Cornelis Van De Kamp

Raoul M Bongers

Frank T J M Zaal

Getting hold of approaching objects --1 Getting hold of approaching objects: In search of a common control of handclosure initiation in catching and grasping

Keywords:

Both in the catching and grasping component of prehension, the hand opens and closes before hand-object contact is made. The initiation of hand closure has to be coordinated with the time course of the decrease of the distance between the hand and the target object, i.e., with the reaching component in prehension or the approach of the target in catching. The authors investigated if this initiation of hand closure could be explained by a common control. For this purpose, they fitted the dynamic timing model to data from the two tasks. In both tasks, participants were asked to get hold of an object approaching along the table top at a constant velocity. In the prehension task, participants could reach out to grasp the object; in the catching task, they were required to keep their hand stationary. In comparison with other accounts, the dynamic timing model performed best in explaining the data. The model proved adequate for the prehension task but not for the current catching task.

Introduction

Prehension, the act of coordinated reaching and grasping, and catching have in common that the hand first opens and subsequently closes to take hold of the target object. In prehension, the initiation of the hand closure has to be coordinated with the reaching (i.e., the movement of the hand towards the target). In catching, the initiation of hand closure has to be coordinated with the arrival of the approaching target. Thus, in both tasks, the gap between the hand and the target closes, and the timing of the initiation of closure of the hand has to be coordinated with this gap closure. The objective of the current study was to see if the control of catching and the control of the grasping component of prehension could be understood within the same framework. More specifically, given the success of the dynamic timing model in the context of prehension [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF], in which first-order time-to-contact information is used to time the initiation of hand closure, we focused on this model to see if it would predict this timing both in the grasping of prehension and in catching equally well.

Two decades of prehension research offered several proposals that described how the initiation of hand closure might be timed. Unfortunately, most of these proposals did not hold very well under experimental scrutiny. Three of these proposals will be briefly presented in the following. In the first proposal, hand closure was thought to be initiated at a fixed time before hand-object contact, independent of task conditions [START_REF] Gentilucci | Temporal coupling between transport and grasp components during prehension movements: effects of visual perturbation[END_REF].

Several empirical studies tested this hypothesis by manipulating reaching amplitude [START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF], object width [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF], object size [START_REF] Wang | Spatial and temporal control of trunkassisted prehensile actions[END_REF], object orientation [START_REF] Rand | Effect of orienting the finger opposition space in the control of reach-to-grasp[END_REF], reach or object velocity [START_REF] Carnahan | Visuomotor control when reaching toward and grasping moving targets[END_REF][START_REF] Rand | Effect of speed manipulation on the control of aperture closure during reach-to grasp movements[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF], and the direction of the object's approach [START_REF] Watson | Time to contact and the control of manual prehension[END_REF], which, taken together, demonstrated that in prehension the initiation of hand closure is not time-invariant over a range of conditions.

A second proposal employed a spatial rather than a temporal variable: The initiation of hand closure should happen at a fixed distance from the hand to the target object (cf. [START_REF] Wang | Coordination among the body segments during reach-to-grasp action involving the trunk[END_REF][START_REF] Wang | Spatial and temporal control of trunkassisted prehensile actions[END_REF]. Recently, Stelmach and colleagues have amended this original proposal [START_REF] Rand | Effect of speed manipulation on the control of aperture closure during reach-to grasp movements[END_REF][START_REF] Rand | Quantitative model of transport-aperture coordination during reach-to-grasp movements[END_REF], after finding that closing distance varied with object orientation [START_REF] Rand | Effect of orienting the finger opposition space in the control of reach-to-grasp[END_REF] and reach velocity [START_REF] Rand | Effect of speed manipulation on the control of aperture closure during reach-to grasp movements[END_REF]. They proposed that the initiation of hand closure occurs at a closing distance that is a function of, among other variables, reaching amplitude and velocity. Although adding these variables improved the data fitting, it made the account less parsimonious, of course.

A third proposal that we will discuss stems from the line of research that considers the timing of interceptive actions to be based on prospective information about the time remaining until the object and the observer meet (e.g., [START_REF] Bootsma | Predictive visual information sources for the regulation of action with special emphasis on catching and hitting[END_REF][START_REF] Lee | A theory of visual control of braking based on information about time-to-collision[END_REF][START_REF] Savelsbergh | Grasping tau[END_REF]. In this proposal, the initiation of hand closure is triggered on the basis of first-order time-to-contact information (tau). In prehension, this first-order time-to-contact information specifies the time for the physical gap between the target object and the grasping hand to be closed under prevailing speed conditions. One way the initiation of hand closure could be based on first-order time-to-contact information is that the hand starts closing at a critical value of tau. In prehension, however, this seems not to be the case. For instance, earlier work showed that at the initiation of hand closure, first-order time-to-contact varied with reaching amplitude [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF] and object velocity [START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF]. However, Zaal and coworkers [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF] showed another way first-order time-tocontact information could be at the basis of the initiation of hand closure, along the lines of [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF] dynamic timing model. We will return to this model later, after having had a look at the catching literature.

As in the literature on prehension, also in the context of catching, researchers have considered the proposal that the initiation of hand closure is timed on the basis of a critical value of first-order time-to-contact. Just as in prehension, little support was found for this proposal. That is to say, ambiguous results concerning values of first-order time-to-contact at the initiation of hand closure were found in studies on catching. Whereas an early study reported no effects of object velocity [START_REF] Savelsbergh | The role of predictive visual temporal information in the coordination of muscle activity in catching[END_REF], later studies showed that firstorder time-to-contact at the initiation of hand closure varied with object velocity [START_REF] Bennett | Timing a one-handed catch I. Effects of telestereoscopic viewing[END_REF][START_REF] Caljouw | Catching optical information for the regulation of timing[END_REF][START_REF] Wallace | The perceptual guidance of grasping a moving object[END_REF]. Also, just as was the case for prehension, it was examined whether the initiation of hand closure is timed on the basis of a critical closing time or closing distance. Findings concerning these hypotheses were inconsistent as well [START_REF] Laurent | The control and coordination of one-handed catching: The effect of temporal constraints[END_REF][START_REF] Mazyn | Reorganization of catching coordination under varying temporal constraints[END_REF][START_REF] Savelsbergh | The role of predictive visual temporal information in the coordination of muscle activity in catching[END_REF][START_REF] Wallace | The perceptual guidance of grasping a moving object[END_REF][START_REF] Wang | Spatial and temporal control of trunkassisted prehensile actions[END_REF].

Together, the proposals introduced so far, do not seem very promising in revealing the control for the separate tasks, let alone for an account for the timing of hand closure common in both prehension and catching. Therefore, we turn to a final proposal, one that has been shown to accurately predict the moment of hand closure in prehension [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF] and might apply to catching as well. In the same vein as the critical-tau (threshold) type of control mentioned before, the dynamic timing model exploits first-order time-to-contact information. However, in this account, the use of tau is more sophisticated than simply waiting until a critical value has been reached. The idea is that grasping behavior is best understood within a dynamical systems approach, in which the hand-opening and hand-closing states are endowed with stability features.

Elaborating on the work of [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF], Zaal andcolleagues (1998, 2004) proposed a formulation of how the stability of the hand-opening state and the stability of the hand-closing state are coupled to first-order time-to-contact. At the beginning of the movement, when first-order time-to-contact between the grasping hand and the target object is long, the hand-opening state is most stable, whereas the hand-closing state is rather unstable. While the reach unfolds, first-order timeto-contact decreases, resulting in a loss of stability of the hand-opening state and a gain of stability of the hand-closing state. At a certain point in time (i.e., the initiation of hand closure), the hand-closing state has become more stable than the handopening state and a transition -which has stability features of its own -from the hand-opening state to the hand-closing state takes place (for more details see [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF][START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF]. A consequence of this nonlinear dynamics alternative for the use of first-order time-to-contact over the threshold type of control is that hand closure does not necessarily occur at a fixed value of first-order time-to-contact. That is to say, in the situation of non-constant velocities, whereas values of first-order time-to-contact at the moment of the initiation of hand closure might vary across experimental conditions, the dynamic timing model might be able to accommodate these variations (cf. [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF]. Furthermore, the dynamical regulation of the initiation of hand closure makes the system resistant to perturbations, for instance, when the hand needs to be retracted or the target object changes position. [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF] considered the standard prehension task of having participants pick up stationary target objects, and showed that closing time, closing distance, and first-order time-to-contact at the moment of hand-closure initiation all varied with factors such as distance and the size of the objects, but no effects were seen when looking at the differences between the model predictions and the actual moments of hand-closure initiation. These differences had been computed on a trial-to-trial basis. In an earlier study, Zaal and colleagues (1998) had studied the picking up of objects that either remained stationary or moved away from the participants in the experiments. They inspected the performance of the dynamic timing model, using compound time-to-contact time series rather than individual ones, and showed that the model predicted the moments of initiation of hand closure quite accurately. Here, we will present an experiment in which participants were asked to either reach for and pick up an approaching target (prehension task) or keep their hand still and wait for the object to arrive into the hand that they needed to open and close to get hold of the approaching object (catching task).

Following [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF], we will evaluate the data on a trial-to-trial basis, to investigate if the dynamic timing model also applies to the situation of approaching objects, both in prehension and in catching.

Method

Participants

Seven men and eight women, with an average age of 24 years (ranging from 20 to 41 years) participated in the experiment. All were right-handed and had normal or corrected-to-normal vision. Participants were naive to the exact purpose of the experiment, gave their informed consent and were paid a small fee for their participation.

Apparatus

A cylindrical target object was placed on top of a magnet embedded objectcarrier. This carrier was made to move along a plain white tabletop (2 m x 2 m) by means of a magnetic coupling to a servomotor driven mechanism underneath the tabletop (for a similar setup, see: [START_REF] Schenk | A system for the study of visuomotor coordination during reaching for moving targets[END_REF]. The exact movement path of the target object was computer-controlled through a user interface that was developed for this purpose (LabView, National Instruments). The positions of four infrared light emitting diodes (IREDs) were tracked at a rate of 100 Hz using an Optotrak system (NDI, Waterloo, Ontario, Canada). The IREDs were placed on the center of the target object, the lateral lower corner of the index finger nail, the medial lower corner of the thumb nail, and the skin immediately proximal to the styloid process of the radius at the wrist.

Procedure and design

We used two tasks (prehension and catching) in which the target object (diameter: 3 cm, height: 1.5 cm) approached the participants with one of three constant velocities (20, 40 and 60 cm/s) and starting from one of two initial distances (75 and 100 cm). With a set acceleration of 600 cm/s 2 it took the object at most 6 cm (100 ms) to reach the constant velocity. Participants sat alongside the table, with their right side touching the table edge and their sagittal plane parallel to the table edge. The object approached along the participants' sagittal plane, some 30 cm away from the edge of the table. At the start of each trial, the right hand, with the tips of the thumb and index finger touching, was positioned on a Plexiglas span; this span allowed the object to pass underneath. In the prehension task, participants were to reach for and grasp the object. As soon as the object started to move towards the participant, he or she was free to choose the moment to start the reach to grasp movement. This way, the pick-up location was left to the participant; the only instruction was to carry out a continuous, fast but accurate reaching-tograsp movement. In the catching task, the participants were required to catch the approaching object between the pads of the thumb and index finger while keeping the position of their hand fixed, resting on the Plexiglas span. After liftoff, the object was to be placed on the tabletop somewhere around the pickup location. During each trial headphone-delivered white noise was played.

The order of the catching and prehension tasks was counterbalanced across participants. Presenting the 6 randomized conditions (3 object velocities x 2 initial distances) in 12 blocks for the two tasks resulted in a total of 144 trials per participant.

Data analysis

A total of 2080 trials were used for the data analysis. In 3 trials the object was unintentionally dropped, while in 46 trials, either some IRED data was missing or we encountered problems with the object carrier.

Position data was smoothed using a low-pass recursive Butterworth filter with a cutoff frequency of 10 Hz. Speed was computed using a three-point finite difference algorithm. Hand position was defined as the average position of the thumb and index-finger IRED. The start of the reaching movement was defined as the moment at which the tangential hand speed rose above a threshold of 2 cm/s. Hand aperture was defined as the three-dimensional distance between the thumb and the index finger IREDs. The start and end of the grasping movement were defined as the moment when hand opening and closing speed rose above or dropped below a threshold of 2 cm/s, respectively.

To determine the initiation of hand closure we looked back from the moment of peak closing velocity and detected the first moment that the hand closing speed passed a threshold of 2 cm/s. Closing distance was defined as the distance (projected along the dimension of the object approach) between the object and the hand at the initiation of hand closure. Closing time was the time from the initiation of hand closure until the end of the grasping movement. First-order time-to-contact 1 TC 1 (D) at the moment of hand-closure initiation corresponded to the time it would take to make contact with the object if conditions would prevail (i.e., constant velocity) and was computed by dividing the distance (projected along the dimension of the object approach) between the object and the hand by the momentary speed at which this distance was closed.

An important part of our analyses examined the accuracy with which the initiation of hand closure was predicted by the dynamic timing model [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF][START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF]. For each trial, we compared the predicted moment of hand-closure initiation with the experimentally observed moment of hand-closure initiation. To arrive at the model prediction, we numerically simulated the model's set of differential equations (see Appendix) using a Runge-Kutta algorithm, with a fixed time step of 0.01s, equal to the time step of the kinematic data. All parameters of the model were set at a fixed value (α = 10; ω = 10; γ = 10; β = 90; σ = 0.75; r crit = 0) except the parameter c vision , which was allowed to vary across participants and tasks (as discussed later). The parameter c vision represents the strength of the contribution of the first-order time-to-contact variable (see Appendix; cf. [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF][START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF]. We optimized the values of the c vision parameter to have a best fit between the average model predictions and the average observed initiation moments by finding the c vision parameter setting that resulted in a minimum sum of squared prediction errors. In a first pass, we allowed c vision to vary only across participants (see Table 1 for the values of c vision ). In a subsequent analysis, c vision was allowed to vary across participants but also between tasks (prehension vs. catching; see Table 1). *** Table 1 about here *** Model accuracy was evaluated in terms of a temporal prediction error, which was defined as the time difference of the experimentally observed moment of handclosure initiation and the moment of hand-closure initiation as predicted by the dynamic timing model; a positive difference meant that the model prediction preceded the observed moment of initiation of hand closure.

A problem for the comparison of the four dependent variables that we identified before (closing distance, closing time, first-order time-to-contact, and dynamictiming-model prediction error) is that they are defined along different dimensions.

To arrive at dependent measures that are defined along the same dimension for each dependent variable, we computed temporal prediction errors, analogous to the dynamic-timing-model prediction error, for each variable that we considered in our comparison. To this end, we assumed that these variables (closing time, closing distance, or first-order time-to-contact) were to be kept constant at a specific value in the control of grasping. We took this value (which was allowed to vary across participants) to be the value at which the sum of squared prediction errors was minimal, just as we had done for the dynamic-timing-model error. Next we inspected for each trial when this value was reached, and computed the difference in time between the latter moment and the moment of hand-closure initiation. We did so for the variables of closing time, closing distance, and firstorder time-to-contact. The resulting dependent variables were all along the dimension of real time, just as the prediction error of the dynamic timing model. Thus, a fair comparison among all four of the temporal prediction errors was possible.

Each dependent variable (temporal prediction errors of closing distance, closing time, first-order time-to-contact, and the dynamic timing model) was analyzed with a separate repeated-measures analyses of variance (ANOVAs) with task (catching vs. prehension), object velocity (20, 40, or 60 cm/s), and initial distance (75 vs. 100 cm) as within-participant factors. Greenhouse-Geisser corrections of degrees of freedom were used when sphericity assumptions were violated. For every statistically significant effect, we calculated effect sizes using generalized etasquared values (cf. [START_REF] Bakeman | Recommended effect size statistics for repeated measures designs[END_REF]. These effect sizes were interpreted according to [START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF] recommendation of .02 for a small effect, .13 for a medium effect, and .26 for a large effect. In post-hoc analyses, we applied Bonferroni corrections to control Type-I errors.

Results

Fig. 1 shows typical examples of hand-aperture profiles. Figs. 1A andB present the hand aperture as a function of actual time to contact (i.e., the time until the end of the grasping movement), equivalent to the most familiar representation of hand apertures in prehension, and given in the majority of the studies of grasping and catching (e.g., in the context of prehension: [START_REF] Castiello | The neuroscience of grasping[END_REF][START_REF] Jeannerod | The timing of natural prehension movements[END_REF][START_REF] Jeannerod | The neural and behavioural organization of goal-directed movements[END_REF][START_REF] Marteniuk | Functional relationships between grasp and transport components in a prehension task[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF]; but see Bongers, Zaal, & Jeannerod, submitted; and in the context of catching: [START_REF] Mazyn | Planning and on-line control of catching as a function of perceptual-motor constraints[END_REF][START_REF] Savelsbergh | Grasping tau[END_REF][START_REF] Savelsbergh | The visual guidance of catching[END_REF]. In Figs. 1C andD, hand aperture is plotted as a function of actual distance to contact (i.e., hand-object distance), a less familiar representation, although given in a subset of prehension studies that stress the role of distance rather than time (e.g., see [START_REF] Haggard | Coordination of hand aperture with the spatial path of hand transport[END_REF][START_REF] Rand | Effect of speed manipulation on the control of aperture closure during reach-to grasp movements[END_REF]Wallace, Stevenson, Spear, & Weeks, 1994;[START_REF] Wing | The contribution of the thumb to reaching movements[END_REF]Zaal & Bootsma, 2000). Finally, Figs. 1E andF give hand aperture as a function of first-order time-to-contact (TC 1 (D); cf. [START_REF] Lee | A theory of visual control of braking based on information about time-to-collision[END_REF].

Closing time, closing distance, and first-order time-to-contact

Although we will perform our inferential statistics on the temporal prediction errors that we defined before, for the sake of comparison of the present data with results of previous papers, Table 2 gives the values of the three variables of closing time, closing distance, and TC 1 (D), at the moment of the initiation of hand closure.

Table 2 presents the averages and average standard deviations of these three variables, for the three object-velocity conditions. As we explained before, we did not analyze these averages per se, but inspected temporal prediction errors, to allow a proper comparison among the variables (see Methods section for details). *** Fig. 1 and Table 2 The ANOVA on the closing-distance prediction error revealed a large effect of object velocity, F(1.13, 15.77) = 119.42, p < .001, 2 G = .503 (see Table 3). The post-hoc tests showed that all levels of object velocity differed from each other (p < .01). Furthermore, the ANOVA showed a small to medium Task x Object-velocity interaction, F(1.32, 18.49) = 7.67, p < .05, 2 G = .068.

Finally, we found a small to medium effect of object velocity on the prediction error of first-order time-to-contact at the moment of hand-closure initiation, F(1.21, 16.91) = 13.33, p < .005, 2 G = .125 (see Table 3). Post-hoc tests indicated that all differences between the means, except for the difference between the two highest object velocities, were statistically significant (p < .05).

*** Table 3 about here ***

Dynamic-timing model

As we mentioned in the Methods section, we first evaluated the model predictions using the same c vision parameter setting for the prehension and the catching tasks. Inspection of the average prediction errors learned that in the catching task, the model, on average, was some 15.5 (SD = 23.4) ms too late, whereas in the prehension task, it was some 8.5 (SD = 29.4) ms too early.

Although these differences were rather small, a large task effect was found, F(1, 14) = 50.34, p < .001, 2 G = .418. Furthermore, we found a small to medium effect of object velocity, F(1.25, 17.53) = 7.95, p < .01, 2 G = .117 (see Table 3). Post-hoc tests showed that the average prediction error of the lowest object-velocity condition differed from the other object-velocity conditions (p < .05). Finally, the ANOVA revealed a small to medium Task x Object-Velocity interaction, F(1.19, 16.67) = 9.59, p < .01, 2 G = .039.

Given the two effects that include the factor of task, we were interested to see if allowing the c vision parameter to vary, not only across participants, but also across tasks, would result in a situation in which the model would predict the initiation of hand closure in both tasks accurately, albeit with different strengths of the optical information on the intrinsic dynamics of hand opening and closing. The ANOVA on the prediction errors revealed a same pattern of effects as we had found when we used the same values of the c vision parameter for both tasks, although with smaller effect sizes. The analysis showed a medium to large effect of object velocity, F(1.23, 17.28) = 7.89, p < .01, 2 G = .142 (see Table 3). Again, the prediction errors in the slowest object-velocity condition differed from those in the two other objectvelocity conditions (p < .05). Furthermore, we found a small effect of task, F(1, 14) = 7.55, p < .05, 2 G = .028, M (SD) = -3.6 (22.9) and 1.1 (29.5) ms for catching and prehension, respectively. Finally, the ANOVA revealed a small Task x Object-Velocity interaction effect, F(1.24, 17.42) = 6.11, p < .05, 2 G = .028 (see Table 4).

To unpack this interaction effect, we performed two ANOVAs, with factors of object velocity and initial distance, for the prehension task and catching task separately.

Whereas we found no statistically significant effects of any of both factors for the prehension task, the analysis of the catching data revealed a large object-velocity effect, F(1.22, 17.13) = 9.53, p < .010, 2 G = .248, due to different prediction errors when catching the objects that approached at the lowest velocity (p < .05).

*** Table 4 about here ***

Discussion

The main purpose of the current study was to see if the timing of closure of the hand was controlled similarly in prehension and catching, both with approaching objects. Given the earlier successes [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF] of modeling the initiation of hand closure in prehension with [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF] dynamic timing model, with the hand-closure initiation timed on the basis of first-order time-to-contact information, we took this model as our starting point of the study. [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF] reported, for prehension of stationary objects, effects of distance, size, and width on the TC 1 (D) values in their experiment but that the dynamic timing model accommodated all these variations: They found no significant effects on the average prediction errors. The present study showed that this was not the case in all the conditions that we tested, in which we asked our participants to either reach for and grasp an object approaching at a constant velocity or catch it (with a stationary hand). We found effects of both object velocity and task on the quality of the dynamic timing model's prediction of the moment of hand-closure initiation. Importantly, we also found an interaction effect of these two factors. It turned out that dynamic timing model had difficulty fitting particularly the condition of catching the object approaching at its lowest velocity; In the other conditions, the prediction errors were less than 5 ms, on average.

In line with previous studies, which had shown that closing time [START_REF] Laurent | The control and coordination of one-handed catching: The effect of temporal constraints[END_REF][START_REF] Rand | Effect of orienting the finger opposition space in the control of reach-to-grasp[END_REF][START_REF] Rand | Effect of speed manipulation on the control of aperture closure during reach-to grasp movements[END_REF][START_REF] Rand | Quantitative model of transport-aperture coordination during reach-to-grasp movements[END_REF][START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF] and closing distance (e.g., [START_REF] Carnahan | Visuomotor control when reaching toward and grasping moving targets[END_REF][START_REF] Rand | Effect of orienting the finger opposition space in the control of reach-to-grasp[END_REF][START_REF] Rand | Effect of speed manipulation on the control of aperture closure during reach-to grasp movements[END_REF][START_REF] Wang | Spatial and temporal control of trunkassisted prehensile actions[END_REF][START_REF] Watson | Time to contact and the control of manual prehension[END_REF][START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF] varied with factors such as object velocity, object orientation, object width, object size, reaching amplitude, and reaching velocity, we found a large effect of the velocity with which the object approached on these two variables. The effects of object velocity on these two factors that came out of our experiment corroborated the conclusion that closing time or distance are not being kept constant, and, thus, do not act as control variables in the coordination of reaching and grasping or in the timing of catching in the situation that the hand is not moving.

Our results showed that the dynamic timing model did not perform well for the current catching task of a stationary hand and an object approaching at constant velocity, especially when the target arrived at a low speed. From this, one might conclude that the model was not appropriate to explain the timing of hand-closure initiation in catching. An alternative conclusion might be that our choice of designing the catching task in such a constrained way might have brought our participants in a rather unnatural situation, in which they were invited to show behavior that they would not show in natural catching. We have planned experiments to explore this possibility. In these experiments, objects arrive at nonconstant velocities or we allow the hand to move when objects do arrive at constant velocities.

When we looked at the different object-velocity effects in our data, we found that the dynamic timing model most closely fit the patterns of results of the current experiment. Except for the condition of catching approaching objects at the lowest speed used in the current study, temporal prediction errors were less than 5 ms (cf.

Table 4). The dynamic timing model did explain the results just a little better than a model with the timing of the initiation of hand closure at a critical value of first-order time-to-contact, and much better than models in which closure distance or time were to be kept constant. Temporal prediction errors of the constant-time-to-contact account were roughly twice as big as those of the dynamic timing model, except for the prediction errors of the lowest object velocity (see Table 3). This lowest object velocity proved to be problematic, particularly for the current catching task (we did not find a significant object-velocity effect when we considered the data of the prehension task separately). Taken together, the dynamic timing model accounted for much of the variability seen in closing time and closing distance, and a small amount of the variability seen in first-order time-to-contact. For now, we conclude that the dynamic timing model does a fine, albeit not a perfect, job in predicting the moment of the initiation of hand closure on the basis of first-order time-to-contact, at least for the task of prehension. As to the task of catching, the jury is still out.

Footnote

1 Following [START_REF] Bongers | On the informationbased regulation of movement: What Wann (1996) may want to consider[END_REF], we distinguish the optical time-to-contact information (τ) from the organism-environment property that it specifies, TC 1 (D). TC 1 (D) is defined as the first-order time-to-contact, the time that distance gap D will be closed when closing velocity would remain constant.

Although our ultimately interest is in the information, technically speaking, it is the physical first-order time-to-contact TC 1 (D) that we manipulated in the current study, and that we used in our model simulations.

Appendix: Model equations

The dynamic timing model is a set of differential equations, originally formulated by [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF], and adopted for the situation of the grasping of prehension by Zaal and colleagues [START_REF] Zaal | The use of time-to-contact information for the initiation of hand closure in natural prehension[END_REF][START_REF] Zaal | Coordination in prehension: Information-based coupling of reaching and grasping[END_REF]). In the model, a state variable x is mapped onto the hand-opening and hand-closing regimes of prehension. The model equations combine so-called intrinsic dynamics --the dynamics that give the state variable x its stability properties --and the contribution of the visual information:

d dt x y = f grasp + f vision (1)
In the model three attractors in state space are defined. That is to say, there are two fixed-point attractors, for the hand-opening regime (x open ) and the hand-closing regime (x close ), respectively, and a limit-cycle attractor passing through these two fixed-point attractors:

f grasp = f osc + f open + f close (2a) f osc (x, y) = α ω -ω α x y -γ (x 2 + y 2 )x (x 2 + y 2 )y (2b) f open (x,y) = -β int f range x -x open y -y open (2c) f close (x, y) = -β int f range x -x close y -y close (2d) f range (x, y, x i ,y i ) = exp - (x -x i ) + (y -y it ) 2σ 2 (2e)
The contribution of the visual variable r(D), which is the inverse of TC 1 (D) in our case, is defined:

f vision (x, y, x open ,y open , x close ,y close ) = β vision (D) + f range (x,y, x open , y open ) x -x open y -y open -f range (x, y, x close ,y close ) x -x close y -y close (3a) β vision (D) = c vision (r(D) -r crit ) (3b)
A closer inspection of Equation (3a) shows large similarities with Equations (2c) and (2d). A growing value of the visual variable r(D) leads to an increase of the variable β vision , resulting in a decrease in the strength of attraction of the point-

attractor at x open and an increase in the strength of attraction of the point attractor at

x close . For a more detailed introduction to the model and its equations, we refer the reader to [START_REF] Schöner | Dynamic theory of action-perception patterns: The time-beforecontact paradigm[END_REF].

Table 1

Values of the cvision parameter that were used in the simulations when the same parameter was used for both tasks and when different parameters were used for the two tasks of catching and prehension. Means and average within-participant standard deviations (between brackets) of closing time, closing distance, and first-order time-to-contact TC 1 (D), at the moment of initiation of hand closure, as a function of object velocity.

cvi
object velocity (cm/s) 20 40 60

Closing Time (ms) 122.4 (32.0) 107.9 (24.8) 100.1 (20.7)

Closing Distance (mm) 12.5 (6.6) 18.0 (6.7) 25.3 (8.2)

TC 1 (D) (ms) 54.8 (16.0) 47.9 (11.5) 46.3 (10.5)

Figure caption

Fig. 1 .

 1 Fig. 1. Typical examples of hand-aperture profiles of 6 different trials of one and the same participant, of the catching task (left column) and of the prehension task

Fig . 1

 1 Fig .1

  about here *** We found a large effect of object velocity on the closing-time prediction error,

	F(1.40,19.59) = 28.77, p < .0001, 2 G = .269 (see Table 3). Post-hoc tests learned
	that all means were different from each other (p < .05). In addition, the ANOVA
	showed a large task effect, F(1, 14) = 19.51, p < .0005, 2 G = .432, M (SD) = 12.1

(22.0) and -11.6 (29.7) ms for catching and prehension, respectively.

Table 2

 2

	participant	sion parameter
		same	different
			catching	prehension
	1	4.21	5.29	3.90
	2	4.27	4.23	4.27
	3	4.67	4.85	4.59
	4	4.37	5.37	4.14
	5	3.75	4.82	3.62
	6	4.32	4.70	4.18
	7	3.87	4.72	3.63
	8	3.96	5.89	3.59
	9	3.86	4.95	3.53
	10	6.04	6.10	5.80
	11	4.40	4.57	3.86
	12	4.11	4.14	3.97
	13	4.89	5.00	4.89
	14	3.50	3.85	3.23
	15	3.80	4.64	3.73

Table 3

 3 Means and average within-participant standard deviations (between brackets) of the model prediction errors (ms) as a function of object velocity.

	object velocity (cm/s)
	20	40	60

Table 4

 4 The significant Object-Velocity x Task interaction effect of the dynamic-timingmodel prediction errors (ms); Means and average within-participant standard deviations (between brackets).

	Task	object velocity (cm/s)
		20	40	60
	Prehension	-3.2 (36.3)	3.6 (26.2)	2.8 (26.1)
	Catching	-13.7 (33.5) 2.4 (17.2)	1.5 (18.1)

Acknowledgment

This research was supported by a VIDI grant (452-03-356) from the Netherlands Organization for Scientific Research (NWO) awarded to Frank Zaal. We thank Henry van de Crommert, Wim Kaan, and Wolter de Goede for developing the setup that we used in the study, and two anonymous reviewers for their helpful comments on an earlier draft of the current paper.


Paul Mcardle

Robert Young

Toby Quibell

David Moseley

Rob Johnson

Early intervention for at

come

Child psychiatric disorders and their associated suffering, functional impairment and costs to society have become common in developed societies. [START_REF] Costello | Impaired but undiagnosed[END_REF][START_REF] Fergusson | Educational, psychosocial, and sexual outcomes of girls with conduct problems in early adolescence[END_REF][START_REF] Roza | Stable prediction of mood and anxiety disorders based on behavioral and emotional problems in childhood: a 14-year follow-up during childhood, adolescence, and young adulthood[END_REF][START_REF] Prosser | The changing mental health of children and adolescents: evidence for a deterioration?[END_REF] This has led to a search for effective prevention technologies, often now classified as universal and targeted. [START_REF] Offord | The state of prevention and early intervention[END_REF] The former focus on 'all children in a geographic area or setting…', the latter on those belonging to a high risk group such as children of families dependent on welfare (selective) or who have already developed symptoms (indicated). [START_REF] Offord | The state of prevention and early intervention[END_REF] An example of an early 'universal' intervention comprised parent training, a behaviourally based classroom curriculum and work in small groups for 1 st grade children from a poor area. [START_REF] Hawkins | Reducing early childhood aggression: results of a primary prevention program[END_REF] Compared to school as usual, this was associated with reduced externalizing scores on teacher scales a year later. A more recent example, the 'LIFT' program, evaluated a brief parenting and school based intervention conducted over ten weeks involving all children in 1 st or 5 th grade in schools drawn from areas of higher than average delinquency. [START_REF] Reid | Description and Immediate Impacts of a Preventive Intervention for Conduct Problems[END_REF] The authors reported reduced playground-observed aggression in the year after intervention.

Both interventions showed effects that were most striking among those parents and children whose initial scores were most deviant. [START_REF] Hawkins | Reducing early childhood aggression: results of a primary prevention program[END_REF][START_REF] Reid | Description and Immediate Impacts of a Preventive Intervention for Conduct Problems[END_REF][START_REF] Reid | Enhancing a classroom social competence and problemsolving curriculum by offering parent training to families of moderate-to high-risk elementary school children[END_REF][START_REF] Stoolmiller | Detecting and describing preventive intervention effects in a universal school based randomized trial targeting delinquent and violent behavior[END_REF] Others have demonstrated a similar phenomenon. [START_REF] Webster-Stratton | Preventing conduct problems and improving school readiness: evaluation of the Incredible Years Teacher and Child Training Programs in high risk schools[END_REF][START_REF] Kellam | The course and malleability of aggressive behavior from early first grade into middle schools: results of a developmental epidemiologically based preventive trial[END_REF][START_REF] Vazsonyi | Evaluation of a school-based universal violence prevention program: low-medium-and high-risk children[END_REF] Although often effective, universal interventions have the disadvantage that most children participating are low-risk and do not require intervention.

Targeted and especially indicated interventions have the advantage of excluding asymptomatic children and, in the evaluation phase, of focusing on smaller numbers. The Montreal Prevention Experiment is 'a prime example'. [START_REF] Tremblay | A bimodal preventive intervention for disruptive kindergarten boys: its impact through mid-adolescence[END_REF][START_REF] Earls | Conduct and oppositional disorders[END_REF] This study randomized 250 7-9 year old boys who were screenpositive for disruptiveness according to teacher ratings (drawn from 1161 screened boys) to a two year combined school-based group work and parent training intervention or to a no-intervention control. [START_REF] Vitaro | Disruptive behavior, peer association and conduct disorder: testing the developmental links through early intervention[END_REF][START_REF] Vitaro | Preventive intervention: assessing its effects on the trajectories of delinquency and testing for meditational processes[END_REF] Intervention was associated with reduction in conduct disorder and delinquency five and seven years respectively following intervention, a greater likelihood of high school graduation and a lower rate of criminality fifteen years post intervention. [START_REF] Boisjoli | Impact and clinical significance of a preventive intervention for disruptive boys: 15-year follow-up[END_REF] The similar Coping Power intervention deployed school-based children's and parent's groups, for boys screened positive for aggression over a 15 month period also showed promising results. At one year follow-up compared to no-intervention control intervention boys showed reduced behaviour problems at school, 'covert delinquency' and parent-reported substance use. [START_REF] Lochman | the Coping Power programme for preadolescent aggressive boys and their parents: outcome effects at the 1 year follow-up[END_REF] A larger, more elaborate intervention (additional school tutoring, group work, home visiting, parent-child sessions and a universal social and emotional curriculum sustained over five school years) was associated with only a modest reduction in behavioural symptoms. [START_REF]Using the fast track randomized prevention trial to test the early starter model of the development of serious conduct problems[END_REF] These and other data suggest that a limit could exist beyond which an intervention becomes too complex for some vulnerable populations. [START_REF] Area | A cognitive-ecological approach to preventing aggression in urban settings: Initial outcomes for high-risk children[END_REF] Whether as part of a universal or targeted intervention, the effectiveness of behavioural interventions, particularly in school, has been known for many years and has been replicated. [START_REF] Williford | Using mental health consultation to decrease disruptive behaviors in preschoolers: adapting an empirically supported intervention[END_REF][START_REF] Flannery | Initial behavior outcomes for the Peacebuilders Universal school-based violence prevention program[END_REF][START_REF] Kolvin | Help Starts Here[END_REF] However, use of even the relatively simple techniques involved is not widespread or routine reflecting the difficult issue of non-dissemination of evidence-based techniques into everyday prevention practice. [START_REF] Jensen | Commentary: the next generation is overdue[END_REF] This has been attributed to many factors that increasingly appear to be intractable. [START_REF] Offord | The state of prevention and early intervention[END_REF][START_REF] Jensen | Commentary: the next generation is overdue[END_REF] Focusing on the evaluation of interventions that are already routinely implemented should in principle avoid this problem. Adapting an insight from the treatment field, evaluating and identifying what interventionists actually do successfully may be an alternative route to building effective and sustainable interventions. [START_REF] Westen | The empirical status of empirically supported psychotherapies: Assumptions, findings, and reporting in controlled clinical trials[END_REF] A major example from the UK is the nationally implemented Sure Start program which focused on enhancing the development of preschool children from poor areas. [START_REF] Belsky | The National Evaluation of Sure Start: does area-based early intervention work[END_REF] The program shares characteristics with other pre-school interventions that appear to have shown lasting benefits. [START_REF] Anderson | Uncovering gender differences in the effects of early intervention: a reevaluation of the Abecedarian, Perry preschool, and Early Training projects[END_REF][START_REF] Reynolds | Effects of school-based early intervention on adult health and well being[END_REF] However, the project as a whole proved difficult to evaluate as it lacked a control group and differed in content from centre to centre. [START_REF] Belsky | The National Evaluation of Sure Start: does area-based early intervention work[END_REF][START_REF] Melhuish | Variation in community intervention programmes and consequences for children and families: the example of Sure Start local programmes[END_REF] Further, such studies are rare: only 13% of the prevention studies in a metanalysis were routine interventions among which the authors retrieved only one randomized trial. [START_REF] Lipsey | The effectiveness of school based violence prevention programs for reducing disruptive and aggressive behavior[END_REF] A further issue concerns what precisely is the active ingredient in interventions. [START_REF] Jensen | Commentary: the next generation is overdue[END_REF] For instance, whether alone or within a more complex intervention group work is common to many intervention studies and is widely deployed in routine interventions. [START_REF] Wilson | The effects of school based intervention programs on aggressive behavior: a meta-analysis[END_REF][START_REF] Taylor | Going Round in Circles: Implementing and Learning from Circle Time[END_REF] However, groups may use a variety of techniques, such as educational [START_REF]Using the fast track randomized prevention trial to test the early starter model of the development of serious conduct problems[END_REF] , nurture or play [START_REF] Kolvin | Help Starts Here[END_REF] , behavioural [START_REF] Kellam | The course and malleability of aggressive behavior from early first grade into middle schools: results of a developmental epidemiologically based preventive trial[END_REF] or cognitive-behavioural [START_REF] Clarke | A randomized trial of a group cognitive intervention for preventing depression in adolescent offspring of depressed parents[END_REF][START_REF] Dadds | Early intervention and prevention of anxiety disorders in children: results of a two year follow up[END_REF][START_REF] Mcardle | Group therapy prevention of child mental health problems[END_REF] that are rarely compared.

Consequently, whether it is the precise theory-based content or the group intervention format that influences outcome is unknown.

This paper reports the two and three-year outcomes of a randomized controlled trial of a routine school based targeted intervention and includes a reanalysis of the initial fifteen months of the study. The study compares two group interventions: drama group therapy (DGT) with small group teaching of mathematics and English (curriculum studies -CS) on the self-, parent-and teacher-ratings of adjustment of symptomatic boys and girls. A charitable organization based in the North-East of England (http://www.learningchallenge.org.uk/) developed and trained teachers in the DGT techniques for routine use in a number of schools. In the evaluation, the DGT and CS interventions were delivered by a drama group therapy trained teacher. The study is therefore an evaluation of the effectiveness of existing routine practice. We hypothesized that compared to the CS intervention, the DGT would have a greater immediate and long term effect on behavioural and emotional symptoms as rated by teachers, parents and children. In the initial publication [START_REF] Mcardle | Group therapy prevention of child mental health problems[END_REF] at one year follow-up DGT was associated with a significantly greater decline in overall teacher-reported symptoms than waiting list or CS. Curriculum studies (CS) was initially conceived of simply as a control intervention but the data showed that it was more effective than waiting list control i.e. that it was a potentially active intervention.

Method

Participant selection

In order to identify children in mainstream schools at risk of emotional and behavioural problems, a series of criteria were agreed through discussion with teachers. These included one or more of the following criteria: (i) scholastic under-performance, (ii) known major family problems, (iii) ill-nourished or poorly cared for appearance, (iv) impaired peer relationships and (v) behavioural or emotional difficulties; a mixture of selective (i-iii) and indicated (iv-v) criteria. [START_REF] Offord | The state of prevention and early intervention[END_REF] The children attended primary, middle and high schools in a predominantly working class area in the North of England. Once identified, the schools approached parents to explain the intervention and the evaluation and to seek agreement for their children's participation. Contact was made by letter and included the offer of an interview with a designated member of staff. In addition meetings in each school provided forums for parents to hear more details and to discuss the project. None of the parents refused their child's initial participation. The Newcastle-upon-Tyne research ethics committee approved the project.

Numbers, randomisation and follow-up

Teachers identified 122 children meeting the selection criteria, approximately 40% of the total population available, from three primary (up to year 6), one middle (spanning year 5 through 8), and one high school (from year 7). The children were allocated by the school-teacher who knew them (but who did not know to which intervention they would be allocated) on the basis of a broadly equivalent mix of problems and severity of disturbance to blocks of eight. All groups included males and females. A member of the intervention team who did not know the children randomly allocated each block to DGT or CS. In order to ensure an adequate mix of problems or to avoid a particularly volatile grouping, in consultation with teaching staff, a total of eight children were exchanged post allocation between DGT and CS. The process of 'balancing' the groups was regarded by the research team as crucial in order to ensure viable groups. Both types of group occurred within each school. At the time of selection there were no significant differences between the problem scores of the two intervention groups (CS and DGT) according to teacher, parent or self-reports. The ages of the children in the CS (mean age 11.3 years, SD 1.3) and DGT groups (mean age 11.4 years, SD 1.1) were similar.

Over the course of the study the research team maintained links with the schools. Contact was maintained with parents through school meetings, letters and home visits. The interventions were presented as an additional optional service offered within the school. Parents were offered the opportunity to freely opt out which none did at the initial stage. Over the life of the project teachers were approached to complete questionnaires by key contacts in the schools, often the year head. The same process was used with the children who were approached discreetly to complete the questionnaire during the school day. Teachers were reimbursed for their time, parents and children received no incentives.

Blind ratings

Since the same therapist was responsible for both interventions, it is likely that teachers and parents were blind to intervention status. By one year follow-up most children had changed school or had changed teachers. For the two and three year follow-up it is likely that all teachers were blind to intervention status.

Intervention

The intervention was staggered in cohorts across three school terms. The DGT and CS groups each ran for an hour a week over a 12 week period. Groups were conducted during school hours. No child refused to participate in the groups. Interventions comprised drama group therapy -DGT 'essentially play, guided ... to ... the acquisition of personal and social skills', [START_REF] Stephenson | Use of drama[END_REF] comprising reflective discussion and role play conducted by a drama therapy trained teacher. Role play was central to the drama therapy and involved mini-plays using themes suggested by the children themselves. These included disciplinary episodes involving teachers or police, family difficulties and peer issues, seeking alternative practicable solutions that appeared more satisfactory from the perspective mainly of the children themselves. Sessions were structured through activity designed to reflect the stages of group formation and the perceived needs of the children. Each group therefore had a slightly different content, but all were structured towards the therapeutic re-enactment of 'live' situations reflecting the experience of the group. These brief role-plays were usually preceded by exercises of controlled emotional expression and with the intention of creating a holding structure for experience. Following the role play, the group would reflect on its dramatic qualities and 'truth'. Finally, and importantly, the role play was re-enacted to establish positive alternative outcome scenarios. These alternative scenarios comprise the analysis and psychoeducation of the group. The curriculum studies (CS) also took place in school. The children were encouraged in the formation of a positive peer group that focused on developing skills in mathematics and English. During these sessions no attempt was made to investigate any emotional/behavioural issues, but the pedagogy was strongly inclusive and supportive, with clear expectations for conduct. The trained teacher used focused praise for achievement of task outcome and for collaborative behaviours shown by the group.

Procedures for follow-up

Three major outcome domains were identified: i. The Teacher Report Form (TRF) is a widely used questionnaire in child and youth mental health clinical practice and research, generating data across areas of positive as well as negative adjustment. Scores are generated on eight subscales within two broad areas of externalizing (e.g. 'delinquent' or 'aggressive behavior') and internalizing behaviors (e.g. 'withdrawn, anxiousdepressed, somatic complaints'). [START_REF] Achenbach | Integrative Guide for the[END_REF] ii. The Youth Self-Report (YSR) is similar to the TRF, focusing on child-report symptoms. [START_REF] Achenbach | Integrative Guide for the[END_REF] iii. The parent-completed Child Behavior Checklist (CBCL) identifies parent perceptions of child symptoms. These questionnaires show good reliability and validity in US and European populations. [START_REF] Achenbach | Integrative Guide for the[END_REF][START_REF] Koot | Prevalence of problem behavior in Dutch children aged 2-3[END_REF][START_REF] Verhulst | Mental health in Dutch adolescents: self-reported competencies and problems for ages 11-18[END_REF] Attrition Figure 1. represents the numbers of children traced and for whom parent, teacher or self-report data were available. There were no significant differences in initial TRF, CBCL or YSR scores, caseness, gender, or school type between participants successfully traced at two and three years and those lost to follow-up. At two year and three year follow-up, some youths missing at an earlier phase were successfully re-traced. This meant that not all the cases seen at three-years had necessarily been involved in the two-year follow-up, and similarly for the two year follow-up data. If a parent refused to complete a follow-up questionnaire, we did not seek teacher or self reports, taking refusal to mean withdrawal from the trial. Of the 120 children initially entering the trial, 27 were lost to follow up due to moving away (8), leaving school (12) or refusing (7). Of those remaining, teacher data (the most complete) was available on 82% at two years and 69% at three years follow up.

Analyses

The current paper presents an analysis of intervention and three year follow-up data using latent growth modelling of questionnaire data via AMOS 16. [START_REF] Arbuckle | AMOS Version[END_REF] In contrast to the traditional statistical approach - A LGM with no covariates is called an unconditional model. However, as with standard regression, variables can be added to explain the variability in both intercept and slope outcomes -termed a covariate LGM. For example, if being male was positively associated with the intercept this would indicate a higher score at initial assessment among males; if positively associated with the slope, this would indicate that males' scores increased over time at a faster rate than girls'.

An unconditional LGM was fitted initially in order to assess the trajectory of internalizing and externalizing problems over the 3-years of the trial. To assess the impact of the intervention a second model with intervention status and two other relevant covariates, gender and school-type, were entered as dummy variables. Separate analysis was conducted using the total, internalizing, externalizing and all other questionnaire subscores. However because the subscore results were so similar to the internalizing /externalizing findings they are not reported here, but are available upon request from one of the authors (RY). Several measures of fit were used to assess the models; including the chi-square and the comparative fit index (CFI). [START_REF] Bentler | Comparative fit indexes in structural models[END_REF] Adjustments for missing data were incorporated by using the full information maximum likelihood procedure for estimation. [START_REF] Arbuckle | AMOS Version[END_REF] The TRF data were the most complete and analysis focused on this outcome. In LGMs, loadings (regression paths) from each TRF measurement to the intercept are fixed to 1; this provides an estimate of the underlying baseline TRF score. The paths between TRF measures and the slope are used to model the expected trajectory over time. Conventionally in a linear LGM these paths are fixed at intervals which reflect the passage of time. For example, the time-gap between the 2 nd and 5 th wave is 1-year and that between 2 nd and 6 th wave 2-years, typically we would fix these paths to 1 and 2 respectively in order to scale the slope to represent the expected change over a year; a relatively easy to interpret scale. Time-1 (beginning of waiting list) and time-2 (immediately pre-intervention) slope paths (also called loadings) should be fixed at zero, since both measures took place before the intervention, and were taken only a month apart. This approach suggests the most appropriate fixed slope loadings for t1 to t7 would be, 0, 0, 0.25, 0.5, 1, 2, 3 respectively. However, initial modelling of the TRF trajectories using these fixed parameters produced poor results, indicating that children's internalizing and externalizing problems followed a distinctly non-linear trajectory.

In order to incorporate this non-linearity the slope parameter estimates for loadings at year-one and slope were fixed at 1 and pre-intervention time-points loadings fixed at zero, and the remaining slope loading parameters freely estimated. Thus, the overall slope estimate remained interpretable on a changeper-year scale. The discrepancy between the obtained values of the freely estimated path loadings and their expected linear values can be used to evaluate the shape of the trajectory, or more specifically how it differs from a linear course. For example, the loading for 6-months should be approximately 0.5 on a linear trajectory -to indicate half a year's passage of time: if the estimate obtained is 0.75, this indicates an accelerated trajectory roughly 50% faster than that predicted from a linear course. Unconditional LGM's produced models only trivially short of conventional acceptability (CFI, 0.88-0.89). Given the small sample, multiple time-points, non-linearity and missing data, this was deemed acceptable. A generic path diagram with conventional LGM notation for the unconditional and covariate

LGM is shown in fig. 2a and2b respectively.

Place figure 2a,2b. here

RESULTS

Unconditional Growth model

Total TRF score: The results for the unconditional LGM are reported in table 2. In relation to baseline scores a mean (SD) intercept of 36.84 (2.64) TRF points was estimated, with the slope indicating that scores significantly decreased over time by approximately -17.59 (0.65) points per year. Caution must be used when interpreting these results because the decline over time is distinctly non-linear, with an accelerated decline from time zero to three months (from beginning to end of the intervention) equivalent to that expected from half-a-year's linear trajectory (i.e. a slope parameter estimate of 0.45 vs. 0.25 expected for a 3 months linear decline, table 2, 2 nd column, last row). The slope begins to level out at one year with no further substantive change. Thus, a more accurate description of the model would be a rapid decline of -17.59 points over the first year which stabilizes so that after year-one the trajectory is relatively flat. This is illustrated clearly in fig- 3, which shows the average TRF t-score over time for both DGT and CS groups.

There was a strong negative correlation (r=-0.79) between slope and intercept, indicating that individuals with higher baseline TRF scores demonstrated a more rapid reduction in mental health problems over the following years.

Place table 3 (if included) and figure 3 here

Covariate Growth Model

Total TRF score: adjusting for covariates (see table 2) explained a significant amount of the variation in baseline TRF scores, reducing the variance of the intercept from 791.07 (109.42) to 549.06 (77.61). Including covariates reduced the mean TRF intercept to 23.28 (4.32) points, which is interpretable as the baseline TRF score for girls in elementary school who did not take part in DGT. Results suggest that being male and in high school was associated with a significant increase in baseline score of 13.36 (4.45) and 27.31 (5.26) TRF points respectively, while being in middle school significantly lowered baseline scores by -12.12 (5.03) points. As would be expected if assignment were random, intervention status was not associated with baseline scores.

Including the covariates in the LGM explained much (but not all), of the change in TRF score over time, reducing the variance of the slope from 661.12 (114.19) to 413. 45 (78.45) and the mean slope from to -17.59 (0.65) to -11.15 (4.17) points decline per year. This suggests a significant proportion of the decline in TRF scores over time can be explained by age, since gender was unrelated to the slope. In comparison to primary school, beginning the intervention in middle school was associated with a significant increase in slope of 21.16 (4.90) TRF points, whereas starting in high school resulted in a significant decrease of -19.05 (5.10) points. In other words, during middle school teachers report children show worsening, while those in high school show improving mental health, independent of any intervention effects. The DGT intervention had a significant effect on the slope, reducing it by -9.67 (4.26) points. Again almost all of this reduction took place over the first year then stabilized over the following two years. When the analyses were repeated for externalizing and internalizing subscores, the effect of DGT intervention significantly reduced only externalizing scores by -3.96 (1.74) points over the first year. The intervention findings from analysis of parent and self-report questionnaires were not statistically significant.

DISCUSSION

A previous paper described this randomized controlled trial and presented waiting list, intervention and 12 month follow-up data. The current paper presents a reanalysis and includes the two and three years follow up data. This trial contrasted two targeted routine small-group group interventions: a drama group therapy (DGT) intervention and a curriculum studies group (CS) both led by a drama group trained teacher.

The format (small groups conducted in schools) and the dose (weekly sessions over a school term) were similar and both were conducted by the same teacher but the group work content differed. Out hypothesis that DGT would have a greater immediate and long term effect on behavioural and emotional symptoms was only partially proved.

Analyses show a decline in TRF scores during and immediately after intervention for the sample as a whole and that low scores were maintained through follow-up. They also confirm a more rapid decline in TRF scores in the DGT compared with CS groups. These findings show that this routine DGT intervention was effective in rapidly improving behavioural adjustment in schools. In contrast to subject teaching, the DGT emphasis on structured interaction, problem solving and role play initiated by the participants, characteristics similar to those of 'cognitively-oriented programs', was a primary difference between the two interventions and may have contributed to this accelerated change. [START_REF] Lipsey | The effectiveness of school based violence prevention programs for reducing disruptive and aggressive behavior[END_REF] Also, consistent with an intervention effect, the DGT effect appeared only subsequent to the waiting list control period. In addition, natural history studies show that externalizing problems follow a chronic or gradually declining course and not the relatively abrupt desistence reported here. [START_REF] Nagin | Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency[END_REF][START_REF] Bongers | Developmental trajectories of externalizing behaviours in childhood and adolescence[END_REF][START_REF] Odgers | Female and male antisocial trajectories: from childhood origins to adult outcomes[END_REF] The data show that after their initial decline from just below the clinical cut-off (t-score 70), scores for both groups remained within the non-clinical range throughout follow-up. This common long term outcome suggests that that the effects were partly attributable to aspects that were shared by the interventions. These shared group processes are likely to include routine meetings of a diverse group of young people that might not otherwise have repeatedly met, structured supervised interaction, limited elements of self-expression, befriending facilitated by an interested adult with characteristics previously shown to be of therapeutic value, empathy but also assertiveness. [START_REF] Kolvin | Help Starts Here[END_REF] Although it was not detected by the instruments used, it is possible some enhancement of social skills may have contributed. [START_REF] Mytton | School-based secondary prevention programmes for preventing violence[END_REF][START_REF] Bernat | The Early Risers preventive intervention: testing for six-year outcomes and mediational processes[END_REF] However, long term reductions of externalizing symptoms following group interventions have also been attributed to sustained change in peer networks, that post-intervention include non-disruptive peers, mediating sustained reduction of externalizing behavior. [START_REF] Vitaro | Disruptive behavior, peer association and conduct disorder: testing the developmental links through early intervention[END_REF] This peer group process could explain the greater group work effect on externalizing than internalizing behavior reported here and elsewhere. [START_REF] Kolvin | Help Starts Here[END_REF] The data also showed that the higher the initial problem score, the steeper was the decline, consistent with findings elsewhere. [START_REF] Hawkins | Reducing early childhood aggression: results of a primary prevention program[END_REF][START_REF] Reid | Description and Immediate Impacts of a Preventive Intervention for Conduct Problems[END_REF][START_REF] Reid | Enhancing a classroom social competence and problemsolving curriculum by offering parent training to families of moderate-to high-risk elementary school children[END_REF][START_REF] Stoolmiller | Detecting and describing preventive intervention effects in a universal school based randomized trial targeting delinquent and violent behavior[END_REF][START_REF] August | An integrated components preventive intervention for aggressive elementary school children: the early risers program[END_REF] This may be attributable to so-called 'discrepancy proportional peer influence' so that the behaviour of those at the extremes is drawn toward the group mean. [START_REF] Boxer | Proximal peer-level effects of a small group selected prevention on aggression in elementary school children: an investigation of the peer contagion hypothesis[END_REF] If this is true, it requires the presence of some less overtly deviant peers in the therapeutic group. Their absence could explain the failure of an intervention involving a group of children all of whom scored at or above the 95 th centile for teacher-rated aggression. [START_REF] Smolkowski | Schools and homes in partnership (SHIP): long terms effects of a preventive intervention focused on social behavior and reading skill in early elementary school[END_REF] Indeed, if association with non-deviant peers is important in sustaining change, it seems crucial that all the participants do not show a high level of aggression.

Irrespective of which type of intervention received, starting the trial in high school was associated with a larger (but still negative) slope i.e. a more rapid decline in symptoms. In light of meta-analytic data showing that older children and especially older higher risk children showed lessened effects of preventive intervention, [START_REF] Lipsey | The effectiveness of school based violence prevention programs for reducing disruptive and aggressive behavior[END_REF] this is surprising and could represent unmeasured differences between schools. [START_REF] Rutter | Fifteen thousand hours: secondary schools and their effects on children[END_REF] In keeping with meta-analyses, there was no effect of gender on outcome. [START_REF] Wilson | The effects of school based intervention programs on aggressive behavior: a meta-analysis[END_REF] No effect was found on parent or self-report. This may be in part because parent-and self-report data were less complete so that statistical power was less. It may also mean that the teacher-reported effect was not strongly mediated by changes in home-life or in subjective-well being. This fits with the view that effects of intervention tend to be 'domain specific' and that more complex, intensive or sustained intervention may be required to effect change elsewhere than at school. [START_REF]Using the fast track randomized prevention trial to test the early starter model of the development of serious conduct problems[END_REF] It is consistent too with the possibility that early-adolescence is not the optimal time for intervention if it is to have widespread effects, unlike action taken pre-school. [START_REF] Anderson | Uncovering gender differences in the effects of early intervention: a reevaluation of the Abecedarian, Perry preschool, and Early Training projects[END_REF] Nevertheless, as peer associations appear central to so many adverse outcomes, if alterable by group interventions, multiple benefits beyond the school could potentially accrue, even at a relatively late age and even if not detected in this small trial.

Finally, group work can use a variety of techniques and, as demonstrated here, young people can benefit from quite disparate group techniques. As long as attention is paid to fundamentals such as group mix, the capacity of a reliable group leader to engage young people, and perhaps dose (i.e. the number of sessions), this means that effective group interventions may be a particularly accessible and flexible means of targeted intervention. In an era of great difficulty for young people, they should be more widely deployed.

Limitations of the study

This study identified approximately 40% of the population as at-risk, reflecting our hybrid selective and indicated criteria. Because of the difficulty maintaining teachers blind to interventions, especially in small primary schools, it is possible that teachers were not blind to intervention status at initial one year follow-up. However, since the same therapist had conducted both interventions, it is likely that teachers were blind even at this stage. Nevertheless, we repeated the analyses for the last cohort who received their intervention during the final term of the school year. For this cohort the six months follow-up evaluation was conducted by their following year teacher who was likely to have been blind to whether the child had received DGT or CS or indeed any intervention. The findings remained essentially unchanged. It is very unlikely that by two and three year follow-up teachers were aware of intervention status. Change could also be partly attributable to regression to the mean. However, the absence of significant change over the waiting list period (McArdle et al 2002) suggests that this is unlikely to have made a significant contribution.

Conclusions

This study shows that a routine intervention achieved significant and sustained change in behavioural adjustment in school and that the greatest effect was evident for those with the most abnormal initial scores. It showed that a comparison group intervention with different content and process was associated with a slower onset of action but convergent outcomes three years post intervention. The findings are consistent with the possibility that, irrespective of differences in group process, sustained change following group therapy is partly or largely due to aspects of group intervention that are shared. Nevertheless, in the short term, the DGT exerted a more rapid onset of effect, suggesting that a more explicitly empathic, problem-solving and interactive process facilitates more rapid change. The findings suggest that professional-led school-based brief group work is an important public health technique.

1

  Early intervention for at risk children: three year follow up Paul McArdle, Robert Young, Toby Quibell, David Moseley, Rob Johnson, Ann LeCouteur.

  which compares group averages of an outcome at each time point -the latent growth model (LGM) focuses on an individual's trajectory over a period of time. It summarises this trajectory into two basic components; an individual's underlying baseline score at the beginning of the process -termed the intercept, and an individual's rate of increase or decrease in score over a period of time -termed the slope. Thus an individual's progress at any point can be summarized by knowing their initial intercept, slope (rate of change per unit of time) and the time since baseline assessment. Further, because these estimates are based on multiple measures at multiple time-points measurement error is reduced. More complicated LGMs can accommodate non-linearity in the slope component, for example it is possible to model exponential growth or step changes in trajectory sometimes seen in developmental processes.

Figure 3 .

 3 Figure 3. Teacher Report Form score over time by intervention.

  Table 1 shows the gender and age of the children at different follow-up points.

	Place figure 1. here
	Place table 1. here
	Place table 2 here

(Time 1-7)[START_REF] Costello | Impaired but undiagnosed[END_REF] 1 Expected linear slope estimates for t1 to t7 (0, 0, 0.1, 0.25, 0.5, 1, 2, 3). Note: Bold fixed values, other parameters are estimated. CFI = Comparative fit index. * p ≤ .05, ** p ≤ .01, *** p ≤ .001

Acknowledgements: Northern and Yorkshire NHS R&D; the authors are grateful to the schools, teachers, young people and their families without whose support this study would not have been possible. Robert Young is supported financially by Medical Research Council of Great Britain as part of the Youth and Health Programme (WBS U.1300.00.007) at the Social and Public Health Sciences Unit.