Andy Cockburn

Carl Gutwin
email: gutwin@cs.usask.ca

Philippe Palanque
email: palanque@irit.fr

Yannick Deleris
email: yannick.deleris@airbus.com

Catherine Trask

Ashley Coveney

Marcus Yung

Karon Maclean
email: maclean@cs.ubc.ca

Open Archive Toulouse Archive Ouverte

Keywords: Touch interaction, turbulence, aviation. ACM Classification H.5.m. Information interfaces and presentation (e.g., HCI): Miscellaneous

dials, keypads and wheels. While computer-based flight instrument displays are becoming increasingly prevalent (i.e., the 'glass cockpit'), these displays are almost exclusively used for data output, and user input to displayed objects is dependent on a separate indirect device. When a cursor is incorporated in the display, a trackball is typically used for item selection.

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Touch interaction in commercial cockpits offers potential advantages to pilots, airlines, and aircraft manufacturers. Pilots may gain from familiar, expressive, and direct means for interaction in certain tasks. Airlines and manufacturers could gain from reduced hardware installation complexities. Replacing hard-wired physical controls with touchscreens could therefore ease development, facilitate upgrades, reduce weight, and improve pilot interaction. However, turbulence is a challenge for cockpit touchscreen deployment. There are risks that the pilot's ability to interact with the touchscreen may be substantially impaired or eliminated during periods of heavy cockpit vibration. Previous studies have shown that vibration can be a factor in touch input, but there are few studies that look at different types of interactive touch tasks or that consider how the problem of turbulence might be reduced.

We therefore examined users' ability to interact with various types of interactive objects at different vibration levels. We used a motion platform to expose participants and a large touchscreen to three levels of simulated turbulence -none, low, and high. All tasks were completed using three different means for input -a trackball (used in many commercial aircraft), a 21.5 inch touchscreen, and the same touchscreen augmented with a guiding stencil overlay. The stencil

ABSTRACT

Touchscreen input in commercial aircraft cockpits offers potential advantages, including ease of use, modifiability, and reduced weight. However, tolerance to turbulence is a challenge for their deployment. To better understand the impact of turbulence on cockpit input methods we conducted a comparative study of user performance with three input methods -touch, trackball (as currently used in commercial aircraft), and a touchscreen stencil overlay designed to assist finger stabilization. These input methods were compared across a variety of interactive tasks and at three levels of simulated turbulence (none, low, and high). Results showed that performance degrades and subjective workload increases as vibration increases. Touch-based interaction was faster than the trackball when precision requirements were low (at all vibrations), but it was slower and less accurate for more precise pointing, particularly at high vibrations. The stencil did not improve touch selection times, although it did reduce errors on small targets at high vibrations, but only when finger lift-off errors had been eliminated by a timeout. Our work provides new information on the types of tasks affected by turbulence and the input mechanisms that perform best under different levels of vibration.

INTRODUCTION

Commercial aircraft cockpits are replete with physical controls, including many forms of switches, knobs, levers, overlay consisted of a 3mm-thick transparent Lexan sheet that entirely covered the touchscreen (see Figure 1). Holes were cut through the sheet to permit interaction with underlying widgets. The sheet's 3mm thickness eliminated capacitive sensing of the finger outside the cut-out regions.

The stencil overlay provided two potential benefits for turbulent touchscreen interaction. First, it might offer stabilization benefits because users can place their fingers or hand-edge on the stencil, without contact registration, and subsequently move their finger to the target (by sliding over the stencil and 'popping' into a hole). Second, once within a hole, users can further stabilize movement by pushing against the stencil's edge. Results include the following findings: touch interactions were faster and preferred at low vibrations; the stencil assisted touch selection of small targets at high vibrations; the stencil increased accidental lift-off errors, but this can be accommodated by using a short timeout between touch selections; drag-based selections were highly inaccurate during vibration with touch and trackball; and multi-touch pan-and-zoom interactions were much faster than the equivalent trackball method, even at high vibrations.

We make three main contributions: we demonstrate that different tasks have very different tolerance to turbulence; we provide empirical data on the effectiveness of touch and trackball input at high levels of vibration; and we test the value of a novel stabilizing stencil overlay for touchscreens. Overall, we provide new information for designers who must accommodate high vibration in touch-input.

BACKGROUND Aviation Cockpits and Controls

The lifespan of a passenger aircraft is typically 20-30 years, operating between 35,000 and 110,000 cycles dependent on longhaul or shorthaul use. During its lifetime, much of an aircraft's componentry will be replaced and updated, including power-units, jet engines, and parts of the airframe.

Updates to cockpit controls are also desirable, particularly when new technologies become available. However, cockpit updates can be prohibitively expensive because controls predominantly rely on mechanical switches, dials, and levers with wiring redundancy (in case of failure on one channel). The reliance on hardware stems from the aviation regulatory environment, including conformance with standards.

While physical controls retain many advantages for certain flight-critical tasks, there is substantial industry interest in moving functions to cockpit display systems. Examples include the Lockheed Martin F-35 Lightning II stealth fighter, which has a large touchscreen display [START_REF] Fletcher | Panoramic cockpit displays for tactical military cockpits[END_REF], Garmin International's patent for dual touch/cursor control of a CDS [START_REF] Komer | Avionics control and display unit having cursor control mode of operation[END_REF], Rockwell Collin's cockpit touchscreens [START_REF] Hardinger | Built to touch[END_REF] for light jets and turboprops, and the Thales Group "Avionics 2020" vision for the cockpit of the future [START_REF] Zammit-Mangion | A Single Interactive Display Concept for Commercial and Business Jet Cockpits[END_REF]. See Kaminani [START_REF] Kaminani | Human computer interaction issues with touch screen interfaces in the flight deck[END_REF] and Hamon [START_REF] Hamon | Multi-Touch interactions for control and display in interactive cockpits: issues and a proposal[END_REF] for reviews.

In empirically examining touchscreen interaction in the cockpit, Stanton et al. [START_REF] Stanton | To twist, roll, stroke or poke? A study of input devices for menu navigation in the cockpit[END_REF] analyzed the effectiveness of four different input devices (trackball, rotary controller, touch pad and touchscreen) for menu navigation in non-turbulent environments. They concluded that different devices have different strengths, and that the touchscreen had the highest number of 'best' scores. Barbé et al. [4] and Huseyin et al. [START_REF] Huseyin | Designing touch-enabled electronic flight bags in SAR helicopter operations[END_REF] examined ergonomic aspects of touchscreen positioning in airplanes and helicopters. Wang et al. [START_REF] Wang | The Effect of Touch-key Size and Shape on the Usability of Flight Deck MCDU[END_REF] recently examined the influence of the size and shape (square versus rectangular) of touchscreen targets. Targeting time results were consistent with Fitts' law, errors were higher with rectangular targets, and participants preferred larger targets to small ones. In another cockpit-directed targeting study, Lewis et al. [START_REF] Lewis | The Effect of Handedness on use of Touch Screen versus Touch Pad[END_REF] compared left-handed targeting performance when using a touchscreen and trackpad (lefthanded performance was studied based on the assumption that the right hand would be on a control stick, which is not the case for the captain in a passenger aircraft). The main finding was that participants were faster when using the touchscreen than they were with the trackpad. Neither of these studies examined the influence of turbulence.

In addressing issues arising from turbulence, at least two patent applications disclose methods to facilitate the use of touchscreens in the cockpit. Kolbe [START_REF] Kolbe | Method for determining valid touch screen inputs[END_REF] describes a method for determining whether touchscreen events are invalid following detection of a 'movement indicative of turbulence' by analyzing the characteristics of the contact data. The method of Komer et al. [START_REF] Komer | Avionics control and display unit having cursor control mode of operation[END_REF] instead assures that an alternative cursor-based indirect method of input (such as a trackball) is available to supplement touchscreen control.

In an early empirical study of cockpit touch interaction, Bauersfeld [START_REF] Bauersfeld | Effects of turbulence and activation method on touchscreen performance in aviation environments[END_REF] compared two different touchscreen activation modes for confirming selection actions -confirm on contact versus confirm on lift-off (i.e., when the finger is released from the surface). These modalities were compared at two different levels of simulated turbulence, using a NASA-Ames flight simulator. Results confirmed the higher accuracy of lift-off observed in prior findings in nonturbulent environments [START_REF] Potter | Improving the accuracy of touch screens: an experimental evaluation of three strategies[END_REF]. More recently, Hourlier et al. [START_REF] Hourlier | Testing Touch Screens in Realistic Aeronautic Turbulent Conditions (Light to Severe)[END_REF] examined the perceived realism of cockpit turbulence simulations. Participants used an 11 point rating scale (from 0 for "not realistic at all" to 10 for "extremely realistic") to assess the perceived realism of turbulence that was simulated using a hexapod robot similar to that used in our experiment. Mean ratings ranged from 6 for the simulation of light turbulence to ~9 for the simulation of severe turbulence.

Huseyin et al. [START_REF] Huseyin | Future Flight Decks: Impact of +Gz on Touchscreen Usability[END_REF] recently reported results of a Fitts' Law study examining the influence of simulated constant Gforces on touchscreen target acquisition. Participants wore a weighted bag on their wrist while acquiring 55px (15mm) and 75px (20mm) targets. Results showed that acquisition times increased with wrist weight. Finally, Dodd et al. [START_REF] Dodd | Touch Screens on the Flight Deck: The Impact of Touch Target Size, Spacing, Touch Technology and Turbulence on Pilot Performance[END_REF] examined the impact of moderate levels of simulated flight turbulence on touchscreen virtual-keypad input, with buttons of different sizes and separations. Data entry times, error rates, fatigue, and perceived workload were all higher at moderate turbulence than in a no-turbulence condition.

Touchscreen interaction and input stabilization

Abundant research has examined the efficiency and accuracy of different methods for interacting with various forms of touchscreens. Early studies compared user performance (time and error rate) attained when different methods are used to terminate touch selections, such as first-contact, slide-over, and lift-off [START_REF] Mack | A Benchmark Comparison of Mouse and Touch Interface Techniques for an Intelligent Workstation Windowing Environment[END_REF][START_REF] Potter | Improving the accuracy of touch screens: an experimental evaluation of three strategies[END_REF][START_REF] Ren | Improving selection performance on pen-based systems: a study of penbased interaction for selection tasks[END_REF]. Dragging actions with different input devices have also been closely examined (e.g., [START_REF] Cockburn | Understanding performance in touch selections: Tap, drag and radial pointing drag with finger, stylus and mouse[END_REF][START_REF] Forlines | Direct-Touch vs. Mouse Input for Tabletop Displays[END_REF]). In general, studies suggest that direct finger pointing is fast but relatively imprecise unless augmented with a cursor-like indicator (e.g., [START_REF] Vogel | Shift: A Technique for Operating Pen-Based Interfaces Using Touch[END_REF]), that stylus input is fast but can be inaccurate, and that indirect pointing methods can be highly accurate. Zhai et al. [START_REF] Zhai | Foundational Issues in Touch-Surface Stroke Gesture Design -An Integrative Review[END_REF] provide a comprehensive review of work on gestural touch interaction.

Several studies have examined methods for enabling interaction by users with motor impairments, such as uncontrolled tremor. Barrier Pointing [START_REF] Froehlich | Barrier pointing: using physical edges to assist target acquisition on mobile device touch screens[END_REF] proposed stylusbased methods for target acquisition using bezel edges and corners of a mobile device to facilitate input stabilization. Similarly, EdgeWrite [START_REF] Wobbrock | EdgeWrite: A stylus-based text entry method designed for high accuracy and stability of motion[END_REF] used the ridge edges of a small trackpad region to improve text-entry movements for users with limited motor control. Related studies have examined trackball and joystick input for similar objectives [START_REF] Wobbrock | An Alternative to Push, Press, and Tap-tap-tap: Gesturing on an Isometric Joystick for Mobile Phone Text Entry[END_REF][START_REF] Wobbrock | Trackball Text Entry for People with Motor Impairments[END_REF].

The work most closely related to our current study concerns computer input in motion environments. In an early study of ship motion, McLeod et al. [START_REF] Mcleod | The influence of ship motion on manual control skills[END_REF] showed that joystick input was negatively affected by motion across a variety of tasks. Hill and Tauson [START_REF] Hill | Soldier performance issues in C2 "On the move[END_REF] showed that soldiers' computer input in vehicles (such as an armored personnel carrier) is negatively influenced by vehicle vibration. Similar findings have been shown for mouse input on trains [START_REF] Narayanamoorthy | Error analysis of task performance with laptop in vibration environment[END_REF].

Yau et al. [START_REF] Yau | Effects of Input Device and Motion Type on a Cursor-Positioning Task[END_REF] examined user performance in abstract target selection activities using three types of trackball and a touchscreen in five different settings based on ship movement (static, 'heave', 'roll', 'pitch' and 'random', all at 0.3Hz, with a maximum vertical displacement of ±100mm). Results indicated the superiority of the mouse in vibration environments, as well as showing that touchscreen errors increase with vibration. Similarly, Lin et al. [START_REF] Lin | The performance of computer input devices in a vibration environment[END_REF] based their studies on ship motion, comparing the influence of lowfrequency vibrations on touchscreen, mouse, and trackball input using a Fitts' Law target selection methodology. Their maximum vibration condition used root mean square (RMS) accelerations of 0.34m/s 2 , corresponding to the lower end of 'a little uncomfortable' on the ISO vibration discomfort index for low frequency vibration (ISO 2631-1). Their findings indicated that the touchscreen allowed fast but inaccurate input and that the trackball was slow but accurate.

Neither of the ship studies examined the higher frequency vibrations and much higher accelerations incurred during flight. While the study of Dodd et al. [START_REF] Dodd | Touch Screens on the Flight Deck: The Impact of Touch Target Size, Spacing, Touch Technology and Turbulence on Pilot Performance[END_REF] did examine touch input during simulated flight turbulence, they did not examine other types of input activity beyond keypad entry (such as dragging or multitouch pan/zoom operations), nor did they compare touch with other input methods. None of the studies examines methods for remediating touch input problems during turbulence.

TOUCHSCREEN STENCIL OVERLAYS

The success of edge-based touch stabilization for users with motor impairments suggests that related methods might improve touch input in turbulent environments. In particular, there are opportunities to assist stabilization on interface widgets by placing a transparent stencil overlay on top of the touchscreen. The idea of using a stencil or template to allow users to feel touchable parts of a touch panel was first described by Buxton et al. [START_REF] Buxton | Issues and techniques in touch-sensitive tablet input[END_REF]. Like Buxton's template, our stencil contains holes that match the geometry of underlying touchscreen widgets, offering two potential interaction benefits. First, users may be able to stabilize their hand and fingers during target approach by placing fingers or palm on top of the overlay, without triggering touch registrations on the touchscreen. Users could then slide their finger into a hole to acquire a widget. Second, once one or more fingers rest within a stencil hole, users could press against the stencil edge to stabilize their control over dragging actions (e.g., when operating a slider). However, it is possible that the stencil may slow certain interactions. For example, Avrahami [START_REF] Avrahami | The Effect of Edge Targets on Touch Performance[END_REF] showed that touch target acquisition times were slower by up to ~90 ms when targets were located close to the edge of a displayed border.

Stencil Thickness, Material, and Friction

Multitouch touchscreens predominantly use projected capacitive sensing to detect charged objects on, or in close proximity to, the surface. Objects are not sensed beyond a few millimeters from the display -a 3mm-thick sheet of Lexan polycarbonate prevented touches except where holes were cut. Lexan has excellent optical clarity, is easily worked, has good abrasion resistance, and is widely used in the aerospace industry for aircraft window dust covers.

It is likely that there is a friction sweet-spot that best affords contact stability without compromising control due to 'stick-slip' effects. Robinson et al. [START_REF] Robinson | Physical Touch Aspects of the Touch Interface for Flight Deck Applications[END_REF] examined surface friction in cockpit touchscreen environments, and Levesque et al. [START_REF] Levesque | Enhancing Physicality in Touch Interaction with Programmable Friction[END_REF] examined methods for programmatically varying touchscreen contact friction.

Hole Size and Edge Profile

The shape of the edge of the stencil holes, and the size of the holes with respect to the underlying widget, will influence several aspects of interaction, including comfort, accuracy, friction, and stability provided by the edge. We considered various edge profiles, including those shown in Figure 2.

The bevel edge shown in Figure 2a should be relatively comfortable, but it is likely to provide low levels of edge stability. The square edge shown in Figure 2b should be more stable, less comfortable, and may create a small gap between the stencil edge, finger, and contact surface. The ∑ profile edge shown in Figure 2c, is likely to be yet more stable, but uncomfortable, and with a small possible gap between stencil base and finger at the contact surface. The blade edge of Figure 2d is also likely to be stable and uncomfortable, with a potentially large gap from stencil base to finger contact. Our experiments used the square edge shown in Figure 2b, representing a compromise between comfort, stability, and potential stencil-finger surface gap. To account for the anticipated gap, we made the stencil holes approximately 0.5 mm wider on each side than the target widgets (i.e., 1 mm diameter larger for circular targets).

Supporting Mode Transitions

Cockpit display systems often support modes that present different interfaces, such as control panel (ECAM) in one mode and navigation information in another. Various techniques could be used to reconfigure the overlay when mode changes occur. If stencils were fabricated from a flexible material, the overlays could be stored on a roller housed beneath the display. When a mode change occurred, actuators could roll the required stencil into place. Alternatively, actuators might pop-up a new stencil from a 'toast-rack' of candidates when needed, leaving the flight crew to snap the stencil into a housing.

Input Redundancy

While stencil overlays may assist touch input during turbulence, the need for safety critical operation requires high redundancy in the cockpit. Therefore, redundant methods of input would need to be supported, such as the availability of a trackball and cursor to augment or replace a touchscreen in the event of hardware failure. This need is directly addressed by a patent to Komer et al. [START_REF] Komer | Avionics control and display unit having cursor control mode of operation[END_REF].

EVALUATING INPUT METHODS IN TURBULENCE

We conducted an experiment to better understand the influence that simulated turbulence has on interaction. Participants completed a series of five interactive task types using each of three different input methods (trackball, touchscreen, stencil-touchscreen) at each of three different vibration levels (none, low, and high). Trackball performance serves as a baseline comparator, representing current industry practice for cockpit interaction with CDSs.

Turbulence was simulated using a motion platform.

Participants wore a safety harness connected to the ceiling and sat on a seat mounted on the platform. The harness did not encumber arm movement, and was present only to prevent falling. The display, also mounted on the motion platform, was placed approximately 45cm in front of the participant's chest, with the top edge of the display ~25cm below the user's eye-line. This arrangement facilitates easy touch interaction without interfering with the pilot's view from the cockpit windows [START_REF] Barbé | Physical ergonomics approach for touch screen interaction in an aircraft cockpit[END_REF].

The objectives of the study were to compare and examine users' performance (task accuracy and error rates) with the three devices for various activities during vibration.

Task Types

Five different types of experimental tasks were used with each input device at each vibration level. These five tasktypes were as follows.

Target selection -participants tapped/clicked on targets as quickly and accurately as possible. There were three movement amplitudes (96, 256 and 512 px) and two target sizes (32 and 64 px). Figure 3a shows the arrangement of targets, with the annotated braces showing the target sequence -successive movement directions were N, E, S, N, W, S for each of small, medium and long distances. Each target was highlighted green, reverting to blue when successfully tapped. Each successful tap advanced to the next target, highlighting it green. Any error caused the background to turn red for 500 ms.

Keypad tasks involved entering and confirming each of four different three-digit numerical values. Each target value was cued at the top of the window (Figure 3b). Once the three digits were entered, the participant pressed "OK" to confirm the value. Each button in the keypad was 104 px wide. Keypad interaction is a common activity for pilots, such as entering radio frequencies or waypoint coordinates.

Feedback for each keypress was shown immediately below the target value. Incorrect entries (on pressing 'OK') resulted in the cue and entered numbers highlighting red for 500 ms; users then had to correct their error (using the 'Del' key to delete erroneous digits) and correct their entry. Correct values were highlighted green for 500 ms, then the next target value was shown. Each three-digit target value comprised two digits from the edge of the keypad (1, 2, 3, 4, 6, 7, 9) and the digit '5' (centre of the keypad).

Slider tasks involved setting a slider to a target value (Figure 3c). This task emulates a CDS version of setting a physical lever to a target value. The target value was shown above the slider, and a slider permitted selecting values between 0 and a maximum value shown at the right of the slider (see Figure 3c). Lifting from the display, or releasing the trackball button, completed the selection. Correct selections were confirmed by green highlighting for 500 ms, then the next trial began automatically. Incorrect selections caused red highlighting for 500 ms, and the trial continued.

The slider length was 512 px. The number of candidate items within the slider varied across trials to control the precision required. The full set of trials comprised three repetitions at each of the following resolutions: 2, 8, 32, 128 candidate items in the slider. For each trial, the target value was randomly selected between 1 and the maximum value; and the slider was initially set to 0 for each trial.

Dial tasks involved setting a dial needle to a highlighted target in the dial (see Figure 3d). The needle could be set to a location either by tapping/clicking in the target area (suitable when the target segment was large) or by dragging the needle with the finger/cursor. Software checked the correspondence between the needle and intended target when the finger was lifted from the display (or trackball button released). If correct, the dial flashed green for 500 ms and the next target segment was highlighted, with the needle initially at due north. If incorrect, the dial flashed red for 500 ms and the trial continued. The resolution of task precision was controlled across trials by varying the number of candidate items in the dial. The full set of trials consisted of two repetitions at each of the following resolutions: 4, 8, 32, 128. Pan-zoom tasks required moving a yellow highlighted target circle into the view and manipulating scale until the circle was sufficiently large to become selectable (it turned green). The task emulates the activity of selecting destinations or waypoints on a map. The display contained two circles labelled 'left' (on the left) and 'right' (horizontally right of the left target). The first target (left) was displayed at a scale factor of 0.5. Touchscreen and stencil users zoomed into the target through multitouch manipulation of the surface ('pinch to zoom' or its bimanual equivalent). Single finger contacts panned the display at its current scale factor.

Trackball users panned by dragging the ball with the button depressed, and zoomed by rotating the scrollwheel, with the cursor location serving as the scale orientation point.

The target turned green and became selectable at scale factor 1.3 or greater. Having selected the left target by tapping/clicking on it, it turned grey and the right-hand target was highlighted. Users then zoomed out to view the next target, and combined pan and zoom actions to bring it to a selectable scale factor. Software implemented maximum and minimum scale factors of 3.0 and 0.04. Software prevented panning to illogical regions, such as moving the left target beyond the right display edge. The full set of trials consisted of four target selections, beginning with an initial selection of 'Left' (data discarded, as unlike other trials it began with the target in view).

Procedure

On arrival, participants were briefly shown the motion platform, harness, and apparatus. They were informed that the objective of the study was to compare the effectiveness of different means for input in turbulent environments.

Having put on the harness, they sat on the platform seat, and the ceiling-mounted securing cables were adjusted.

For familiarization, participants completed a training set of trials consisting of all input methods and task type. The motion platform was turned off during this familiarization.

Participants then advanced to the main experimental tasks, which progressed according to the following algorithm:

for each vibration ∈ {none, low, high}: for each input device ∈ {trackball, touch, stencil} for each task ∈ {targets, keypad, slider, dial, panzoom} complete task trials complete subjective experience questionnaire Participants could orally request that any trial be abandoned.

The experimenter then pressed a control key to advance to the next task, and the abandonment was logged.

Order of vibration level and input device was counterbalanced. Tasks types were always completed in the order targets, keypad, slider, dial, panzoom.

Having completed all tasks with all three input devices at one vibration level, participants completed a subjective experience questionnaire, recording NASA TLX measures for physical workload and frustration, subjective preferences, and other comments. While the participant completed these worksheets, the experimenter loaded the next vibration profile into the motion platform.

Apparatus

Software for the experiment was written in Python 3.4, using the kivy package for multitouch interaction in the pan-zoom tasks. Software ran on a Dell S2240T 21.5 monitor running at 1920  1080 pixel resolution (4.03 px/mm). The monitor and participant's seat were mounted on a Mikrolar R-3000 hexapod motion platform. The experimenter sat alongside the motion platform, configuring changes to motion profile and input device when instructed by the software.

To avoid continually switching between different stencils for the five task-types, a single stencil was used for all task types. Each task type therefore inhabited a different region of the display -target tapping was top-left; keypad top-right; dial top-centre; slider bottom-centre (the transparent stencil overlay is visible in Figure 1). During pan-zoom tasks with the stencil, participants were instructed to use the dial hole for controlling interaction.

The motion pattern sent to the hexapod platform was designed to induce non-periodic vertical displacements with a mean motion frequency of 3.1 Hz (max 5 Hz). Movement amplitudes were configured to produce weighted RMS accelerations in the 'very uncomfortable' range (ISO2631-1, 1.25 < aT < 2.5 m/s 2 ) for the high vibration condition and 'uncomfortable' for low vibration (0.8 < a T < 1.6 m/s 2 ). Accelerometer sampling on the seatpan confirmed x, y, z and total (a T ) acceleration values of 0.15, 0.05, 1.08, and 1.10 m/s 2 respectively for the low vibration condition, and 0.72, 0.07, 2.11 and 2.15 m/s 2 for the high vibration condition. These frequency and acceleration levels are much higher than previous studies of ship motion (e.g., [START_REF] Lin | The performance of computer input devices in a vibration environment[END_REF][START_REF] Yau | Effects of Input Device and Motion Type on a Cursor-Positioning Task[END_REF]).

Trackball input was provided through a Logitech M570 device, which includes a scrollwheel that was used for controlling zoom-level in the pan-zoom tasks.

Participants

Eighteen volunteer participants were recruited for the study -11 female, aged 19-40 (mean 27). All were familiar with touchscreen input methods, reporting a mean daily use of ~2 hours. Participation in the experiment lasted for 1 hour, and was compensated with a $10 payment.

Design

Data from each task type was separately analyzed for dependent measures trial time and error rate.

Target selection data was analyzed using a 3×3×3×2 repeated-measures analysis of variance (RM-ANOVA) for factors vibration (none, low, high), device (touch, stencil, trackball), distance (96, 256, 512 px) and size (32, 64 px). Keypad and Pan-zoom data was analyzed using a 3×3 RM-ANOVA for factors vibration and device. Dial and Slider data was analyzed using a 3×3×4 RM-ANOVA for factors vibration, device, and resolution (4, 8, 32, 128 items for Dial; 2, 8, 32, and 128 items for Slider).

RESULTS

The 18 participants produced data for a total of 7244 successful trials across the five task types. Only 5 trials were abandoned (0.07%), all in high vibration.

Target Selection Tasks

Selection time analysis RM-ANOVA showed a significant effect of vibration level on error free selection time (Figure 4): F 1.33,22.5 = 45.1, p < .001, η 2 G = 0.16; Greenhouse-Geiser correction applied for sphericity violation. Mean selection times more than doubled as vibration increased from none (mean 1163 ms, sd 677), through low (1648 ms, sd 973), to high vibration (2360 ms, sd 1798). Distance had a significant effect on selection time (F 2,34 = 4.3, p < .05), but the magnitude of its effect was small (η 2 G = 0.009), with mean times increasing through 1610, 1684 and 1860 ms for 96, 256 and 512 px distances. Distance was not a factor in any significant interactions.

Target size had a stronger and significant effect on selection time: F 1,17 = 139.7, p < .001, η 2 G = 0.12. Mean selection times for small (32 px) and large targets (64 px) were 2137 and 1299 ms, respectively.

There was no significant main effect of device: F 2,34 = 1.15, p = .33. Mean selection times were similar with stencil and touch, at 1648 and 1686 ms respectively, and slightly higher for trackball (1820 ms). However, there was a significant device × size interaction (F 2,34 = 3.46, p < .05). This effect was largely caused by the relatively similar mean selection times for the three devices with small targets (2089, 2193, and 2128 ms for stencil, touch, and trackball respectively), whereas with large targets, the trackball was slower (mean 1510ms) than stencil and touch (1207 and 1180 ms).

There was also a significant three way interaction of device × size × vibration: F 4,68 = 3.27, p < .05. For small targets (Figure 4, left), when there was no vibration, mean selection times with stencil were slower than touch, but as vibration increased stencil selections became faster than touch. However, for larger targets (as shown in Figure 4, right), the benefits of the stencil over touch at high vibrations did not emerge; presumably because the targets were large enough to be easily acquired without a stabilizing edge. Also, the benefits of self-stabilization with the trackball become apparent at high levels of vibration, particularly with small targets. In summary, there is cross-over effect for small targets (that is largely absent with large targets)unstabilized touch is fastest when vibration is absent, but it becomes slowest when vibration is present.

Error analysis

Two different interaction events could be interpreted as errors: 1. the user misses the target, instead hitting 'dead space' between active elements in the interface; 2. the user hits an incorrect target. Our error analysis focuses on incorrect target selections for two reasons. First, the stencil essentially eliminates the possibility for hitting interface dead-space because the holes are only present over active elements. Second, interactions on dead-space are likely to have lower consequences than incorrect selections. Incorrectly or accidentally hitting a target could have serious flight consequences, such as unknowingly changing a mode, whereas hitting dead-space has no effect.

Across all trials, an incorrect target was selected in 2.9% of trials. There were zero incorrect target selections with the trackball, even at high vibration.

There were a total of 45 errors with the stencil (rate 4.67%) and 40 with touch. Figure 5 summarizes error rate across conditions. One surprising observation from this analysis was that stencil errors were higher than touch.

Timing analysis showed the cause of this anomaly. It revealed that a large proportion of stencil errors were caused by a second selection of the target shortly after its correct selection. Figure 6 shows the cumulative proportion of stencil and touch errors occurring within 500 ms of a correct selection -64% of stencil errors and 28% of touch errors occurred within 200 ms of correctly hitting a target. These errors occur during the 'lift off' action, and can be attributed to vibration causing accidental activation during 'lift off'. Dragging the finger up the stencil edge during lift-off appears to increase the incidence of these errors.

These stencil accuracy problems are easily remediated by discarding contacts that occur within 200 ms of a previous contact. Discarding these 'bounce' contacts within 200 ms of a previous touch reduces the overall mean error rate for the stencil to 1.6% (and 2.8% for touch); and for small targets at high vibration, the stencil error rate was 2.5%, compared to 10.4% for touch.

To confirm that stencil errors can be eliminated by addressing 'bounce' errors on lift off, we conducted a small subsequent study of stencil and touch Target Selection with four participants at high vibration. We used the same stencil as the original study, but increased the target sizes to 54 (small) and 96 px (large) to make them easier to acquire with touch (subjective comments, reported later, indicated touch was too hard at high vibration). Each participant completed four repetitions of the same target pattern used in the original experiment, producing data for 576 trials (4 participants, 2 devices (stencil and touch), 4 repetitions, 18 targets). With the 200 ms delay between contacts, there were 11 incorrect target selections with stencil (3.8%), compared to 8 with touch (2.8%). If all touch contacts off the intended target were interpreted as errors (including all wrong target and dead-space selections), the total number of errors with stencil was 31 (mean 0.11 errors/selection) compared to more than six times as many with touch (198, 0.69 errors/selection).

Keypad Tasks

RM-ANOVA of keypad tasks showed significant effects of device (F 2,34 = 38.8, p < .001) and vibration (F 2,34 = 20.0, p < .001) on the mean time to enter and confirm a three-digit target number. There was no significant device × vibration interaction (p = .53). As shown in Figure 7, mean performance was very similar between stencil and touch (3398 and 3371 ms, respectively), and trackball was substantially slower (5166 ms). The similarity of task time for stencil and touch suggests that users were able to confidently hit the large key targets without relying on the stencil edge, even when vibration was high.

Error rates were low. An incorrect number was submitted (by pressing the 'OK' button) 12 times from 542 tasks (2%): 6 times with touch, 4 times with trackball, and twice with stencil. The 'DEL' key was used to correct an incorrectly entered digit 48 times (from 1626 required digit entries) -24 with touch, 4 with trackball, and 20 with stencil.

Slider Tasks

The performance of slider tasks was dominated by errors. Errors were high in all conditions, but particularly so for precise selections in high vibration settings. Figure 8 shows the mean number of errors per trial, which ranged from 0.019 for trackball selections with 2 candidate items and low vibration, through to a maximum of 1.97 for touch selections with 128 candidate items and high vibration. There were no significant effects on task time. Mean selection times were 1.9, 1.8 and 1.8 s with stencil, touch and trackball.

Even when there was no vibration, all of the devices produced high error rates (means of 0.37, 0.54 and 0.7 for touch, trackball, and stencil). The particularly high error rate for stencil (~1.5 per trial) with 128 items and no vibration may be due the finger contact forming unanticipated shapes (triggering unintended movement) during lift-off due to pressure against the stencil edge.

At high vibrations, the error rates were extremely high (overall mean of 0.9 errors per trial, ranging from 0.6 for trackball to 1.12 with touch). One factor contributing to these high error rates is the mechanical difficulty of maintaining contact pressure on a touch-surface (or trackball button) to continue the dragging state while undergoing vibration. The target selection and keypad tasks allowed users to make discrete 'stabs' at the surface or button, reducing the temporal window for vibrations to influence interaction. Dragging actions, in contrast, necessarily have a prolonged duration, increasing the opportunity for vibration to adversely affect execution.

Setting values via sliders therefore seems to be an inadvisable interaction for a cockpit environment, where accuracy and error-free interaction is required.

Dial Tasks

Like slider tasks, dial tasks required maintaining a dragging state while manipulating the location of the arrow. Any unintended lift-off caused an error. In light of the slider results, it is therefore unsurprising that dial task performance was also dominated by errors, with an overall mean of 0.24 errors per trial. There were no significant effects on task time (means of 3.8, 3.5 and 3.4 s for stencil, touch and trackball).

Mean errors per trial were lower with the trackball (0.06) than stencil (0.34) or touch (0.32). At high vibrations with 128 candidate items, all devices had high error rates: 0.33 with trackball, 1.89 with stencil, and 1.69 with touch.

Pan-Zoom Tasks

Pan-zoom tasks are interesting because completing them on a touchscreen requires manipulating dragging actions (i.e., pinch to zoom and pan). However, unlike slider and dial tasks, lift-off actions are anticipated as part of successful task completion -users are expected to repeatedly clutch their pinching/depinching gestures to attain the required zoom level. Also, completing zoom actions with the trackball did not require maintaining a dragging state because zooming could be directly controlled by rotating the scrollwheel. 2) and high (11.9 s, 5.0). There was no device × vibration interaction (p = .3).

Subjective Responses

Analysis of subjective responses reinforce the objective data reported above. We used Friedman tests to analyze participants' ratings of physical workload and frustration (reported on 7-point Likert scales) for target selection tasks across vibration and device. As shown in Figure 10 (left), Participants' comments further emphasized that unstabilized touch interaction was difficult, whereas the trackball and stencil were easier to control in high vibration:

S1: "Stencil help to control where I want to point out on the screen … touchscreen is hardest to control especially for the small object." S6: "Stencil was best because its borders offered a control or support for the tasks that were difficult under vibration (e.g., the tapping). The touchscreen was worst because even a little vibration threw me off." S12: "The touchscreen was impossible in the 'tapping' run. My arm was out of my control … The stencil made it hard to mess up." S13 "trackball [best for high vibration] because it was easiest, least jerky (i.e., most stable), most precise". S18 "I found the stencil helps a lot when I want to press exact places. It's a combination of benefits of using touchscreen (handy) & improves the exactness."

Several participants commented that when vibration was absent, touch interaction was the easiest input method and that the trackball was slow and awkward. S1: "trackball takes time to move the pointer to where I want" S13 "trackball was slowest and least user-friendly."

One participant commented that the stencil was distracting:

S17 "the limitations in space [with the stencil] made it more difficult to concentrate on the task. It's like doing more things at once."

Participant Strategies

Participants were not instructed on how best to use each of the input devices, nor on how to stabilize their input. Although we had anticipated that participants would rest their fingers, palm, or hand on the stencil surface to stabilize their input, only one participant did so (S11). In hindsight, this is understandable -the Lexan sheet minimally influenced the appearance of widgets on the display, and consequently, participants were likely resistant to making contact with the sheet to avoid accidental activations. Explicit training that the Lexan sheet would prohibit contact registration may have influenced participants' strategies and improved performance with the stencil. Most participants quickly learned that when vibration was present they needed to stabilize their hand and finger in order to complete tasks with touch and stencil. Typically they did so by placing their fingers or thumb on the screen edge (off the contact surface) and spanning their hand to make contact with the target on the display. Target selection tasks were located on the left side of the display (see the holes in Figure 1). Most users therefore placed the thumb of their right hand on the left or top display edge, and spanned to the targets with their index or middle finger. The same strategy was used for both touch and stencil.

Slider tasks were located near the bottom of the display (Figure 3). This placement allowed most participants to grasp the screen edge with their thumb and span their index finger to the slider widget. Keypad tasks were located near the right edge of the display, and some participants grasped the right edge of the display with the fingers of their right hand and spanned their thumb to the targets. However, keypad targets were sufficiently large for most participants to complete selections without stabilization.

DISCUSSION

Seven key findings emerge from the results:

1. There was a wide variance in the different tasks' overall tolerance to vibration. Tasks with low pointing precision requirements (e.g., Keypad) could be accomplished successfully with any device under all vibration conditions; others (particularly Slider) were difficult in all cases (even without vibration). 2. When pointing precision requirements were low, touchbased interactions (stencil or touch) were generally faster than trackball input. This includes the selection of large targets (e.g., the 104 px wide keys in the keypad task) and coarse pan-zoom actions, even at high vibration levels. 3. However, when selecting small targets in a vibrating environment, trackball input was faster than unstabilised touch-based input. 4. Trackball input was accurate, even at high vibrations. 5. Drag-based selections, in which lift-off/release confirmed selection, were inaccurate with touch and trackball. Errors were high when there was no vibration, and became extremely high when vibration was present. 6. The stencil was generally unsuccessful. The device × size × vibration interaction in target selection tasks (Figure 4) suggests that the stencil may improve touch interaction with small targets at high vibrations, but further work is needed to confirm this effect. Also, the stencil relies on a timeout after each selection (~200 ms), without which it is susceptible to finger lift-off errors during vibration. Froehlich et al. [START_REF] Froehlich | Barrier pointing: using physical edges to assist target acquisition on mobile device touch screens[END_REF] used a similar timeout method to eliminate 'bounce' errors by users with tremor. 7. Multi-touch pan and zoom selections were much faster than equivalent interactions with a trackball (panning and scrollwheel zooming) at all vibration levels.

Touch vibration tolerance

Prior to conducting the study, we suspected that participants would have extreme difficulty in completing tasks in the touch condition at high vibration. However, results showed that participants were resilient, maintaining their ability to interact by stabilizing their hand on the screen edge.

Our experimental conditions facilitated this hand stabilization because the smallest targets (and therefore hardest to acquire) were located near the screen edges. Based on our observations of strong reliance on hand stabilization, we suspect that performance with touch would have been much slower and more error prone if the target widgets had been placed nearer the center of the display. Central placement would reduce the availability (or effectiveness) of the hand-spanning strategy we observed, with the fingers or thumb placed on the screen edge.

However, flight displays could easily be constructed to facilitate hand stabilization. Each display could be small, or on a large display, touch-inactive horizontal and vertical bezel ridges for finger/thumb placement could be overlaid on the display, with the distance between the ridges configured to permit easy hand-spanning to targets. There are also interesting opportunities for augmenting touchscreens with graspable or tactile widgets that could be positioned on or near the display (e.g., [START_REF] Harrison | Providing dynamically changeable physical buttons on a visual display[END_REF][START_REF] Jansen | Tangible remote controllers for wall-size displays[END_REF][START_REF] Miruchna | Localized Tactile Feedback Through Thin, Programmable Gel[END_REF][START_REF] Tuddenham | Graspables revisited: multi-touch vs. tangible input for tabletop displays in acquisition and manipulation tasks[END_REF][START_REF] Ullmer | structures for composing tangibles and complementary interactors for use across diverse systems[END_REF]).

Lessons from the stencil

Results from the stencil are interesting, suggesting both strengths that can be deployed, and weaknesses that should be avoided. In terms of strengths, several participants referred to the intended benefits of stabilization. Also, while raw wrong-target errors were increased by the stencil, timing analysis indicated that once lift-off errors were eliminated (by disabling touch for 200 ms following a selection), the stencil had similar wrong-target error rates as unstabilised touch, and dramatically fewer off-target errors.

However, participants seemed reluctant to place their fingers or hand onto the stencil surface, possibly because they mistakenly believed that their contacts would be registered there. Additionally, at least one participant referred to visual 'distraction' caused by the overlay. For these reasons, it seems that the extra bezel ridges described above, may be preferable to the transparent overlay examined in our study.

Study limitations and further work

To our knowledge, this is the first study of touch interaction in cockpits that covers a range of important task types and different vibration levels, as well as examining a potential remediation technique. As touchscreens are increasingly used in aircraft, there is need for further research on the problems of turbulent input and their remediation.

Our study used a limited range of vibration profiles, partially due to the maximum displacement (150 mm) attainable by our motion platform. Further studies could examine different vibrations, such as higher amplitudes and varied frequencies.

Another obvious area for further study concerns the impact of target distribution on performance with the various devices (e.g., sparsely versus densely packed targets).

Participants in our study were predominantly university students. While this demographic is familiar with touchscreen interaction, none were pilots, and they were not explicitly trained on how to best exploit the evaluated technologies. There are therefore opportunities to study whether comparative user performance with stencils and trackballs changes when users are well trained. Similarly, testing with real pilots and real cockpit display systems will help validate findings for commercial use. Also, our trackball input used default control-display gain functions provided by the device manufacturer, which are likely to differ from the proprietary functions used by aircraft manufacturers.

Finally, we plan to test these results in other settings where touchscreens are being introduced into environments that can experience turbulence. For example, touchscreens are becoming common in cars and motorbikes -and here, the added concern of visual attention becomes an important issue. In further studies we will test whether the tactile feedback provided by the stencil reduces the amount of time users must attend to the display during vibration episodes.

CONCLUSION

Touchscreen interaction in the cockpits of commercial aircraft offers potential advantages to aircraft manufacturers, airlines, and pilots. However, turbulence is a challenge for their deployment. Our studies of touch interaction in simulated turbulence indicate that users can successfully stabilize discrete touch input (i.e., target selections) by touching the side of the screen with their fingers or thumbs.

For small targets, a stencil overlay assisted users in hitting targets. Drag-based selections that confirmed selection on lift-off/release were highly error prone during simulated vibration, both with touch and trackball. However, multitouch pan-and-zoom selections were much faster with touch than trackball.

Figure 1 :

 1 Figure 1: Experimental set up. Participants sat on a motion platform and made selections using a touchscreen or trackball. A transparent stencil (shown) was overlaid on the touchscreen.

Figure 2 :

 2 Figure 2: Candidate stencil edge profiles, from most (left) to least (right) comfortable (and least to most stable).

Figure 3 :

 3 Figure 3: Target Selection (left), Keypad (right), Slider (bottom) and Dial (top) tasks shown in their approximately location within the experimental display. The Target Selection figure is augmented with braces showing the target sequence.

Figure 4 .

 4 Figure 4. Error-free selection times for small (left) and large (right) targets by device and vibration. Error bars ±1 s.e.m.

Figure 5 .

 5 Figure 5. Error rates for stencil and touch for differently sized targets at each vibration level. Error bars show ±1 s.e.m.

Figure 6 .

 6 Figure 6. Cumulative proportion of errors with stencil and touch across time since correct selection.

Figure 7 .Figure 8 .

 78 Figure 7. Mean time (± 1 s.e.) to enter a 3-digit number. 0 1000 2000 3000 4000 5000 6000 7000 None Low High Mean number entry time (ms) Stencil Touch Track

Figure 9 .

 9 Figure 9. Mean selection times in pan-zoom trials for each input device at each vibration level. Error bars show ±1 s.e.m.

Figure 10 .

 10 Figure 10. Mean responses (±1 s.e.) for physical workload and frustration by vibration (7 point Likert scales, 1 low, 7 high).

  ' ratings of physical workload increased significantly as vibration increased, from a mean of 2.35 at no vibration, through 3.83 at low vibration, to 4.56 at high vibration -2 = 39.8, p < .001. There was no significant effect of device on physical workload (p = .09). Frustration ratings (Figure10, right) showed significant effects of vibration ( 2 = 34.7, p < .001) and device ( 2 = 16.1, p < .001) with touch having the highest overall frustration rating (4.0), followed by stencil (3.2) and track (2.9).

	Mean error count per selection	2 2.5	Stencil	Touch	Track
		2	8 32 128 2	8 32 128 2	8 32 128
			None	Low		High

participants


L Tusell

A Legarra

M García-Tomás

O Rafel

J Ramon

M Piles

Different ways to model biological relationships between fertility and pH of the semen in rabbits 1

Keywords: (co)variance component, fertility, pH of semen, rabbit, recursive model

This work aimed to study the relationship between pH of the semen and fertility (Fert, defined as the success or failure of conception), which is of special interest because pH of the semen can be considered a global marker of the expression of some seminal quality traits. Different methods used to model the relationship between Fert and pH are presented here: 1) ignoring genetic and environmental correlations and including pH either as a covariate or as a cross-classified effect on fertility, 2) a bivariate mixed model, and 3) recursive bivariate mixed models. A total of 653 pH records and 6,365 Fert records after AI were used. Crossbreed does from 2 maternal lines were artificially inseminated with buck semen from a paternal line in a commercial environment. A negative, and almost linear, effect of pH on Fert was detected. The posterior median of pH and Fert heritabilities, and the highest posterior density interval at 95% (in parentheses) were approximately 0.18 (0.05, 0.29) and approximately 0.10 (0.02, 0.20) across all the models, respectively. Genet-ic correlations between traits were negative, but the highest posterior density interval at 95% included zero [i.e., -0.31 (-0.91, 0.33) in the bivariate mixed model and -0.17 (-0.99, 0.48) and -0.44 (-0.99, 0.10) in the recursive bivariate mixed models including pH as a covariate or as a cross-classified effect, respectively]. All models predicted Fert data reasonably well (i.e., 76 and 62% correct predictions for success and failure, respectively). No differences in the prediction of the EBV for male fertility were encountered between models, showing a good concordance in the animals ranked by their EBV (the correlation between EBV in all models was close to 1). Thus, no differences in results were obtained considering, or not considering, genetic and environmental correlations between pH and Fert and assuming, or not assuming, recursiveness between each trait. This is because the magnitude of the effect of pH on Fert was not large enough; therefore, the same results were obtained even though the models were of different complexity.

INTRODUCTION

Male fertility (Fert) is one of the most economically interesting traits in rabbit breeding, especially with the use of AI [START_REF] Alvariño | Reproductive performance of male rabbits[END_REF]. However, because of the low heritability of this trait in both natural and AI mating systems [START_REF] Piles | Genetic parameters of fertility in two lines of rabbits with different reproductive potential[END_REF][START_REF] Tusell | Genotype × AI conditions interaction for male effect on fertility and prolificacy[END_REF], finding seminal quality traits to be used as criteria for indirect selection to increase male Fert has been paramount [START_REF] More O'ferrall | Relationship between pH, other semen traits and fertility in rabbits[END_REF][START_REF] Bencheikh | The effect of frequency of ejaculation on semen characteristics and sperm output in the rabbit[END_REF][START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF][START_REF] Lavara | Do parameters of seminal quality correlate with the results of on-farm inseminations in rabbits?[END_REF]. The hydrogen-ion concentration in semen (pH) is a by-product of sperm metabolism associated with the number and activity of spermatozoids. Thus, it can be considered an indicator of seminal quality traits [START_REF] More O'ferrall | Relationship between pH, other semen traits and fertility in rabbits[END_REF][START_REF] Bencheikh | The effect of frequency of ejaculation on semen characteristics and sperm output in the rabbit[END_REF]; therefore, it is of special interest to determine its relationship to Fert.

Fertility and pH have a complex biological relationship. The pH of semen could affect the phenotypical expression of Fert but, contrary to some environmental effects (e.g., type of extender, AI technician, farm), pH is not an external effect of the animal, so, in turn, pH could also have genetic and permanent effects. Recursive models can accommodate this kind of biological relationship.

A recursive multitrait model is a particular case of structural equation models, which [START_REF] Gianola | Quantitative genetic models for describing simultaneous and recursive relationships between phenotypes[END_REF] applied in a quantitative genetic context. These models are useful in describing biological relationships between traits that have simultaneity or recursiveness between their phenotypes, leading to a better interpretation of results. [START_REF] Gianola | Quantitative genetic models for describing simultaneous and recursive relationships between phenotypes[END_REF] also pointed out that, in the presence of feedback and recursiveness, biased (co)variance estimates can be obtained if these relationships are not taken into account. Since then, several authors have been using these models to describe biological relationships between several traits in livestock species (de los [START_REF] De Los Campos | A structural equation model for describing relationships between somatic cell score and milk yield in dairy goats[END_REF][START_REF] López De Maturana | Analysis of fertility and dystocia in holsteins using recursive models to handle censored and categorical data[END_REF][START_REF] Varona | Analysis of litter size and average litter weight in pigs using a recursive model[END_REF][START_REF] Wu | Exploration of lagged relationships between mastitis and milk yield in dairy cows using a Bayesian structural equation Gaussian-threshold model[END_REF].

In this work, we studied the relationship between semen pH and male Fert by using different models: 1) ignoring genetic and environmental correlations between each trait, 2) a classical multitrait model, and 3) recursive multitrait models. Models were compared according to their ability to predict Fert and the across-model EBV correlations. Ratios for genetic and environmental sources of variation were also estimated for both pH and Fert.

MATERIALS AND METHODS

The research protocol was approved by the animal care and use committee of the Institut de Recerca i Tecnologia Agroalimentàries.

Animals and Experimental Design

Bucks came from the Caldes line, selected for growth rate during the fattening period [START_REF] Gómez | The Caldes strain (Spain). Options Méditerr[END_REF]. They were bred and reared on an experimental farm from the Institut de Recerca i Tecnologia Agroalimentàries in Caldes de Montbui (Barcelona, Spain). After weaning at 32 d, males were housed in cages of 8 individuals with a photoperiod of 16 h of light/d. Animals were fed a commercial diet ad libitum (15.5% CP, 2.3% fat, 17.2% fiber; DM basis) until d 60. After this period, they were individually housed and feed was restricted to 180 g/d of another commercial diet (16% CP, 4.3% fat, 17% fiber; DM basis). Fresh water was always available. All males began the training to an artificial vagina at 4.5 mo of age. One ejaculate per male and per week was collected for 2 wk. Their reproductive life began at 5 mo. At this age, 2 ejaculates per male and per week were collected, with an interval of 30 min between collections. From 5 to 9 mo of age, all males were evaluated at 3 different times for seminal quality traits and Fert score after AI of crossbred females on a commercial farm.

All semen evaluations and preparations of the AI doses were performed in a laboratory located beside the experimental farm of the Caldes line. Ejaculates were stored in a dry bath at 35°C until evaluation, for no more than 15 min after collection. Ejaculates containing urine and calcium carbonate deposits were discarded, and gel plugs were removed. Individual motility of the ejaculate was measured in aliquots (25 μL) under a light microscope (Nikon, Lewisville, TX) at 400× magnification according to a subjective scale from 0 to 5 (0, 1, 2, 3, 4, or 5 = 0 to 10, <10 to 25, <25 to 50, <50 to 70, <70 to 90, or <90 to 100, respectively, of the motile spermatozoa showing progressive movement). A small preselection of ejaculates was performed, discarding for AI only those with individual motility scores less than 2 and a percentage of dead spermatozoa greater than 50%.

Semen pH was determined using a 507 Crison pH meter (Crison Instruments, SA, Barcelona, Spain). Preselecting good-quality ejaculates could have biased the sample. However, most of the rejections (60.5%) were due to the presence of urine or calcium carbonate deposits from the bladder, which are not part of a normal ejaculate and could also affect the pH measurement. Semen was immediately prediluted 1:1 with a commercial extender (Cunigel, IMV Technologies France, L'Aigle, France). The semen from each buck obtained on the same day was pooled, and cell sperm concentration was measured by using a sperm cell counter (NucleoCounter SP-100, ChemoMetec A/S, Allerød, Denmark). The resultant pool was divided into 2 parts, which were diluted to 10 × 10 6 and 40 × 10 6 spermatozoa/mL, respectively, to obtain AI doses at 2 different sperm concentrations. Semen doses were stored in straws of 0.5 mL at 18°C for 24 h until their use.

Artificial insemination was performed on crossbred does [P × V; V line: [START_REF] Estany | Mixed model methodology for the estimation of genetic response to selection in litter size of rabbits[END_REF]; P (Prat) line: [START_REF] Gómez | A genetic study of a line selected on litter size at weaning[END_REF] on a commercial farm. Females followed a semi-intensive reproductive rhythm: first mating at about 4.5 mo of age, with subsequent 42-d reproductive cycles. All females were treated 48 h before AI with 15 IU of eCG (subcutaneously; Foligon, Intervet International B.V., Booxmeer, the Netherlands), and ovulation was induced immediately after AI with 0.02 mg of gonadorelin (intermuscularly; Fertagyl, Intervet International B.V., Booxmeer, the Netherlands).

Data

Diagnosis of pregnancy was made by palpation, 14 d after AI, confirming the result at parity. The assigned Fert score was 1 when the female was diagnosed as pregnant and 0 otherwise. A total of 6,613 Fert records, involving 243 males and 2,293 females, were obtained between November 2006 and July 2007. Different Fert records had the same pH measurement. From the to-tal amount of data, Fert records without a corresponding pH measurement were not included in the analysis, leading to a final figure of Fert data of 6,363.

The pH was measured separately in each ejaculate and pooled whenever 2 ejaculates per male and day were obtained, as follows:

pH Vol V ol Vol Vol pH pH = - × + × ( ) × + ( )           - - - log 10

Model and Statistical Analysis

Semen pH was assumed to be normally distributed and was analyzed jointly with Fert in a bivariate Gaussian-threshold model [START_REF] Foulley | Prediction of genetic merit from data on binary and quantitative variates Figure 2. Posterior predictive distributions of data on the mean of a fertility success [E(ŷ| y = 1)] and the mean of a fertility failure [E(ŷ| y = 0)] estimated in the following models for fertility and pH of the semen: a mixed model without genetic and environmental correlations including pH as a covariate in the model of fertility (model UMM cov ), a bivariate mixed model (BMM 0 ), and a recursive mixed model including pH as a cross-classified effect in the model of fertility (RMM cross ). with an application to calving difficulty, birth weight and pelvic opening[END_REF]. In a threshold model, the observed Fert y Fert is considered the expression of an underlying continuous variable l, often called the liability [START_REF] Falconer | The inheritance of liability to certain diseases, estimated from the incidence among relatives[END_REF], which is rendered discrete by a fixed threshold that divides the observed response into 2 categories [START_REF] Wright | An analysis of variability in number of digits in an inbred strain of guinea pigs[END_REF]: the failure or the success of conception. The probability that an observed Fert data value (y i,Fert ) falls into 1 of these 2 categories given the liability is

p y l p l I y p l I y i Fert i i i Fert i i Fert ( | ) ( ) ( ) ( ) ( ). , , , = > = + ≤ = 0 1 0 0
The threshold being fixed at 0, I(.) is an indicator function that takes the value of 1 or 0.

A bivariate recursive model postulates that trait 1 has an effect on trait 2, but that trait 2 has no effect on trait 1. In the case of recursiveness (but not for simultaneity), the model presented by [START_REF] Gianola | Quantitative genetic models for describing simultaneous and recursive relationships between phenotypes[END_REF] can be expressed as a classical multitrait model in which trait 1 is included as a systematic effect in the model for trait 2 (López de [START_REF] López De Maturana | Analysis of fertility and dystocia in holsteins using recursive models to handle censored and categorical data[END_REF]. In this work, pH is trait 1 and Fert is trait 2. Thus, in our case, the jth pair of records for an individual i had the following linear relationship:

y l ij pH ij Fert ij pH ij Fert , , , ,               = ′ ′            x x 0 0                   +               ← b b pH Fert ij pH Fert p y 0 0 0 0 , l H H i pH i Fert i pH i Fert u u pm pm               +               + , , , ,                +               + 0 pf pmd pmd m Fert ij pH ij Fert , , ,                +               e e ij pH ij Fert , ,
.

[1]

The coefficient λ Fert←pH denotes the phenotypical rate of change of the liability of Fert with respect to pH (i.e., it models the recursive effect of pH on Fert). The different λ Fert←pH assumed in the models were always estimated as systematic effects in Eq. [1]. The terms y ij,pH and l ij,Fert were the jth data value for the observed pH and the unobserved liability for male i. The term β pH was the vector of systematic effects affecting pH, including day of ejaculate collection (27 levels) and male age (4 levels at about 1 mo intervals from 5 to 9 mo old). The term β Fert was the vector of systematic effects for Fert, including concentration of the AI dose (2 levels: 10 or 40 million spermatozoa/mL), physiological status of the female (3 levels: nulliparous does, multiparous does in lactation, and multiparous does not in lactation at AI), a combined effect between day and inseminator (19 levels at 14-d intervals between November 2006 andJuly 2007), and a combined effect between the age of the male and the building (9 levels, about 1 mo intervals from 5 to 9 mo of age combined with the 2 buildings on the commercial farm where AI was performed). The term x ij,k (k = pH, Fert) was the corresponding row of an incidence matrix relating data with systematic effects. The term u i,k was the male genetic additive effect, pm i,k was the male permanent environmental effect, pf m,Fert was the mth female effect for the Fert trait, pmd ij,k was the permanent environmental effect resulting from the combination between male and day of AI, and e ij,k was the random residual effect. Note that for pH, pmd ij,pH is also a residual component, and it can be separated from the residual only because it is related to pmd ij,Fert of the Fert model. This residual decomposition increases the data connectivity and permits the estimation of a possible environmental correlation between the traits.

A Bayesian approach was adopted for inference. Note that Ω = {β, λ Fert←pH , u m , p m , p f , p md , G, P m , P f , P md , R} is the vector including all the unknown parameters in the model. The term β is the vector of systematic effects, u m is the vector of male genetic additive effects, p m is the vector of the male nonadditive genetic plus permanent environmental effects, p f is the vector of female effects, and p md is the vector of permanent environmental effects resulting from the combination between male and day on which the AI was performed. The terms G, P m , P f , and P md are the different (co) variance matrices of the corresponding random effects defined above, and R is the residual (co)variance matrix. The joint posterior distribution of all parameters was

p p p p l I y pl I y pH Fert pH i n i i i i W W , | , | l y y y ( ) ∝ ( ) × ∏ > ( ) = ( ) + ≤ ( ) = =1 0 1 0 0 0 ( ) { } × ( ) × ( ) p p l | . W W
The prior distributions for the parameters of the model were p(β)

~ k; p(u m |G) -N(0, A ⊗ G); p(p m |P m ) -N(0, I ⊗ P m ); p(p f |P f ) -N(0, I ⊗ P f ); p(p md |P md ) -N(0, I ⊗ P md ); and p(e|R) -N(0, I ⊗ R),
where k is a constant and A is the numerator relationship matrix. Bounded uniform prior distributions were assumed for β, λ Fert←pH , and the components of G, P m , P f , P md , and R. The R was a diagonal matrix with the residual variance for Fert set to 1. Data augmentation was used to deal with the missing Fert data [START_REF] Sorensen | Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics[END_REF].

Three sets of models to describe the pH and Fert relationship were used. Within each model, different types of λ Fert←pH were assumed (i.e., null, covariate, or cross-classified effect) to accommodate null, linear and nonlinear recursive effects (López de [START_REF] López De Maturana | Exploring biological relationships between calving traits in primiparous cattle with a Bayesian recursive model[END_REF] of pH on Fert. Table 1 shows the summary statistics for the different levels of λ Fert←pH as a crossclassified effect in the Fert model.

Univariate Mixed Models

In the univariate mixed models (UMM), the genetic and environmental correlations between each trait were set to zero. This implies that the phenotypic recursion is the only cause of correlation between any of the random effects (López de [START_REF] López De Maturana | Modeling relationships between calving traits: A comparison between standard and recursive mixed models[END_REF]. The effect of pH on Fert (λ Fert←pH ) was estimated as a covariate in the UMM cov model or as a cross-classified effect of 8 categories (as described in Table 1) in the UMM cross model.

Bivariate Mixed Model

In the bivariate mixed model (BMM), the 2 traits were genetically and environmentally correlated. In the BMM 0 model, the genetic and environmental relationships between each trait were accounted for by the covariances and no recursive effect was assumed (λ Fert←pH = 0).

Bivariate Recursive Mixed Models

To take into account the phenotypical influence of pH on Fert, and also the genetic and environmental relationships between each trait, 2 recursive Gaussianthreshold (mixed) models (RMM) were proposed. First, model RMM cov had λ Fert←pH as a covariate. Some identification problems appeared in this model. Because the p md effect is also a residual component in the pH model, the restriction that P md was diagonal was added to ensure likelihood identification [START_REF] Varona | Analysis of litter size and average litter weight in pigs using a recursive model[END_REF]. Second, model RMM cross had λ Fert←pH as a cross-classified effect of 8 categories (as described in Table 1). Table 2 shows the type of structural coefficient λ Fert←pH and the structure of the (co)variance component matrices used in each of the models.

The Gibbs Sampler

Procedures developed by [START_REF] Sorensen | Bayesian-inference in threshold models using Gibbs sampling[END_REF] and extensions of them, based on Markov chain Monte Carlo methods, allow the univariate and joint analysis of categorical and continuous traits. Marginal posterior distributions of the parameters of interest were approximated using the Gibbs sampler algorithm (using the TM software developed by [START_REF] Legarra | TM Threshold Model[END_REF]. Fully conditional posterior distributions of the model parameters needed for the implementation of this algorithm can be found in [START_REF] Sorensen | Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics[END_REF]. Single chains of 500,000 iterations were run for all the models, discarding the first 50,000 iterations of each chain and saving 1 of every 10 samples. The number of samples discarded in the burn-in was, in all the analyses, much larger than the value recommended by [START_REF] Raftery | How many iterations in the Gibbs sampler? Pages[END_REF] and [START_REF] Geweke | Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments[END_REF] for assessing convergence. The sampling variance of the chains was obtained by computing Monte Carlo SE [START_REF] Geyer | Practical Markov chain Monte Carlo[END_REF].

Recursive Model as an Alternative Parameterization of a Classical Bivariate Model

Following [START_REF] Varona | Analysis of litter size and average litter weight in pigs using a recursive model[END_REF], a recursive model can have an equivalent parameterization in a classical bivariate model. This equivalence is as follows:

Λ -1 H*(Λ -1 )′ = H, where L = -               ← 1 0 1 l Fert pH
is the matrix of structural coefficients, and H* and H are the different (co)variance components of the recursive and the bivariate mixed model, respectively. In our study, H corresponded to the G, P m , P f , P md , and R described above, and this equivalence was assessed between the RMM cov and BMM 0 models.

Data Prediction Ability and EBV Comparison

Posterior predictive distribution of Fert data was calculated in all the models. Thus, the ability to predict success or failure in each Fert record was averaged as follows:

E M n M Fert Fert Fert i Fert i i i n ˆ| , , , ˆ| , , , , , y y l y
y l W W ( ) = ( ) = ∑ 1 1
where ŷFert is the vector of predicted data on the observed scale, y Fert is the observed Fert data, Ω is the vector of unknown parameters, l is the vector of liabilities, M is the model used, and n is the total number of iterations. Each predicted Fert value was assumed to be correct when the absolute value of the difference between the observed Fert and the posterior mean of the posterior distribution did not differ by more than 0.25. Thus, good data prediction was achieved when the probability of correctly predicting data was ≥75%:

y Ey M Fert i F ert i Fert i , , , (ˆ| , , , ) . . - ≤ - y l W 0 25
Correlations between the posterior means of the EBV for Fert obtained in the models were calculated to evaluate possible differences in the estimation of male additive genetic effects. The average EBV of the top 10% animals was also calculated in each of the models.

RESULTS AND DISCUSSION

The average semen pH was 7.43 (SD = 0.42), well within the range of values obtained previously in the same line and in other rabbit breeds [START_REF] Bencheikh | The effect of frequency of ejaculation on semen characteristics and sperm output in the rabbit[END_REF][START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF]García-Tomás et al., 2006a;[START_REF] Brun | Paramètres génétiques des caractéristiques de la semence de lapin. 13èmes[END_REF]. The current Fert (52%) was less than had been observed previously in the nucleus of selection of this paternal line over purebred females when us-ing either natural mating (86.2%; [START_REF] Piles | Genetic parameters of fertility in two lines of rabbits with different reproductive potential[END_REF] or AI (71.7%, [START_REF] Tusell | Genotype × AI conditions interaction for male effect on fertility and prolificacy[END_REF]. The AI conditions of this experiment (smaller sperm dosage and 24-h storage period of the doses before AI on a commercial farm over crossbred females) could be more unfavorable than natural mating conditions and the AI conditions in the nucleus of selection over purebred females (high sperm dosage, no storage period). To our knowledge, there is only 1 published research paper reporting Fert results after AI using homospermic doses [START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF]. After rejecting a larger number of ejaculates, [START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF] obtained similar Fert rates (ranging from 49.4 to 63.6% in 2 purebred lines and their reciprocal crosses). However, female receptivity was not artificially induced and could contribute to reducing Fert even after strong sperm quality selection.

Effect of Structural Coefficients of pH on Fert

Table 3 shows the estimates on the liability scale of the λ Fert←pH effects in each model. Estimates of each level within each cross-classified effect λ Fert←pH were obtained as deviations from the estimated marginal posterior distribution (EMPD) of the mean Fert liability. Transformations from the underlying scale to the observed scale of the effects of the structural coefficients on Fert are plotted in Figure 1. The estimated values of λ Fert←pH as a covariate or as a cross-classified effect were consistent across all the models, indicating that an increase in pH leads to a decrease in Fert, and this relation seems to be almost linear. The recursive models were the ones that had a shallower slope in this decrease, probably because part of the pH effect was included in the covariances between the random Table 2. Structural coefficients (λ Fert←pH ) and (co)variance component matrices for the male additive effects (G), the male nonadditive plus permanent environmental effects (P m ), the female effects (P f ), the permanent environmental effects of male and day of AI (P md ), and the residual effects (R) for pH of the semen and fertility (Fert) assumed in the different models

0 0                 s s u pH u Fert , , 2 2 0 0                 s s s s u pH upH Fert u Fert pH uFert , , , , , , 2 2 
               s s s s u pH upH Fert u Fert pH uFert , , , , , , 2 2 
               s s s s u pH upH Fert u Fert pH uFert , , , , , , 2 2 
               P m s s P pH P Fert m m , , 2 2 0 0                  s s P pH P Fert m m , , 2 2 0 0                  s s s s P pH P pH Fert P Fert pH P Fert m m m m , , , , , , 2 2 
                s s s s P pH P pH Fert P Fert pH P Fert m m m m , , , , , , 2 2 
                s s s s P pH P pH Fert P Fert pH P Fert m m m m , , , , , , 2 2 
                P f 0 0 2 0 s P Fert f ,                  0 0 2 0 s P Fert f ,                  0 0 2 0 s P Fert f ,                  0 0 2 0 s P Fert f ,                  0 0 2 0 s P Fert f ,                  P md s s P pH P Fert md md , , 2 2 0 0                  s s P pH P Fert md md , , 2 2 0 0                 
                  s s P pH P Fert md md , , 2 2 0 0                  R s e pH , 2 0 1 0                s e pH , 2 0 1 0                s e pH , 2 0 1 0                s e pH , 2 0 1 0                s e pH , 2 0 1 0                1
UMM cross/cov = mixed model without genetic and environmental correlations for pH and fertility including pH as a cross-classified effect (cross) or as a covariate (cov) in the model of fertility; BMM 0 = bivariate mixed model for pH and fertility; RMM cross/cov = recursive mixed model for pH and fertility including pH as a cross-classified effect (cross) or as a covariate (cov) in the model of fertility.

effects of the traits. The observed negative relationship between pH and Fert agrees with previous studies that evidenced a negative correlation between pH of the ejaculate and Fert [START_REF] Coffey | Androgen action and the sex accessory tissues[END_REF][START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF] and also with litter size [START_REF] More O'ferrall | Relationship between pH, other semen traits and fertility in rabbits[END_REF]. The greater the concentration and motility of spermatozoids in ejaculates, the lower the pH because of a greater production of lactic acid [START_REF] Hulet | A fertility index for rams[END_REF][START_REF] Coffey | Androgen action and the sex accessory tissues[END_REF][START_REF] Bencheikh | The effect of frequency of ejaculation on semen characteristics and sperm output in the rabbit[END_REF][START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF]García-Tomás et al., 2006b). Thus, pH can be an indicator of semen quality, offering AI centers an easy way to select ejaculates and males for AI to improve Fert. Although the relationship between pH and Fert is almost linear, the inclusion of λ Fert←pH as a crossclassified effect allows checking nonlinearity without an apparent loss of accuracy or computing problems.

(Co)Variance Components

Table 4 shows features of the posterior distributions of phenotypic variances, ratios of variances, and correlations between traits. The posterior median of pH heritability (h 2 ) was equal in all the models (0.18) but was greater than had been reported previously [START_REF] Brun | Paramètres génétiques des caractéristiques de la semence de lapin. 13èmes[END_REF]. The difference between the h 2 values could be because in our work the analyzed trait was the pH corresponding to the pooled semen obtained from each male in the day of collection, whereas [START_REF] Brun | Paramètres génétiques des caractéristiques de la semence de lapin. 13èmes[END_REF] used pH measurements from individual ejaculates. Greater h 2 estimates are obtained if the trait consists of means of several records than if it corresponds to an individual record [START_REF] Ducrocq | Genetic characteristics and evolution of semen production of young Normande bulls[END_REF][START_REF] Wolft | Genetic correlations between production and semen traits in pig[END_REF]. The posterior median of pH repeatability (r pH ) was approximately 0.23, with the highest posterior probability density interval at 95% (HPD 95% ) being approximately [0.13, 0.34] in the models presented. Previous studies in rabbits showed similar values of r pH : 0.07 to 0.24 by [START_REF] Bencheikh | The effect of frequency of ejaculation on semen characteristics and sperm output in the rabbit[END_REF], who compared groups with different frequencies of ejaculate extraction, and 0.17 by [START_REF] Brun | Paramètres génétiques des caractéristiques de la semence de lapin. 13èmes[END_REF]. García-Tomás et al. (2006b) obtained an r pH of 0.38 in 2 paternal lines of rabbits (one of them being the Caldes line) and its reciprocal crosses.

The posterior median of h 2 for male Fert was 0.10, and its repeatability (r Fert ) was approximately 0.19 with HPD 95% being approximately [0.12, 0.26] across all the models. Both estimates were greater than had been reported previously (HPD 95% of h 2 = [0.004, 0.024], r Fert = 0.044; [START_REF] Piles | Genetic parameters of fertility in two lines of rabbits with different reproductive potential[END_REF]. The probability of h 2 >0.02 was greater than 96% in all the models. The AI procedure used in this work (stringent ejaculate selection, reduced spermatic concentration, and a storage period) were probably optimal to detect Fert differences between males [START_REF] Amann | Detection of differences in fertility[END_REF]. This fact could lead to obtaining a greater genetic variability of this trait than the one obtained after natural mating. This was probably due to the observation of other genetic effects in underlying Fert that were masked in optimal conditions of AI or after natural mating [START_REF] Tusell | Genotype × AI conditions interaction for male effect on fertility and prolificacy[END_REF].

Biased trait parameters could be obtained if the selection criterion (growth rate) was correlated with the analyzed traits [START_REF] Gianola | Bayesian methods in animal breeding theory[END_REF]). To our knowledge, these correlations have not been estimated. Nevertheless, genetic correlation between female Fert and growth rate was low (-0.13; [START_REF] Tusell | Genetic analysis of fertility and average daily gain in rabbit[END_REF]). Because the genetic correlation for male and female Fert seems to be positive in this line [START_REF] Piles | Genetic parameters of fertility in two lines of rabbits with different reproductive potential[END_REF], it is UMM cross/cov = mixed model for pH and fertility without genetic and environmental correlations including pH as a cross-classified effect (cross) or as a covariate (cov) in the model of fertility; RMM cross/cov = recursive mixed model for pH and fertility including pH as a cross-classified effect (cross) or as a covariate (cov) in the model of fertility. expected that the genetic correlation between male Fert and growth rate is negligible as well.

Although estimates were very imprecise, the posterior median of genetic correlation (r g ) between pH and Fert was moderately negative in all the models [i.e., P(r g < 0) of 0.83, 0.68, and 0.91 for BMM 0 , RMM cov , and RMM cross , respectively]. Semen pH is also genetically correlated with several motility parameters that, in turn, are related to Fert [START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF][START_REF] Brun | Paramètres génétiques des caractéristiques de la semence de lapin. 13èmes[END_REF].

Recursive Model as an Alternative Parameterization of a Classical Bivariate Model

Following the method of [START_REF] Varona | Analysis of litter size and average litter weight in pigs using a recursive model[END_REF], the expression that defines the equivalence for the phenotypic (co)variance components between a recursive and a classical bivariate model is

L L RMM 1 RMM R MM 1 BMM cov c ov cov 0 V V - - ( )( ) ′ =
. The matrix of structural coefficients containing the regression coefficient of pH on Fert is

L RMMcov = -           1 0 0 15 0 07 1 . ( . )
(Table 3; RMM cov model). The term V RMM cov denotes the phenotypic (co)variance matrix estimated with model RMM cov (Table 4). Therefore, . ( . ) . .

L L RMM 1 RMM R MM 1 cov c ov cov V - - ( )

Model Comparison

The ability to predict Fert data was similar across all the models: 76% of correctly predicted successes and 62% of correctly predicted failures. Figure 2 shows the histograms of the mean of posterior predictive distributions for success and failure Fert data estimated using models UMM cov , BMM 0 , and RMM cross .

Correlations between posterior means of the EBV for male Fert across models were close to 1. No differences were encountered in the average EBV of the top 10% animals among the models. Thus, after performing a truncated selection of the best 10% ranked by its EBV, the same the genetic gain would be obtained by using any of the models studied. The highest correlations were between models that differed only in the type of λ Fert←pH included in the model (covariate or cross-classified effect), confirming the linear effect of pH on Fert.

Despite differing in complexity, the models did not differ in terms of Fert predictions. This was due to both the imprecision of parameter estimation and the low magnitude of those parameters.

Effect of AI Dose Concentration and Physiological Status of the Female on Fert

For simplicity, only results from model UMM cov are presented. The amount of sperm per dose had an important effect on Fert. The EMPD of differences in Fert percentage between AI at 40 or 10 × 10 6 spermatozoa/ mL was 10.0 (HPD 95% = [8.0, 11.8]). It is known that increasing the concentration can compensate for some seminal deficiencies associated with low Fert [START_REF] Farrell | Relationship of semen quality, number of sperm inseminated, and fertility in rabbits[END_REF][START_REF] Alvariño | Effects of semen concentration on rabbit artificial insemination with fresh or 24 hours stored semen[END_REF][START_REF] Viudes-De-Castro | Effect of sperm count on the fertility and prolificity rates of meat rabbits[END_REF][START_REF] Saacke | Relationship of seminal traits and insemination time to fertilization rate and embryo quality[END_REF].

Lactation had a negative effect on Fert. The EMPD of differences in Fert percentage between lactating and nonlactating females was -5.2 HPD 95% = [-8.1, -2.5]. This result agrees with previous estimates obtained after natural mating in the same line [START_REF] Piles | Genetic parameters of fertility in two lines of rabbits with different reproductive potential[END_REF] and in another breed in which receptive and lactating females had a smaller kindling rate than the ones not in lactation [START_REF] Brun | The relationship between rabbit semen characteristics and reproductive performance after artificial insemination[END_REF]. Lactation produces decreases in female receptivity, ovulation rate, and ovulation frequency, and also increases in the number of embryo deaths and postimplantation mortality [START_REF] Théau-Clément | A study on relationships between receptivity and lactation in the doe and their influence on reproductive performances[END_REF][START_REF] Fortun | Les effets de la lactation sur les performances de reproduction chez le lapine[END_REF]. Although, in this work, ovulation was hormonally induced, some negative effects of lactation on female Fert had not been totally suppressed.

Conclusions

There is a quasi-linear negative effect of semen pH on Fert in rabbits. This effect could be equally estimated by using either recursive or classical multivariate models. Both types of models predicted Fert data reasonably well. No differences in the prediction of the EBV for male Fert were encountered between models, showing a good concordance when the animals were ranked by their EBV and in their average EBV of the top 10% best animals. Thus, from the point of view of selection, irrespective of the model of choice, small changes would be encountered in the evaluation of the animal for male Fert. The fact that the models were almost equivalent despite differing in complexity may have been due to small recursiveness effect of pH on Fert and the low precision obtained for the parameter estimates.

The pH of semen could be used to select qualitatively better ejaculates to increase Fert. However, despite the moderate value of heritability obtained for this trait, it does not seem to be advisable to use semen pH as a selection criterion to improve male Fert by indirect selection because the genetic correlation between the 2 traits might not be sufficiently high.

Figure 1 .

 1 Figure 1. Effect of pH on fertility on the observed scale in the different models for pH of the semen and fertility: a mixed model without genetic and environmental correlations including pH as a covariate or as cross-classified effect in the model of fertility (models UMM cov and UMM cross , respectively), and recursive mixed models including pH as a covariate or as a cross-classified effect in the model of fertility (RMM cov and RMM cross , respectively).

  These values are almost equal to the phenotypic (co) variance matrix obtained with the BMM 0 model:

Table 1 .

 1 Summary statistics of the cross-classified effect of pH (1 to 8) included in the fertility model

	Item	1	2	3	4	5	6	7	8
	Range	≤7	(>7, 7.13)	(>7.13, 7.25)	(>7.25, 7.38)	(>7.38, 7.5)	(>7.5, 7.75)	(>7.75, 8)	(>8, 9)
	Mean	6.80	7.06	7.19	7.31	7.44	7.60	7.88	8.21
	No. of fertility records	796	710	915	1,074	881	1,001	601	387

Table 3 .

 3 Means (SD) of the posterior distributions of the recursive effect of pH in the liability of fertility, with λ and λ i (for i = 1, . . . 8) as a covariate or as a cross-classified effect

	Model 1	λ	λ 1	λ 2	λ 3	λ 4	λ 5	λ 6	λ 7	λ 8
	UMM cross	-	0.42 (0.16)	0.28 (0.16)	0.17 (0.15)	0.08 (0.14) -0.03 (0.15) -0.09 (0.14) -0.41 (0.16) -0.49 (0.19)
	UMM cov	-0.63 (0.11)	-	-	-	-	-	-	-	-
	RMM cross	-	0.19 (0.21)	0.14 (0.18)	0.07 (0.16)	0.03 (0.14) -0.02 (0.15) -0.01 (0.15) -0.25 (0.20) -0.19 (0.28)
	RMM cov	-0.15 (0.07)	-	-	-	-	-	-	-	-
	1

Table 4 .

 4 Medians (highest posterior density interval at 95%) of the posterior distributions of the phenotypic variance (σ 2 additive effect), m (environmental plus genetic nonadditive variation attributable to the male), f (variation attributable to the female), and md (environmental variation attributable to the male and day of AI).

	), phenotypic correlation (r), male

1 i = g (male

Corresponding author: llibertat.tusell@irta.cat Received June 11, 2010. Accepted January 10, 2011.

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Diagonal = the limits of the diagonal are within a set of 2 columns and within a model.Tusell et al.

Modeling pH of the semen and fertility relationships

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1 This research was supported by Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria (INIA, Madrid, Spain) project RTA2005-00088-CO2. Part of it was carried out during a visit by the first author to INRA, Toulouse. Llibertat Tusell Palomero received a fellowship from the INIA. The authors dedicate this work to Josep Terrades i Colom for his valuable cooperation with IRTA and his dedication and love of the rabbit industry during his life. The authors also thank Anna Pérez, Carme Requena, and Oscar Perucho (Unitat de Cunicultura, IRTA, Caldes de Montbui, Barcelona, Spain), and Daniel Mozer Schonborn and Sinela Mozer Schonborn (Montmajor, Barcelona, Spain) for their invaluable cooperation during the experimental phase of this work and Rod Cantlay (Sant Feliu de Codines, Barcelona, Spain) and Jules Hernández (Genètica i Millora Animal, IRTA, Lleida, Spain) for the English revision of the manuscript.