and Publication rates a) Total sample (n=143). As shown in Tables

  study sponsor and/or one, or more, GSK employees were included among study authors in all published manuscripts.

2 (

 2 Results of 21 trials were neither published nor posted on CSR, most of them (13; 62%) belonging to cancelled projects Time to publication a) Total sample (n=97/143): Median (range) time from study completion (last visit of the last subject) to publication was 27.3 (6.0-61.9) months. Time (median) to publication by therapy area ranged between 21.3 (Anti-infectives) and 38.5 months (Vaccines). The median (range) window time available for publication (from first study reaching last subject/last visit -or its equivalent-up to cut-off date) was 5.5 (2.5-8.24) years.b) Clinical trials (n=57/94): Median (range) time from trial completion to publication was 28.4 (6.0-61.9) months. Phase 1 studies were published 16.7 months (median) after study completion. Remarkably, phase 2 (26.4 months) took about 10 or 6 months more (median) for publication than phase 3 (36.0 months) or 4 (30.4 months) trials, respectively. Time (median) to publication by therapy area ranged between 18.Neurosciences) and 39.1 months (Vaccines). Trials with positive and negative results were published 26.9 (6.0-61.9) and 36.5 (8.2-55.8) months after study completion, respectively.Impact factorThe results of 97 studies were published in 56 different peer-reviewed journals (89 papers in 51 international journals). All journals, except two, are indexed in PubMed; 91% (51/56) indexed by ISI Web of Science. The median impact factors of all published

( a )

 a Publication of full study results in a peer-reviewed journal. Includes paper (n=94) and on-line (n=3) publications. (b) Impact factor of the year of publication. For papers published on 2008 or 2009, 2007 impact factor is used (c) No of full trials' results posted at GSK's CSR that have not been published yet. Hence, if the results of a trial were published in a journal, it is not included in this figure, regardless if it is, or not, posted in the CSR N: No of studies C, M & R: Cardiovascular, Metabolism and Respiratory R, U & G: Rheumatology, Urology and Gastroenterology

( a )

 a Published or posted on GSK Clinical Study Register (CSR) (b) And an additional epidemiological longitudinal prospective study N: No of clinical trials

Table 1

 1 Study sample (n=143). Public availability of study results: Publications and study results posted on GSK's Clinical Study Register (CSR), by study type.

	o r
	P
	e
	e r
	R
	e
	v i e
	w

(a) Longitudinal prospective, 5; cross-sectional, 1; retrospective, 2; case-control, 1 (b) Longitudinal prospective, 4; retrospective, 2; cross-sectional, 4 (c) In vitro research, 12: surveillances, 6; animal models, 2 (d) Health Scales Validation, 4; Pharmacoeconomy, 2; Systematic Review, 1; Mathematical Model, 1; Clinical trial on devices and modelling, 1; Health care quality assessment , 1 (e) Publication of full study results in a peer-reviewed journal. Includes paper (n=94) and on-line (n=3) publications. (f) No of full trials' results posted at GSK's CSR that have not been published yet. Hence, if the results of a trial were published in a journal, it is not included in this figure, regardless if

Table 2 .

 2 Study sample (n=143). Public availability of study results: Publications and study

	results posted on GSK's Clinical Study Register (CSR), by therapy area
			N	A: Publication a N	B: Non-published	A+B:
			(%)	(%) / Impact factor-	but posted on	Publication +
				median b	CSR c N (%)	CSR. N (%)
	Therapy	Neurosciences 22	13 (59) / 3.13	1 (5)	14 (64)
	Area		(15)
		C, M & R	24	16 (67) / 3.00	5 (21)	21 (88)
			(17)
		Anti-	47	38 (81) / 3.89	6 (13)	44 (94)
		infectives	(33)
		Oncology	13	7 (54) / 13.60	2 (15)	9 (69)
			(9)
		Vaccines	19	15 (79) / 3.09	3 (16)	18 (95)
			(13)
		R, U & G	18	8 (44) / 3.20	0 (0)	8 (44)
			(13)
		Total	143 97 (68) / 3.61	17 (12)	114 (80)
	if it is, or not, posted in the CSR
	N: No of studies
	C, M & R: Cardiovascular, Metabolism and Respiratory
	R, U & G: Rheumatology, Urology and Gastroenterology

(a) Publication of full study results in a peer-reviewed journal. Includes paper (n=94) and on-line (n=3) publications. (b) Impact factor of the year of publication. For papers published on 2008 or 2009, 2007 impact factor is used (c) No of full trials' results posted at GSK's CSR that have not been published yet. Hence, if the results of a trial were published in a journal, it is not included in this figure, regardless

Table 3 .

 3 Clinical Trials (n=94). Cancelled projects and fate of results.

	Cancelled projects, 20	16 for lack of efficacy: 12 in phase 2, 1 in phase
		2/3 and 3 in phase 3; comprising 17 trials b , of all
		therapy areas except vaccines
		2 for safety reasons: 1 in phase 1, due to animal
		toxicological findings, 1 in phase 2 ; comprising
		3 trials of neurosciences and anti-infectives
		2 for manufacturing issues: 1 in Phase 2, and 1
		in phase 3 ; comprising 2 vaccine trials
	Trials associated to cancelled projects -N (%)	22 (23)
	Published / Publicly available a	4 (18) / 9 (41)
	Not published / Non Publicly available	18 (82) / 13 (59)
	Trials associated to non-cancelled projects -N (%)	72 (77)
	Published / Publicly available a	53 (74) / 64 (89)
	Not published / Non Publicly available	19 (26) / 8 (11)
	"Project" is defined as the group of studies comprising the product development plan for a
	given indication.
	(a) Published or posted on GSK Clinical Study Register (CSR)
	(b) And an additional epidemiological longitudinal prospective study
	N: No of clinical trials

Table 4 .

 4 Clinical Trials (n=94). Main characteristics, fate of results and impact factors. Clinical trials conducted with marketed products, i.e. conducted with products that were marketed at the time the trial was run or with a product that was marketed (for the indication, dosage, etc) as of the cut-off date (March 31 st , 2009).

	N (%) (Unless otherwise
	stated)

(f) All trials were published or posted on GSK CSR, except one which results were posted on Bayer's Website. N: No of clinical trials (where appropriate)

Table 5 .

 5 Factors predicting publication and public availability. Odds Ratios (OR) and 95% Confidence Intervals (95% CI) from multivariate logistic regression models.

				Clinical Trials;
				Publication as	Clinical Trials; Public
		All studies a	outcome		Availablility as outcome
		OR
		(95%		OR (95%
		CI)	P	CI)	P	OR (95% CI)	P
	Factor Therapy Area		0.488		0.772		0.101
	Study associated to	0.069
	a cancelled project	(0.02 -		0.075 (0.016
		0.24) 0.000	-0.343)	0.001	0.052(0.007 -0.382)	0.004
	Clinical trial	1.33
		(0.45 -
		3.93) 0.606
	Sample size	1.00
		(0.99 -		1.00 (0.999
		1.01) 0.979	-1.001)	0.964	1.00 (0.999 -1.001)	0.823
	Positive trial result			1.028 (0.286
				-3.691)	0.966	1.118 (0.210 -5.942)	0.896
	Duration of	1.004
	experimental phase	(0.96 -		0.996 (0.945
		1.05) 0.878	-1.050)	0.885	1.064 (0.962 -1.171)	0.231
	Constant	1.17	0.826	2.306	0.373	1.529	0.720
	(a) All studies (n= 119) are included, except those belonging to Microbiology (n=20),
	Systematic Review, (n=1); Mathematical Model (n=1); Pharmacoeconomy (n=2)
	OR = Odds ratio; 95% CI = 95% confidence interval

Table 6 .

 6 Factors predicting time to publication. Coefficients (B) and standard errors (SE) from multivariate linear regression models.

		All studies a		Clinical Trials
	B c	SE	P	B c	SE	P

  Publication bias is well recognized[1] with important implications. One of them is that researchers have no access to all the data regarding the intervention of interest. The conclusions of their analyses are therefore bound to be biased but may nevertheless still influence treatment guidelines and decisions. Since it has been shown that positive

	studies are more likely to be published than negative or inconclusive ones [1-13],
	reviews tend to overestimate the effects of the intervention. Publication of results could
	be influenced by investigators, sponsors, journal editors and regulations [14]. Many
	papers have addressed the publication rate and time to publication of studies and factors
	influencing these. Most of them are based on abstracts presented at scientific congresses
	[6,7,12,15-18,24], studies approved by Research Ethics Committees (REC; or
	Institutional Review Boards in the US) [2-4,8,13, 19-22], studies funded by public F agencies [5], or clinical trials submitted to Regulatory Agencies [10,11,23]. To our o r knowledge, none of them, however, has so far used internal data from a pharmaceutical
	company.	P
		e
	was 80%(114/143) for all studies and 78%(73/94) for clinical trials; publication rates were e r
	68% and 61%, respectively. The median time to publication for all studies and trials R was 27.3 and 28.4 months, respectively. 'Study associated to a cancelled project' was e the only significant factor associated to lower publication rate for all studies (OR:0.069; 95%CI 0.02-024;p<0.001) and trials (OR:0.075;95%CI 0.016-0.343;p=0.001) and a lower public availability rate (OR:0.052;95%CI 0.007-0.382;p=0.004) for trials' results. v i e
	Therapy area,sample size,positive trial results,duration of experimental phase and being a clinical trial,did not predict publication or public availability. w
	Conclusions
	80% of studies included in this analysis are publicly available. Website posting
	increases clinical trial results' public availability rate from 61% to 78%.Cancellation of
	projects is the single factor negatively influencing publication and public availability
	rates.
	Key words: clinical trials, results, disclosure, sponsor, pharmaceutical company
		3

cohort comprised 143 studies (94 clinical trials, of these,87 were included in international products' clinical development plans). Public availability rate

  This is a retrospective cohort study based on all scientific studies managed by GSK's Medical Department in Spain. All studies initiated (i.e. first visit of the first subject -or

its equivalent, e.g. first in vitro test performed; first clinical history reviewed) in 2001 or later, and completed (i.e. last visit from the last subject -or its equivalent) up to December 31 st , 2006, were included. Studies managed by international contract research

  point, was published. Reason for not publication was captured. "Project" is defined as the group of studies comprising the product development plan for a given indication. When no statistical test was performed, the results were considered as 'positive' if classified by investigators as "important" or "striking", and as 'negative', if classified as of "moderate or "little importance" or "not striking"[28].

	Confidence Interval (CI) excluding the pre-specified difference) were considered RESULTS
	'positive'. Publication was defined only as an original article in a peer-reviewed journal, issued up
	to March 31 st , 2009 (cut-off date). For 'time to publication', only the month/year of
	publication were considered; an on-line article was included only if no paper publication
	was available. Journals' impact factors were obtained through ISI Web of Science
	(http://admin-apps.isiknowledge.com/JCR/JCR) for the year of each publication,
	F o r between April 1st and 15 th, 2009. Data management and statistical analyses F o r
	P e Quality control was conducted to ensure 100% accuracy and completeness for primary P outcomes and main explanatory variables, certifying <1% error in secondary data. e
	e r e r
	R R
	e e
	v i e v i e
	w 'Impact factor' which was Ln transformed. Non-standardized B coefficients were w
	obtained. Only studies with data in all predicting factors were included in the
	"Cancelled projects" (i.e. those prematurely terminated) and the reason for such multivariate models. All candidate factors were maintained in the final models; no
	decision were also recorded. stepwise procedures were used for selection. Factors were considered significant if P
	value <0.05. SPSS statistical software version 15.0 was used.
	Results of trials were classified as 'positive' if the protocol-defined hypothesis (primary
	end-point) was confirmed (i.e. statistically significant difference in favour of the
	5 6

experimental arm), or 'negative' if the hypothesis did not reach statistical significance, (i.e. not significant or significant in favour of the control arm). For non-inferiority trials, results meeting the protocol definition (below the pre-specified significance level, or a Standard descriptive statistics were used for discrete and continuous data. A multivariate logistic regression model was performed to identify factors predicting publication. Two additional models with publication and public availability as outcomes were used only for clinical trials. Factors considered were 'Therapy Area', 'Study being associated to a cancelled project', 'Clinical trial', 'Sample size', 'Positive trial result' (only for clinical trials models) and 'Duration of experimental phase'. Odds Ratio and their 95% CI were calculated. Two multivariate linear regression models were used to identify potential factors explaining time to publication for studies and for clinical trials, respectively. Candidate factors were the same considered above plus the inclusion of

Table 3

 3

	All study protocols comprising

), were prematurely cancelled, most frequently because of lack of efficacy (16/20; 80%), and usually in phase 2 (15/20, 75%). An additional project was cancelled for lack of efficacy on an animal model, and after conducting an epidemiological longitudinal prospective study.

Table 1

 1 Study sample (n=143). Public availability of study results: Publications and study results posted on GSK's Clinical Study Register (CSR), by study type.

	The time lag to publication analysis was applied to a completed set of studies (those
	with last subject/last visit date <1 Jan,2007). No additional censored data from studies
	finalizing beyond this date were considered and, therefore, Cox regression models
	simultaneously considering public availability and time lag to this endpoint were not
	employed; it should be noted that this model does not allow invariant calculation of
	publication and public availability rates. Although Cox regression analysis is used when
	assessing publication rate because of the nature of the datasets considered, we believe,
	as others did before [26], that our approach for a complete dataset over a defined time
	period is more informative.
	Conclusion	o r
	Eighty percent of studies managed by GSK in Spain are publicly available. When P clinical trials are considered, this figure is 78%, comprising a 61% journal publication e rate plus an additional 17% of not published trial results posted on CSR -hence,
	substantially increasing public availability rate. As 93% of these clinical trials are e r multinational, it seems could be regarded as reasonably representative of what GSK
	activity is worldwide. Cancellation of projects is the single factor influencing a lower R publication and public availability rates. There is, however, room for improvement for e attaining a complete public availability of study results conducted by pharmaceutical v e i companies.
		w
		12

Table 2 .

 2 Study sample (n=143). Public availability of study results: Publications and study

	results posted on GSK's Clinical Study Register (CSR), by therapy area
			N	A: Publication a N	B: Non-published	A+B:
			(%)	(%) / Impact factor-	but posted on	Publication +
				median b	CSR c N (%)	CSR. N (%)
	Therapy	Neurosciences 22	13 (59) / 3.13	1 (5)	14 (64)
	Area		(15)
		C, M & R	24	16 (67) / 3.00	5 (21)	21 (88)
			(17)
		Anti-	47	38 (81) / 3.89	6 (13)	44 (94)
		infectives	(33)
		Oncology	13	7 (54) / 13.60	2 (15)	9 (69)
			(9)
		Vaccines	19	15 (79) / 3.09	3 (16)	18 (95)
			(13)
		R, U & G	18	8 (44) / 3.20	0 (0)	8 (44)
			(13)
		Total	143 97 (68) / 3.61	17 (12)	114 (80)

Table 3 .

 3 Clinical Trials (n=94). Cancelled projects and fate of results. Cancelled projects, 20 16 for lack of efficacy: 12 in phase 2, 1 in phase 2/3 and 3 in phase 3; comprising 17 trials b , of all

	therapy areas except vaccines
	2 for safety reasons: 1 in phase 1, due to animal
	toxicological findings, 1 in phase 2 ; comprising
	3 trials of neurosciences and anti-infectives
	2 for manufacturing issues: 1 in Phase 2, and 1
	in phase 3 ; comprising 2 vaccine trials

"Project" is defined as the group of studies comprising the product development plan for a given indication.

Table 4 .

 4 Clinical Trials (n=94). Main characteristics, fate of results and impact factors. Clinical trials conducted with marketed products, i.e. conducted with products that were marketed at the time the trial was run or with a product that was marketed (for the indication, dosage, etc) as of the cut-off date (March 31 st , 2009).(f) All trials were published or posted on GSK CSR, except one which results were posted on Bayer's Website.

	N (%) (Unless otherwise
	stated)

N: No of clinical trials (where appropriate)

Table 5 .

 5 Factors predicting publication and public availability. Odds Ratios (OR) and 95% Confidence Intervals (95% CI) from multivariate logistic regression models.

				Clinical Trials;
				Publication as	Clinical Trials; Public
		All studies a	outcome		Availablility as outcome
		OR
		(95%		OR (95%
		CI)	P	CI)	P	OR (95% CI)	P
	Factor Therapy Area		0.488		0.772		0.101
	Study associated to	0.069
	a cancelled project	(0.02 -		0.075 (0.016
		0.24) 0.000	-0.343)	0.001	0.052(0.007 -0.382)	0.004
	Clinical trial	1.33
		(0.45 -
		3.93) 0.606
	Sample size	1.00
		(0.99 -		1.00 (0.999
		1.01) 0.979	-1.001)	0.964	1.00 (0.999 -1.001)	0.823
	Positive trial result			1.028 (0.286
				-3.691)	0.966	1.118 (0.210 -5.942)	0.896
	Duration of	1.004
	experimental phase	(0.96 -		0.996 (0.945
		1.05) 0.878	-1.050)	0.885	1.064 (0.962 -1.171)	0.231
	Constant	1.17	0.826	2.306	0.373	1.529	0.720
	(a) All studies (n= 119) are included, except those belonging to Microbiology (n=20),
	Systematic Review, (n=1); Mathematical Model (n=1); Pharmacoeconomy (n=2)
	OR = Odds ratio; 95% CI = 95% confidence interval

European Journal of Clinical Pharmacology

Acknowledgements

We thank Dolores Asenjo, Teresa Compañy, Arturo Fueyo, Pilar García Corbeira, José Emilio Martín, Elena Morejón, Felipe Rodríguez-Alcántara and Josefa Samper for data collection, and Pilar Vicente for providing the impact factor of all manuscripts. We also thank Drs A Benbow (London, United Kingdom) and X Carne (Barcelona, Spain) for their critical reading and suggestions on the manuscript.

Financial Disclosure

This study required no funding.

Acknowledgements

We thank Dolores Asenjo, Teresa Compañy, Arturo Fueyo, Pilar García Corbeira, José Emilio Martín, Elena Morejón, Felipe Rodríguez-Alcántara and Josefa Samper for data collection, and Pilar Vicente for providing the impact factor of all manuscripts. We also thank Drs A Benbow (London, United Kingdom) and X Carne (Barcelona, Spain) for their critical reading and suggestions on the manuscript.

Financial Disclosure

This study required no funding.

Competing Interests

At the time of conducting the analysis all authors were GlaxoSmithKline SA employees and own stock in GSK.

Abbreviations

REC, research ethics committee.

GSK, GlaxoSmithKline

CSR, clinical study register CI, confidence interval

Competing Interests

At the time of conducting the analysis all authors were GlaxoSmithKline SA employees and own stock in GSK.

Abbreviations