Damien Lemaitre
email: damien.lemaitre@grenoble-inp.fr

Benoît Sarrazin

Thierry Brincourt

Alexis Derbey

Yohan Wanderoild

Yves Lembeye

Wireless Power Transfer Losses Redefined by C0G Capacitor

Technology of resonant capacitors generally used in wireless power transfer system (WPTS) are commonly polypropylene film capacitors. Although it provides many interesting features on temperature, ESL value, energy density, self-reparation, etc... Their equivalent series resistance (ESR) induce important losses higher than 30% of total losses in WPTS with power higher than kW. Multilayer ceramic C0G capacitor show very low ESR allowing to divide losses by more than ten compared to film capacitors. This article proposes the use of C0G capacitor as an alternative of film capacitor for high power, and propose solutions on pcb design to improve C0G pcb performances for high power applications.

Introduction

The development of electric vehicles (EVs) is leading to a growing need for chargers. EVs are largely charged with plug-in chargers. The safety of no exposed conductor, the galvanic isolation, the resistance to vandalism, weathering and the user comfort make wireless power transfer system (WPTS) attractive. The possibility of being connected to the grid without human intervention is also an essential asset in vehicleto-grid (V2G) systems. Maximising efficiency is one of the main objectives of WPTS to compete with plug in chargers. Efficiency consideration is even more important in V2G chargers related to a greater amount of energy transferred in front of the unidirectional chargers. The principle of the WPTS is presented in Fig. 1. The electrical circuit of the system is given in Fig. 2 considering a series-series (SS) resonant topology. At the same time, for WPTS above kW, resonant capacitors losses represent one third or even half of the total losses when using film capacitors [START_REF] Lemaitre | Efficiency-Oriented Design of Litz Wire for Several kW Power Experimented on 20 kW Prototype[END_REF]- [START_REF] Bosshard | Modeling and $\eta $ -$\alpha $ -Pareto Optimization of Inductive Power Transfer Coils for Electric Vehicles[END_REF]. For systems of 10 kW or more, the commonly used conduction cooled polypropylene film capacitors (CCPFC) are required but expensive compared to the metallized polypropylene film capacitors (MKP) used for applications of a few kW. To reduce this cost while keeping a good efficiency for the WPTS, this paper proposes to consider the C0G/NP0 multi-layer ceramic capacitors (MLCC) typically used for few hundred watts systems to systems with output power up to several tens of kW. Part 1 is a summary of the capacitor technologies used for the resonant compensation of WPTS. The interest of C0G capacitors in high power systems will be detailed in part 2. Experiments are carried out in part 3 to validate previous analysis. Part 4 proposes a study on the printed circuit board (PCB) design in order to improve the cooling by natural convection. This section first introduces the important capacitor characteristics required for resonant applications such as the WPT.

Specifications for WPT

Inductive chargers use capacitors to compensate the reactive energy of the winding inductances, thus improving the power factor. In the series-series (SS) resonant topology, knowing the resonant frequency f and the winding inductance L, the capacitance C is sized according to formula (1) below.

f = 1 2. 𝜋. √𝐿. 𝐶 (1)
A slight variation can have harmful impacts on the system, particularly on the loss of zero voltage switching (ZVS), the appearance or amplification of bifurcation and frequency splitting phenomena. Class 1 capacitors are therefore preferred because of their high stability and low losses due to a low ESR. One of the criteria for belonging to this class is to have a low temperature coefficient allowing the capacitance to be stable with temperature. The self-resonant frequency of the capacitor is very high compared to the switching frequency f , resulting in negligible inductive behaviour. The electrical circuit equivalent to a capacitor can thus be simplified to a capacitor C in series with the ESR as shown in Fig. 3. A very low ESR capacitor is required to protect the capacitors from high temperatures, thus increasing their lifetime and simplifying thermal management. The manufacturer's recommended self-heating limit of the capacitors ΔT is thus more easily respected.

The ESR is obtained from equation [START_REF] Sarrazin | Bidirectional Wireless Power Transfer System with Wireless Control for Electrical Vehicle[END_REF], where tan(𝛿) is the dissipation factor, C is the capacitance and ω is the electrical pulse.

𝐸𝑆𝑅 = tan(𝛿) 𝐶. 𝜔

(2)

Commonly used technologies

Class 1 metallized polypropylene film capacitors (MKP) often offer an excellent compromise between the above characteristics and are therefore often selected for systems up to several kW. Class 1 MLCCs of the C0G/NP0 type may be preferred over MKP film capacitors thanks to their better capacity stability, especially as a function of temperature as shown by their temperature coefficient [START_REF] Tdk | Features of High voltage MLCCs with C0G Characteristics and Replacement Solutions Overview[END_REF], [START_REF] Avx | Multilayer Ceramic Leaded Capacitors[END_REF].

In order to simplify the designation of capacitors, the terms "MKP" and "C0G" will be used hereafter instead of "MKP film capacitors" and "C0G/NP0 MLCC ".

C0Gs have an ESR up to 9.5 times lower than MKPs for the same capacity, see Table 1. Their 65 times higher energy density and surface mounted device (SMD) characteristics are additional advantages. However, their maximum voltage and capacity values are reached more quickly than for MKPs [START_REF] Tdk | Features of High voltage MLCCs with C0G Characteristics and Replacement Solutions Overview[END_REF]. MKPs therefore have a great advantage over C0Gs if the resonant capacitor specification includes a large capacitance value and a relatively low current so that losses in MKPs are acceptable.

When WPTS power ratings reach 10 kW or more, conduction-cooled polypropylene film capacitors (CCPFC) are used. The latter allow to carry higher currents for the same volume, as shown in Table 1. This is possible thanks to a much lower ESR than conventional MKPs. In order to reduce the cost of resonant capacitors, a series and parallel arrangement of MKPs for high power WPTS. This solution is, however, limited by the number of components, considering that it is a through component. In addition, this would represent a very constraining volume compared to the volume of CCPFCs, which is why the latter are preferred for high powers [START_REF] Lemaitre | Efficiency-Oriented Design of Litz Wire for Several kW Power Experimented on 20 kW Prototype[END_REF], [START_REF] Bosshard | Multi-Objective Optimization of 50 kW/85 kHz IPT System for Public Transport[END_REF]. A C0G assembly, on the other hand, has a much smaller volume and weight than CCPFCs and MKPs for the same electrical constraints. This is detailed in the next section.

COG capacitors for WPTS

The limitations of C0Gs are related to their limited voltage withstand and capacitance values. A solution is proposed to overcome these limitations and will be illustrated by an example in part 2.3.

C0G Assembly benefits

C0Gs have an ESR up to 10 times lower than MKPs for the same capacity and voltage, see Table 1. Their higher capacitance stability, 50 times higher energy density and SMD characteristics are of great interest. A parallel series assembly of C0G on PCB can be an alternative to the efficient but expensive, bulky and heavy CCPFCs. The C0G hold a few tens of nF maximum when holding voltages in the kV range. This corresponds to the value of the resonant capacitor used for WPTS of several kW, ten kW with a SS topology. A series-parallel assembly increases the voltage and current withstand and thus meets the specifications. The assembly on PCB also makes it possible to improve the design of the latter in order to improve thermal dissipation.

CCPFCs are large with a limited choice of capacitances compared to C0Gs. However, the capacitance of the resonant capacitor resulting from the design must be achieved with a certain accuracy. For this purpose, a certain amount of CCPFCs in a parallel series assembly may be required to achieve the desired capacitance value, although the electrical constraints of the specification can be met with fewer capacitors. This can lead to significant additional costs due to the high unit cost of these components. With a C0G assembly, the desired capacitance can be achieved at a lower additional cost. This is due to the diversity of capacitance and a greater parallel series association.

The following section presents the approach followed to select the C0G capacitors and the series-parallel combination to be achieved.

Design approach

The C0G PCB that forms the resonant capacitor must meet the equivalent capacitance, voltage and current requirements of the specifications. These can be easily obtained from the WPTS design. Getting close enough to the theoretical capacitance is an additional constraint. The circuit diagram of the C0G PCB is shown in Fig. 4, with a parallel connection of N capacitor branches, each consisting of N capacitors in series. The considered circuit diagram of a C0G is that shown in Fig. 3.

The equivalent capacitance and resistance of the PCB are denoted by C and R respectively. The parasitic resistances, inductances and capacitances related to the PCB tracks are negligible considering the working frequency, this is experimentally validated in part 4. By noting I and V respectively as the current and voltage across each C0G, it is important to ensure that these quantities do not exceed the limits set in the datasheet. The selection approach is summarised in Fig. 5.

The numbers of series and parallel capacitors N and N given in equations ( 3) and (4) must be observed.

𝑁 ≥ 𝑁 = 𝑉 𝑉 (3) 𝑁 ≥ 𝑁 = 𝐼 𝐼 (4)
The choice of C0G with high capacitance, voltage and current withstand allows to limit the number of N capacitors on the PCB, as well as to reduce the ESR, which is inversely proportional to the capacitance, see [START_REF] Sarrazin | Bidirectional Wireless Power Transfer System with Wireless Control for Electrical Vehicle[END_REF]. It is then possible to estimate N and N and thus the desired capacitance C , using equation [START_REF] Tdk | Features of High voltage MLCCs with C0G Characteristics and Replacement Solutions Overview[END_REF].

𝐶 ≈ 𝑁 𝑁 . 𝐶 (5)
Fig. 5: Choice of C0G capacitor for assembly.

An analytical comparison between an assembly of C0G, MKP and CCPFC is proposed in the following section.

C0G PCB design for a 20 kW WPTS

The prototype considered for testing a C0G assembly is a 20 kW WPTS presented in [START_REF] Lemaitre | Efficiency-Oriented Design of Litz Wire for Several kW Power Experimented on 20 kW Prototype[END_REF]. It is a WPTS with a 210 mm air gap, sized at a resonant frequency of 81 kHz. At rated power, the primary resonant capacitor operates at a voltage V of 5.3 kV and a current I of 34 A

. The capacitance of C from equation ( 1) is 11.4 nF. Following the approach presented above, the selected C0G capacitors are C5750C0G3A333J280KC. The same selection method is used for the choice of the MKPs R76UN23304040J. The CCPFCs compared in Table 2 to the previous two types of capacitors are those used in the previous work [START_REF] Lemaitre | Efficiency-Oriented Design of Litz Wire for Several kW Power Experimented on 20 kW Prototype[END_REF]. The assemblies of C0Gs, MKPs and CCPFCs in the format N × N = N are 7×18=126, 9×23=207, 1×8=8 respectively. A minimum margin of 20% is taken on the maximum RMS current from the datasheet and presented in Table 1. The better current handling of the C0Gs in front of the MKPs reduces the total number of components N . The values of capacitance, RMS voltage, ESR and capacitance tolerance for the three capacitor technologies used are available in the "Capacitor characteristics" section of Table 2. Those of the capacitor sets are presented in the "Capacitor set characteristics" section. C0Gs are easier to mount as small SMD components. CCPFCs need to be screwed to the PCB although they are SMDs, which increases the complexity of assembly in front of the C0G PCB mounting. MKPs are through-hole components, they do not allow oven mounting of the PCB, so assembly time is increased. All three technologies allow for 5% accuracy on capacity tolerance. C0Gs allow losses to be divided by a factor of 9.5 for a price twice high as the MKPs. The interest of C0Gs thus lies in WPTSs with high currents, and therefore high powers.

The voltage levels are in fact imposed by the grid and the EV battery in the absence of additional converters allowing a modification of the voltage levels.

The currents consequently tend to increase in proportion to the power transferred to the WPTSs dedicated to EV charging.

The very low ESR of the C0Gs also makes it possible to reduce cooling requirements, or even eliminate them, as is the case for the C0G PCB tested in part 3. This simplification of the thermal aspects brings a gain in volume, weight and robustness due to the absence of cooling systems such as a fan. Furthermore, only the prices of the components are considered here. Consideration of the cost of assembly would be to the great advantage of C0Gs, especially in the case of mass production. Automatic assembly of SMD components such as MLCCs are also of great interest for industrial applications.

The following section aims at experimentally validating the theoretical comparison between the C0Gs and the CCPFCs achieved at Table 2.

Experimentations

The C0G PCB presented in sections 2.3 and 2.4 has been made in order to measure its performance. These will be compared to the CCPFC capacitors used in previous works and commonly used for WPTS of several tens of kW.

Setup

The WPTS tested and shown in Fig. 6 has been sized for a nominal power of 20 kW, at a resonant frequency of 81 kHz. Full system characteristics are available in paper [START_REF] Lemaitre | Efficiency-Oriented Design of Litz Wire for Several kW Power Experimented on 20 kW Prototype[END_REF].

The SS resonant topology is used with CSM 150/200 CCPFCs. In particular, the primary-side resonant capacitor consists of 8 capacitors arranged in series as described in section 2.4 and shown in Fig. 7. The PCB assembly of C0G shown in Fig. 8 was made to be used as a primary-side resonant capacitor, allowing comparison with CCPFCs. Two configurations are thus tested. The resonant capacitor on the secondary side is a CCPFC assembly in both cases. Configuration 1 uses CFLCCs on the primary side, while configuration 2 uses the C0G PCB on the primary side. Tool measurements used are current probes LeCroy CP150, voltage probe LeCroy HVD3605, WaveRunner RO 66Zi 600 MHz oscilloscope and thermal camera. The working frequency at the nominal operating point of 20 kW is slightly above 76 kHz.

Results

It should be noted that for the selected resonant frequency of 81 kHz and a primary winding inductance of 338 µH, the primary resonant capacitor must have a value of 11.4 nF. The choice of components being limiting, the total capacitance achieved is 12.5 nF and 12.83 nF for the equivalent CCPFC and C0G resonant capacitor respectively. This slight difference is sufficient to change the operating point at a given frequency. As a result, the current through the primary resonant capacitor is not the same for the WPTS with C0Gs as the one with CCPFCs at equal power transferred. Fig. 9 shows that the current measured on the primary side of the coupler at nominal power is 33.2 A for CCPFCs, compared to 35.9 A for C0Gs. The ESRs presented in Table 3 consider only the resistance of the components. Modifications were made to the C0G PCB to allow current measurements in each leg, which increases the PCB resistance. The measured efficiencies shown in Fig. 10 agree with the ESR values presented in Table 3 on the best performance for C0G compared to CCPFC, especially from 8 kW of transferred power. The voltages measured across several capacitors and the currents measured in each branch of the PCB show relative differences in mean values of less than 1%. The distribution of electrical stresses is homogeneous. A theoretical comparison of a C0G and CCPFC assembly for the secondary resonant capacitor design is also presented in Table 3. This comparison is interesting because the secondary specifications differ in terms of the voltage/current constraints to be met of 2.3 kV / 54 A , for a target capacitance of 48 nF. If the number of capacitors of CCPFC remains equal to that of the primary in order to comply with this new specification, this is not the case for the C0G PCB equal to 77. The ESR of the C0G PCBs compared to the CCPFCs is about twice as small in the primary and twice as large in the secondary. The efficiency obtained with C0G at the primary and secondary sides of the WPTS reaches the efficiency obtained with CCPFCs. Thus, the use of C0G allows gains on the overall primary and secondary sides presented in Table 3. This while ensuring an efficiency as high as that provided by CCPFCs. The tests below aim to validate and test limits of design to conclude on the best design for a C0G PCB according to thermal and electrical constraints.

Prototypes

Five PCB that have been realized are present on the Fig. 11. Equivalent capacitance value C is equal to 15.7 nF, which is the order of magnitude for EV charger working near the SAE recommended frequency of 85 kHz. The capacitors used are C5750C0G3A223J280KC from TDK. Versions V , V , V , V have the same size while V is two times smaller. First one called V is the reference PCB with a classical design. Connections between parallel branches are added on PCB V to improve the electrical constraints distribution. To better homogenise capacitors temperature, V PCB central branches are more distant from each other while those on the periphery are closer to keep same size as PCB V . PCB V aims to increase the loss dissipation by reducing thermal resistance of the PCB between top and bottom. Much more loss are so dissipated the bottom side in that way, thanks to the use of thermal vias of 0.2 mm diameter. Last PCB V is two times smaller than others, allowing to measure and quantify the increase of temperature due to a compact design. If the parasitic inductance of the PCBs is measured in the next section, the resistance of the PCB tracks can be neglected now. Considering a track width equal to the 5 mm width of the C0G pad, the resistance of one branch is 1 mΩ. This is negligible compared to the equivalent resistance of 23 mΩ of the 7 capacitors in series of a branch.

Experiments

Each prototype has been tested at the PCB nominal power. The working frequency is 75 kHz, the total current is 16 A and total voltage is 2.2 kV . These electrical constraints may correspond to those held by the resonant capacitors of a 7-kW prototype, class WPT2 of the J2956 standard. Voltage and current measures show a maximum difference under 1% of nominal values, for each PCB. V version does not bring improvements on the homogenisation of electrical constraints regarding the very good results of reference version V . Study of the V design on electrical constraints distribution compared to V considering aging is an interesting outlook. The capacitance and resonance frequency of the PCBs were measured at the impedance bridge. The PCB capacitance measurements available Table 4 differ by a maximum of 2.1 % with the equivalent capacitance C of 15.7 nF. The self-resonant frequency of a capacitor is 15.2 MHz, so the parasitic inductance is 5 nH. The self-resonant frequency of PCBs is between 3.38 MHz and 3.74 MHz, due to a parasitic PCBs inductance ranging from 116 nH to 142 nH. The self-resonant frequency is still much higher than the working frequency. The inductive behaviour of PCBs is therefore quite negligible.

Fig. 1 : 2 :

 12 Fig. 1: WPTS principle of an EV charging Fig. 2: Schematic of a series-series resonant DC-DC converter

Fig. 3 :

 3 Fig. 3: Considered electrical circuit of a capacitor.

Fig. 4 :

 4 Fig. 4: Detailed electrical circuit of a capacitor (a) and equivalent circuit of a PCB implementing C0G (b).

Fig. 6 :Fig. 7 :Fig. 8 :

 678 Fig. 6: 20 kW WPTS used for experimentations

Fig. 9 :

 9 Fig. 9: DC-to-DC efficiency of the WPTS with CCPFC (Blue) and C0G PCB (Orange) on the primary side as function of the transferred power.

Fig. 10 :

 10 Fig. 10: DC-to-DC efficiency of the WPTS with CCPFC (Blue) and C0G PCB (Orange) on the primary side as function of the transferred power.

Fig. 11 :

 11 Fig. 11: Comparison of five PCB design: classical design (a), add of connections between parallel branches (b), add of space between center branches (c), add of thermal vias (d) and a more compact design (e).

Table 1 :

 1 Characteristics comparison of the C0G C5750C0G3A333J280, MKP R76UN23304040J, and CCPFC CELEM 150/200. C0G and CCPFC will be compared theoretically and experimentally in part 2 and 3.

		Capacitance.	𝑉	ESR (100 kHz)	Max Current (100 kHz)	Capacitance Tolerance	Volume Ratio/C0G	Energy density
	C0G	33 nF	353 V	2 𝑚𝛺	6.2 𝐴	± 5 %	1	52 µJ/𝑚𝑚
	MKP	33 nF	495 V	19 𝑚𝛺	4.9 𝐴	± 5 %	48	0.8 µJ/𝑚𝑚
	CCPFC	100 nF	1000 V	1.2 𝑚𝛺	250 𝐴	± 5 %	772	2.1 µJ /𝑚𝑚

Table 2 :

 2 Comparison of resonant capacitors characteristics using C0G, MKP R76UN23304040J and CCPFC. Design made for the primary side of 20 kW prototype[START_REF] Sarrazin | Bidirectional Wireless Power Transfer System with Wireless Control for Electrical Vehicle[END_REF]. * ESR are given for 100 kHz.

		Capacitor characteristics			Capacitor set characteristics
		Capacitance	𝑉	ESR*	Ease of mounting	Cap number	Price Ratio	ESR Ratio	Volume Ratio
	C0G	33 nF	353 V	2 𝑚𝛺	+++	126	1	1	1
	MKP	33 nF	495 V	19 𝑚𝛺	+	207	0.5	9.5	79
	CCPFC	100 nF	1000 V	1.2 𝑚𝛺	++	8	3.2	1.9

Table 3 :

 3 Comparison of resonant capacitors characteristics using C0Gs, MKPs and CCPFCs. Design made for the primary and secondary side of 20 kW prototype[START_REF] Sarrazin | Bidirectional Wireless Power Transfer System with Wireless Control for Electrical Vehicle[END_REF]. * ESR are given for 100 kHz.

			Cap. Number Weight Ratio Volume Ratio Price Ratio	𝐸𝑆𝑅*	𝐸𝑆𝑅 Ratio
	Primary	C0G	126	1	1	1	5.14	1
	Side	CCPFC	8	40	49	3.2	9.6	1.9
	Secondary	C0G	77	1	1	1	4.0	1
	Side	CCPFC	8	66	80	5.3	2.4	0.6

Table 4 :

 4 Capacitance, resonant frequency, the maximum self-heating and the maximum temperature difference between the C0Gs of the prototypes 𝐕 𝐑𝐄𝐅 , 𝐕 𝐂𝐎𝐍 , 𝐕 𝐒𝐏𝐀𝐂𝐄 , 𝐕 𝐕𝐈𝐀𝐒 and 𝐕 𝐃𝐄𝐍𝐒𝐄 measured with impedance bridge.

		C (nF)	Fr (MHz)	ΔT	ΔT
	V	15.33	3.42	5°C	12°C
	V	15.33	3.39	5°C	12°C
	V	15.33	3.38	3°C	12°C
	V	15.35	3.54	4°C	9°C
	V	15.53	3.74	6°C	15°C

The maximum self-heating ΔT defined by equation [START_REF] Avx | Multilayer Ceramic Leaded Capacitors[END_REF] and measured with the thermal camera is 12°C for PCBs V , V , V . Thermal vias improved the loss dissipation and limited the maximum self-heating to 9°C for PCB V

. Dividing the PCB size by 2 results in a higher ΔT of 15°C for V . Thermal vias allowed without additional cost a reduction of 3°C with a natural air cooling and without heatsink.

The maximum temperature difference between the C0Gs defined by equation ( 7) was measured, the results are presented in Table 4.

The maximum difference of capacitors temperature over PCB is 5°C for V compared to 3°C with V . The latter allows to better homogenise temperature constraints over capacitors, that has a positive effect on components aging. In the light of these results, a combination of the V , V and V PCB designs appears to be the best solution. In addition, the V and V designs allow for a better homogenisation of thermal stresses while reducing them.

Conclusion

The measurements carried out on a 20-kW prototype allow to validate the theoretical analysis. The data from Table 3 show that the C0G capacitor PCB achieves same performances than the conduction cooled polypropylene film capacitors commonly used for its very low ESR, and high voltage/current withstand, while being around 4 times cheaper, with a weight 53 times lighter and a volume 65 times smaller. The efficiency of the C0G technology as well as the proposed improvements in the design of the PCB allow its use in high power WPTS up to several tens of kW, leading to a minimal weight and volume. The excellent C0G thermal coefficient and the possibility to operate with natural convection air cooling make the system more robust. In addition, the rapidly increasing performances and decreasing prices of MLCC capacitors are creating even more opportunities for this capacitor technology.


Liquet Benoit
email: benoit.liquet@isped.u-bordeaux2.fr

Commenges Daniel

Computation of the p-value of the maximum of score tests in the generalized linear model; application to multiple coding

Keywords: Generalized linear model, multiplicity, p-value, Score test

We propose a method to correct the significance level for a series of tests corresponding to several transformations of an explanatory variable in generalized linear model. Correlation between score test are derived to apply the proposed method.

Introduction

In many applied studies it is common that the model used is not completely fixed in advance and that several possibilities are tried. We consider the case where a regression model belonging to the family of the generalized linear models is used. In epidemiology for instance it is quite common that the study focusses on one particular risk factor; the problem is to analyse whether this risk factor has an influence on the risk of a disease or on a biological trait. To answer the question a regression model is used in which the risk factor will be represented by a continuous variable X, and allowing to adjust on p -1 already known risk factors of the studied trait. The analysis focusses on the test of "β = 0", where β is the coefficient representing the effect of the risk factor of interest. However the form of the effect (or the dose-effect relationship) is not known in advance.

While a non-parametric approach to this problem may be useful [START_REF] Hastie | Generalized Additive Models[END_REF], most often a simpler approach is used: several transformations of the original variable g k (X), k = 1, . . . , K, may be tried. Examples of such transformations are dichotomizations of the original continuous variable, where K cutoff points may be used, and Box-Cox tranformations, where K different powers of the original variable are tried (the Box-Cox family also includes the log transform). For each of the transforms tried a test of "β = 0" is performed.

Generally the most significant test is retained and often given without corrections in publications. Of course this leads to increased type I error risk [START_REF] Miller | Simultaneous Statistical Inference[END_REF].

For instance in studying the effect of minerals in drinking water on the risk of a disease, threshold effect models have been currently used: higher (or lower) risk is expected for a concentration above a certain level. However this threshold value is generally unknown, leading the epidemiologist to perform several trials.

Multiplying the tests and reporting the most significant one without correction is able to pervert the whole inference process in applied science and particularly in epidemiology.

The p value should be corrected to take account of the multiplicity of the test.

The simplest correction is to apply the Bonferoni rule; however this leads to a conservative test if the original tests are positively correlated as is the case here. [START_REF] Efron | The length heuristic for simultaneous hypothesis tests[END_REF] proposed a correction taking account of the correlation between two consecutive tests, if there is a natural order between the tests with high correlation between adjacent tests. [START_REF] Liquet | Correction of the P-value after multiple coding of an explanatory variable in logistic regression[END_REF] and [START_REF] Hashemi | Correction of the p-value after multiple tests in a Cox proportional hazard model[END_REF] proposed a more exact correction taking into account the whole correlation matrix, for score tests obtained in logistic regression and proportional hazards models respectively.

Here, we propose a correction of the p value when multiple transformations of an explanatory variable have been tried in a generalized linear model. We construct K score tests corresponding to the K coding of the explanatory variable.

We thus have a vector of test statistics T = (T 1 , . . . , T K ) for the same null hypothesis H 0 which have asymptotically a standard normal distribution. Rejecting H 0 if one of the test T k is larger than a critical value c is equivalent to rejecting H 0 if T max > c, where T max = max(T 1 , . . . , T K ). To cope with the multiplicity problem, we need to compute the probability of Type I error for the statistic T max under the null hypothesis H 0 :

p value = P (T max > t max ) = 1 -P (T 1 < t max , . . . , T max < t max ).
The asymptotic joint distribution of T 1 , . . . , T max is a multivariate normal distribution with zero mean and a certain covariance matrix Σ that we will estimate in the next section. Then we will be able to compute P (T 1 < t max , . . . , T max < t max ) using numerical integration [START_REF] Genz | Numerical computation of multivariate normal probabilities[END_REF].

2 Correlation between tests in generalized linear model

f Y (Y i , θ i , φ) = exp {[θ i Y i -b(θ i )]/a(φ) + c(Y i , φ)}, with E[Y i ] = µ i = b (θ i ), var[Y i ] = b (θ i )a(φ).
We want to test the influence of a variable X i , adjusted on a vector of explanatory variables Z i . We consider the case where we do not know the form of the effect of X i so we may consider K transformations of this variable X i (k) = g k (X i ), k = 1, . . . , K. The model for transformation k can be obtained by modeling the canonical parameter θ i as:

θ i (k) = Z i γ + X i (k)β k , i = 1, . . . , n,
where

Z i = (1, Z i 1 , . . . , Z i p-1
) and γ = (γ 0 , . . . , γ p-1 ) T is a p × 1 vector of coefficients. In the sequel,we will denote by X(k) the vector (X 1 (k), . . . , X n (k));

µ = (µ 1 , . . . , µ n ) T and l k the log-likelihood of Y = (Y 1 , . . . , Y n ) T for the model with X(k).

Score test

For all the K transformations, H 0 : "β k = 0" is the same null hypothesis, given by θ i (k) = Z i γ. This defines a unique probability measure under which the distribution of the score tests (T k ) can be computed as standardized versions of the score statistics U (k) which are asymptotically normal distributed:

U (k) = ∂l k ∂β k (β = 0) = 1 a(φ) X T (k)[Y -μ] = 1 a(φ) X T (k) R,
where R is the vector of residuals Ri = Y iμi computed under the null hypothesis. Asymptotically, the variance of the score test can be computed as (see [START_REF] Cox | Theoretical Statistics[END_REF]:

var U (k) = I β k β k -I β k γ I -1 γγ I γβ k , where I β k β k = -E ∂ 2 l ∂β k ∂β k , I β k γ = -E ∂ 2 l ∂β k ∂γ , I γγ = -E ∂ 2 l ∂γ∂γ .
Another method is to calculate directly var U (k) by :

var U (k) = 1 a(φ) 2 X T (k)var[ R]X(k).
The term, var[ R], will be also necessary to determine the correlation between score tests.

Estimation of var[Yμ] = var[ R]

A Taylor expansion of Yμ about its values in γ, the real parameter value, gives

Y -μ = Y -µ - ∂µ ∂γ T (γ)(γ -γ) + o p (n -1/2 ), (1)
where ∂µ ∂γ T (γ) = 1 a(φ) V Z and Z is a n × p matrix with row Z i , i = 1, . . . , n ; V is the diagonal matrix with diagonal terms v ii = var Y i . Expanding the first derivatives of the log likelihood around γ yields

∂l ∂γ (γ) = ∂l ∂γ (γ) + ∂ 2 l ∂γ 2 (γ)(γ -γ) + o p (n -1/2 ), (2)
where

∂ 2 l ∂γ 2 (γ) = -1 a(φ) 2 Z T V Z and ∂l ∂γ (γ) = 0 since γ is estimated nullifying the score U γ = ∂l ∂γ (γ).
With (2) we find :

γ -γ = a(φ) 2 (Z T V Z) -1 U γ + o p (n -1/2 ),
where U γ = 1 a(φ) Z T (Yµ). So, replacing in (1) we find:

Y -μ = Y -µ -( 1 a(φ) V Z)a(φ)(Z T V Z) -1 Z T (Y -µ) + o p (n -1/2 ) = (I -H)(Y -µ) + o p (n -1/2 ),
where H is the matrix

H = V Z(Z T V Z) -1 Z T .
var U (k) = 1 a(φ) 2 X T (k)(I -H)V X(k).
In practice, we use an estimator of var U (k) defined by :

var U (k) = 1 â(φ) 2 X T (k)(I -H) V X(k), (3)
where â(φ) and V are the estimator of a(φ) and V .

Correlation between two tests

Let T k and T l be two score test statistics associated with the transformations X(k) and X(l):

T k = U (k) var U (k) ; T l = U (l) var U (l) .
Neglecting the covariance between the estimators of the variances of U (k) and U (l), the correlation between the two tests is :

ρ kl = corr(T k , T l ) cov(U (k), U (l)) var U (l) var U (l) = 1 â(φ) 2 X T (k)var[Y -μ]X(l)
var U (l) var U (l) .

Using (3) we finally obtain an expression which does not depend on a(φ):

ρ kl = X T (k)[(I -H) V ]X(l) X T (k)(I -H) V X(k) X T (l)(I -H) V X(l)
.

Then we can compute P (T 1 < t max , . . . , T K < t max ) for determining the p-value associated to the test T max . This is done by integrating the normal density over a quadrant and this can be done quite accurately for K < 20 using the subregion adaptive algorithm proposed by [START_REF] Genz | Numerical computation of multivariate normal probabilities[END_REF]. Note that this result allows a correction of the p-value if one knows the vectors of the transformed variable X(k), k = 1, . . . , K, the matrix of explanatory variables Z and the estimated variances V ; that is, there is the possibility to design a unique program for any model in the family and for any transformation.

Simulation

Simulations for studying the effect of the correction on the type I error risk were carried out with a Poisson model (a

(φ) = 1, b(θ i ) = e θ i , µ i = E[Y i ] = e θ i )
consisting of two explanatory variables: Z 1 an adjustment variable and X the variable of interest. We considered the following models

log E[Y i ] = θ i (k) = γ 0 + γ 1 Z i 1 + βX i (k), i = 1, . . . , n.
where X i (k) were dichotomized versions of a continuous variable X. For the cut-off points we chose the median for one dichotomous transformation (K = 1), the first and the second terciles for two dichotomous transformations (K = 2), the quartiles values for three transformations (K = 3) and so on. Z 1 and X were generated according to a uniform distribution U [0, 1]. The sample size was set to be 100. We used 10,000 replications for the simulation.

Study of type I error rate

In this simulation, we took γ 0 = 1, γ 1 = 10 and β = 0. For a replication, the rejection criterion of H 0 (β = 0) was a p value less than 0.05. Thus, for a simulation, the empirical type I error rate was the proportion of p value less than 0.05. Figure 1(a) shows the type I error rate for dichotomous transformations, as a function of the number of coding (K) tried. The naive approach which retains the test with the smallest p-value had, as expected, a type I error rate which increased with the number of cutpoints tried: this is exhibited by the "no correction" curve in Figure 1(a). The error rate calculated by the Bonferroni method decreased with the number of cutpoints, leading to a conservative test.

The exact calculation proposed here gave a type I error rate very close to the nominal 0.05 value.

Figure 1 approximately here

Power

In this simulation, we took γ 0 = 1, γ 1 = 5 and β = 5. We studied the power for a threshold effect model with a cutpoint value at the first tercile ("threshold effect model"). Figure 1(b) gives the power as a function of the number of cutpoints. With any method we expect that the power will be substantially less than that of a test that would be done knowing the cutpoint value: this gives an upper bound for the power (represented by a horizontal line in Figure 1(b)). As expected the exact calculation provides more power than the Bonferroni method.

Each time the true cutpoint value appears in the values tried the power increases; however these oscillations decrease with the number of values tried. When the good cutpoint value does not appear in the values tried, the power of the exact

  Note that this expression is exact for the normal linear model. With an approximation error of order o p (n -1 ), we have var[Yμ] = var[(I -H)(Yµ)]. Using the idempotence property of (I -H), it can be seen after some computation that var[(I -H)(Yµ)] = (I -H)V . Finally we find :

method tends to increase with the number of values tried: the power is slightly higher using deciles (9 values tried) than using quartiles (3 values tried), and this is in contrast with the Bonferroni method.

LIST OF FIGURE