As with the PCA results, cows appeared fairly evenly spread along this linear object, with no clear clustering to suggest social cohesion amongst large or temporally persistent subgroups. Comparing these results with the embedding of the permutated queue records , no clear geometric features were recovered from data simulated from a purely random queueing strategy. This reinforced that the linearity of the observed records was not simply an artifact of the physical linearization of cows as they enter the parlor single-file, but a reflection of a consistent pattern in queue formation that might be driven by some underlying behavioral or biological mechanism. While the diffusion map embeddings convey a clear linear geometry, there was also unexplained curvature in the band along which cows were projected. This proved not to be an inherent feature of the data itself but a harmonic artifact imposed by the spectral value decomposition of the graph Laplacian used to deduce the shape of the underlying network between cows . Such a mathematical operation has several physical interpretations. As a result, each axis of the subsequent embedding contains an element of the harmonic series, producing the curvature seen in these milk order visualizations.
Fortunately, this artifact can be described by closed form equations and imposed onto the data to aid in discerning authentic geometric features of the data . Thus, trim trays while diffusion map did provide a clearer geometric representation of the inherent linearity of this dataset than PCA, this dataset al.so reinforces that modern manifold learning techniques are also not infallible in recovering the underlying geometry of high dimensional data. While such embedding techniques may provide helpful insights into the underlying structure of large datasets, a conservative approach to visual interpretation of such results is still warranted.Independent visualization of parlor entry records from each individual cow revealed that the majority of animals in this sample were surprisingly stationary in their queueing position. Animals that frequented the front and end of the queue, being more consistent in their entry position, provided clearer visual evidence for a lack of temporal trend. Cows in the middle of the queue showed far greater variability in their entry positions, making it more difficult to visually discern temporal trend from stochastic fluctuations. Only two animals were identified as having a clearly visible trend: cow 13467, who had no recorded health events, and cow 13826, who was diagnosed with metritis during the enrollment phase early in the trial.
Both cows showed similar trajectories, starting nearer the end of the herd and moving progressively forward towards the front, but neither change in queue position coincided with the shift to overnight pasture access. This consistency in queue position was further reflected in a clear linear association between median entry quantiles from overnight pen and pasture subperiods . A slightly wider spread was discernable amongst cows occupying the middle ranks, but for the majority of animals, median entry quantile values did not change more than r0.2. Among the handful of animals demonstrating a more extreme shift, these jumps tended to be in the forward direction towards the head of the queue. Overall, fewer extreme shifts were seen in this dataset than in a similar bivariate meansplot provided in Beggs et al. , though this may simply be a reflection of the longer subperiods over which median entry positions were assessed. Correlations between these values were also quite high, with a Pearson correlation estimate of 0.91 and a Kendal Tau estimate of 0.74 . These values are, as expected, higher than the estimates of consistency reported for individual milk order samples , but on par with results using subperiod averages on similar time scales . Given the extreme shift in management routine spanning these two subperiods, however, this level of stability in parlor entry positions was an unexpected result. Such resilience to changes in overnight housing environment and the subsequent distance traversed to access the parlor could suggest that milking order is largely determined in the crowd pen, a result supported by early observations by Soffie , who reported little correlation between the order of cows exiting the home pen and entering the parlor past the first few animals.
Collective assessment of entry quantile records using data mechanics visualizations did, however, reveal additional temporal features not identified using independent visualizations of cow records or collective assessment of aggregate records. The first and perhaps must surprising insight was that, with finer granularity in number clusters applied to the temporal axis, data mechanics identified several days with anomalous queuing patterns. In Figure 5, a total of 8 column clusters are imposed without any social stratification on the subset of cows with no health events. If these records were completely stationary with no temporal effects, we would expect days to be randomly partitioned into these eight categories. Instead four days are isolated from the remaining observations. Days 85 and 91 are separated into clusters of size n = 1, and 89 and 91 are also isolated into their own cluster of size n = 2. Looking from left to right along the heat map to identify temporal heterogeneity, it is easy to see that on these observation days animals typically occupying the extremes of the queue appear to have been pushed towards the center and animals typically found in the center of the herd were either pushed towards the extremes or inverted their tendency to stay towards the front or end of this middle section of the queue. While some of the entry quantile values encompassed by these observation days would likely be identified as outliers for individual cows, other values would likely be deemed irregular but not worthy of exclusion. These clustering results, on the other hand, suggest that either transient environmental or internal social factors have disrupted the entire herd and caused them to collectively respond with highly irregular queuing patterns. As the row axis is stratified to allow for nonhomogeneous temporal responses across subsets of animals, these same days are consistently isolated from the remainder of the dataset, reinforcing that these observations constitute an outlier that should be excluded from any downstream analyses.Looking next at the coarser stratifications of the temporal axis, we also see that pen and pasture observations are not equally dispersed among the column clusters. As the animal axis is more finely stratified to allow for social heterogeneity within the herd, the source of the temporal heterogeneity between these two environments comes into resolution. In Figure 6, which contains entry quantile observations on both sick and healthy animals, pen and pasture observations are perfectly stratified across only two column clusters. Looking at the subsets of animals who consistently entered at the front and rear of the herd, entry quantile values appear quite homogenous in color between the two temporal clusters. Scanning from left to right among the subgroups of animals that frequented the center of the queue, on the other hand, trimming tray systematic fluctuations in daily entry quantile values can be seen even without finer temporal stratification. This pattern is clearest in the cluster which contains both cow 13467 and cow 13826 ±the two animals identified by independent inspection of cow entry quantile plots to show evidence of non-stationarity. In this subgroup, cows showed a tendency to frequent the latter half of the queue when coming to the milking parlor from the home pen, but during the pasture period showed progressively greater proclivity to enter in the front half of the queue.
Where this shift is the most uniform in the latter half of the pasture period, we also see a compensatory patternin the subgroup directly above, where cows shifted from nearer the front to the back half of the queue. Whether these results reflect the coordinated movement of relatively small social subgroups or just a common response to environmental conditions is impossible to say from this data alone. These results do make it clear, however, that not only are the cows occupying the center of the queue less consistently in their entry position, they are also less stable in their entry pattern. Further, these visualizations underscore that these divergent dynamics in the pen and pasture subperiods cannot be captured by a simple fixed effect term. The simplest option would be to drop from the analysis the animals that show the strongest nonstationary patterns. With such a large group, this would still leave ample observations to maintain statistical power, but could risk biasing the subsequent inferences. Alternatively, by specifying a heterogeneous variance model between animals, as was deemed necessary in the original entropy plots, the influence of these cows on the fitted model may be reduced sufficiently that deviations from the assumption of stationarity in this subgroup might not unduly destabilize the final model. Finally, some preliminary insights can be gleaned from the cow attributes added to the row margins of both heat maps. In Figure 6, animals with documented health events appear fairly evenly dispersed across subsections of the queue. A slightly lower rate of illness might be attributed to animals that consistently occupied the very front of the queue, and perhaps a marginally higher rate of transition diseases was seen in the animals at the very rear of the queue, but these patterns appear subtle at best and thus likely not the only determinant of queue position. This result was somewhat surprising, as previous research has suggested that sick animals tend to populate the rear of the queue . If this previously reported trend is driven by a reluctance among animals in the acute phases of a disease to move, it is possible that the daily health checks prescribed in this experimental trial succeeded in identifying and removing sick animals from the herd sufficiently early that this behavioral mechanism was not at play in this dataset.This might suggest that inclusion of these additional animals into subsequent analyses might not unduly bias subsequent behavioral inferences. Of perhaps greater concern to subsequent modeling is the lack of clear color gradients among cows attribute values across the queue, which could indicate that underlying associations may either be weak or that there are complex interaction effects creating a nonuniform trend.In contrasting the results of these two models, the loss of significant association between entry quantile values and peak yield with the addition of sick animals is perhaps not surprising. If a disease challenge early in the trial curtailed peak lactation in these cows but did not cause chronically deficient production, then the 95-th quantile value of milk yield used here to estimate peak lactation level maynot adequately reflected the overall productivity of these animals across the duration of this extended trial, obscuring the underlying biological mechanism. The emergence of a nearly significant association between entry quantile and age with the addition sick animals, however, is more difficult to explain. In either case, a relatively small number of animals may be unduly influencing statistical inferences. Visual examination of predicted queue positions plotted against age and peak yield for both the full dataset and healthy subset seem to confirm these misgivings. Looking first at age, the first lactation heifers, being evenly spread across the center of the queue, cannot be driving this linear effect. Among the multi-parous animals, the five cows seen consistently in the front of the queue are indeed among the oldest in the herd, but if this handful of animals and their corresponding queue positions are ignored, a clear gradient is not visible among the remaining cows. Results of the mutual conditional entropy tests confirm this suspicion. For the full dataset, where the linear effect is marginally significant, the MCE test does not . This suggests that either that age effect is only discernable after adjusting for peak yield or that the association is not robust.This result suggests that the MCE tests may also be used in mixed modeling analyses to recover nonrandom patterns that are not well-represented by linear trends. Such a nonlinear trend here could belie more complex interaction effects between these or other unmeasured biological drivers of queue position.Contextualizing these results within the existing base of literature underscores the inconsistency in drivers of queuing behaviors. With respect to milk yield, several studies have found no significant association , but among those that have, most have reported high yielding cows frequent the front of the queue. In early studies, cows were offered concentrate in the milking parlor, which may have increased the motivation of high yielding animals with greater energy deficiencies to enter the parlor.