How to use (and not to use) movement‐based indices for quantifying foraging behaviour

Abstract Movement‐based indices such as moves per minute (MPM) and proportion time moving (PTM) are common methodologies to quantify foraging behaviour. We explore fundamental drawbacks of these indices that question the ways scientists have been using them and propose new solutions. To do so, we combined analytical and simulation models with lizards foraging data at the individual and species levels. We found that the maximal value of MPM is constrained by the minimal durations of moves and stops. As a result, foragers that rarely move and those that rarely stop are bounded to similar low MPM values. This implies that (1) MPM has very little meaning when used alone, (2) MPM and PTM are interdependent, and (3) certain areas in the MPM‐PTM plane cannot be occupied. We also found that MPM suffers from inaccuracy and imprecision. We introduced a new bias correction formula for already published MPM data, and a novel index of changes per minute (CPM) that uses the frequency of changes between move and stop bouts. CPM is very similar to MPM, but does not suffer from bias. Finally, we suggested a new foraging plane of average move and average stop durations. We hope that our guidelines of how to use (and not to use) movement‐based indices will add rigor to the study of animals’ foraging behaviour.

expose fundamental drawbacks of MPM that seriously question the ways scientists have been using it for more than three decades, and propose guidelines to avoid these pitfalls.

| The range of MPM values is determined by PTM
The first and most fundamental drawback of MPM is that sit-andwait animals that rarely move and active foragers that rarely stop    (Sibly, Nott, & Fletcher, 1990;Yeates, Tolkamp, Allcroft, & Kyriazakis, 2001).

| MPM is inherently biased
Moves per minute suffers from another methodological drawback of intrinsic inaccuracy due to the fact that movements have a continuous duration, but the number of movements is discrete. That is to say, the number of discrete movements counted in a given observation may include just fractions of movement bouts, at the beginning and end of the observation. This leads to MPM values that never converge to the true movement frequency, regardless of the sample size. To clarify this issue, let us consider a species that has a distinct movement pattern of two brief stops during 5 min (as in Box 1). This means that in 5 min this species conducts two short breaks and three movements; in 10 min it conducts four breaks and five movements; in 20 min-eight breaks and nine movements, etc. Therefore, the estimate of MPM, M PM, of the 5-min observations (0.6) neither equals to that of the 10min (0.5) nor to that of the 20-min observations (0.45) (See Figure S1 for numerical example of M PM causes of bias).
Deriving a model for the bias, we found that the relative bias in MPM is given by the formula: where AM is the average move duration and OD is the observation duration. Please consult appendix III for detailed explanations and derivations of this expression, and Figure 3a,b for simulations confirming it. This formula indicates that the relative bias is independent of the average stop duration (AS) and linearly dependent on AM and on 1/OD. Hence, the bias will be considerable for short observations of animals with long move durations.
To test whether this bias indeed occurs in real foraging data, we used our records of the movement sequences of lacertid lizards. We In (b), we use S = 20 and study the effect of M and OD on relative bias. In (c), we calculated the ratio of the variance of CPM/2 to the variance of MPM. The bold black line is the contour 1, above which MPM has lower variance, which rarely happens. For each parameter combination we generated a long movement sequence of 2·10 6 s and calculated its MPM, then sampled from it 10 4 short sequences and calculated their MPM. The variance of these MPMs was used as the error in (c) and their mean was compared to the MPM of the long sequence to determine bias in (a)  Reported MPM values are expected to suffer not just from intrinsic bias but also from imprecision. As mentioned above, the inherent MPM bias largely depends on the OD. Yet, OD may differ substantially between studies (Perry, 2007; see appendix IV). Moreover, in many studies, researchers have pooled observations of different durations to characterize the foraging behaviour of a single species, and reported only the minimal observation duration or the average duration (e.g., Cooper, Vitt, Caldwell, & Fox, 2001;Huey & Pianka, 1981;Sales & Freire, 2015). Pooling together MPM estimations that are based on various ODs, hence including different biases, may increase the errors in the estimation of summary statistics such as species-average MPM.
Together, the problems of inaccuracy and imprecision add yet another question mark to the validity of MPM as a reliable index for foraging behaviour. Moves per minute is often used complementarily with PTM to reveal variation in foraging behaviours that cannot be identified using PTM alone (Cooper, 2005b;Perry, 2007;Perry et al., 1990). An acknowledged limitation of PTM is that animals can have identical PTM values while using very different foraging behaviours (Perry et al., 1990). The histogram of these ratios for the 134 sequences is shown. Note that CPM/2 is unbiased and has lower variance than MPM, and MPM′ is also unbiased with roughly the same distribution or Fig. 1 in Cooper (2005b) ("frequent, very brief movements relative to pauses" and "frequent, brief movement-briefer pauses"), are not possible ( Figure 5). This means that the absence of species from these areas of the MPM-PTM plane does not reflect data deficiency (as was previously suggested, see Cooper, 2005b), or selective pressures against these strategies, but rather it stems from an intrinsic methodological limitation.

| IMPLICATIONS OF MPM DRAWBACKS TO DATA INTERPRETATION
As we show in Appendix III, the bias in MPM estimations is particularly substantial for short observation of animals with long move durations ( Figure 3). Many studies have used observation durations that were shorter than 5 min, at times as short as 1 min. For example, our thorough literature review of lizards' foraging behaviour revealed that in 83 of 118 studies for which minimal OD was reported, OD min was equal or smaller than 3 min (Median OD min = 2.18; Appendix IV). Perry (2007) found that shorter observations often produce relatively high values of MPM and greater variability than longer observations. He explained this finding by claiming that short observations only sample part of the behavioural repertoire of the animal, and suggested conducting longer observations, especially for species with intermittent locomotion. We add that higher MPM values in shorter observations could result purely from the way the index is being calculated, and that this problem may be relevant especially for highly active species (see Figure 3).

| Bias-corrected estimator for MPM
Our second goal was to suggest ways to resolve some of the above- Since the harmonic mean is never larger than the arithmetic mean, this correction factor will always correct some (or all) of the bias, but never create extra bias. Our numerical simulations confirmed this result (Appendix III). We can, therefore, recommend using the MPM′ correction even when only study-level averages are available.
We applied the MPM′ correction to all published studies on lizards' foraging behaviour from which we were able to extract the relevant data (see Appendix IV for details). While in many cases MPM′ differs from MPM in <1% (as we expected, since most lizard species for which MPM, PTM and OD are currently available are sit-and-wait foragers), in 9 of 98 studies the difference is >10% and in one study the difference is 38%. The corrected MPM′ values for all published studies on lizards' foraging behaviour are provided in appendix IV. We encourage future comparative studies on lizards' foraging behaviour to use these MPM′ values rather than MPM values (Appendix IV).
Our MPM bias correction relies on several assumptions, and particularly that there is no correlation between the duration of a move or stop and other moves and stops. Thus, we recommend using this expression only to correct published results for which the movements'

| A new movement frequency index-CPM
Changes per minute is calculated by dividing the number of ob-

| AM-AS plane
While MPM′ and CPM seem to resolve the intrinsic inaccuracy of MPM, they cannot resolve the inherent triangle-like limitations of the MPM-PTM plane, and their implications for subsequent analyses.
Thus, we suggest using a plane of average move duration (AM) and average stop duration (AS) as an alternative approach. Cooper (2005a) examined AM (i.e. AD in the original paper) as an additional foraging index to MPM or PTM. We, instead, focus on the strengths and weaknesses of using the AM-AS foraging plane. Figure 2 illustrates the correspondence between the MPM-PTM and AM-AS planes using simulated, species level (based on Cooper, 2005a) and individual foraging data. It is important to note that the AM (calculated by Cooper, 2005a) and AS values at the species levels were calculated using reported averages of MPM, PTM and OD. Consequently, these rough estimations of AM and AS may suffer from inflated inaccuracy and imprecision that may render biological analyses that use them invalid. Nonetheless, we decided that with no other data in hand, these species-level estimations can still be valuable for exploring the pros and cons of the AM-AS plane. First, as opposed to MPM-PTM, the AM and AS axes are methodologically independent. Hence, foraging strategies can be assigned to any part of the plane, as depicted in Figure 5. Thus, any correlation found between AM and AS may reflect meaningful biological information. We think that the AM-AS plane is favourable because AM and AS are easy to interpret, and do not suffer from inherent constraints or intrinsic biases. This analytical approach is useful especially for exploring evolutionary-ecological aspects of movement behaviour across related taxa by methods of cluster analyses.
Despite these advantages, we want to emphasize three limitations of the AM-AS plane approach. First, there is no single axis that defines foraging strategies and is capable of replacing PTM. Species-level data in Figure 2 show that while sit-and-wait and active foragers (defined as below and above PTM of 10%, respectively, as in Cooper et al., 2001) are distinct on the AM-AS plain, neither of these variables alone separates them and can be used as a stand-alone index in comparative analyses. Second, since the number of moves and stops is usually not very large even for active foragers, and since it is necessary to drop the edges of the sampling sequence (because the entire duration of these first/last move/stop is not sampled), sample size for these variables is smaller than for PTM, which considers every unit of sampled time.
Third, as opposed to PTM and MPM, the AM-AS plane approach uses absolute values rather than standardized values. Thus, this approach may impede comparisons between unrelated taxa that differ in body size or the environment they inhabit.

| GUIDELINES FOR FUTURE USE OF MPM
To encourage better use of MPM in future behavioural studies, we provide guidelines of how to use (and not to use) this index.  (Dias, Rangel-Negrín, Coyohua-Fuentes, & Canales-Espinosa, 2009;Hawlena et al., 2006).

5.
Last, the values of M min and S min as determined by the observer along with the exact OD should always be reported.

| CONCLUDING REMARKS
Movement-based indices, such as MPM and PTM, are simple, intuitive and easy to measure. Thus, these indices have been used extensively to depict and study foraging behaviour across species and systems. Yet, MPM suffers from major drawbacks that must be acknowledged to prevent misuse. We demonstrated that MPM values are constrained by the minimal move and stop durations, leading to similar low values for both active and sit-and-wait foragers.
Also, we showed that this index suffers from intrinsic inaccuracy and imprecision. To assist avoiding these pitfalls, we developed a new bias correction formula for already published MPM data.
When raw data on moves and stops are available, we proposed using a novel index of changes per minute (CPM) that is very similar to MPM, but does not suffer from bias and inflated error. It is important to note that MPM′ and CPM are similar in their interpretation and converge to the same value as MPM when the observation duration increases (up to the bias of MPM). Hence, previously published results using MPM (corrected using MPM' when possible) can be easily compared with new results using CPM/2. We also suggested a new foraging plane of average move and average stop durations that resolves some of the inherent limitations of the MPM-PTM plane. We want to emphasize that our goal is certainly not undermining the use of frequency-based indices to study foraging behaviour. On the contrary, we believe that such simple and comparable movement-based indices are still very useful to explore ecological and evolutionary aspects of foraging behaviours, especially in comparative studies. We hope that our work will add rigor to these attempts by assisting researchers to avoid common methodological pitfalls that can seriously affect further development of this important field.