Christian Mestres
email: christian.mestres@cirad.fr

Mark Taylor

Gordon Mcdougall

Santiago Arufe

Thierry Tran

Ephraim Nuwamanya

Dominique Dufour

Mariam Nakitto

Karima Meghar

Dominique Rinaldo

Lea Ollier

Romain Domingo

Jhon Larry Moreno

Luis Fernando Delgado

Hermann Antonin Kouassi, I N'

Nan Afoué

Sylvie Diby

Didier Mbeguie-A-Mbeguie

Noël Akissoe

Laurent Adinsi

Agnès Rolland-Sabate

Qualisud

Contrasting effects of polysaccharide components on the cooking properties of roots, tubers and bananas

Keywords: cooking behaviour, texture, dry matter, multivariate statistics, galacturonic acid, amylose, pectins

BACKGROUND: Consumer preferences for boiled or fried pieces of roots, tubers and bananas (RTBs) are mainly related to their texture. Different raw and cooked RTBs were physiochemically characterized to determine the effect of biochemical components on their cooking properties. RESULTS: Firmness in boiled sweetpotato increases with sugar and amylose contents but no significant correlation was observed between other physicochemical characteristics and cooking behaviour. Hardness of boiled yam can be predicted by dry matter (DM) and galacturonic acid (GalA) levels. For cassava, no significant correlation was found between textural

INTRODUCTION

Roots, tubers and bananas (RTB) play an essential role as staple foods in the tropics and subtropics, particularly in Africa. As a result of their adaptability in different ecosystems and high yields compared to local cereals, they are a primary and reliable source of calories. [START_REF] Dufour | Integrating end-user preferences into breeding programmes for roots, tubers and bananas[END_REF] Breeding programmes for RTB crops initially gave priority to yield and disease/pest resistance, but now end-product quality traits and processor and consumer preferences must be considered for improving varietal adoption. [START_REF] Thiele | A review of varietal change in roots, tubers and bananas: consumer preferences and other drivers of adoption and implications for breeding international[END_REF] The main characteristics preferred by RTB consumers have been described recently. [START_REF] Dufour | Integrating end-user preferences into breeding programmes for roots, tubers and bananas[END_REF] Generally, important RTB quality traits for consumers are texture attributes such as rapid softening and/or the development of friability or mealiness and/or short cooking time for boiled products [START_REF] Banda | Analysis of ⊎-amylase gene (Amy⊎) variation reveals allele association with low enzyme activity and increased firmness in cooked sweetpotato (Ipomoea batatas) from East Africa[END_REF][START_REF] Honfozo | Boiled yam end-user preferences and implications for trait evaluation[END_REF] ; smooth and stretchy dough for pounded products (matooke, pounded yam, eba) [START_REF] Otegbayo | End-user preferences for pounded yam and implications for food product profile development[END_REF] ; softness and texture in the mouth for cassava-derived products [START_REF] Iragaba | Identification of cassava quality attributes preferred by Ugandan users along the food chain[END_REF] ; and softness of cooked plantain. [START_REF] Newilah | Analysis of consumer-oriented quality characteristics of raw and boiled plantains in Cameroon: implication for breeding[END_REF][START_REF] Amah | End-user preferences for plantain food products in Nigeria and implications for genetic improvement[END_REF] However, reliable means of phenotyping textural properties and an understanding of the biochemical and genetic basis of textural variation are not available for most RTB.

Progress has been made in understanding the molecular basis of cooking time in some RTB crops. In potato (Solanum tuberosum) Ducreux et al. [START_REF] Ducreux | Expression profiling of potato germplasm differentiated in quality traits leads to the identification of candidate flavour and texture genes[END_REF] showed that tubers from the Phureja Group have much shorter cooking time than those from the Tuberosum Group. [START_REF] Ross | Discerning intra-tuber differences in textural properties in cooked Solanum tuberosum group Tuberosum and group Phureja tubers[END_REF] Expression profiles in tubers from Tuberosum and Phureja types were significantly different for genes involved in cell wall modification, which could contribute to textural differences. [START_REF] Ducreux | Expression profiling of potato germplasm differentiated in quality traits leads to the identification of candidate flavour and texture genes[END_REF] In particular, the differentially expressed genes included a pectin methyl esterase gene (PME; EC 3.1. 1.11) involved in the modification of pectin structure. PME activity was significantly lower in Phureja genotypes and transgenic experiments demonstrated a linkage between PME gene expression and tuber texture and cooking time. [START_REF] Ross | Pectin engineering to modify product quality in potato: pectin engineering in potato[END_REF][START_REF] Ross | Potato tuber pectin structure is influenced by pectin methyl esterase activity and impacts on cooked potato texture[END_REF] PME may extend cooking time in Tuberosum group tubers by strengthening of cell walls through the removal of methyl esters from pectin in a blockwise fashion. [START_REF] Limberg | Analysis of different de-esterification mechanisms for pectin by enzymatic fingerprinting using endopectin lyase and endopolygalacturonase II from A. niger[END_REF] These demethylated pectin chains can then chelate calcium ions to form egg box structures that strengthen the cell wall. [START_REF] Grant | Biological interactions between polysaccharides and divalent cations: the egg-box model[END_REF] The textural properties of sweetpotato are complex but not directly linked to total starch and amylose contents. [START_REF] Banda | Analysis of ⊎-amylase gene (Amy⊎) variation reveals allele association with low enzyme activity and increased firmness in cooked sweetpotato (Ipomoea batatas) from East Africa[END_REF] The joint effects of starch and pectin breakdown during cooking appear to be the major factors in cooking time and may explain the soggy texture of cooked roots [START_REF] Valetudie | Influence of cooking procedures on structure and biochemical changes in sweet potato[END_REF] of some genotypes. Sweetpotato ⊎-amylase catalyzes the hydrolysis of starch to oligomers during cooking and contributes to the mealy texture of cooked roots. [START_REF] Nakamura | Maltose generation by beta-amylase and its relation to eating quality of steamed storage roots of sweet potato cultivars, including recently developed varieties in Japan[END_REF] A negative correlation between ⊎-amylase activity and firmness for boiled roots has been reported. [START_REF] Banda | Analysis of ⊎-amylase gene (Amy⊎) variation reveals allele association with low enzyme activity and increased firmness in cooked sweetpotato (Ipomoea batatas) from East Africa[END_REF] In addition, an increase in cell wall thickness has been observed in cooked roots, particularly in precooked roots at 70 °C. [START_REF] Valetudie | Influence of cooking procedures on structure and biochemical changes in sweet potato[END_REF] This may be linked to PME activity, which can modify the cell wall during cooking at 100 °C and at 70 °C. [START_REF] Binner | Cell wall modifications during cooking of potatoes and sweet potatoes[END_REF] Precooking sweetpotato root has been recommended to reduce enzymatic pectin hydrolysis. [START_REF] Valetudie | Influence of cooking procedures on structure and biochemical changes in sweet potato[END_REF] Preparation of each type of yam product requires specific quality attributes, but texture attributes are always important. For boiled yam, friability is the most important, [START_REF] Honfozo | Boiled yam end-user preferences and implications for trait evaluation[END_REF] whereas pounded yam must be stretchable, mouldable, smooth, fairly firm, soft and sticky, [START_REF] Honfozo | Boiled yam end-user preferences and implications for trait evaluation[END_REF][START_REF] Otegbayo | End-user preferences for pounded yam and implications for food product profile development[END_REF][START_REF] Egesi | Genetic diversity of organoleptic properties in water yam (Dioscorea alata L)[END_REF] and 'amala' should be elastic, soft and nonsticky. [START_REF] Akissoé | Physical, chemical and sensory evaluation of dried yam (Dioscorea rotundata) tubers, flour and amala, a flour-derived product[END_REF] Several studies have shown a link between starch content, starch structure or composition and the texture of the desired products. 'Amala' stickiness was associated with soluble amylose, starch gelatinization temperature and enthalpy changes, [START_REF] Akissoe | Prediction of the sensory texture of a yam thick paste (amala) using instrumental and physicochemical parameters[END_REF] whereas pounded yam firmness was associated with DM, soluble starch and amylose content. [START_REF] Akissoe | Microstructure and physico-chemical bases of textural quality of yam products[END_REF] Several studies found correlation between yam cooking properties with intrinsic starch properties. [START_REF] Akissoé | How blanching and drying affect the colour and functional characteristics of yam (Dioscorea cayenensis-rotundata) flour[END_REF][START_REF] Otegbayo | Pasting characteristics of fresh yams (Dioscorea spp.) as indicators of textural quality in a major food product -'pounded yam[END_REF][START_REF] Kouadio | Effect of resistant starch on the cooking quality of yam (Dioscorea spp.) and cassava (Manihot esculenta) based paste products international[END_REF][START_REF] Brunnschweiler | Structure-texture relationships of fresh pastes prepared from different yam (Dioscorea spp.) varieties[END_REF] In addition, the structure of cell walls must play a role in texture. The extent of cell disintegration appears to be linked to firmness, [START_REF] Brunnschweiler | Structure-texture relationships of fresh pastes prepared from different yam (Dioscorea spp.) varieties[END_REF] as well as the thickness of cell walls. [START_REF] Akissoe | Microstructure and physico-chemical bases of textural quality of yam products[END_REF] Also, studies with chelating agents (phytates) indicate the possible importance of pectins and the level of lignins have also been implicated. [START_REF] Medoua | Hard-to-cook defect in trifoliate yam Dioscorea dumetorum tubers after harvest[END_REF] The texture and cooking quality of boiled cassava is highly variable depending on cultivar, [START_REF] Kouadio | Water absorption as an evaluation method of cooking quality for yam (Dioscorea alata) and cassava (Manihot esculenta crantz)[END_REF][START_REF] Borges | Evaluation of cassava varieties for human consumption[END_REF][START_REF] Tran | Correlation of cooking time with water absorption and changes in relative density during boiling of cassava roots[END_REF] age at harvest and environmental conditions. [START_REF] Franck | Effects of cultivar and harvesting conditions (age, season) on the texture and taste of boiled cassava roots[END_REF] Although sensory-based phenotyping is available, limitations exist. [START_REF] Uchendu | Genome-wide association study of root mealiness and other texture-associated traits in cassava[END_REF] No clear link between starch and texture of cooked cassava roots has been found, [START_REF] Borges | Evaluation of cassava varieties for human consumption[END_REF] but several studies have indicated possible effects of intercellular adhesion, cell wall components and pectins. [START_REF] Favaro | The roles of cell wall polymers and intracellular components in the thermal softening of cassava roots[END_REF] Moreover, a recent genome-wide association study provided the first insights into understanding the underlying genetic basis of boiled cassava roots texture and pointed to the putative role of ⊍-amylase and PME inhibitors, which impact the structure and properties of starch and pectins, respectively. [START_REF] Uchendu | Genome-wide association study of root mealiness and other texture-associated traits in cassava[END_REF] The texture and cooking behaviour of cooking and dessert bananas depend on variety, ripening stage and cooking method. [START_REF] Newilah | Analysis of consumer-oriented quality characteristics of raw and boiled plantains in Cameroon: implication for breeding[END_REF][START_REF] Qi | Effect of cooking on banana and plantain texture[END_REF][START_REF] Gibert | A kinetic approach to textural changes of different banana genotypes (Musa sp.) cooked in boiling water in relation to starch gelatinization[END_REF][START_REF] Kouassi | Textural and physicochemical predictors of sensory texture and sweetness of boiled plantain[END_REF] For boiled banana and plantain, Gibert et al. [START_REF] Gibert | A kinetic approach to textural changes of different banana genotypes (Musa sp.) cooked in boiling water in relation to starch gelatinization[END_REF] established a correlation between the initial DM content of raw crops (positively correlated to starch content) and firmness, as well as a strong contribution of starch gelatinization to thermal softening. Thermal softening in the early stages of boiling has also been related to middle lamella dissolution causing cell wall separation. [START_REF] Qi | Effect of cooking on banana and plantain texture[END_REF] However, to our knowledge, no data has been published on relationships between polysaccharides and fried plantain textural characteristics.

Robust standard operating procedures for assessing biochemical components and their relationship with textural characteristics of boiled or fried pieces for whole RTBs are described in the present study. Investigations across a broad spectrum of RTB products would provide novel insights possibly suggesting that a diversity of biochemical traits is important for textural traits in RTB crops. Robust phenotyping methods and the development of genetic markers will accelerate breeding efforts to develop varieties with improved traits that are important to the consumer.

MATERIALS AND METHODS

Sweetpotato

Seventeen sweetpotato genotypes were grown in replicated field trials in Uganda in 2019 using standard agronomic practices. Roots were harvested and firmness were measured after cooking cubes of 2.5 cm 3 at 85 °C for 15 min, [START_REF] Banda | Analysis of ⊎-amylase gene (Amy⊎) variation reveals allele association with low enzyme activity and increased firmness in cooked sweetpotato (Ipomoea batatas) from East Africa[END_REF] with 300 g of sample cooked in 2 L of water. Fresh samples were freeze-dried for 48-72 h for further analyses: DM was assessed by weight difference after freeze-drying 3 ; free sugars were measured by HPLC [START_REF] Mestres | SOP for Sugars by HPLC[END_REF] ; starch was predicted by near infrared spectroscopy; amylose content, starch gelatinization temperature and enthalpy were determined by DSC, 20,37 PME 38 and ⊎-amylase (BETAMYL-3® METHOD) activities; and temperature, peak and final viscosities for 7% flour dry basis (db) suspension were evaluated with a Rapid Visco Analyser (RVA) (PerkinElmer, Waltham, MA, USA) in 25 mL of water with or without inhibitor. [START_REF] Dufour | Differentiation between cooking bananas and dessert bananas. 2. Thermal and functional characterization of cultivated Colombian Musaceae (Musa sp.)[END_REF] In addition, cell walls were prepared after starch hydrolysis and protein solubilization [START_REF] Mcdougall | Preparation of cell wall material from sweetpotato roots. Biophysical characterization of quality traits[END_REF] and monosaccharide analysis of cell walls was carried out according to McDougall et al., (2021). [START_REF] Mcdougall | Analysis of monosaccharide composition of sweet potato cell wall material/polysaccharides after acid hydrolysis by high performance anion exchange chromatography (HPAEC). Biophysical Characterization of Quality Traits[END_REF] Fourier transform infrared (FT-IR) spectroscopy was used to investigate the degree of esterification of pectin, a major cell wall component, McDougall et al. [START_REF] Mcdougall | Fourier transform infra-red spectroscopy analysis of cell walls from sweetpotato roots[END_REF] Yam Yam tubers were harvested in 2021 and 2022 at INRAE and CIRAD stations (five and six genotypes, respectively) in Guadeloupe (France) and others collected from different markets in 2022 in Benin (three cultivars). Each cultivar (three tubers) was washed and peeled; proximal, central and distal parts were analyzed separately. [START_REF] Adinsi | Sample preparation and cooking time for texture analysis of boiled yam[END_REF] Hardness and DM were determined on fresh and stored yams, [START_REF] Adinsi | Sample preparation and cooking time for texture analysis of boiled yam[END_REF][START_REF] Adesokan | SOP for determination of dry matter content[END_REF] whereas starch and galacturonic acid (GalA) contents were determined on freeze dried samples. [START_REF] Holm | A rapid method for the analysis of starch[END_REF][START_REF] Mestres | SOP for the determination of galacturonic content[END_REF] Cassava Two populations of cassava roots were evaluated. Firstly, a panel of 200 cassava genotypes from a NaCRRI population was evaluated at Namulonge and Serere (Central and Eastern Uganda, respectively). Second, a panel of 29 cassava landraces with good to poor cooking behaviour were grown and harvested from the same field at CIAT (Palmira, Colombia) at three different ages: 9, 10 and 12 months after planting (MAP). DM of fresh roots was determined by NIRS (at both NaCRRI and CIAT [START_REF] Nuwamanya | NIRS acquisition on fresh cassava roots using the ASD Quality Spec (QST) and relating spectra to root dry matter content by oven method. High-throughput phenotyping protocols (HTPP)[END_REF] ) and confirmed by oven drying (105 °C overnight). Texture of boiled roots (18 min) was determined using the same standard operating procedure (SOP) at NaCCRI and CIAT [START_REF] Tran | SOP for characterization of cooking time and texture of boiled cassava: texture-extrusion[END_REF] ; and maximum force (g), area (g × mm) and initial gradient (g/mm, after 1 mm of extrusion) were calculated. For the NaCRRI population, amylose content also was determined using a colorimetric assay, [START_REF] Hovenkamp-Hermelink | Rapid estimation of the amylose/amylopectin ratio in small amounts of tuber and leaf tissue of the potato[END_REF] and, for the CIAT population, water absorption (WA) after 30 min boiling [START_REF] Tran | Correlation of cooking time with water absorption and changes in relative density during boiling of cassava roots[END_REF][START_REF] Escobar | Standard operating protocol for characterization of water absorption, cooking time and closing angle of boiled cassava[END_REF] and total pectin content (as GalA equivalent [START_REF] Mestres | SOP for the determination of galacturonic content[END_REF] ) were determined in triplicate. Calcium content of freeze-dried cassava roots from CIAT was determined by X-ray fluorescence spectrometry [START_REF] Marguí | Application of X-ray fluorescence spectrometry to determination and quantitation of metals in vegetal material TrAC[END_REF][START_REF] Tezotto | Simple procedure for nutrient analysis of coffee plant with energy dispersive X-ray fluorescence spectrometry (EDXRF)[END_REF] on duplicates (harvests at 9 and 12 MAP). To test the effect of calcium on cooking behaviour, fresh cassava root pieces were marinated in solutions of calcium chloride (CaCl 2 , concentrations from 0 to 8 g Ca 2+ L -1 ) at room temperature for 24 h, then boiled in the marinating CaCl 2 solutions for 30 min and WA measured.

Fried plantain

Seven plantain varieties (Corne 1, Saci, Bita 3, Pita 3, Zakoi, FHIA 21) were grown under conventional growing conditions. Fruits were harvested at commercial maturity corresponding to the appearance of the first yellow finger on the bunch and kept to ripen at room temperature. They were sampled at four ripening stages: green (G), yellow tip green (GT), yellow (Y), and yellow tiger (YT). Aloco was prepared as described before [START_REF] Kouassi | Sensory evaluation of fried plantain (Aloco). Biophysical characterization of quality traits[END_REF] and the sensory analysis performed by a trained panel. Three texture attributes were evaluated directly on a scale of 1 (very weak) to 10 (very strong) for firmness and stickiness. For chewiness, the chew count varies greatly from one panellist to another. Therefore, the number of chews was previously computed in a non-dimensionalized form for each panellist and then converted to a scale value between 1 and 10 (1 being the lowest dimensionless value of all products and all panellists and 10 being the highest). DM and starch were determined in the raw samples as described [START_REF] Holm | A rapid method for the analysis of starch[END_REF] and on alcohol insoluble solids using the total starch enzymatic assay kit (K-TSTA Megazyme, Wicklow, Ireland), respectively. Pectin content in Gal A equivalents was determined in both raw and fried form as described by Mestres et al. [START_REF] Mestres | SOP for the determination of galacturonic content[END_REF] Measurements were performed in triplicate.

Statistical analysis

Analysis of variance, mean comparison tests (Tukey), and one way or two ways correlations tests (using Pearson's test) were performed.

RESULTS

Boiled sweetpotato

Roots exhibited a wide range of firmness (from 1.1 to 3.5 kg) (see Supporting information, Table S1) with a significant effect of cultivar. Several physico-chemical properties of sweetpotato did not vary much between cultivars (e.g. starch characteristics and pasting properties with inhibitor), whereas others displayed very wide variation such as ⊎-amylase (from 0.09 to 2.87 μmol s -1 g -1 ) or PME (0.51-2.92 pmol s -1 g -1 db) activities (see Supporting information, Table S2). The cell walls (CW) from the different genotypes (approximately 10% db) all yielded similar main monosaccharide products, with arabinose and galactose as major constituents (more than 50% of CW). The amount of GalA is indicative of pectin content; and it varied considerably from 0.9 to 10.1% of CW (i.e. 0.03-0.47% of root db). The ratio of FT-IR signals at 1730/1625 cm -1 and 1415/1235 cm -1 of CW preparations can be used to assess the relative level of pectin methylation and the intensity of these signals varied considerably between the different genotypes (see Supporting information, Fig. S1).

Among the 17 cultivars, the roots exhibited a wide range of pasting properties measured by RVA, with or without inhibitor (see Supporting information, Table S1). As expected, we observed a negative correlation between starch content and free sugars (r = -0.83); and positive correlations between starch content and pasting viscosities (particularly with inhibitor; see Supporting information, Table S3); and also between DSC gelatinization temperature and pasting peak temperature. There was a dramatic decrease of pasting viscosities when inhibitor was omitted, which was surely linked to amylase activity. CW yield was negatively correlated with pasting viscosities with inhibitor (see Supporting information, Table S3), although there was no significant correlation between PME activity, or FTIR ratio with peak viscosities or flour pasting behaviour.

No evidence of a direct correlation between biophysical characteristics and firmness was observed (see Supporting information, Table S3). However, a multiple regression performed on 15 samples (because some data were lacking for two samples) identified two characteristics that can be used to predict firmness (Fig. 1), with firmness increased with sugars and amylose contents. On the other hand, no significant correlation between firmness and beta-amylase activity was found (see Supporting information, Table S3), as already noted for sweetpotato, [START_REF] Banda | Analysis of ⊎-amylase gene (Amy⊎) variation reveals allele association with low enzyme activity and increased firmness in cooked sweetpotato (Ipomoea batatas) from East Africa[END_REF] and/or with PME activity and pectin methylation as already observed for potato. [START_REF] Ross | Pectin engineering to modify product quality in potato: pectin engineering in potato[END_REF][START_REF] Ross | Potato tuber pectin structure is influenced by pectin methyl esterase activity and impacts on cooked potato texture[END_REF] Boiled yam DM content varied between 25.7% and 40.5% and starch content, between 55.9 and 82.5 g per 100 g db (see Supporting information, Table S4). Both ranges similar to those previously published. [START_REF] Akissoe | Microstructure and physico-chemical bases of textural quality of yam products[END_REF][START_REF] Brunnschweiler | Structure-texture relationships of fresh pastes prepared from different yam (Dioscorea spp.) varieties[END_REF][START_REF] Otegbayo | Characterizing genotypic variation in biochemical composition, anti-nutritional and mineral bioavailability of some Nigerian yam (Dioscorea spp.) land races[END_REF] GalA content varied widely from 0.62 up to 3.18 g per 100 g db (see Supporting information, Table S4), which was higher than reported earlier [START_REF] Brillouet | Alterations in cell wall constituents of yams Dioscorea dumetorum and D. rotundata with maturation and storage conditions. Relation with post-harvest hardening of D. dumetorum yam tubers[END_REF] (i.e. 0.11-0.29 g GalA per 100 g for cell walls from Dioscorea rotundata and Dioscorea dumetorum). Hardness after boiling varied between 1.5 and 14.5 N.

DM and starch contents were not significantly different between the different tuber sections (see Supporting information, Table S4). However, GalA levels were significantly higher in the proximal sections for most cultivars. In parallel, a significant section effect was observed for the hardness for two cultivars (61F and 74F) with the proximal section always the hardest.

Considering the results obtained from all tubers, hardness of boiled yam was found to be related to DM and GalA (Table 1). However, it is important to note that correlations were not significant when hardness was low (between 1.5 and 5.9 N, 80% of total dataset). This led to the conclusion that 20% of the dataset, with hardness > 6 N, made the correlation significant. Hence, further work is required with samples of more contrasting hardness to confirm or rule out this tendency.

Nevertheless, considering DM and GaIA contents as predictive variables, a model for hardness was developed by linear regression (Fig. 2). This model satisfactorily predicts hardness (r 2 = 0.76 and root mean square error = 0.37) for this particular set. Again, such a model must be tested and adjusted using a larger set of samples exhibiting larger differences in terms of hardness after cooking.

Boiled cassava Significant differences were observed among cassava genotypes for DM, amylose, and total pectins), and cooking quality traits (force, area, and gradient; WA). Among the NaCRRI breeding population, the average DM (28% wb) (Fig. 3) was lower than the CIAT population (average 36.4%) (Fig. 4). The amylose content of NaCRRI genotypes ranged from 20% to 30% (db), which is slightly higher than typical literature values, [START_REF] Sánchez | Screening of starch quality traits in cassava (Manihot esculenta Crantz)[END_REF] possibly because the perchloric acid extraction procedure could overestimate amylose content as already observed with potato starch. [START_REF] Hovenkamp-Hermelink | Rapid estimation of the amylose/amylopectin ratio in small amounts of tuber and leaf tissue of the potato[END_REF] DM was not directly correlated with cooking quality of boiled cassava, as determined by WA and texture measurements of both NaCRRI and CIAT populations (Figs 3 and4, respectively), which confirmed previous observations. 28,29 A significant negative relationship was observed between amylose content and textural properties of boiled roots (Fig. 3). A higher total pectin content was related to harder texture after boiling and lower WA (Fig. 4). However, the coefficients of determination between gradient, for example, were low with amylose and pectins (r 2 = 0.20 and 0.38, respectively), indicating that other factors may be at play in determining cooking quality.

Ca 2+ content appeared highly correlated with texture (r 2 = 0.53, gradient = 0.81 × Ca 2+ + 0.81), possibly because of pectin and Ca 2+ complexation resulting in a stiffer pectin network. [START_REF] Grant | Biological interactions between polysaccharides and divalent cations: the egg-box model[END_REF] The increased WA of root pieces marinated and then boiled in CaCl 2 solutions provided additional indirect evidence that pectins are involved in determining cooking quality (see Supporting information, Fig. S2). As the Ca 2+ concentration increased and strengthened the pectin network through complexation, [START_REF] Grant | Biological interactions between polysaccharides and divalent cations: the egg-box model[END_REF] WA30 decreased towards the same lower limit (2-3%) across several genotypes of contrasting cooking quality. That lower limit may correspond to the point when the pectin network became saturated with Ca 2+ ions. Additionally, higher pectin content appeared related to lower DM and higher calcium content in fresh cassava roots (r 2 = 0.25 and 0.47, respectively), which may reflect both the water retention properties of pectins and their chelation of calcium ions.

Fried plantain

For raw plantains and regardless of ripening stage, DM varied from 18.6 to 40.4 g per 100 g WW and total pectin fr0m 0.65 to 3.38 g of GalA per 100 g DW (see Supporting information, Table S5). At the yellow stage (the favoured consumption stage), starch content in fried plantain varied from 17.2 to 55.7 g per 100 g dry solid and GalA between 1.99 and 2.81 g per 100 g dry solid. A large increase of pectin content was observed during ripening of four varieties (Fig. 5a), combined with a slight decrease of DM (Fig. 5b). DM and starch contents agreed with previous values in the literature [START_REF] Qi | Effect of cooking on banana and plantain texture[END_REF][START_REF] Gibert | A kinetic approach to textural changes of different banana genotypes (Musa sp.) cooked in boiling water in relation to starch gelatinization[END_REF][START_REF] Shiga | Ripening-associated changes in the amounts of starch and non-starch polysaccharides and their contributions to fruit softening in three banana cultivars[END_REF] whereas total pectin contents were significantly higher than values in the literature, [START_REF] Qi | Effect of cooking on banana and plantain texture[END_REF][START_REF] Shiga | Ripening-associated changes in the amounts of starch and non-starch polysaccharides and their contributions to fruit softening in three banana cultivars[END_REF] possibly as a result of differences in the extraction methods used.

In parallel, three texture attributes (firmness, chewiness, and stickiness) of fried plantain were evaluated. The overall analysis of the data set, regardless of variety and harvest stage, indicated a correlation between DM and two sensorial attributes (positive with firmness and negative with stickiness). However, no correlation between sensorial attributes and pectin content was observed (Table 2; see also Supporting information, Fig. S3).

DISCUSSION

The RTB samples in the present study represent the broadest range of textural properties of cooked products previously examined, and the relationships with physicochemical parameters (DM, starch, amylose, cell wall, pectin and Ca 2+ content, and PME activity) (Table 3) provide an opportunity to test hypotheses about the biochemical parameters that control such textural variation.

Impact of dry matter, starch, and amylose on cooking properties

Theoretically, hardness should increase with DM of RTB samples because water can plasticize the system. It is largely observed in pasty products, such as in pounded yam, [START_REF] Akissoe | Microstructure and physico-chemical bases of textural quality of yam products[END_REF] but positive correlations were only found between DM of raw samples and the hardness of boiled yam (as already reported [START_REF] Akissoe | Sensory and physicochemical quality of pounded yam: varietal and storage effects[END_REF][START_REF] Adinsi | Characterising quality traits of boiled yam: texture and taste for enhanced breeding efficiency and impact[END_REF] ) and fried (see Supporting information, Fig. S3) and boiled plantain. [START_REF] Gibert | A kinetic approach to textural changes of different banana genotypes (Musa sp.) cooked in boiling water in relation to starch gelatinization[END_REF] No significant correlation was however observed between DM and texture of boiled sweetpotato or cassava; and direct correlation between texture of boiled cassava and DM is indeed rarely observed. [START_REF] Kouadio | Water absorption as an evaluation method of cooking quality for yam (Dioscorea alata) and cassava (Manihot esculenta crantz)[END_REF][START_REF] Padonou | The quality of boiled cassava roots: instrumental characterization and relationship with physicochemical properties and sensorial properties[END_REF] Starch is the major component of most fresh RTBs, except for ripe plantain (Table 3), but no clear correlation was found between DM and starch or amylose contents for sweetpotato, yam (Table 1) or cassava (Fig. 3), as already reported. [START_REF] Padonou | The quality of boiled cassava roots: instrumental characterization and relationship with physicochemical properties and sensorial properties[END_REF][START_REF] Maraphum | Modified specific gravity method for estimation of starch content and dry matter in cassava[END_REF][START_REF] Wholey | A comparison of simple methods for estimating starch content of cassava roots[END_REF] However, there was a positive correlation between starch content and DM for plantain at the yellow ripening stage (r 2 = 0.87, n = 7), as already noted. [START_REF] Gibert | Differentiation between cooking bananas and dessert bananas. 1. Morphological and compositional characterization of cultivated Colombian Musaceae (Musa sp.) in relation to consumer preferences[END_REF] Although starch or amylose were not directly correlated with the texture of boiled sweetpotato (see Supporting information, Table S3) or yam (Table 1), amylose content was negatively correlated with hardness of boiled cassava (Fig. 3). This negative correlation between amylose content and hardness cannot be explained by lower cassava starch swelling with amylose. [START_REF] Zhao | Development of waxy cassava with different biological and physico-chemical characteristics of starches for industrial applications[END_REF] The role of starch or amylose on the texture of boiled RTB may also be masked by the more prominent role of other components and, indeed for sweepotato, amylose was a variable for predicting texture within a two-variables model (Fig. 1).

Starch-degrading enzymes can modulate the role of starch. For example, ⊎-amylase plays a key role in starch degradation during plantain ripening [START_REF] Gao | Analysis of resistant starch degradation in postharvest ripening of two banana cultivars: focus on starch structure and amylases[END_REF] and may also be active during plantain cooking lowering firmness through reducing starch-swelling potential. In fried plantain, ⊎-amylase is rapidly inactivated by the high temperatures reached during frying, but may have a greater impact on texture when cooking at lower temperatures. [START_REF] Banda | Analysis of ⊎-amylase gene (Amy⊎) variation reveals allele association with low enzyme activity and increased firmness in cooked sweetpotato (Ipomoea batatas) from East Africa[END_REF][START_REF] Nakamura | Maltose generation by beta-amylase and its relation to eating quality of steamed storage roots of sweet potato cultivars, including recently developed varieties in Japan[END_REF] ⊎-amylase activity appeared to influence pasting properties of sweetpotato flours through hydrolysis of starch chains (see Supporting information, Table S3), but no effect on firmness of boiled sweetpotato was found in this set of samples. Because ⊎-amylase primarily catalyzes the hydrolysis of linear glucan chains of starch 66 such as in amylose, the relationship between amylose content and firmness may also be modulated by differential ⊎-amylase activity and preferential hydrolysis of amylose during sweetpotato cooking. Indeed, the positive impact of amylose on the firmness of cooked sweetpotato may be related to lower cell separation as a result of lower starch swelling, in turn caused by lower amylose content.

Impact of pectin on cooking properties

A direct correlation between pectin content and textural properties was found for boiled yam and cassava. For the latter, the Ca 2+ content of raw roots was he most significant variable for predicting texture after boiling, and boiling with added calcium ions reduced WA (see Supporting information, Fig. S2), which clearly implicates pectins in determining cooking properties. In plantain, some biochemical traits linked to pectin structure, such as levels of PME activity (Table 3), were also correlated with textural properties. Sweetpotato, yam, and plantain exhibited various levels of PME activity (Table 3). Sweetpotato firmness was not predicted by PME activity in the investigated genotypes, probably because of the very low activity, whereas the high level of PME in plantain suggested that it could have a greater impact on cooking behaviour. This would be in line with the positive relationship previously observed between PME activity and optimal cooking time of potatoes. [START_REF] Ross | Pectin engineering to modify product quality in potato: pectin engineering in potato[END_REF][START_REF] Ross | Potato tuber pectin structure is influenced by pectin methyl esterase activity and impacts on cooked potato texture[END_REF] Overall, this indicated that different mechanisms of pectin modification during cooking were at play in these different crops. In yam, the mechanism of texture development (and thickening of cell walls) during cooking [START_REF] Akissoe | Microstructure and physico-chemical bases of textural quality of yam products[END_REF] is unlikely to be connected with endogenous enzymatic activities during cooking because no PME activity was detected. PME, however, could play a role during Note: For seven yams and plantain, cell walls were extracted using the corresponding SOP [START_REF] De La Rochère | Sample preparation for cell wall polysaccharides analysis of raw and boiled yam and plantain. Biophysical characterization of quality traits[END_REF] and PME activity were determined as described previously. [START_REF] Ribas-Agustí | Towards the use of biochemical indicators in the raw fruit for improved texture of pasteurized apricots[END_REF] Abbreviations: CW, cell wall content; DM, dry matter; GalA, galacturonic acid content; NA, not available; PME, pectin methylesterase. boiling of plantain by demethylating pectins, which could then strengthen the cell-cell adhesion in the middle lamella through cross-linking with calcium ions, resulting in cell wall thickening and a firmer texture. Moreover, the loss of pectin content observed after plantain frying, possibly related to the decrease of firmness, may be the result of ⊎-elimination reaction. This reaction occurs on methylated pectin at pH higher than 4.5 and is increased at temperatures > 80 °C. Such pectin degradation during cooking would agree with the middle lamellae dissolution observed during boiling. [START_REF] Qi | Effect of cooking on banana and plantain texture[END_REF] This nonenzymatic hydrolysis of pectin could therefore be a driver of softening in fried plantain.

Interaction between starch, pectins, and water absorption

Starch is a major contributor to DM of raw samples in major RTBs (see above) but pectins could also contribute to textural differences as they have a higher water retention capacity than nongelatinized starch (maximum native starch water content of approximately 45% wb) and raw RTB had a moisture content of 55-80% (Table 3). A negative correlation was observed between pectins and DM content of fresh samples for raw plantain and cassava, but not for yam and sweetpotato. Interactions with other compounds, such as amylose, cell walls, lignins and soluble sugars (or a different pectin structure) may explain this discrepancy between RTB crops.

In the present study, the hardness of cooked samples increased with the DM of raw samples for yam and plantains only (see above). This common behaviour could be explained by different mechanisms. The DM of raw plantain is the main driver of firmness after cooking (Table 2; see also Supporting information, Fig. S3) with starch playing a major role in increasing firmness (see above) and pectin content influencing texture in a opposite fashion by increasing water content (Fig. 5). In yam, however, both pectin and DM content seemed to increase hardness in cooked tubers independently of starch content (Fig. 2 and Table 1). Moreover, hard cooking yam varieties absorb more water and release less soluble DM after cooking compared to mealy yam varieties. [START_REF] Kouadio | Water absorption as an evaluation method of cooking quality for yam (Dioscorea alata) and cassava (Manihot esculenta crantz)[END_REF] This behaviour may be a result of the higher content of lowmethylated pectin in hard cooking yam varieties leading to a preferential WA by pectins (rather than pectin solubilization) and strengthening cell walls through calcium bridges. Kouadio et al. [START_REF] Kouadio | Water absorption as an evaluation method of cooking quality for yam (Dioscorea alata) and cassava (Manihot esculenta crantz)[END_REF] showed that WA is the main parameter determining cooking quality (hard versus mealy cooking) of cassava and yam, even though they exhibit different behaviours. The lower WA observed for hard cooking cassava (Fig. 4; see also Supporting information, Fig. S2) was influenced by higher calcium fixation, which is also known to increase firmness. Despite the particularly high starch and low CW content in cassava (Table 3), pectins played a major role through their WA capacities. By contrast, sweetpotato cooking behaviour appeared also to be driven by WA capacity but mainly through its starch/amylose swelling properties (see above). Furthermore, the observed decrease of firmness and stickiness of fried plantain after ripening (see Supporting information, Table S5) may be linked to both pectin solubilization and starch degradation commonly observed during ripening. [START_REF] Shiga | Ripening-associated changes in the amounts of starch and non-starch polysaccharides and their contributions to fruit softening in three banana cultivars[END_REF] These phenomena may underly the acceptability of fried plantain as preliminary studies have identified the optimum consumption at the yellow ripening stage. [START_REF] Kouassi | Sensory evaluation of fried plantain (Aloco). Biophysical characterization of quality traits[END_REF] The development of new SOPs and access to diverse RTB samples with wide ranges of cooking properties and biochemical characteristics has allowed the present study to confirm tendencies previously suggested and identified new and specific relations between biochemical characteristics and cooking behaviour. Nevertheless, additional studies are necessary to unravel the combined and synergistic impact of starch and pectin metabolism on the cooking properties of RTB crops. Also, obtaining genotypes with highly contrasting compositions and textural/ cooking behaviours could lead to a better understanding of the determinants and main reactions in play during cooking. Finally, this would help the breeders to target traits for the development of new RTB varieties with good consumer acceptability.

Figure 3 .Figure 2 .

 32 Figure 3. Distribution of amylose (% db), DM (% wb), force (g), area (g. mm), and initial gradient (g mm -1 ) among 200 genotypes of cassava (NaCRRI population), and correlation analysis. Values indicate the coefficients of correlation (r). Asterisks (***) indicate statistically significant correlations (P < 0.05).

Figure 4 .Figure 5 .

 45 Figure 4. Effect of dry matter and total pectin content of fresh cassava roots on cooking quality parameters water absorption and gradient among 29 genotypes of cassava (CIAT population). Each point represents the average of each genotype across three MAPs (9, 10, and 12 months), or two MAPs (9 and 12 months) when Ca 2+ is measured.

Table 1 .

 1 Pearson correlation between hardness, dry matter, starch and GalA contents measured for 24 yam samples Model parameters for predicting firmness of sweetpotato and plot of predicted versus observed firmness.

	Variables	DM	Starch	GalA
	Starch	0.223 (0.30)	1
	GalA	-0.082 (0.70)	-0.524 (0.01)	1
	Hardness	0.694 (0.00)	-0.265 (0.21)	0.464 (0.02)

Note: Probability levels are shown within parenthesis. Bold value represent the correlation is significant. Abbreviations: DM, dry matter; GalA, galacturonic acid. © 2023 Bill and Melinda Gates Foundation. Journal of The Science of Food and Agriculture published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry. J Sci Food Agric 2023

Table 3 .

 3 Overall biochemical characteristics of studied raw crops

	Raw crop	CW (% db)	GalA (% db)	Starch (% db)	DM (% wb)	Amylose content (%)	PME activity (pmol s -1 g -1 db)
	Sweet potato	7-14	0.2-4.8	56-72	30-42	11-14	0.51-2.92
	Yam	3-13	0.6-3.2	45-83	26-45	2-35 67,68	0
	Cassava	2-5 32,69	0.7-1.2 (0.56 69 )	6 9 -80 70	20-45	0-40	NA
	Plantain	5-11	1.5-2.4	17-56	20-40	20-27 39,65	2.4-5.6 × 10 5
	(yellow stage)

Table 2 .

 2 Pearson correlation between texture attributes and biochemical parameters (n = 17 samples) in fried plantain Values in bold are statistically significant (P < 0.05) and the probability levels are given in parenthesis. Data used were collected from 17 samples of fried plantain (aloco) obtained from seven varieties including Bi-gt and Bi-yt: Aloco from Bita 3 variety taken at the yellow tip green and yellow tiger stage; Co-y and Co-yt: Aloco from Corne 1 variety taken at the yellow and yellow tiger stages; Fh-y and Fh-yt: Aloco from FHIA 21 variety taken at the yellow and yellow tiger stages; Sh-gt, Sh-y and Sh-yt: Aloco from SH3640 variety taken at the yellow tip green, yellow and yellow tiger stages; Pi-gt, Pi-y and Pi-yt: Aloco from Pita 3 variety taken at the yellow tip green, yellow and yellow tiger stages; Sa-gt and Sa-y: Aloco from Saci variety taken at the yellow tip green and yellow stage; Za-y, Za-gt and Za-yt: Aloco from Zakoi variety taken at the yellow and yellow tiger stages.

	Texture attributes and	Texture attributes of fried plantain		Total pectin (GalA g per 100 g db)
	chemical analysis results	Firmness	Stickiness	Chewiness	DM	Fresh plantain	Fried plantain
	Firmness_fried plantain	1 (0.00)
	Stickiness_ fried plantain	-0.465 (0.60)	1 (0.00)
	Chewiness_fried plantain	0.617 (0.00)	0.068 (0.00)	1 (0.00)
	DM (% wb)	0.898 (0.00)	-0.554 (0.02)	0.341 (0.18)	1 (0.00)
	Total pectin_fresh plantain	-0.349 (0.20)	-0.216 (0.48)	-0.521 (0.07)	-0.561 (0.01)	1 (0.00)
	(GalA g per 100 g db)
	Total pectin_ fried plantain	-0.088 (0.78)	-0.342 (0.28)	-0.212 (0.51)	-0.213 (0.51	0.685 (0.04)	1 (0.00)
	(GalA g per 100 g db)
	Note:

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ACKNOWLEDGEMENTS

We are grateful for the grant opportunity INV-008567 (formerly OPP1178942): Breeding RTB Products for End User Preferences (RTBfoods), to the French Agricultural Research Centre for International Development (CIRAD), Montpellier, France, by the Bill & Melinda Gates Foundation (BMGF): https://rtbfoods.cirad.fr. We acknowledge Hernán Ceballos for the comments that significantly improved the manuscript, as well as Aliénor Dutheil de la Rochère, Caroline Garcia, Nouha Bouaziz (INRAe Avignon, France), Jean-Luc Irep, David Lange (INRAe Guadeloupe, France), Thierry Bajazet and Mathieu Lechaudel for experimentations, data treatments, and supplying fresh yam. Important contributions to experimental work by Luis Londoño, Jorge Luna, Maria Alejandra Ospina and Cristian Duarte (Alliance Bioversity-CIAT, Cali, Colombia) and Pham Thi Ha Giant (CIRAD) are gratefully acknowledged. Catherine Ebah Bomoh, Anne-Marie Tatiana Beugré and Traoré Siaka (CNRA, Abidjan, Côte d'Ivoire) are also acknowledged for data treatments, experimentations, and supplying fresh plantain.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

AUTHOR CONTRIBUTIONS

TT, AR-S, CM, MT and DD were responsible for conceptualization. TT, AR-S, SA, DR and EN were responsible for data curation. AR-S, CM, SA, DM-A-M, MT, HAK, EN and ASDN&N were responsible for formal analysis. DD was responsible for funding acquisition. SA, KM, HAK and DR were responsible for investigations. AR-S, CM, HAK, TT, MT and DM-A-M were responsible for methodology. DD was responsible for project administration. DR, MN, JLM, HAK and NA were responsible for resources. AR-S, CM and DM-A-M were responsible for supervision. AR-S, CM, SA, DM-A-M, MT, TT and EN were responsible for writing the original draft. AR-S, CM, SA, HAK, ASDN&N, GMcD, DM-A-M, MT and DD were responsible for reviewing and editing.

SUPPORTING INFORMATION

Supporting information may be found in the online version of this article.


David Causeur
email: david.causeur@agrocampus-rennes.fr

François Husson

Power of double-sampling tests for General Linear Hypotheses

Keywords: Auxiliary covariate, Double-sampling, F-test, Missing data, Optimal sampling design MSC: 62D05, 62J10, 62K05

In this paper, testing procedures based on double-sampling are proposed that yield gains in terms of power for the tests of General Linear Hypotheses. The distribution of a test statistic, involving both the measurements of the outcome on the smaller sample and of the covariates on the wider sample, is first derived. Then, approximations are provided in order to allow for a formal comparison between the powers of double-sampling and single-sampling strategies. Furthermore, it is shown how to allocate the measurements of the outcome and the covariates in order to maximize the power of the tests for a given experimental cost.

Introduction

Since the first works on the so-called regression estimator of a total or a mean by Cochran [START_REF] Cochran | Sampling Techniques[END_REF][START_REF] Cochran | Sampling Techniques[END_REF], double-sampling is widely used in the context of sample survey to reduce the variance of the estimation by use of measurements of an auxiliary covariate that are available on a wider sample. The same idea can be found, for instance in Conniffe and Moran [START_REF] Conniffe | Double sampling with regression in comparative studies of carcass composition[END_REF] , transposed to the estimation of the parameters of a multivariate normal regression model. In the agricultural applications handled by Conniffe and Moran [START_REF] Conniffe | Double sampling with regression in comparative studies of carcass composition[END_REF] and later by Engel and Walstra [START_REF] Engel | Increasing precision or reducing expense in regression by using information from a concomitant variable[END_REF], a surrogate and also cheaper version of the outcome is expected to be a beneficial auxiliary covariate for use in a double-sampling design.

The purpose is then to find the optimal allocation of the measurements of the outcome and of its approximate version, namely the double-sampling scheme that minimizes the variance of the estimators of the regression coefficients subject to a given experimental cost. In the presence of many auxiliary covariates, Conniffe [START_REF] Conniffe | Estimating regression equations with common explanatory variables but unequal numbers of observations[END_REF], and more recently Causeur and Dhorne [START_REF] Causeur | Finite-sample properties of a multivariate extension of double-regression[END_REF], show the enhancements that are made possible by accounting for the joint distribution of the concomitant variables when optimizing the sampling design. Furthermore, aiming at a more detailed definition of the optimal sampling plan by accounting for every marginal experimental cost, some extensions based on multiple-phase monotone sampling designs were also described by Causeur [START_REF] Causeur | Optimal sampling from concomitant variables for regression problems[END_REF]. Analogous ideas can also be found in Breslow et al. [START_REF] Breslow | Large sample theory for semi-parametric regression models with two-phase, outcome-dependent sampling[END_REF], where the properties of estimation in non-parametric models are investigated in a double-sampling framework.

Probably because they are mostly motivated by regression examples in which the main issue is to derive a prediction formula, the previously cited works focus on the improvement of estimation that is made possible by a double-sampling design. On the contrary, very less is said about the testing procedures that traditionally complete the estimation results in a regression analysis. Such issues are encountered for instance when comparing types of soils with respect to some physical or chemical property when a few number of soil samples can be used for a reliable analysis of the property under study whereas much more numerous indirect measurement of the same property can be obtained by remote sensing. Especially in this case involving a qualitative explanatory variable, explicit expressions for the estimators of the parameters of the analysis of variance model are straightforward deduced from the existing works but, here, testing procedures for the effects of the different factors are usually much more important. The first aim of this paper is therefore to extend the double-sampling analysis by testing strategies for General Linear Hypotheses. By analogy with the well-known desirable properties of the double-sampling estimator, it is shown in the present paper how to include the measurements of the auxiliary covariates in the test statistic in order to improve its power relative to the traditional F-test used in the single-sampling context. Furthermore, in such situations where the main goal is the test of an effect rather than the estimation of parameters, a strategy is proposed for an optimal allocation of the measurements of the outcome and the auxiliary covariate, which consists, for a given experimental cost, in maximizing the power of the test with respect to the sample sizes. Calculation tools to derive asymptotic approximations of the power of the double-sampling test are provided in order to make this approach numerically possible.

In section 2, the paper introduces the sampling framework together with notations for the different models involved in the procedure. Some results on the functional relationships between the parameters of the models are recalled and the case of qualitative explanatory variables is also specifically examined in this section. Section 3 is devoted to the definition of a testing procedure for General Linear Hypotheses that takes advantage of the doublesampling framework. An asymptotic approximation of the distribution of the test statistic is given. In section 4, strategies to obtain the most powerful sampling design for a given experimental cost are presented.

Regression settings in the double-sampling framework

First, the main model describes linearly the conditional distribution of a continuous outcome Y given p predictors x = (x (1) , x (2) , . . . , x (p) ), with p ≥ 1:

Y ∼ N β (0) y + xβ y ; σ 2 y ,
where β (0) y is the constant of the model and β y = (β (1) y , . . . , β (p) y ) is the p-vector of the slope parameters.

When a double-sampling scheme is used, auxiliary covariates Z = (Z (1) , . . . , Z (q) ) are introduced in the model by means of two auxiliary regression models. In the following, in order to improve the readability of the arguments, the presentation will focus on the case q = 1. An extension to the multivariate context can be achieved on the basis of Causeur and Dhorne [START_REF] Causeur | Finite-sample properties of a multivariate extension of double-regression[END_REF]. However, accounting for multiple-phase monotone sampling schemes that could be inspired by Causeur [START_REF] Causeur | Optimal sampling from concomitant variables for regression problems[END_REF] is more complex and out of the purpose of this paper.

First, common observations of the outcome, the predictors and the auxiliary covariates are supposed to be observable on a sample of size n, and the following linear model is assumed for the distribution of the outcome given the predictors and the auxiliary covariate:

Y ∼ N β (0) y|z + xβ y|z + γ Z; σ 2 y|z ,
where β (0) y|z is the constant of the model, β y|z = (β (1) y|z , . . . , β (p) y|z ) is the p-vector of the slope parameters for the predictors and γ is the slope parameter for the covariate. Now, common observations of the covariate and the predictors are also available on a sample of size N ≥ n, including the first sample, and the following linear model is assumed for the distribution of the covariate given the predictors:

Z ∼ N β (0) z + xβ z ; σ 2 z ,
where

β (0) z is the constant, β z = (β (1) z , . . . , β (p)
z ) is the p-vector of slope parameters and σ 2 z is the residual variance.

A closed-form expression of the maximum-likelihood estimators is now based on a kind of transitiveness relationship that enables to express the parameters of the main model as a combination of the parameters of the auxiliary models:

β y = β y|z + γβ z , σ 2 y = σ 2 y|z + γ 2 σ 2 z .
Although such a double-sampling procedure has mainly been used in situations where the predictors are quantitative variables, the preceding definitions are formally valid in analysis of variance contexts where the explanatory variables are qualitative.

In an illustrative purpose, let us examine the case a one way analysis of variance model with a factor taking three levels. The double-sampling design is summarized in table 1.

The preceding presentation of the model still holds where the explanatory variables are now dummy variables for the factor: for the ith level of the factor

Y ∼ N μ y + α (i) y ; σ 2 y , (1)
with, for instance, the cornerstone restriction on the main effects: α (1) y = 0. Table 1. An example of a double sampling design for a one way analysis of variance model with a factor taking three levels. Missing values for the response variable are identified by ?.

Sample sizes

Y Z X n 1 firstst level N 1 -n 1 ? n 2 Second level N 2 -n 2 ? n 3 Third level N 3 -n 3 ?
Similarly, given the covariate and the factor's level, it can be tempting to assume an analysis of covariance model with interaction:

Y ∼ N μ y|z + α (i) y|z + (η + δ (i) )Z; σ 2 y|z ,
with α (1) y|z = 0 and δ (1) = 0. However, it is straightforward checked that the assumption of homoscedasticity in model ( 1) holds if and only if all the interaction parameters δ (i) are zero.

In that case, the analogy between the analysis of variance model and the regression settings can be used to express the parameters of the main model as a function of the parameters of the auxiliary models:

μ y = μ y|z + ημ z , α y = α y|z + ηα z , σ 2 y = σ 2 y|z + η 2 σ 2 z .
Simple conditions for the estimability of the parameters of interest (μ y , α (i) y , σ 2 y ) are deduced from the above functional relationships. Indeed, to ensure that all parameters in model ( 2) are estimable, both response Y and covariate Z should be observed on a sample of size n ≥ 4, namely the number of levels plus 1, with n = n 1 + n 2 + n 3 and no n i should be zero.

Moreover, as mentioned in the next section, the sampling fractions n i /N i will be assumed to have the same limit for large values of the sample sizes.

Testing procedures

In the settings introduced in the previous section, we are interested in the following test of a General Linear Hypothesis:

H 0 : β y = 0 H 1 : β y = 0 .
Results for tests of a particular subset of coefficients in β y can be deduced from those that are presented thereafter.

Call Y n and Z n the n-vectors of the observations of the outcome and the covariate respectively on the sample of size n. Call also Z N the N -vector of the observations of the covariate on the whole sample of size N . Finally, call X (n) and X (N ) the matrices with n and N rows respectively and p columns, containing the observations of the predictors on the samples of size n and N respectively. Up to now, the same notation will be used as a sub or super scripts for other quantities, such as estimators, when they are derived on the small or large sample of size n or N respectively. Furthermore, X (n) is assumed to be full rank and such that the corresponding variance-covariance matrix

S (n) = (1/n)X (n) (I n -(1/n)J n )X (n)
, where J n is the n × n matrix with all entries equal to 1, tends to a positive definite matrix x for large values of n. It is also assumed that S (N ) = (1/N )X (N ) (I N -(1/N )J N )X (N ) tends to the same limit for large values of N .

A useful way of defining the model in a double-sampling context consists in working with the (n + N)-vector obtained by concatenating Y n and Z N . Under the assumptions of section 2, U = (Y n , Z N ) is normally distributed with the following expectation and variance:

E(U ) = 1 n X (n) 0 n 0 n,p 0 N 0 N,p 1 N X (N ) ⎛ ⎜ ⎜ ⎜ ⎝ β (0) y β y β (0) z β z ⎞ ⎟ ⎟ ⎟ ⎠ , = Xβ, Var(U ) = ⎡ ⎢ ⎣ σ 2 y I n γ σ 2 z I n 0 1,N -n γ σ 2 z I n σ 2 z I N 0 N-n,1 ⎤ ⎥ ⎦ , = V , ( 2
)
where 1 k and 0 k stand for the vectors of size k which all entries equal 1 and 0 respectively and 0 k,p stands for the k × p matrix with all elements equal to 0. First, let us examine the testing procedure in the situation of known variance parameters before extending the results to the general case of unknown variance parameters. Note that assuming the parameters in V are known implies not only that σ y and σ z are known but also γ .

Exact test for known variance parameters

According to Causeur and Dhorne [START_REF] Causeur | Finite-sample properties of a multivariate extension of double-regression[END_REF], the generalized least squares estimator β of β can be explicitly derived as follows: It can be deduced, for instance from Conniffe [START_REF] Conniffe | Estimating regression equations with common explanatory variables but unequal numbers of observations[END_REF], that the former test statistic can also be expressed as follows:

β = (X V -1 X) -1 X V -1 U, = β(n) y + γ β(N) z -β(n) z β(N)
SS β y = β(n) y σ y + ρ β(N) z -β(n) z σ z ρ 2 N S -1 (N ) + 1 -ρ 2 n S -1 (n) -1 β(n) y σ y + ρ β(N) z -β(n) z σ z ,
where ρ = γ σ z /σ y is the partial correlation coefficient between Y and Z given the predictors.

The test statistic appears therefore as a continuum indexed by ρ between the classical standardized sum-of-squares derived on the small sample for ρ = 0 and on the wider sample for ρ = 1. Therefore, it can be expected that the present testing strategy will always be at least as good as the usual F-test based on the available measurements of the outcome and always worse than the F-test that could be derived if the outcome was observed on the whole sample. According to well-known results in the theory of linear models, the former quadratic form is distributed according to a noncentral χ 2 p distribution with non-centrality parame-

ter β y [(ρ 2 /N )S -1 (N ) + ((1 -ρ 2 )/n)S -1 (n) ] -1 β y /σ 2 y
, which enables a classical testing procedure based on the exact distribution of the test statistic. Var( βy

) -1 = N σ 2 y ρ2 I p + 1 -ρ2 ϕ ρ2 S (N ) S -1 (n) -1

S (N ) .

Indeed, replacing S (N ) S -1

(n) by its asymptotic limit I p , at least provided that the distribution of the predictors has two finite first moments, leads to the asymptotic approximation of Var( βy ) -1 by nS (N ) / σ 2 y K( ρ2 ). Moreover, the asymptotic expansion of the distribution of βy given in Causeur [START_REF] Causeur | Exact distribution of the regression estimator in double-sampling[END_REF] shows that it is not markedly different, even in small sample conditions, of the distribution of βy , which proves the proposition.

Note that, when n = N, K( ρ2 ) = 1. Both SS β y and SS * β y coincide then with the classical F -statistic used in the single-sampling framework.

The following proposition takes advantage of the preceding results to give an approximation for the distribution of SS β y in terms of classical distributions.

PROPOSITION 2 Let us define the random variate T (n,N ) (δ (n,N ) , ρ 2 ) as follows:

T (n,N ) (δ (n,N ) , ρ 2 ) = n 1 + ϕ ρ 2 1 -ρ 2 T 1 T 2 + ϕ 2 T 3 ,
where T 1 , T 2 and T 3 are mutually independent. In addition, T 1 is distributed according to a non-central χ 2 p with non-centrality parameter δ (n,N ) /K(ρ 2 ) with δ (n,N ) = nβ y S (N ) β y /σ 2 y and T 2 is distributed according to a χ 2 n-p-2 distribution. Suppose now that B and S are independent random variates following respectively a beta distribution B([np -1]/2, [Nn]/2) and a χ 2 N -p-1 , then T 3 is conditionally distributed, given B and S, as the ratio between a non-central chi-square variable with 1 degree of freedom and non-centrality parameter [ρ 2 /(1ρ 2 )]BS and B.

SS β y and T (n,N ) (δ (n,N ) , ρ 2 ) have the same limiting distributions when n and N are large.

Proof First, it is deduced from proposition 1 that the asymptotic distributions of SS β y and SS * β y are the same. Moreover, SS *

β y = n 1 + ϕ ρ 2 1 -ρ 2 T 1 T 2 + ϕ 2 T 3 , with T 1 = n β y S (N ) βy /σ 2 y K(ρ 2 ), T 2 = [Y (n) (P (n) -P * (n) )Y (n) ]/σ 2 y (1 -ρ 2 )] and T 3 = [Y (n) P * (n) Y (n) /σ 2 y (1 -ρ 2 )]/[Z (n) P (n) Z (n) /Z (N ) P (N ) Z (N ) ].
In the former definitions, P (n) and P (N ) are the orthogonal projectors onto the linear spaces orthogonal to the spaces spanned by the columns of X (n) and X (N ) respectively and P *

(n) = P (n) Z (n) Z (n) P (n) /Z (n) P (n) Z (n) .
With the same arguments that were used to achieve proposition 1, T 1 is shown to follow asymptotically a noncentral χ 2 p with non-centrality parameter δ (n,N ) /K(ρ 2 ). The former statistic depending only on estimators of expectation parameters, it is independent of T 2 and T 3 . Moreover, the independence of T 2 and T 3 is made obvious by considering their conditional joint distribution given the covariate and the predictors and the distribution of T 2 is straightforward deduced from the classical theory of the distributions of quadratic forms. Note finally that [Y

(n) P * (n) Y (n) /σ 2 y (1 -ρ 2 )] is distributed as a χ 2 1 variate with non-centrality param- eter [ρ 2 /(1 -ρ 2 )][Z (n) P (n) Z (n) /σ 2 z ]
. Finally, it is deduced from Johnson et al. [11, p. 349] that B = Z (n) P (n) Z (n) /Z (N ) P (N ) Z (N ) and S = Z (N ) P (N ) Z (N ) /σ 2 z are independently distributed according to a B([np -1]/2, [Nn]/2) and a χ 2 N-p-1 respectively.

The former proposition provides calculation tools to derive approximations of the quantiles or the probability distribution function of SS β y by means of Monte-Carlo simulations. This is therefore very useful to maximize the power of the present test with respect to the sample sizes. This point is addressed in the next section. Note that δ (n,N ) is a non-centrality parameter, which expression is similar to the equivalent parameters encountered when testing the global significance of the parameters in the classical framework. By the way, if n = N , the distribution of (1/p)T (n,N ) (δ (n,N ) , ρ 2 ) coincides with the non-central Fisher distribution involved for the same test in a single sampling context.

Example

In the simulation feature introduced above, the quantiles of the distribution given in proposition 3.2 are plotted in figure 2 against the theoretical quantiles of a χ 2 2 distribution, together with the quantiles of SS β y . Obviously, this figure confirms the good approximation of the distribution of SS β y by the distribution of T (n,N ) (σ 2 y , ρ 2 ) even in small-sample conditions.

Optimal sampling designs

In the single-sampling context, where the sample size is say m, the power of the F-test is defined with respect to a given standardized distance from zero δ = β y S (m) β y /σ 2 y of the vector of expectation parameters. This distance from zero, that appears in the non-centrality parameter of the F-distribution under the alternative hypothesis, is chosen as the minimal value over which the test is expected to reject the null hypothesis with a high probability. The task consisting in giving a relevant value for δ is by the way much easier in the case of a one-way analysis of variance model where it can be expressed more intuitively in terms of the sum of the squares of the marginal effects parameters.

Let c y be the unitary experimental cost for the joint measurements of the outcome and the predictors. It is interesting to keep in mind that double-sampling designs are beneficial in the situations where c y is much larger than c z , namely the unitary experimental cost for the measurement of the covariate and the predictors. The global cost for a regression experiment involving a single sample, which size is m, is therefore mc y . Under these experimental conditions, the power of the F-test for the significance of β y is denoted π m (δ). Now, let c yz denote the unitary experimental cost for the joint measurements of the outcome, the covariate and the predictors. Note that, in many situations where the covariate is an intermediate measurement of the outcome, c yz = c y . The global cost for a double-sampling design is therefore c yz n + c z (Nn). The issue is now to provide the double-sampling design which maximizes the power π (n,N ) (δ) of the test subject to the following restriction on the subsequent experimental cost:

c yz n + c z (N -n) = mc y .
In other words, the object is to seek for the optimal sampling design among those which experimental cost is equivalent to a single-sampling strategy with m experimental units.

given in table 2. Here again, the results show that, provided c z is low enough, the doublesampling approach can help saving experimental cost relative to the single-sampling plan.

Discussion

The present paper gives the asymptotic distribution of a test statistic for General Linear Hypotheses in a double-sampling context. Such a result is first useful after a double-sampling experiment to complete the analysis based on estimates by tests of the significance of the predictors. Moreover, in situations where measurements of the outcome are very expensive but can be approximated by a cheaper and highly correlated version, sampling designs aiming either at a reduction of the global experimental cost or at an improvement of the power of the test can be deduced from a closed form expression of the asymptotic distribution of the test statistic in terms of classical distributions.

The double-sampling framework and its multivariate monotone extension are particularly studied patterns of missing data because they enable explicit expressions for the maximumlikelihood estimators of the parameters of a joint multinormal distribution.

The present results take advantage of this desirable property to achieve testing procedures, at least in the case of a two-phase sampling. For convenience, the results are indeed presented in the case of only one covariate but they can straightforward be extended to a multivariate context on the basis of Causeur and Dhorne [START_REF] Causeur | Finite-sample properties of a multivariate extension of double-regression[END_REF]. However, for more complex sampling schemes in the presence of many covariates such as the monotone designs proposed by Causeur [START_REF] Causeur | Optimal sampling from concomitant variables for regression problems[END_REF], the joint distribution of the variance parameters is far more tedious to be obtained in terms of known distributions. Equivalent studies in the former context are however worthwhile to face specific applications for example to time series or growth curve modelling where monotone patterns of missing data are encountered.

  y , β(n) z and β(N) z are ordinary least squares estimators of β y and β z . Now, call SS β y = SS β y (γ , σ 2 y , σ 2 z ) the sum-of-squares traditionally involved when testing the nullity of β y : SS β y = β y Var ( βy ) -1 βy .

Figure 1 .

 1 Figure 1. Empirical quantiles of SS βy versus the quantiles of SS βy .

Figure 2 .

 2 Figure 2. Approximation of the quantiles of SS βy by the quantiles of T (n,N ) (σ 2 y , ρ 2 ).

Approximate test for unknown variance parameters

According to Causeur and Dhorne [START_REF] Causeur | Finite-sample properties of a multivariate extension of double-regression[END_REF], the maximum-likelihood estimators of the variance parameters are given by the following expressions:

where σ (n) yz , { σ 2 y } (n) , { σ 2 z } (n) and { σ 2 z } (N ) are the usual maximum-likelihood estimators of σ yz , σ 2 y and σ 2 z based on residual sum-of-squares. Note that, due to the functional invariance of the maximum-likelihood estimation, ρ is estimated by ρ = γ σz / σy .

In the general framework of linear mixed models, Kenward and Roger [START_REF] Kenward | Small sample inference for fixed effects estimators from restricted maximum likelihood[END_REF] proposes a small sample inference method to account for the estimation of the variance components in the distribution of the test statistics. We propose an alternative method that is made possible by the specific covariance structure involved in our missing-data problem. Properties of this structure will for instance avoid us some technical assumptions that are used by Kenward and Roger [START_REF] Kenward | Small sample inference for fixed effects estimators from restricted maximum likelihood[END_REF].

We propose hereafter to study the testing procedure based on SS β y = SS β y ( γ , σ 2 y , σ 2 z ) obtained by replacing the variance parameters by their maximum-likelihood estimator in the expression of SS β y .

Example In order to see how estimating the variance parameters modifies the distribution of the test statistic, let us examine an artificial situation. Suppose here that σ y = σ z = 1. Suppose also that p = 2, β y = (0, 0) and β z = (3, 3) . The sample sizes can either be (n = 15, N = 30) or (n = 30, N = 80) and the partial correlation ρ is either 0.2 or 0.8. With such values for the input parameters, 10,000 datasets are simulated as follows: first, predictors x are randomly drawn from a bivariate normal distribution with mean 0 and variance-covariance identity. Then U is randomly simulated 10,000 times according to the normal conditional distribution introduced in expression (2). In figure 1, the empirical quantiles of SS β y are plotted against the theoretical quantiles of a χ 2 2 distribution. Whatever the sample sizes and the partial correlation, this figure shows departures of the distribution of SS β y from the χ 2 2 distribution. As in the single-sampling case where the Fisher distribution replaces the χ 2 -distribution when the variance parameter is estimated, the distribution of SS β y is more heavy-tailed than the χ 2 2 distribution. Moreover, the departures from the χ 2 distribution are more important in smallsample conditions and especially in the case of a poor partial correlation between the outcome and the covariate.

Providing an approximation of the distribution of SS β y by means of a combination of classical distributions is not as straightforward as in the single-sampling framework. First, the test statistic appears to be a quadratic form which kernel matrix Var( βy

is the inverse of a linear combination of matrices with random coefficients. Furthermore, SS β y is also a quadratic function of the random vector βy = β(n)

, which is not On the basis of the approximation provided in proposition 2, the power π (n,N ) (δ) can be evaluated by Monte-Carlo simulations. Given δ, the non-centrality parameter δ (n,N ) that appears in the distribution of T (n,N ) (δ (n,N ) , ρ 2 ) can be approximated by Nδ/K(ρ 2 ) for large N and m. Therefore, as for the optimization of double-sampling designs with respect to the efficiency of estimation, a prior estimation of ρ 2 is also needed here to obtain the optimal sample sizes.

In order to illustrate the derivation of optimal double-sampling designs, consider the following situation: the unitary costs involving the outcome are supposed to be c yz = c y = 20 and c z = 4. For equally spaced value of δ between 0 and 4, the optimal double-sampling design, with the same experimental cost as the single sampling design with m = 10, is calculated. Figure 3 shows the power functions at level α = 0.05 for different values of ρ. For large values of ρ, the figure shows that important gains in terms of power of the test can be expected from a double-sampling strategy here. However, when ρ becomes poorer, the optimal double-sampling design coincides with the single-sampling feature.

Conversely, suppose now that the objective power is fixed to 0.9 for a given δ, say δ = 1. By a single-sampling approach, the problem is quite usual and consists in solving the following equation in m: π m (0.1) = 0.9. Transposed in the double-sampling context, the issue is similarly to find (n, N ) such that π (n,N ) (0.1) = 0.9. In the situations presented above, the solutions are