We evaluated models to fit individual patterns of perceived leg exertion as a function of exercise intensity expressed as proportion of Wmax. Four models were studied, a power model based on the Borg scale, a linear model with a delay, and extension of these two models when introducing a delay in the power model, and a delay model with a linear and a quadratic term. Our main findings were that models incorporating both delay and curvature had the smallest error term (RMSE) and were flexible enough to fit varying individual trajectories. Patterns in the individual trajectories can be illustrated by using functional clusters which identified linear and quadratic patterns, distinguished by the rate of rise in perceived leg exertion and the size of the delay as work became progressively harder.
Borg and Kaijser  also estimated individual curves for perceived exertion but in the case of adults. The mean exponent was 1.2 (SD 0.4), but closer to 1 when an intercept term was used. The individual curves for adults were more homogeneous and mostly linear and thus were comparable between adults. In contrast individual curves for children from our study varied widely and could exhibit a linear trend or rise as a power function with or without a delay.
A delay in the model arises when ratings of perceived exertion stay relatively constant before an estimated threshold at some percentage of Wmax. This may indicate a lack of ability in gradation or serialization for rating perceived exertion in our pediatric population. The intercept can be interpreted as a mathematical construct for a better fit or a threshold for a baseline or resting level of sensation. Both power‐delay and quadratic‐delay models are able to represent linear or quadratic growth in ratings. The exponent in the power model depends on the multiplicative constant to obtain a fit, so that large exponents are balanced with small coefficients for the power function. A quadratic‐delay model simplifies the interpretation, since it has a fixed exponent, and only the coefficients of the quadratic and linear terms need to be estimated. If this is coefficient is zero, then the curve is linear, non‐zero coefficients measure the strength of the curvature. It is also possible to use the quadratic delay model to estimate growth curves with an inflection, for example when ratings increase sharply and then level off. The computational burden for the quadratic delay model was much smaller than for the power‐delay model. In the mixed effects model with varying delays this amounted to 17 minutes versus 100 minutes CPU time. Otherwise the model fits for power‐delay model and quadratic‐delay models were comparable in terms of RMSE, AICc, BIC, DIC. Aggregated reporting in studies relating stimulus with response can obscure individual differences. The choice of mixed effects models have to be carefully considered and are more meaningful when estimating varying delays. However mixed effects models do not allow the estimation of individual curves. Borg and Kaijser , who opted for estimating individual curves in 20 adults, noted one individual with an exponent >2.8 in the respective power function. Individual differences were even more pronounced in our study in a pediatric population compared to differences in individual adult ratings in the aforementioned study. If individual growth functions are comparable as may be the case in adult populations, pooling data is a reasonable choice to estimate a generic function that could be used as a reference. In such cases a the estimated exponents in a power model can be interpretable assuming the coefficients of the power term is similar for individuals. However when individual growth functions exhibit different functional relationships and differ in their maximum work capacity, as was the case in our pediatric subjects, then curvature or delay may be lost in aggregate data when comparing average model parameters.
The median maximum rating for leg exertion among our pediatric subjects was only slightly greater than half the maximum possible value 10. Other investigators using different rating scales of perceived exertion also reported sub maximal ratings [7, 13, 14]. In this regard, it is worth noting that younger children had lower median rating (4, IQR 3‐6) than did children over 13 years of age (median 6, IQR 4‐8). Any explanation why more than half our subjects reported a delay in rating perceived exertion above baseline or resting values must be pure speculation. The period between 8‐12 years corresponds to a developmental level when children learn to differentiate sensations arising from differing parts or regions of their body . Lamb noticed unreliability at lower exercise levels, which he postulated could be due to a number of factors such as motivation and perceptual development . Some children’s ratings were flat with increasing work rather than a continuous rise, perhaps reflecting inability to understand the scale or properly gauge their perceived exertion, a phenomenon also noted by Swain et al. in a study in children ages 7‐11 years . It has been suggested that children at this developmental stage (age range is a generalized approximation) must exercise at a relatively high intensity before they are able to accurately differentiate feelings arising in specific parts or regions of their body such as perceived leg exertion. Moreover, these same authors argued that children at this stage (age being a proxy measure) could distinguish up to four levels of exercise intensity during cycle ergometry . Although it has been observed that adult males experience the same degree of exertion at work maximum , this was not the case for children in this study. An explanation could be that children and adolescents lack the antecedent experiences and required perceptual anchors needed to accurately gauge the greatest imaginable perceived leg exertion at maximal exercise. We also observed children with non‐monotone growth functions, where the ratings may decrease and then increase, typically in children with low ratings. However submaximal rating was also seen in 80% of 460 adults by Killian et al.  who explained this phenomenon by tolerance for discomfort.
A limitation for model fitting was the number of observations with as few as six and at most 12 in this sample of children and adolescents. We believe our subjects exercised to their respective maxima judging from peak exercise values shown in Table 1, but lower work capacity will set a limit on the number of graded measurements one can make in younger children such that this group had fewer measurements than seen in adults. There can be a risk of truncation that affects model fitting since the highest rating is 10. Another limitation is that the functional relationships were estimated from one trial. The level of habitual activity for each subject was not assessed. It would be useful to assess the children in a second trial to observe any changes in the functional relationship and increasing comfort with the process of rating.