Giulia Lioi
email: giulia.lioi@inria.fr

Mathis Fleury

Simon Butet

Anatole Lécuyer

Christian Barillot

Isabelle Bonan

Neurofeedback

Bimodal EEG-fMRI Neurofeedback for Stroke Rehabilitation

BACKGROUND METHODS

Experimental Platform

Neurofeedback (NF) has potential to be applied for stroke rehabilitation [START_REF] Wang | The potential of real-time fMRI neurofeedback for stroke rehabilitation[END_REF], [START_REF] Pichiorri | Brain-computer interface boosts motor imagery practice during stroke recovery[END_REF] however the effectiveness of NF for stroke has not been thoroughly assessed yet.

Bimodal EEG-fMRI NF [START_REF] Zotev | Self-regulation of human brain activity using simultaneous real-time fMRI and EEG neurofeedback[END_REF], [START_REF] Perronnet | Unimodal Versus Bimodal EEG-fMRI Neurofeedback of a Motor Imagery Task To cite this version : Unimodal Versus Bimodal EEG-fMRI Neurofeedback of a Motor Imagery Task[END_REF] is a promising technique to achieve a more efficient and specific self-regulation, which may be crucial for clinical application.

AIMS  Test the feasibility of applying bimodal EEG-MRI NF for stroke rehabilitation in two chronic patients affected by left hemiplegia (subcortical lesion).

 Identify problematics and guidelines in view of a clinical study on stroke patients.

CURRENT AND FUTURE WORKS

 Improve performances and simplify the workflow of the bimodal NF platform.

 Clinical study on Stroke patients to test the efficacy of multisession bimodal NF for rehabilitation.

Figure 4 .

 4 Figure 4. EEG and fMRI NF scores during a NF session. Example from one patient (1).The left column shows the filter and the ROI selected for NF computation during calibration.

Figure 1 .

 1 Figure 1. Bimodal EEG-fMRI NF platform [5] (Neurinfo, CHU Pontchaillou, Rennes). The platform integrates and synchronizes EEG and fMRI subsystems and signal flow for the computation and visualization of the bimodal NF.

Figure 6 .Figure 2 .

 62 Figure 6. fMRI signal regulation as a function of the selected ROI. The bar plots represent BOLD activity in the selected ROI with respect to background (mean+std across blocks) during rest and NF. Relative statistics are showed (Wilcoxon tests, * p<0.05, ** p<0.01)

Figure 5 .

 5 Figure 5. Average BOLD activations maps over the two NF sessions for patient 1 (left) and 2 (right) (TASK>REST; k > 10 voxels).

Figure 3a .

 3a Figure 3a. Lesion and corticospinal tract (CST) of patient 1 (Right ischemic stroke). TheCST was estimated from tractography of diffusion weighted images[START_REF] Lee | Fiber tracking by diffusion tensor imaging in corticospinal tract stroke: Topographical correlation with clinical symptoms[END_REF]. R L


Keywords: Mark-ups, Fixité du capital, Concurrence imparfaite Markups, Capital Fixity, Imperfect Competition JEL Classification: L11, L13, L60

Le traitement des coûts du capital, soit comme coûts fixes ou variables, est essentiel pour l'estimation des mark -ups. Les données penchent nettement pour l'hypothèse de fixité, ce qui explique les hauts niveaux de mark -up relevés dans des études précédentes utilisant la méthodologie de Roeger.

L'estimation directe du ratio de la production sur les coûts variables est préférable.

Introduction

Industrial economics is indebted to [START_REF] Hall | Market structure and macroeconomic fluctuations[END_REF] for estimating markups at sectoral levels.

Improvements have then been proposed by [START_REF] Basu | Intermediate goods and business cycles: implications for productivity and welfare[END_REF] who highlights the quantitative importance of paying greater attention to materials, and by [START_REF] Roeger | Can imperfect competition explain the difference between primal and dual productivity measures? Estimates for U.S. manufacturing[END_REF], who derives a new methodology that circumvents intricate endogeneity issues in Hall's approach. First, it is argued here, that both methodologies should lead to a cautious quantitative interpretation of the respective markup estimates in terms of profit ratios, and second, that the shortcomings in Roeger's estimates reveal specification rather than measurement issues. Roeger's overestimation, identified in previous studies, is elucidated and substantiated to the extent that the returns to scale on the variable factors are decreasing.

Roeger's apparent overestimation reveals misspecification issues

The common framework assumes that identical firms in a given sector have the following homogeneous production function:

) , , ( . M L K F A Y = (1)
where Y is output, K capital, L labor, M materials and A a productivity term. µ denoting the markup over marginal cost and x the returns to scale, first order conditions and Euler's equation lead to:

) . . . ( . . M P L W K R x Y P m + + = µ (2a)
where P is the price of output, and R, W and m P are the respective factor prices of capital, labor and materials. Equation (2a) links the respective factor shares in total output,

M L K a a a , ,
, such that:

µ / x a a a M L K = + + (2b)
dz standing for the logarithm differential of any given Z variable, Hall's approach and its extensions proceed from the derivation of equation ( 1 with dx and dz the respective left-hand side (LHS) and right-hand side (RHS) variables of equation (4Ra). Roeger's equation links the markup to the sensitivity of the capital share to the changes of relative factor shares. In fact, Roeger estimates a specification equivalent to (4Ra) but expressed in terms of the (adjusted

) Lerner index µ / 1 1 ~-= L : dx L dz dx . ) ( = - (4Rb)
What are the comparative advantages of each approach? On the one hand, Hall's methodology allows for the identification of both markup over marginal cost and returns to scale, whereas Roeger's can only estimate their ratio which is the markup over average cost as seen from (2a). Moreover, Hall's does not need any computation of rental capital cost contrary to Roeger's, which requires its logarithmic variation. This second impediment is considered so serious that Roeger's is often disregarded. However, this paper shows that this criticism is exaggerated and misses the main point.

On the other hand, the main difficulty in Hall's, and that Roeger's avoids, comes from the total factor productivity growth term, da, which is correlated to RHS variables in (4H

u COST PY + = . μ (5)
where u are the residuals. Throughout this study, equation ( 5) is called the benchmark equation, and the estimated markup and estimated residual for date t are denoted B µ ~ and t u respectively. Data for this study is from the OECD STAN database and is described in Appendix A.

Table 1 shows, with US manufacturing sectors as an illustrative example, that Roeger's estimates are much greater than the benchmark, for which Durbin-Watson statistics indicate the need to correct for auto-correlation, although once done, the estimates do not change much. 1 Roeger's markups are greater than the benchmark in 11 out of 14 sectors, being perceptibly lower in 1 sector only, and Roeger's average stands at 5.1 points above the benchmark average. Any possibility that the mismeasurement of capital costs is mainly responsible for the differences in these estimates should be discarded for the following three reasons.

First, as can be read from table 2, this issue proves recalcitrant to very different assumptions for computation of capital data, referring to depreciation, interest rates and initial capital stock.

2 Second, the capital share of total output, based on our favored computation of capital stock and rental cost -the first one in table 2 used to produce table 1 -, varies from 3.6% for "Leather products and footwear" to 9.8% for "Basic metals" on average over the period, so that mismeasurment cannot cope with the magnitude of the problem. The apparent positive correlation between the difference in estimates and the average capital share (last row of table 2) is distinct and only gives a clue to the source of the problem.

Third and even more convincingly, Roeger's (apparent) overestimation has been highlighted by other studies based on different database and computations of capital costs, and should therefore be taken as well-established. In other words, differences in estimates from one methodology to another are not mainly due to mismeasurements but rather to misspecifications. Among those studies, [START_REF] Hindriks | Comparative advantages in estimating markups[END_REF] and [START_REF] Hylleberg | A note on the estimation of the markup pricing in manufacturing[END_REF] note that inferred capital shares from Roeger's estimates are unrealistically low as a large proportion run into negative territory. It then appears that the residuals, v, might be correlated with the RHS variable, dz. The OLS Roeger estimate of µ ~ from (4Ra), R µ ~ is therefore given exactly by using (6):

2 1 2 2 ) /( . ) /( . . ~diff diff dz PY dz du dz PY dz drk u B t t t t t t t t t B R + + ≡ + - = ∑ ∑ ∑ ∑ µ µ µ (7)
Equation ( 7) breaks down the numerical difference in the two estimates into two t erms. Table 4 reveals, somewhat surprisingly, that most of it comes from the second term, i.e. from the positive correlation between the first-difference of equation ( 5) estimated residual and the variable dz, itself a complex combination of relative factor shares, a priori negatively related to capital share changes.

Given the correlation displayed in the last row of table 2, the potential negative correlation between equation ( 5) residuals and capital costs might therefore explain most of the difference b etween Roeger's and benchmark markups.

Finally, Hall's estimates are not reported here. Although they are not very sensitive to the various capital stock measures, the OLS estimates suffer two major shortcomings. Firstly, the precision is much poorer than Roeger's. Secondly, Hall's estimates prove extremely sensitive to the price index which is necessary to derive the volume of materials, as the data provides materials in value terms only. Therefore, I have not even tried to overcome the endogeneity issue. Having given several clues in this descriptive analysis, I now come to what most likely explains the differences in estimates.

Capital fixity as the main suspect

Many reasons may invalidate (5). The competition environment in OECD countries has changed substantially over the last thirty years. Assuming markups are constant, as equation ( 2) implies, might be misleading. Indeed, allowing the markup to be time-dependant enriches equation ( 2), as is extensively developed in a related paper, [START_REF] Boulhol | Pro-competitive policies and the convergence of markups[END_REF], but does little to explain the difference between the benchmark and Roeger-type estimates. To see this, consider the specification:

t t t COST PY . μ = with ) ( ~t t X µ µ = (8)
where t X designates exogenous variables representing the competition environment which determines markup levels at time t. µ -based Roeger's equation equivalent to (8) becomes:

µ µ µ / . ~d dz dx t + = (8R)
Estimates in ( 8) and (8R) are compared for a time-dependant markup represented by a polynomial of order up to three. Although the polynomial adds significant explanatory power, i.e. even when accounting for the fact that the steady-state markup µ ~ is not so steady, the apparent overestimation persists. Similarly, the impact of economic cycles and price rigidities prove to matter greatly in estimating markups, as shown in [START_REF] Boulhol | Pro-competitive policies and the convergence of markups[END_REF], but does not help in solving the puzzle.

What does then? Equation (5) holds for COST representing the total cost of the true variable factors used by firms to maximise profits. It is essential at this point to insist that the notion of markup we are interested in is not the tautological definition given by the ratio of output to total costs. Rather, it comes from first order conditions in profit maximisation and captures the idea of market power, i.e. the capacity firms have under imperfect competition to mark up variable costs in setting their prices at the desired level: it reflects the intensity of competition, the market structure, the elasticity of demand, etc.

If capital is fixed, at least in the short run, then costs related to capital will be fixed costs. They will impact overall profitability but will disappear from the markup equation which becomes:

) . . ( . . M P L W Y P m fix + = µ (9)
Naturally in this case, the markup is adjusted with returns to scale on the variable factors only.

3

There is an easy way to check whether equation ( 9) makes more sense than equation ( 5). From the following specification:

u K R h M P L W Y P m + + + = . . . ) . . ( . . µ µ (10)
If the parameter h is not significantly different from 1 then the assumption that capital costs should be considered variable costs and therefore equation ( 5) cannot be rejected; reciprocally, if h is not significantly different from 0 then the assumption that capital is a quasi-fixed factor cannot be rejected.

At the 95% (90% respectively) confidence level, taking the most preferred capital computation method, the parameter h, is significantly positive in only 23% (28% resp.) of the 132 sectors tested: stated differently, the fixity of capital cannot be rejected in 77% (72% resp.) of the cases. Moreover, this result is robust to various measures of capital stock and cost. Also, the assumption that capital costs should be treated as variable costs (H0: h = 1) is rejected in 68% (74% resp.) of the sectors at the 95% (90% resp.) level.

What happens then to Roeger's estimates under the assumption of capital fixity? When equation ( 9) is derived according to Roeger's methodology, the variable drk plays no role, due to fixity, and is removed from variables dx and dz. As explained below, the systematic bias between the estimates coming from the regression of the then-dx variable on the then-dz variable and fix µ ~ from equation ( 9) disappears.

However, there is more: when capital is fixed, Roeger's specification given by (4Ra) -which includes drk ! -is correct after all. 4 This comes from the fact that the variable drk appears in Roeger's equation multiplied by the capital share of variable costs, ) .

( 1

M L a a + -µ
, which equals zero under the assumption of capital fixity leading to equation ( 9). This means that, if capital is truly fixed -as seems to be acceptable in about three quarters of the time series under study -even though Roeger's equation is derived from a specification treating capital as a variable factor, it generates the correct (adjusted for returns to scale on variable factors ) markup over marginal cost, equalling the markup over average cost of the variable factors. This latter is obviously greater than the estimated markup over average total cost, obtained when including capital in the benchmark specification. Although it does not give a direct indication of profitability because fixed costs should be subtracted, it remains relevant for assessing changes in market power through time. Consequently, it is no surprise that Roeger's implied capital shares come out close to zero on average.

Table 5 illustrates these results in the case of the USA. The first two columns repeat estimates from table 1. Column 3 gives the benchmark in the case of capital fixity: as is apparent from the last row, the systematic spread with Roeger's disappears. Then, the estimates of equation ( 10) are successively reported, first bounding h between 0 and 1, and lastly relaxing the constraints. On average, the h parameter takes a value of 0.34 and 0.22 respectively and is almost never significantly different from 0, suggesting a very low speed of capital adjustment to the optimal level. Moreover, the average difference between the unbounded benchmark and Roeger's mostly vanishes, and the average absolute difference (not reported here) is more than halved. Supportively, it is remarkable that when the initial spread between Roeger's and benchmark estimates (first two columns) is high, the corrected benchmark using unbounded estimates, or the very similar capital fixity version (column 3), brings markups closer towards Roeger's.

Finally, Roeger's methodology seems appealing because it is somewhat robust to how capital is treated. However, Roeger's estimates are much less precise than those of the benchmark, as can be seen from table 1, and if capital were truly a variable factor, it is unclear, as discussed extensively by [START_REF] Hindriks | Comparative advantages in estimating markups[END_REF], whether Roeger's specification ought to be preferred to that of the benchmark: it is justified only if the relative change of capital costs over time is considered to be more precisely evaluated than the level, which is dubious.

Conclusion

This study points out that the treatment of capital costs, as either fixed or variable costs, is central in estimating markups. Measurement issues are secondary. Moreover, the data leans clearly towards the assumption of quasi-fixity of capital. Therefore, the usual estimates must be interpreted with caution, as indeed, a markup over unity is required to cover capital costs, even under a long-term zero profit condition. Finally, Roeger's methodology may overestimate markup levels to the extent that the returns to scale on the variable factors are decreasing. To understand this better, consider the Cobb-

Douglas case under constant (overall) returns to scale, b a b a M L K Y - - = 1 . Roeger's estimates will then result in ) 1 /( a R - = µ µ
, and even under perfect competition, Roeger's markups will be greater than unity and increasing with long-term capital shares.

Appendix A: Data description

Two samples have been built covering thirteen OECD countries' manufacturing industries at the twodigit level for the period 1970-2000, using International Standard Industrial Classification (ISIC), third revision. One has more detailed information but is sparse, as some sectors are missing for a number of countries, and is composed of 138 time series (a country-sector crossing). The other contains more aggregated data but is more balanced with 132 annual time series available out of a total of 143.

Sector identification is given in Table A1. Note that the averages across sectors presented in the following tables are unweighted, i.e. treating each equally, because our prime interest lies in the mechanisms at work rather than in the impact for the total economy.

Sectoral data come from the OECD Structural Analysis (STAN) Database. .

Capital

The price of capital, k p , used in the study is the price of investment calculated from the Gross Fixed Capital Formation at current prices (GFCF) and in volume (GFCFK). When data is not available, the price of the GDP deflator (source OECD Economic Outlook) is chosen for k p . The user cost of capital is calculated classically according to: ) .(

a k k p d r p R & - + =
, where r is the interest rate, d the depreciation rate and

a k p &
is the expected relative change in the price of capital. By default, r was chosen as the long-term interest rate (but an alternative with short-term rate was also tested), the depreciation was fixed at 0.05 (but 0.07 was also tested, see below) and a k p & was set at the average of the price change over the last three years.

Net capital stock (NCAPK) is available directly in the data for Belgium and Italy only. For the other countries, I calculated the series based on the Gross Fixed Capital Formation in volume (GFCFK) according to:

t t t GFCFK K d K + - = -1
). 1 (

. Only, the starting point value for the net capital stock is missing to build the series. It was derived differently depending on the countries, due to data availability. For Austria, Finland, Japan, Norway and the USA, I used the Consumption of Fixed

= - = - = ∑ ∑ ∑ ∑ ∑ ∑ ∑ F x M F L F LM M L = + First order conditions imply ) ( / M P WL x PY m LM + = µ , which derivation yields: ] . . .[ / m dp a dwl a x dpy m M L LM + = µ This is equivalent to equation (4Ra) since the coefficient on drk in (4Ra) is )] .( / 1 [ M L LM a a x + -µ
which equals zero under this specification.

In the general case where

F K F x F M F L F K M L / / ) ( - = + ≡ θ
is not constant, after some calculations, whether capital adjusts perfectly or not, one reaches:

θ θ θ µ / ] . . .[ / d m dp a dwl a dpy m M L - + =
This means that Roeger's and benchmark estimates for the case of capital fixity will differ insofar as

F K F x K / - = θ
varies with time and that these time changes are correlated with the RHS variable.

b) Hall's approach

It happens that Hall's approach does not need the equalization of marginal revenue of capital with user cost. Therefore, Hall's equation (4H) will be unaffected whether capital adjusts perfectly or not.

Indeed, derivation of the production function (1) yields:

da dm F M F dl F L F dk F K F dy M L K + + + =
). / ( ). / ( ).

/ (

The marginal productivity of capital is derived from Euler's equation:

da dk x dk dm F M F dk dl F L F da dm F M F dl F L F dk F M F F L F x dy M L M L M L + + - + - = + + + - - = . ) ).( / ( ) )

  standing for the markup adjusted for returns to scale, i.e. the markup over average cost. By substituting the capital share by its expression drawn from (2b) in equations (3H) and (3R), one arrives

	~µ µ ≡	/	x
	at:
			dy	=	µ	.[	a	L	.(	dl	-	dk	)	+	a	M	.(	dm	-	dk	)]	+	x	dk .	+	da	(4H)
			dpy	-	drk		=	.( ~drk m ) .( .[ dp a drk dwl a m M L -+ -µ	)]	which is denoted	dx	=	μ	dz .	(4Ra)
																										) to:
			dy	=	µ	.(	a	K	dk .	+	a	L	dl .	+	a	M	dm ) .	+	da	(3H)
	Roeger's specification can be obtained directly by deriving equation (2a):
			dpy	. ~m dp . . .( a dwl a drk a m M L K + + = µ	)	(3R)

Table 3

 3

					compares effective capital shares with implied benchmark and Roeger's ones from
	a	K	= µ / 1	-	a	L	-	a	M	and indicates the frequency of negative occurrences. Roeger's estimates, because
	they (seem to) overestimate markups, indeed lead to capital shares very close to zero on average and
	negative for around 40% of the 334 observations available for the USA. Note that these results are not
	very sensitive to the choice of the method for computing capital data, even though the two selected for
	illustration purposes -the first and third from table 2 -generate important variations in capital share
	measures. Roeger's original L-based estimates from (4Rb) amplify the differences compared to µ -
	based computations and the explanation of why this is the case is given in Appendix B.
	In understanding the problem, it is convenient to calculate the difference in the two estimates
	numerically. Derivation of equation (5) leads to the exact µ -based Roeger's equation:
			dx	=	dz . ) ~µ u du drk PY u dz ≡ --/ .( / . µ	+	v	(6)

Table A1

 A1

										details the 23
	manufacturing sectors.
	Variables
	PROD: Production (Gross Output) at current prices ( Y P. in the text)
	LABR: Labor compensation of employees ( L W . in the text)
	VALU: Value added at current prices.
	VALUB: Value added at basic prices. When VALUB is available,	Y . P	=	PROD	-	VALU	+	VALUB	.
	Materials:	P m	M .	=	PROD	-	VALU

.

  The parameter θ reflects the ratio of net capital stock to gross capital stock. I ran simulations based on various methodologies (double-decline, geometric, hyperbolic, see OECD,2001) and reasonable values of parameters to arrive at a ratio of between 0.50 and 0.85. I chose

	Capital (CFC) and inferred:	p k	0	.	K	0		=	CFC	0	/	d	for the first date. For Canada, France, the UK, the
	Netherlands and Sweden, I computed		p k	0	.	K	0	=	χ	. VALU	0	θ .	. χ is the average, for each sector over
	time and over countries for which the gross capital stock (CAPK) is available, of	p k	CAPK .	/	VALU	and
	is reported in table A270 . 0 = θ by default, but compared the results with	θ	=	0	55 .	. Finally, as Denmark provides gross capital
	initial capital stock derived from		p k	0	.	K	0	=	CFC	0	/	d	. K2 was calculated similarly but using d = 0.07.
	With the idea of testing extreme assumptions, K3 was bluntly derived from	p	kt	K . = t	CFC	t	/	d	for every
	µ µ 1 ˆ2 2 . . . . 1 ) .( 2 2 2 2 R dz dx dz dx dz dx dz t t t t t dx dz dx dx L t t t t t t -

stock only, I used the constant ratio θ to deduce net capital stock for all dates.

I shall now detail the various computations used for the case of the USA as they appear in table 2. K1 was calculated, as described above, from the investment flows, a depreciation rate d of 0.05 and an date t and d = 0.05. K4 was calculated as K3 but with d = 0.07. I also tested as r, the average of the short-term and the long-term rates, and even a constant for the real interest rate.

(

)

Table 1 : Roeger's apparent overestimation of markups*

 1 First order conditions on labor and materials lead to Hall's equation (4H).

			(						/						/	).		(			/	).	(	/	).
			(				/		.(
	dy	=	µ	.[	a	L	.(	dl	-	dk	)	+	a	M	.(	dm	-	dk	)]	+	x	dk .	+	da

Table 2 : Roeger's apparent overestimation across different measures of capital services and cost*

 2

			benchmark markup estimates	Difference (Roeger's -benchmark)	Average capital share of output
		sector	K1	K2	K3	K4	K1	K2	K3	K4	K1	K2	K3	K4
	usa	15	1.024	1.025 1.028	1.039	0.052	0.046	0.076	0.067	0.041	0.037 0.065	0.056
	usa	16	1.124	1.285 1.090	1.103	0.103	-0.056	0.162	0.150	0.038	0.035 0.066	0.056
	usa	19	1.039	1.038 1.053	1.072	-0.016	-0.034	0.086	0.069	0.036	0.031 0.093	0.078
	usa	20	1.096	1.106 1.029	1.048	0.122	0.127	0.165	0.150	0.059	0.053 0.108	0.094
	usa	21	1.032	1.038 1.015	1.031	0.122	0.127	0.105	0.089	0.086	0.081 0.095	0.082
	usa	22	1.075	1.079 1.078	1.090	0.065	0.050	0.027	0.013	0.053	0.049 0.062	0.052
	usa	23	1.037	1.039 1.030	1.038	-0.010	-0.017	0.045	0.037	0.049	0.045 0.056	0.048
	usa	24	1.143	1.147 1.046	1.076	0.026	0.022	0.123	0.092	0.085	0.078 0.140	0.119
	usa	25	1.019	1.022 1.015	1.025	0.046	0.038	0.020	-0.001	0.059	0.056 0.056	0.048
	usa	26	1.038	1.050 1.027	1.044	0.117	0.115	0.117	0.102	0.085	0.077 0.084	0.072
	usa	27	1.047	1.055 1.000	1.009	0.078	0.080	0.045	0.034	0.098	0.082 0.073	0.063
	usa	28	1.078	1.083 1.040	1.056	0.027	0.019	0.064	0.046	0.052	0.047 0.082	0.069
	usa	34	1.062	1.065 1.019	1.042	0.029	0.033	0.055	0.035	0.056	0.050 0.158	0.136
	usa	35	0.974	0.979 1.003	1.008	-0.048	-0.083	0.018	0.013	0.057	0.050 0.030	0.025
	mean		1.056	1.072 1.034	1.049	0.051	0.034	0.079	0.064 0.061	0.055 0.083	0.071
	difference / capital share correlation		0.40	0.62	0.42	0.34

(*) Capital variables are described in Appendix A. Sector description is given in Table

1

.

Table 3 : Implied average capital shares from

 3

	a	K	= µ / 1	-	a	L	-	a	M

Table 4 : Breakdown of Roeger's / benchmark difference according to equation (7)

 4

				Capital series : K1			Capital series : K3
	country	sector	difference	diff1	diff2	difference	diff1	diff2
	usa	15	0.052	0.010	0.042	0.076	-0.012	0.088
	usa	16	0.103	0.050	0.054	0.162	0.043	0.119
	usa	19	-0.016	0.031	-0.047	0.086	-0.010	0.096
	usa	20	0.122	-0.011	0.134	0.165	0.009	0.156
	usa	21	0.122	-0.020	0.142	0.105	-0.011	0.116
	usa	22	0.065	0.024	0.041	0.027	-0.016	0.044
	usa	23	-0.010	-0.015	0.005	0.045	0.002	0.043
	usa	24	0.026	-0.103	0.129	0.123	-0.042	0.165
	usa	25	0.046	-0.012	0.058	0.020	0.000	0.019
	usa	26	0.117	-0.017	0.134	0.117	0.004	0.113
	usa	27	0.078	-0.064	0.142	0.045	-0.030	0.075
	usa	28	0.027	-0.030	0.057	0.064	-0.019	0.083
	usa	34	0.029	-0.053	0.082	0.055	-0.091	0.146
	usa	35	-0.048	-0.034	-0.014	0.018	-0.023	0.041
	mean		0.051	-0.018	0.068	0.079	-0.014	0.093

Table 5 : Capital fixity as the main suspect

 5

					P	Y	=	W ( . . µ	.	L	+	P m	.	M	)	+	h	. µ	.	K . R	+	u
			bench	Roeger'		benchmark								benchmark	benchmark
			mark	s			AR(2)											AR(2)	AR(2)
			AR(2)
			h = 1				h = 0										0< h < 1	unbounded h
			(1)	(2)				(3)											(4)	(5)
																			Stand.	Stand.
	country sector	µ ~	µ ~	µ ~		Stand. Dev.			µ ~				h	Dev. (h)	µ ~	h	Dev. (h)
	usa	15	1.024	1.076	1.049	0.017		1.043			0.25	0.37	1.043	0.25	0.37
	usa	16	1.124	1.227	1.171	0.034		1.145			0.57	1.10	1.145	0.57	1.10
	usa	19	1.039	1.023	1.050	0.039		1.039			1.00	0.00	1.037	1.26	1.56
	usa	20	1.096	1.219	1.170	0.011		1.170			0.00	0.00	1.187	-0.22	0.39
	usa	21	1.032	1.154	1.141	0.008		1.140			0.00	0.00	1.180	-0.33	0.22
	usa	22	1.075	1.140	1.146	0.004		1.141			0.07	0.51	1.141	0.07	0.51
	usa	23	1.037	1.026	1.027	0.024		1.027			0.00	0.00	1.042	-0.93	0.84
	usa	24	1.143	1.169	1.158	0.038		1.149			0.18	0.35	1.149	0.18	0.35
	usa	25	1.019	1.065	1.038	0.025		1.029			0.20	0.32	1.029	0.20	0.32
	usa	26	1.038	1.155	1.084	0.057		1.084			0.00	0.00	1.098	-0.21	0.37
	usa	27	1.047	1.125	1.073	0.006		1.073			0.00	0.00	1.123	-0.44	0.09
	usa	28	1.078	1.105	1.100	0.022		1.080			0.96	0.33	1.080	0.96	0.33
	usa	34	1.062	1.091	1.110	0.019		1.081			0.56	0.60	1.081	0.56	0.60
	usa	35	0.974	0.925	1.032	0.010		0.974			1.00	0.00	0.932	1.79	1.05
	mean		1.056	1.107	1.096	0.022		1.084			0.34	1.091	0.22

Table A1 : ISIC Rev. 3 Classification

 A1

		Sector desrciption		More aggregated sample
				FOOD PRODUCTS, BEVERAGES AND
	15	FOOD PRODUCTS AND BEVERAGES	15-16	TOBACCO
				TEXTILES, TEXTILE PRODUCTS, LEATHER AND
	16	TOBACCO PRODUCTS	17-19	FOOTWEAR
	17	TEXTILES	20	WOOD AND PRODUCTS OF WOOD AND CORK
				PULP, PAPER, PAPER PRODUCTS, PRINTING
	18	WEARING APPAREL, DRESSING, DYING OF FUR	21-22	AND PUBLISHING
		LEATHER, LEATHER PRODUCTS AND		CHEMICAL, RUBBER, PLASTICS AND FUEL
	19	FOOTWEAR	23-25	PRODUCTS
	20	WOOD AND PRODUCTS OF WOOD AND CORK	26	OTHER NON-METALLIC MINERAL PRODUCTS
				BASIC METALS AND FABRICATED METAL
	21	PULP, PAPER AND PAPER PRODUCTS	27-28	PRODUCTS
	22	PRINTING AND PUBLISHING	29	MACHINERY AND EQUIPMENT, N.E.C.
		COKE, REFINED PETROLEUM PRODUCTS AND
	23	NUCLEAR FUEL	30-33	ELECTRICAL AND OPTICAL EQUIPMENT
	24	CHEMICALS AND CHEMICAL PRODUCTS	34-35	TRANSPORT EQUIPMENT
	25	RUBBER AND PLASTICS PRODUCTS	36-37	MANUFACTURING NEC; RECYCLING
	26	OTHER NON-METALLIC MINERAL PRODUCTS
	27	BASIC METALS
		FABRICATED METAL PRODUCTS, except
	28	machinery and equipment
	29	MACHINERY AND EQUIPMENT, N.E.C.
		OFFICE, ACCOUNTING AND COMPUTING
	30	MACHINERY
		ELECTRICAL MACHINERY AND APPARATUS,
	31	NEC
		RADIO, TELEVISION AND COMMUNICATION
	32	EQUIPMENT
		MEDICAL, PRECISION AND OPTICAL
	33	INSTRUMENTS
		MOTOR VEHICLES, TRAILERS AND SEMI-
	34	TRAILERS
	35	OTHER TRANSPORT EQUIPMENT
	36	MANUFACTURING NEC
	37	RECYCLING

Table A2 : Computation of initial capital stock for each sector: Average over time and countries (Belgium, Canada, Finland, France and Italy) of

 A2

	p k	CAPK .	/	VALU
	sector			χ
	15-16			2.75
	17-19			2.07
	20			3.91
	21-22			2.89
	23-25			3.31
	26			3.15
	27-28			3.14
	29			1.52
	30-33			1.52
	34-35			2.39
	36-37			2.55

Estimates are produced from an AR(2) process for the residuals. Although it corrects for auto-correlation successfully, a more general treatment would have consisted in an error component model, which allows one to distinguish short term from long term dynamics.

I tested more extreme assumptions with a similar outcome overall.

Equation (9) is therefore s trictly correct only if the production function is homogenous in the labor and material inputs.

Again, keep on mind that the estimated markup is the markup adjusted for returns to scale on variable factors. Hall's equation given by (4H) is also correct under the assumption of capital fixity. Appendix C gives the proofs.

Appendix B: Comparison of Roeger's L-based and µ -based estimates

We want to compare Roeger's µ -based markup estimated from (4Ra where

is the R-square from (4Ra). It is easy then to conclude that original Roeger's estimates are higher than µ -based markups, which aggravates the frequencies of negative implied capital shares, all the more so that the fit in equation ( 4Ra) is not perfect:

(the convexity of the relation linking L µ ˆ to L ˆ has a very minor impact).

Appendix C: The case of capital fixity a) Roeger's approach

Roeger's equation cannot be extended to the case of capital fixity because the first order condition on capital is not appropriate in this situation. However, if the production function is homogenous in the variable inputs, then calling LM x the returns to scale on labor and materials:


Introduction

The successful reuse of shared data relies on the existence of easily-available well-described metadata [1]. The metadata, as a rich description of the data, must capture information on how the data was acquired, processed and analyzed. The terms used to describe the data should be chosen with a logical, consistent framework in mind and include definitions to avoid ambiguity. In addition, a lexicon or ontology should reuse terms from existing efforts as much as possible [2].

We report here on recent progress in the construction of two ontologies used to describe experiments, originally focused on neuroimaging, but recently expanded to encompass other modalities such as electrophysiology experiments. The goal of this work is to provide a set of terms that can be used by the Neuroimaging Data Model (NIDM) [START_REF] Auer | [END_REF] and by others. NIDM is comprised of three distinct models: 1) NIDM-Experiment, NIDM-Workflows, and NIDM-Results. These models are being developed under the aegis of the International Neuroimaging Coordinating Facility [START_REF] Ghosh | [END_REF] (INCF). NIDM-Results was developed first and is now available for use [5]. The other two models are under active development.

Methods:

Currently, our efforts are focused on terms describing both a general experiment acquisition and workflow (e.g., "Project" and "Acquisition"), and those that are modality specific (e.g., "T2 Star Weighted Imaging" and "Intracellular Recording"). The NIDM ontology uses the PROV Ontology (PROV-O) [START_REF] Nichols | Oxford Big Data Institute[END_REF] as its base ontology, which expresses the PROV Data Model (PROV-DM) [START_REF] Padhy | [END_REF] using the OWL2 Web Ontology Language (OWL2). The NIDM ontology also incorporates terms from several other ontologies (Fig. 1) that provide terms related to diverse categories such as assessment instruments. Blue entries in Fig. 1 denote products of this project, purple denotes the sources for imported terms.

We have also created a separate DICOM-tag ontology, which is available at https://github.com/incf-nidash/dicom-ontology. Included are terms for each of the DICOM tags, as well as terms that have been extracted from DICOM supplement documents. These terms will be used in the tagging of experimental parameters for data in the DICOM format.

To develop this ontology, we are using GitHub as an environment for openly discussing and defining terms and definitions. The agreed-upon definitions are then referenced by URI in Terse RDF Triple Language (Turtle) documents and expanded into a hierarchical Web Ontology Language (OWL) file. The final OWL file and files containing terms imported from other ontologies are also available on GitHub.

Results:

The NIDM-Experiment ontology is available at: https://github.com/incf-nidash/nidm/ . We are currently adding definitions to the new NIDM namespace terms using a community based workflow. Potential contributors and users are encouraged to request terms by raising a new GitHub issue and participating in the definition process.

As an example, Figure 2 shows the basic element of the model for two activities: informed consent and the acquisition of neuropsychological testing data. The basic interaction in the Prov data model is that an Agent with a specific role is associated with an Activity that generates or uses an entity.

Conclusions:

We have created a set of defined terms for various aspects of the neuro-imaging enterprise for use in the INCF Neuroimaging Data Model as well as more broadly. In future work, we will be adding terms for other imaging modalities as well as other neuro-related domains such as electrophysiology.

Fig. 1

 1 Fig. 1

Fig 2 .

 2 Fig 2.

Acknowledgements:

We would like to acknowledge the work of all the INCF task force members as well as of many other colleagues who have helped the task force. We are particularly indebted to M. Abrams and the INCF secretariat staff. We also acknowledge support of this project by the Wellcome Trust (CM, TEN). J.-B.P. was partially funded by NIH-NIBIB P41 EB019936 (ReproNim) NIH-NIMH R01 MH083320 (CANDIShare) and NIH 5U24 DA039832 (NIF).


Pierre Aucouturier

Fran~oise Duarte

Edith Mihaesco

Nathalie Pineau

Jean-Louis Preud'homme

Jacalin, the human IgA1 and IgD precipitating lectin, also binds IgA2 of both allotypes

Keywords: Jacalin, Lectin, Chromatography, affinity, IgA subclass, IgA2 allotype

The lectin jacalin from jackfruit seeds shows a human IgA-subclass specificity by gel precipitation and Western blotting. However, its reactivity with IgA2 is a matter of controversy. We further studied the immunoglobulin isotype specificity of jacalin by affinity chromatography with myeloma sera and by inhibition of jacalin binding to solid-phase IgA1 by purified monoclonal immunoglobulins. The lectin proved to bind IgA2 of both allotypes with a lower apparent affinity than for IgA1 and IgD.

Introduction

The lectin jacalin from Artocarpus heterophyllus (or A. integrifolia L.) seeds first came to the attention of immunologists because of its mitogenic properties (Bunn-Moreno and Campos-Neto, 1981;[START_REF] Saxon | Jacalin, an IgA-binding lectin, inhibits differentiation of human B ceils by both a direct effect and by activating T-suppressor cells[END_REF]. It was subsequently reported that jacalin could selectively bind human IgA (Roque- [START_REF] Roque-Barreira | Jacalin: an IgA-binding lectin[END_REF]. This lectin thus seemed to be potentially useful for the purification of IgA from biological fluids. It soon became apparent that jacalin predominantly reacts with IgA1 and IgD [START_REF] Kondoh | Jacahn, a jackfruit lectin, precipitates IgA1 but not IgA2 subclass on gel diffusion reaction[END_REF]Aucoutufter et al., 1987;[START_REF] Zehr | Human IgD and IgA1 compete for D-galactose-related binding sites on the lectin jacahn[END_REF] but the IgA2 reactivity of jacalin has been a

Correspondence to: P. Aucouturier, Laboratory of Immunology and lmmunopathology (CNRS UA 1172), CHRU, BP 577, 86021 Poitiers Cedex, France.

Abbreviations: JCE, jackfruit seed crude extract; DEAE, diethylaminoethyl; JBP, jacalin-binding protein; PBS, phosphate-buffered saline; OD, optical density; IEL, immunoelectrophoresis; HRP, horseradish peroxidase. matter of controversy (Table I). We therefore undertook a re-evaluation of jacalin reactivity with purified monoclonal lgA2 of both allotypes. This study confirms the ability of jacalin to bind IgA2 weakly but significantly.

Material and methods

Jacalin

Jacalin crude extract (JCE) was prepared from jackfruit seeds collected in the island La Rrunion. Jacalin was purified by diethylaminoethyl (DEAE) chromatography as previously described (Aucoutufter et al., 1987). Purification was confirmed by the presence of two single bands (the 15 kDa (glycosylated) and 12 kDa (unglycosylated) subunits) after sodium dodecyl sulfate 12% polyacrylamide gel electrophoresis and by thin layer agarose electrophoresis (Paragon, Beckman, Brea, CA). All the following experiments were performed with jacalin fractions with a 3:1 ratio of unglycosylated and glycosylated subunits (Aucou-

Affinity chromatography purification of serum jacafin binding proteins (JBP)

Sepharose 4B (Pharmacia, Uppsala, Sweden) was activated with cyanogen bromide according to [START_REF] March | A simplified method for cyanogen bromide activation of agarose for affinity chromatography[END_REF] and coupled to JCE or purified jacalin (12 mg protein/ml of sedimented beads). In five separate experiments, the yields of coupling were 0.63, 0.53 and 0.54 with JCE and 0.87 on two occasions with purified jacalin. Sepharose-coupled JCE or purified jacalin yielded identical results and hence were used interchangeably. Ammonium sulfate precipitates from myeloma sera containing the proteins Mia (IgA2m(1)X), Gir (IgA2m(1)x), Felg (IgA2m(2)k) or Bel (IgA2m(2)r) (the allotypes were kindly determined by Dr. G. De Lange, Amsterdam), in addition to the controls, Ga (IgMr) and Ja (IgG2x), were adjusted to a concentration of 6.5 mg of protein/ml in 0.01 M phosphate-buffered saline pH 7.4 (PBS) and incubated for 20 rain at 4°C with jacalin-Sepharose (5 ml of settled beads for 10 ml of protein solution). After washing with cold PBS (at least 2 liters with numerous 2 or 3 min incubations) until the optical density (OD) of the effluent at 280 nm was below 0.002, JBP were eluted from the adsorbent by overnight incubation with 20 ml of 0.5 M D-galactose (Merck, Darmstadt, F.R.G.) at 4°C, concentrated by vacuum dialysis against 0.15 M NaC1 and analyzed by standard immunoelectrophoresis (IEL) with polyvalent and anti-immunoglobulin light and heavy chain-specific antisera (Dako, Copenhagen, Denmark) and with JCE (1.8 mg/ml). With the anti-a antiserum, IgA1 spurred over IgA2.

Purified monoclonal human Ig

Six monoclonal Ig were purified for this study; all steps were performed at 4°C and the purified concentrated solutions were stored in the presence of 50% glycerol at -30°C. The IgAlr Min was isolated from serum by ammonium sulfate precipitation, gel filtration on Sepharose 6B (Pharmacia) followed by jacalin-Sepharose affinity chromatography of the 7-8 S fraction. It was pure by IEL at 187 27 mg/ml. The IgA2m(1)x Gir and IgA2m(2)~ Bel were purified by ammonium sulfate precipitation, gel filtration on Ultrogel ACA34 (IBF, Villeneuve-la-Garenne, France) and DEAE-Trisacryl (IBF) chromatography of the 7-8 S fractions in 0.01 M Tris/HC1 buffer pH 7.5 with a linear 0-0.3 M NaC1 gradient. Residual IgG of fast mobility and traces of polyclonal IgA1 were removed by adsorption on protein A-Sepharose (Pharmacia) and on a limited quantity of jacalin-Sepharose. After concentration by vacuum dialysis, the two IgA2 preparations were pure by IEL at 72 and 23 mg/ml respectively. The IgG3K Met has a very low electrophoretic mobility and was easily purified by DEAE-Trisacryl chromatography. The IgMx Mil was purified by ammonium sulfate precipitation, gel filtration on Sepharose 6B and euglobulin precipitation. It was then adsorbed on jacalin-Sepharose to remove small amounts of residual polyclonal IgA. The pure IgDX Car was prepared as previously described (Aucouturier et al., 1987) and stored at -30°C with 5 mM c-aminocaproic acid and 1 mM phenylmethylsulfonyl fluoride in 50 % glycerol. Concentrations of the purified Ig solutions were calculated from their absorbance at 280 nm using A~%cm values of 10.6 for IgA, 17.0 for IgD, 13.5 for IgG and 11.85 for IgM [START_REF] Hudson | Practical Immunology[END_REF][START_REF] Johnstone | Immunochemistry in Practice[END_REF]. The monoclonal IgA2m(2)h Felg and IgA2m(1)X Mia were purified as previously described (Aucouturier et al., 1987). Purity was controlled by IEL and by aminoterminal aminoacid sequencing.

Determination of isotype specificity of jacalin by ELISA

Polystyrene microtitration plates (Nunc, Roskilde, Denmark) were coated with the IgA1 Min at 2/tg/ml in 0.1 M carbonate buffer pH 9.6 (200 #l/well) overnight at 4°C, and saturated with PBS 0.05% Tween 20 (v/v) for 1 h at 37°C. After washing with PBS-Tween, 100 ttl/well of purified Ig at various dilutions were added in triplicate to 100 ttl of purified jacalin conjugated to horseradish peroxidase (HRP, Grade I, Boehringer; Manheim, F.R.G.) by the two-step glutaraldehyde method [START_REF] Avrameas | Peroxidase labeled antibody and Fab conjugates with enhanced intracellular penetration[END_REF]) at a final dilution of 1.04 ttg of jacalin/ml. The plates were incubated for 2 h at room temperature and washed six times with PBS-Tween; enzyme activity was revealed with 0.4 mg/ml o-phenylenediamine 0.012% H202 in 0.05 M sodium citrate pH 5. Absorbance values were read in a Titertek (Flow, McLean. VA) plate reader at a wavelength of 492 nm. As a control the same purified Ig were similarly tested on identical plates with a monoclonal anti-human IgA1 antibody (clone M4D8. kindly provided by Professor I.C.M. MacLennan. Birmingham, U.K.) diluted 1/1000 instead of HRPjacalin. Binding of the antibody to the immobi- Fig. 1. lmmunoelectrophoretic analysis of jacalin-binding proteins (JBP1) and effluent (El) obtained by affinity chromatography of an ammonium sulfate precipitate from the IgA2m(1)r myeloma serum Gir ($1). Anti-NHS: polyvalent anti-normal human serum antiserum. Note (1) that although the bulk of the IgA2 is in the effluent, it is also present in JBP, and ( 2) the spur of polyclonal IgA1 over IgA2 (arrow).

an t i -r ~'c

Western blotting

Purified IgA1 Min, IgA2m(1) Gir and IgA2m(2) Bel at 2.8 mg/ml were analyzed by thin layer agarose electrophoresis (Paragon, Beckman) followed by pressure blotting on nitrocellulose (HAHY, Millipore, Molsheim, France) and detection with either HRP-anti-a antiserum (Institut Pasteur Production, Paris, France, 1/2000) or HRP-jacalin (0.52 #g jacalin/ml) for 90 min at room temperature. The enzymatic activity was developed with benzidine 0.13% H202 0.03% in 0.1 M Tris/HC1 buffer pH 7.6.

Results

Study of proteins isolated by affinity chromatography on jacafin-Sepharose

IEL analysis of JBP from the four IgA2 myeloma sera (Figs. 1 and2) revealed the presence of significant amounts of monoclonal IgA as shown in each case by a typical curvature of the IgA line with the anti-a and relevant anti-light chain antisera, and by a spur of polyclonal IgA1 with the anti-a serum. Indeed, as in our previous experiments (Aucouturier et al., 1987), residual polyclonal IgA1 was considerably enriched in jacalin affinity chromatography-purified material. Jacalin precipitated neither the IgA2m(1) nor the IgA2m(2) present in these JBP, but yielded a faint precipitating line (barely visible on the photographs) corresponding to polyclonal IgA1. In contrast, no traces of monoclonal IgG2r or IgMr could be found in JBP from the serum Ja and Ga respectively.

Competition ELISA

As previously observed, monoclonal IgA1 and IgD strongly inhibited the binding of HRP-jacalin to IgA1 coated plates while IgA2 of both allotypes yielding a weaker but significant inhibition (Fig. 3). The absence of contamination by IgA1 was confirmed by the control assay with the monoclonal anti-al antibody instead of jacalin (Fig. 4), where only IgA1 was found to inhibit the binding of the antibody.

Gel precipitation and western blotting

As previously described and in striking contrast to IgA1, all IgA2 preparations tested failed to react with jacalin in Ouchterlony and IEL analysis or by Western blotting following agarose electro-phoresis and detection with HRP-jacalin (not shown).

Discussion

We have previously reported that jacalin does not react with IgA2 by gel precipitation and western blotting (which makes it very useful for the subclass typing of monoclonal IgA proteins) (Aucouturier and Preud'homme, 1987) but weakly binds to IgA2 (Aucouturier et al., 1987). This is confirmed in the present study of further IgA2 of known allotypes. Indeed, affinity chromatography prepared JBP from IgA2 myeloma sera contained the monoclonal IgA (in contrast to JBP from IgG or IgM myeloma sera which were free of myeloma proteins). Residual polyclonal IgA1 bound jacalin (with a considerable enrichment of the IgA1 compared to the starting sera) which clearly shows the much higher affinity of jacalin for the IgA1 subclass. A large part of the myeloma IgA2 did not bind to jacalin-Sepharose and was recovered free of IgA1 in the effluent, thus permitting easy purification of IgA2, in agreement with the findings of [START_REF] Gregory | Separation of human IgA1 and IgA2 using jacalin-agarose chromatography[END_REF]. Interestingly, even the IgA2 isolated in the JBP fraction did not precipitate jacalin in gel diffusion experiments, which strongly suggests that partial binding of IgA2 to jacalin columns results from low affinity or functional monovalency rather than from the presence of different populations of IgA2 molecules.

All of the IgA2 preparations tested weakly inhibited the binding of jacalin (and not that of an anti-IgA1 monoclonal antibody) to insolubilized IgA1. In addition to the lack of reactivity with this monoclonal antibody, these IgA2 preparations were pure by electrophoretic, immunoelectrophoretic and (in two cases) sequence criteria.

The strong reactivity of jacalin with human IgA1 and IgD is well explained by the presence of several o-linked Gal/31--* 3GalNAc in the hinge region of both heavy chains (Baenziger and Kornfeld 1974a,b;Mellis and Baenziger, 1983) since this disaccharide is the predominant reactive carbohydrate with jacalin [START_REF] Sastry | Analysis of saccharide binding to Artocarpus integrifolia lectin reveals specific recognition of T-antigen (fl-D-Gal(1--* 3)o-Gal-NAc)[END_REF]. Moreover, the location of the binding sites in the hinge region of IgA1 was confirmed by a study of proteolytic fragments [START_REF] Skea | Studies on the specificity of the IgA-binding lectin, jacalin[END_REF]. The situation is not as clear for the other isotypes. Terminal galactose residues exist in all human Ig classes and subclasses, but their density and presentation differ. /~ chains bear five kinds of asparagine-linked oligosaccharides, two of which are of the 'high mannose' type without galactose (Chapman and Kornfeld, 1979a,b), whilst the others have terminal NeuAca2--, 6Galfll 6GlcNAc or Galfll ~ 6GlcNAc [START_REF] Hickman | The structure of the glycopeptide of a human M-immunoglobuhn[END_REF]. The glycoside present on gamma chains is often terminated with Gal/31 ~ 4GlcNAc [START_REF] Mizuochi | Structural and numerical variation of the carbohydrate moiety of immunoglobulin G[END_REF][START_REF] Takahashi | Comparative structural study of the N-linked oligosaccharides of human normal and pathological immunoglobuhn G[END_REF], but its situation inside the pair of CH 2 domains in the entire molecule makes it almost inaccessible [START_REF] Deisenhofer | Cristallographic refinement and atomic models of a human Fc fragment and its complex with fragment B of protein A from Staphylococcus aureus at 2.9 and 2.8 A resolution[END_REF]. The location of IgA2 oligosaccharides has been studied by [START_REF] Toraho | Location and structural significance of the oligosaccharides on human IgA1 and IgA2 immunoglobulins[END_REF]. However, their structures are not yet completely elucidated. Both allotypes seem to share two common N-glycosides with IgA1, one located in the CH3 domain and the other, in the CH2 region, having exposed terminal Galfll ~ 4GlcNAc. IgA-2m(1) and IgA2m( 2) have two and three supplementary asparagine-linked oligosaccharides in contrast to IgA1, but their sequences have not been determined. Thus, the special feature of the carbohydrates found in IgA2 is the existence of at least one terminal Gal/31 --, 4GlcNAc exposed on the surface of the molecule. [START_REF] Sastry | Analysis of saccharide binding to Artocarpus integrifolia lectin reveals specific recognition of T-antigen (fl-D-Gal(1--* 3)o-Gal-NAc)[END_REF] showed that this disaccharide had some affinity for jacalin (although with a low association constant), and we previously showed that D-lactose (Gal/31 ~ 4Glc) inhibits HRP-jacalin binding to IgA1 at higher concentration than GalNAc and Gal (Aucouturier et al., 1987). Although the determination of the structure of the other oligosaccharides on the a2 chain may provide alternative explanations, the presence of Galfll--* 4GlcNAc in the CH2 domain is a possible explanation for IgA2 reactivity with jacalin.

The ability of jacalin to bind both IgA subclasses can be used to remove IgA from biological samples and Ig preparations. However, the complete elimination of IgA2 would require a large excess of jacalin because of its low affinity of interaction. Preliminary experiments with commercially available intravenous IgG preparations showed that the IgA level sharply decreased after a first adsorption on jacalin-Sepharose but that several steps of adsorption were required to re-move all contaminating IgA. Nevertheless, a method based on this principle should be of great value for the preparation of human IgG for therapeutic use, in which the presence of IgA is an important and difficult problem.

Fig. 3 .

 3 Fig. 3. Competition ELISA. Inhibition of HRP-jacalin binding to coated IgA1 by increasing concentration of purified human monoclonal IgA1 (@), IgA2m(1) (n), IgA2m(2) (o), IgD (v), IgG3 (~) and IgM (*).

Fig. 4 .

 4 Fig. 4. Inhibition of binding of the anti-ed antibody M4D8 to coated IgA1 by increasing concentrations of purified human monoclonal IgA1 (O), IgA2m(1) (D), IgA2m(2) ((3), IgD (v), IgG3 (or) and IgM (*).

  anti-Of ..................

	:~ anta-.,a
	anti-O(	......	~ ,,_,~
			aBP2
	jaealin
	Fig. 2. Immunoelectrophoretic analysis of jacalin-binding pro-
	teins (JBP2) and effluent (E2) obtained by affinity chromatog-
	raphy of an ammonium sulfate precipitate from the IgA2m(2)x
	myeloma serum Bel ($2).
	lized IgA1 was revealed by incubation for 1 h at
	room temperature with 200 /~l/well of HRP-
	conjugated rabbit anti-mouse IgG (2 /~g/ml, pre-
	pared in our laboratory).

E2 ......

Acknowledgements

We thank Dr. G. De Lange for typing IgA2 allotypes, Professor I.C.M. MacLennan for the gift of the monoclonal antibody to IgA1, and Dr. J. Mestecky for providing us with the IgA2 Felg.


Camille Maumet

Thomas E Nichols

Choosing a practical and valid Image-Based Meta-Analysis

Introduction

Meta-analysis provides a quantitative approach to summarise the rich functional Magnetic Resonance Imaging (fMRI) literature. When image data is available for each study, the optimal approach is to perform an Image-Based Meta-Analysis (IBMA) [START_REF] Salimi-Khorshidi | Meta-analysis of neuroimaging data: a comparison of image-based and coordinate-based pooling of studies[END_REF] . A number of IBMA approaches have been proposed including combination of standardised statistics (Z's), just effect estimates (E's) or both effect estimates and their standard errors (SE's). While using both E's & SE's and estimating between-study variance should be optimal, the methods are not guaranteed to work for small number of studies. Also, often only standardised estimates are shared, reducing the possible meta-analytic approaches. Finally, because the BOLD signal is non-quantitative care has to be taken in order to insure that E's are expressed in the same units [START_REF] Pernet | Misconceptions in the use of the General Linear Model applied to functional MRI: a tutorial for junior neuro-imagers[END_REF][START_REF] Nichols | SPM plot units[END_REF] , especially when combining data from different software packages. Given the growing interest in data sharing in the neuroimaging community there is a need to identify what is the minimal data to be shared in order to allow for future IBMAs. Here, we investigate the validity of 8 IBMA approaches.

Methods

We studied 8 IBMA methods: Random-effects (RFX) General Linear Model (GLM), contrast permutations (Perm. E) [START_REF] Nichols | Nonparametric permutation tests for functional neuroimaging: a primer with examples[END_REF] , mixed-effects GLM (MFX) [START_REF] Woolrich | Multilevel linear modelling for FMRI group analysis using Bayesian inference[END_REF] , fixed-effects GLM (FFX), Fisher's [6] , Stouffer's [START_REF] Stouffer | The American Soldier: Adjustment During Army Life[END_REF] and weighted Z [START_REF] Zaykin | Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis[END_REF] approaches, Permutations on Z's (Perm. Z). All methods are either approximate in small samples or rely on assumptions that might not be verified in the context of neuroimaging meta-analyses (Table 1).

We used Monte Carlo simulations to investigate the validity of each estimator. We simulated E's and SE's estimates under the null hypothesis with: k ∈ {5,10,25,50} studies; n=20 subjects; between-study variance of 0 (homogeneity) or 1 (heterogeneity); within-study variance equal to n x {0.25,0.5,1,2,4} (homoscedastic) or varying between 1 and α ∈ {2, 4, 8, 16} (heteroscedastic), 10^6 realisations. To simulate units mismatch, {20, 50}% of studies had (E, SE) pairs rescaled to simulate mismatched contrast vectors (4x) or data scaling algorithms (2x).

Results

Fig. 2 presents method performance in terms of P-value distributions under different violations of model assumptions.

When the number of subjects is small, FFX is invalid regardless of the number of studies included in the meta-analysis. MFX is conservative for small number of studies and constant within-study variance. More surprisingly, MFX is invalid in the presence of large variations in the within-study variances, regardless of the number of subjects included in each study. RFX and Perm. E appear robust to heteroscedasticity for all settings studied. For small P-values, Perm. E is conservative as expected due to the discrete nature of its distribution. All fixed-effects methods are invalid under heterogeneity.

When different scaling algorithm are used, Perm. E has a behaviour that is close to nominal. RFX is valid but conservative. MFX is robust to the presence of unit mismatches when studies are homoscedastic. In the presence of strong heteroscedasticity, MFX is invalid (as when the units are matched). In the presence of slight heteroscedasticity, unit mismatches can cause invalidity of otherwise valid MFX. When the contrast are scaled differently, we observe a very similar pattern.

Conclusions [ 678 char]

As expected, fixed-effects methods were invalid in the presence of heterogeneity. In line with fMRI literature [START_REF] Mumford | Simple group fMRI modeling and inference[END_REF] , homoscedastic methods were robust to heteroscedasticity. More surprisingly, MFX was invalid in the presence of strong heteroscedasticity due to its approximations in small samples.

Given the still relatively small sample sizes that can be achieved in IBMA as of today, we recommend using RFX, Perm. E, Z MFX or Perm. Z that do not rely on small sample approximations and are robust to both heterogeneity and heteroscedasticity. Although they are suboptimal [START_REF] Cummings | Meta-analysis based on standardized effects is unreliable[END_REF] , until full metadata are routinely shared, we recommend Z-based methods that are insensitive to units.

Figures Table 1.

Statistics for one-sample meta-analysis tests and their sampling distributions under the null hypothesis H0. Empirical null distributions are determined using permutations with sign flipping. IGE=Independent Gaussian Errors, ISE=Independent Symmetric Errors. Note: Pi =Φ(-Zi), 𝜏^2 is pure between-study variance, σ^2_i is the ith study's variance, and σ_Cˆ2 is the usual one-sample variance.

Fig. 2 .

 2 Fig. 2. Deviation of observed from theoretical P-values (difference of observed and Monte Carlo ('true') -log10 p-value distributions) in one-sample tests under violations of the underlying model assumptions: small sample sizes (A), heteroscedasticity (B), heterogeneity (C) unit mismatch (D). P-values are displayed using a negative log10 scale. Y-axis is the observed cumulative probability minus Monte Carlo cumulative probability for a given (X-axis) -log10 p-value; positive deflections correspond to inflated false positive risk.

Acknowledgments

This work was supported by the Wellcome Trust. The majority of the work was conducted while TEN and CM were at the University of Warwick and used the High Performance Computing cluster of the Department of Statistics, University of Warwick.