Using functional data analysis to summarise and interpret lactate curves

Using functional data analysis to summarise and interpret lactate curves

Computers in Biology and Medicine 36 (2006) 262 – 275 www.intl.elsevierhealth.com/journals/cobm Using functional data analysis to summarise and inter...

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Computers in Biology and Medicine 36 (2006) 262 – 275 www.intl.elsevierhealth.com/journals/cobm

Using functional data analysis to summarise and interpret lactate curves J. Newella,∗ , K. McMillanb , S. Grantc , G. McCabed a Department of Mathematics, National University of Ireland, Galway, Galway, Ireland b Sports Performance Unit, Celtic Football Club, Glasgow, UK c Institute of Biomedical and Life Sciences, University of Glasgow, UK d Department of Statistics, Purdue University, Indiana, USA

Received 28 September 2004; accepted 29 November 2004

Abstract John Tukey [1] used the term exploratory data analysis (EDA) to describe a philosophy for analyzing data where graphical and numerical summaries are used to uncover interesting structures. The applied statistician today has a much more sophisticated set of methods to use when applying the EDA philosophy. One such collection of methods is functional data analysis (FDA), which was used to explore the structure of lactate curves. A principal components analysis and plots of the second derivatives provide new intuitive endurance markers which correlates highly with other numerical summaries of lactate curves that have been suggested in the literature. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Exploratory data analysis; Functional data analysis; Lactate curves; D2 Lmax; Functional principal components

1. Introduction Aerobic fitness is recognised as a very important fitness component in soccer [2]. Traditionally, maximal oxygen uptake (VO2 max) has been viewed as the “gold standard” measurement of aerobic fitness. However, assessment has a number of limitations [3]. More recently, research has indicated that blood lactate variables measured during incremental exercise are better indicators of endurance performance than VO2 max and are more sensitive indicators of changes in training status than VO2 max [4–6]. Indeed, ∗ Corresponding author. Tel.: +353 91 524411x3703; fax: +353 91 750542.

E-mail address: [email protected] (J. Newell). 0010-4825/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compbiomed.2004.11.006

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numerous studies have shown that the blood lactate response to incremental exercise appears to be highly correlated with various types of endurance performance [7–11] and because of these findings, blood lactate assessment is now used regularly on a worldwide basis by exercise physiologists as a predictor of endurance performance. In recent years, the development of reliable rapid-response blood lactate analysers has made blood lactate assessment a quicker and easier physiological assessment method for exercise physiologists and coaches by doing away with the need for complicated and time-consuming methods of blood lactate analysis. The use of these rapid-response blood lactate analysers in the last 20 years has had a huge influence on the amount of research data collected. 2. Description of the data Lactate data were collected for 19 male professional soccer players at six time points across a playing season. Fingertip blood samples were collected at selected running speeds on a treadmill and were analysed for whole blood lactate concentration using an Analox GM7 analyser. Let (yit , tij ) represent the blood lactate reading y for player i at speed t where i = 1, . . . , N players, j = 1, . . . , n speeds. 3. Key physiological lactate curve features Exercise physiologists use certain features of the lactate curve to assess aerobic fitness [12]. A typical curve illustrating these is given in Fig. 1. At low levels of exercise intensity, the curve is fairly flat. Extraneous non-fitness related factors such as glycogen levels [13–16] and caffeine use [17,18] can cause an initial decrease in the blood lactate. As the intensity of the exercise increases, a workload is reached above which the blood lactate response to increasing exercise intensity is curvilinear [3]. Several features of the lactate curve have been suggested as markers for endurance. One of these is the lactate threshold (LT), i.e. the workload corresponding to this sustained increase in blood lactate above resting levels. Given the difficulty in determining the actual

Blood Lactate (mM)

4mM FBLC

4 DMax Lactate Threshold (LT)

0 ----------------> Increasing Work Load ------------>

Fig. 1. A hypothetical lactate curve with typical endurance markers highlighted.

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resting level phase, a variation of the LT has been proposed which corresponds to the workload preceding an increase in lactate concentration of at least 1 mM. An alternative to the LT as a summary measure of endurance is the workload corresponding to a fixed blood lactate concentration (FBLC) [11]. For example, the FBLC of 4 mM is marked on the curve in Fig. 1. Improved endurance is associated with a larger LT value as a player is able to perform at larger workloads. In addition, improved endurance is associated with a smaller slope post-LT resulting in larger FBLC values. An other alternative to the LT is the Dmax marker [19], the speed corresponding to the point that yields the maximum perpendicular from a line joining the first and last lactate measurements to the estimated lactate curve (estimated using polynomial regression, typically of order 3). This marker has been demonstrated to correlate well with endurance [20,21] however given the marker’s dependence on the first and last data points there is the possibility of either observation having an overly influential contribution to the marker’s estimation. 4. Estimation of lactate curve features It is assumed that the lactate curve is generated from an underlying continuous process x(t). Furthermore it is assumed that the underlying lactate curve can be modelled as yij = x(tij ) + ij , where the error term is assumed independently distributed with mean zero and finite variance. 4.1. Lactate resting phase In all, 62.5% of the 114 lactate curves available for analysis showed a decrease from the first lactate measure to the second. A sign test lends support (p = 0.01) to the influence of this previously mentioned extraneous non-fitness related factor in some of these lactate curves. This phenomenon presents a difficulty for analysis of lactate curves. Clearly many curves will show a decrease due to chance fluctuations and it is not clear how to distinguish these from those where the decrease is due to the non-fitness factors. We will see that functional data analysis (FDA) provides a solution to this difficulty. 4.2. Estimating the LT Traditionally the LT is determined subjectively from plots of the lactate concentration versus workload by identifying the treadmill velocity speed that best corresponds to a departure from a linear baseline pattern. There is evidence that blood lactate concentration increase exponentially as work-rate increases [22]. A log transformation of both the workload and blood lactate concentration has been suggested in order to better subjectively determine the LT [23]. Lundberg et al. [24] proposed fitting a linear spline to this log–log relationship. The location of the knot (i.e. the point of intersection between the two linear splines) and the parameters of the lines are estimated by minimizing the sum of the squared differences between the observed log lactate values and the fitted values. The estimated location of the knot is the LT. Similarly, Orr et al. [25] suggested minimizing the residual sum of squares for fitting simple linear regressions to all possible divisions of the data to determine the LT. Note that as lactate measures are collected using blood samples, the number of data points is generally small. Typically 8–9 measures are collected usually equally divided around the LT. The estimated standard

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error resulting from fitting piecewise linear regressions to such few data points can be considerably large resulting in interval estimates that are too wide for practical purposes. Usually, the estimate of the LT is not accompanied by an estimated standard error in published sports science research, although an estimator of the variability of the LT using two linear regressions is available [24]. The estimated LT could be weighted by the standard error for all subsequent analysis, for example in a weighted repeated measures ANOVA for comparing the squad mean LT across the season. 4.3. Estimating the FBLC Estimates of the FBLC are provided by linear interpolation between the data immediately above and below the FBLC of interest while refinements using curves [26] have also been suggested. Alternatively, regression using a high order polynomial can be fitted to the data and inverse prediction methods can be used. 5. FDA applied to lactate curves FDA is an important new statistical methodology for the analysis of data that can be modelled as continuous functions. In FDA, samples are collected at discrete time points on the same subject. These multivariate observations are then represented as continuous curves that are viewed as the raw data for all further statistical analysis. These curves are termed ‘functional data,’ and the statistical methods are called ‘FDA’ [27–29]. The underlying philosophy is to consider the observed lactate functions x(t) as single entities, rather than a sequence of individual observations. It is assumed that a single function x(t) is being observed which satisfies reasonable continuity conditions on the bounded interval of interest T and that derivatives D M x(t) of order M can be evaluated. The first stage of a FDA is to represent the observations (yit , xit ) in functional form x(t) using a suitable basis function. We assume that the raw observations can be approximated with sufficient accuracy by a finite linear combination of k basis functions k (t), x(t) =

K 

ck k (t),

k=1

where the coefficients of the expansion ck are determined by minimising the least-squares criterion  2 n K   yj − ck k (tj ) SMSSE(y|c) = j =1

k=1

and derivatives of order M are evaluated as ˆ = D M x(t)

K 

ck D M k (t).

k=1

The choice of basis function can be guided by the nature of the underlying function being estimated. In the case of lactate data, a regression spline basis function using polynomial splines of degree four joined smoothly at the distinct speeds recorded was deemed suitable.

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In order to extract the systematic component of the lactate curve and control for the error in the recorded data, a roughness penalty approach was used to smooth x(t). This approach penalises the total curvature in x(t) by finding the function x(t),M that minimises the penalised residual sum of squares   2 PENSSE = [x(t) − x(t),M ] +  [D M x(t)]2 dt conditional on  and M. Choosing an appropriate  is an important step in practice. A very small  will yield an estimate practically equivalent to the data, while a larger  that is will produce an estimate practically equivalent to the linear regression estimate of the data. Cross-validation (CV) and generalised cross-validation (GCV) [30] are popular approaches for choosing an appropriate . These procedures have been criticised however for choosing s that are “too small” [31] and other approaches have been proposed, for example Wahba’s [32] generalised maximum likelihood (GLM) criterion. An alternative approach involves choosing the effective degrees of freedom, approximated by a simple function of the trace of the smoothing matrix S [31]. In the examples presented in this paper, the smoothing parameter was initially chosen using CV [27] however a value of  equivalent to 4 df was considered more suitable. Note that this value of  was equivalent in most players to n/2. The value of M, the degree of curvature penalised, was chosen as four also as this resulted in smooth first and second derivative estimates. The functions x1 (t), x2 (t), . . . , xN (t) corresponding to each of the N subjects in a data set can be used to generate simple functional summary statistics such as the mean, variance, covariance and correlation coefficient. In addition, techniques for functional linear modelling and functional adaptations to classical multivariate variance reduction procedures such as principal components analysis (PCA) and canonical correlation have been investigated [27]. PCA is a useful technique for extracting the important features from multivariate data. The functional form of PCA extends this notion to extract the key features of data functions. Each principal component is specified by a weight function (t) that represent dominant features of variation in the curves by choosing those that maximise the mean square of the scores zi where  zi = (t)xi (t) dt, subject to the constraint    j (t)1 (t) dt = j (t)2 (t) dt = · · · = j (t)j −1 (t) dt = 0. This constraint insures mutually orthogonal components and bounded variance for the first j components. If necessary, a roughness penalty may be incorporated in order to smooth the extracted functional components [27].

6. Results and discussion A plot of the smoothed lactate curves for all players tested midway through the playing season is given in Fig. 2. The midpoint of the season is chosen for illustrative purposes only. Due to injury, lactate

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Lactate (mM)

15

10

5

0 10

12

14

16

Speed (km/h)

Fig. 2. Plots of smoothed lactate curves for all players available mid-season (using  = 0.001).

Mean Lactate Response (mM)

8 7 6 5 4 3 2 10

12

14

16

Speed (km/h)

Fig. 3. Plot of the mid-season smoothed mean lactate curve.

data were available for only 14 of the 19 players at this time point. We can see considerable player-toplayer variability at the initial lactate reading and around the 4 mM FBLC. This may be an important consideration for the Dmax endurance marker given its dependence on the first and initial reading. The lactate curve evident in Fig. 2 that is considerably ‘poorer’ in terms of endurance when compared to the rest of the squad corresponds to the goalkeeper. The mean and standard deviation curves (Figs. 3 and 4) provide interesting summary information. The mean curve dips slightly during the baseline period due to the non-exercise related factors present in some

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Lactate Std. Dev. (mmol/l)

3.0

2.5

2.0

1.5

1.0

10

12 14 Treadmill Speed (km/h)

16

Fig. 4. Plot of the smoothed standard deviation of mid-season lactate curves.

Pre Season Oct Dec Jan Apr June

Lactate mM

8

6

4

2 10

12

14

16

Speed (km/h)

Fig. 5. Plot of the smoothed mean lactate curves at different time points across a season.

of the individual curves. The squad average LT is about 12.5 km/h, followed by a curved rise in the blood lactate response. The squad average 4 mM FBLC is about 14.4 km/h. The standard deviation curve shows relatively large variation in the baseline period, low variation around the LT and increasing variation after the LT. The two periods of high variation are caused by two different phenomena. The baseline variation is due to the non-exercise related factors. Much of the post-LT variation is due to variation in the location of the LT. A plot of the mean curve across the six testing points illustrates the change in mean endurance across the season (Fig. 5). As the season progresses the lactate curve resting phase tends to lengthen and the

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6 Pre Season Early Season

5

Week 10 Week 25

Week 16 Lactate mM

4

Week 21

3 2 1 0 9

10

11 12 Speed (km/h)

13

14

Fig. 6. Smoothed lactate curves for a particular player before injury and 10, 16, 21 and 25 weeks post-injury.

lactate curve tends to flatten out. A plot of the estimated lactate curve can provide useful feedback to players rather than providing them with a single estimated endurance marker. For example, a series of lactate curves for the same player at different times before and after an injury is given in Fig. 6. The pre-season curve is displayed in bold as reference. In the early season improvement in aerobic fitness is seen by a shift of the curve to the right. The player’s inactivity due to the injury is evident: 10 weeks after the injury, the curve is shifted to the left. The plots for 16, 21, and 25 weeks after the injury show the gradual recovery back to the early season level. Note the changes in the LT and the 4 mM FBLC. When the LT is higher, the player can exercise at a greater intensity while maintaining approximately baseline lactate. Similarly, a higher FBLC means that a player can exercise at a greater intensity for the given lactate concentration. Plots of an individual lactate curve with estimated first and second derivatives provide useful information with regard to estimating the traditional lactate curve endurance markers discussed earlier. For example, the smoothed lactate curve for a particular player, in addition to a plot of the residuals and first two derivatives is given in Fig. 7. The modified LT corresponding to the workload preceding an increase in lactate concentration of at least 1 mM can now be estimated from the plot of the smoothed lactate curve while the maximum of the second derivative will provide information on the workload corresponding to the point of maximum lactate acceleration. This new endurance marker, the D2 Lmax may prove useful as a reproducible and interpretable endurance marker in the future. An interval estimate of the true D2 Lmax (or an estimate of it’s standard error) can be provided using a simple normalised residual resampling bootstrap procedure [33,34]. As an example, a plot of the bootstrap lactate curves and a histogram of the estimated D2 Lmax for each resampled curve are given in Fig. 8 from which an interval estimate of D2 Lmax or an estimate of the standard error can be calculated [33]. A PCA of the lactate curves was performed. For each component, this analysis produces a principal component function. An individual observed curve is approximated by the mean curve plus a linear combination of these principal component curves. If we approximate an individual curve by the mean plus a constant times a particular component curve, the area between this curve and the mean curve is

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J. Newell et al. / Computers in Biology and Medicine 36 (2006) 262 – 275 Residuals

Lactate Curve 4.0

0.05

3.5 3.0 2.5

-0.05

2.0

-0.15 11

12 13 Speed (km/h)

14

11

12 13 Speed (km/h)

14

D2L

D1L 1.5 1.0

1.0 0.5

0.5

0.0

0.0

-0.5 11

12 13 Speed (km/h)

14

11

12 13 Speed (km/h)

14

Fig. 7. Smoothed lactate curve, and plots of the first and second derivatives for a particular player using  = 0.002, resulting in D2 Lmax = 12.89. 4

Lactate (Mm)

3

2

1

0 11

12

13 Speed (km/h)

14

Fig. 8. Bootstrap lactate curves for the player illustrated in Fig. 7 (with the estimated lactate curve illustrated in white) and histogram of D2 Lmax estimated empirical sampling distribution.

the principal component score. These scores are analogous to the principal component scores discussed in multivariate analysis and quantify the distance between the mean curve and the approximation of the individual curve in the dimension of the particular component. To interpret the principal components, Ramsay and Silverman [29] recommend using a plot for each component. The mean function is plotted with curves obtained by adding and subtracting a multiple of the component curve. We have used a multiple based on two standard deviations in the component scores. For

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Blood Lactate (mmol/l)

10

8

6

4

2

271

+ ++ ++ ++ + ++ ++ ++ + + ++ ++ ++ + + ++ ++ ++ + ++ ++ ++ + ++ +++ +++ + --+ +++ --+ + + +++ -+++ ----++++ + + + +++ ---++++++ ---++++++++++ ++++++++++++++++++++++++++++++++++++++++++ -----------------------------------------------------------------------------------10

12

14

16

Treadmill Speed (km/h)

Fig. 9. Plot of first principal component (78.6).

Blood Lactate(mmol/l)

8

6

4

2

----- +++++ - ++ -- ++++ -+-++++ -++ +-+-+ +-+-++ + ++++-+++---+ + ++ -+++ -+++ --+ ++++++ + + +++++++ +++ -++++++++ +++++ -++++++++ +++++ -----+ + ++++++++++ + + ++++++++++++++++++++++++++ ---------------------------------------------------------------10

12 14 Treadmill Speed (km/h)

16

Fig. 10. Plot of second principal component (15.7).

the lactate curves, the first two components accounted for 79 and 16 percent of the variation, respectively, and we will confine our discussion to these two. The plot for the first principal component is given in Fig. 9. The centrally located mean curve is the curve plotted in Fig. 4. The upper and lower curves correspond to component scores that are two standard deviations above and below the mean, respectively. Note that this component is fairly insensitive to values in the baseline period. High and low values of the component scores give curves that are close to the mean curve in this region. On the other hand, this component is very sensitive to the variation in the curves at

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the higher treadmill speeds. It appears to capture the overall fitness information that is of physiological interest, with higher scores corresponding to higher LTs and greater 4 mM FBLCs. The variability in the initial baseline values is captured by the second component (see Fig. 10). High scores are associated with high initial values in the baseline. This component appears to isolate the non-exercise factors that are not of primary interest to exercise physiologists. Given that the first principal component explains the majority of the variation in the estimated lactate curves and ‘ignores’ the effect of baseline, this component should be a good measure of “lactate-based endurance”. This view is reinforced by considering the correlations between the first principal component scores and the respective estimated LT and FBLC scores measures. These correlations are −0.91 (p < 0.001) and −0.89 (p < 0.001), respectively (using piecewise linear regression and inverse prediction based on a cubic polynomial regression to determine the LT and 4 mM FBLC).

7. Conclusion We have analysed a set of lactate curves using FDA techniques. First, the raw data were transformed to smooth curves using polynomial splines. Using plots of the second derivative of the estimated smooth lactate curve, a new lactate marker, D2 Lmax was suggested which corresponds to the workload associated with the maximum of the second derivative. The smooth curves were then approximated by principal components curves. As a result of these analyses, an additional new endurance component was separated from a component that was sensitive to non-exercise factors in the baseline period. These results illustrate how these computationally intensive techniques can be used for EDA. FDA methods such as these may also prove to be useful for the further study of lactate curves. For example, an important application concerns the characterisation and analysis of lactate curves under changing conditions.

8. Summary Lactate data typically consist of lactate readings collected from athletes during incremental exercise at fixed time points. A natural approach is to represent the set of readings for each individual as a curve. In practice this approach is applied using regression techniques and from this simple lactate summaries are estimated (e.g. the lactate threshold, the Dmax marker or some fixed blood lactate concentrations). The analysis proceeds by reducing what are intrinsically functional responses to single lactate markers, which are then analysed to draw conclusions. As a result, a great deal of potentially informative data is ignored. Functional data analysis (FDA) is an emerging field in statistics that focuses on treating an entire sequence of measurements for an experimental unit as a single curve. FDA appears to be very suitable for analysing lactate data. In this paper, we introduce FDA methods as exploratory tools to analyse a set of aerobic fitness data. Blood lactate measurements were collected for 19 male professional soccer players. Samples were taken at fixed levels of exertion on a treadmill. The level of blood lactate versus treadmill velocity is called the lactate response to exercise, with the response generally being curvilinear. Once the raw lactate data are transformed to smooth curves using smoothing splines, FDA methods were used to explore the structure of lactate curves and to identify key features. Plots of the second derivative (i.e. lactate

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acceleration) provide a new intuitive endurance marker while a principal components analysis of the curves reveals a first component with 79 percent of the variation that isolates an overall fitness curve. The corresponding principal component score correlates very highly, about 0.9, with other numerical summaries of lactate curves that have been suggested in the literature. References [1] J. Tukey, Exploratory Data Analysis, Addison-Wesley, Reading, MA, 1977. [2] T. Reilly, Science and Soccer, London, E and FN Spon, 1996. [3] A. Weltman, The blood lactate response to exercise, in: Current Issues in Exercise, Monograph Number 4, Human Kinetics, Champaign, IL, 1995. [4] P.A. Farrell, J.H. Wilmore, E.F. Coyle, et al., Plasma lactate accumulation and distance running performance, Med. Sci. Sport. Exerc. 11 (1979) 338–344. [5] I. Jacobs, P.A. Tesch, O. Bar-Or, J. Karlsson, R. Dotan, Lactate in human skeletal muscle after 10 and 30 s of supramaximal exercise, J. Appl. Physiol. 55 (1983) 365–367. [6] S. Grant, I. Craig, J. Wilson, T. Aitchison, The relationship between 3 km running performance and selected physiological variables, J. Sport. Sci. 15 (1997) 403–410. [7] D.G. Allen, D.R. Seals, B.F. Hurley, A.A. Ehsani, J.M. Hagberg, Lactate threshold and distance running performance in young and older endurance athletes, J. Appl. Physiol. 58 (1985) 1281–1284. [8] E.F. Coyle, M.E. Feltner, S.A. Kautz, et al., Physiological and biomechanical factors associated with elite endurance performance, Med. Sci. Sport. Exerc. 23 (1991) 93–107. [9] L. Fay, B.R. Londeree, T.P. Lafontaine, M.R. Volek, Physiological parameters related to distance running performance in female athletes, Med. Sci. Sport. Exerc. 21 (1989) 319–324. [10] R. Fohrenbach, A. Mader, W. Hollmann, Determination of endurance capacity and prediction of exercise intensities for training and competition in marathon runners, Int. J. Sports Med. 8 (1987) 11–18. [11] H. Heck, A. Mader, G. Hess, S. Mucke, R. Mulle, W. Hollmann, Justification of the 4 mmol/l lactate threshold, Int. J. Sports Med. 6 (1985) 117–130. [12] P. Bourdon, Blood lactate transition thresholds: concepts and controversies, in: Physiological Tests for Elite Athletes, Australian Sports Commission, Human Kinetics, Champaign, IL, 2000. [13] E.F. Hughes, S.C. Turner, G.A. Brooks, Effects of glycogen depletion and pedaling speed on “anaerobic threshold”, J. Appl. Physiol. 52 (1982) 1598–1607. [14] A. Weltman, The blood lactate response to exercise, in: Current Issues in Exercise, Monograph Number 4, Human Kinetics, Champaign, IL, 1995. [15] I. Jacobs, N. Westlin, J. Karlsson, M. Rasmusson, B. Houghton, Muscle glycogen and diet in elite soccer players, Eur. J. Appl. Physiol. 48 (1982) 297–302. [16] T. Yoshida, Effect of exercise duration during incremental exercise on the determination of anaerobic threshold and the onset of blood lactate accumulation, Eur. J. Appl. Physiol. 53 (1984) 196–199. [17] J. Bangsbo, Metabolism in soccer, Abstract from the European Congress on Football Medicine, Stockholm, 1992. [18] S. Dodd, E. Brook, S.K. Powers, et al., The effects of caffeine on graded exercise performed in caffeine naı¨ve versus habituated subjects, Eur. J. Appl. Physiol. 62 (1991) 424–429. [19] B. Cheng, H. Kuipers, A.C. Snyder, A. Keizer, et al., A new approach for the determination of ventilatory and lactate thresholds, Int. J. Sports Med. 13 (1992) 518–522. [20] D. Bishop, D.G. Jenkins, L.T. Mackinnon, The relationship between plasma lactate parameters, Wpeak and endurance cycling performance, Med. Sci. Sport. Exerc. 30 (8) (1998) 1270–1275. [21] D. Bishop, D.G. Jenkins, M.F. Carey, M. McEniery, The relationship between plasma lactate parameters and muscle characteristics in female cyclists, Med. Sci. Sport. Exerc. 32 (6) (2000) 1088–1093. [22] M. Yeh, T. Gardner, F.G. Adams, G. Yanowitz, R. Carpo, Anaerobic threshold: problems of determination and validation, J. Appl. Physiol. 55 (1983) 1178–1186. [23] W.L. Beaver, K. Wasserman, B.J. Whipp, Improved detection of lactate threshold during exercise using a log–log transformation, J. Appl. Physiol. 59 (1985) 1936–1940.

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[24] M.A. Lundberg, R.L. Hughson, K.H. Weisiger, R.H. Jones, G.D. Swanson, Computerized estimation of lactate threshold, Comput. Biomed. Res. 19 (1986) 481–486. [25] W. Orr, J. Green, L. Hughson, W. Bennett, A Computer Linear Regression Model to Determine Ventilatory Anaerobic Threshold, The American Physiological Society, 1982, pp. 1349–1352. [26] L. Laurencelle, A. Quirion, S. Nadeau, Lactate threshold determination: A Monte Carlo comparison of two interpolation methods, Arch. Int. Physiol. Biochim. Biophys. 1994 (102) (1993) 43–49. [27] J.O. Ramsay, C. Dalzell, Some tools for functional data analysis (with discussion), JRSS 53 (1991) 539–572. [28] J.O. Ramsay, B.W. Silverman, Functional Data Analysis, Springer, New York, 1997. [29] J.O. Ramsay, B.W. Silverman, Applied Functional Data Analysis: Methods and Case Studies, Springer, New York, 2002. [30] P. Craven, G. Wahba, Smoothing noisy data with spline functions, Numer. Math. 31 (1979) 377–403. [31] T.J. Hastie, R.J. Tibshirani, Generalized Additive Models, Chapman & Hall, New York, 1990. [32] G.A. Wahba, Comparison of GCV and GML for choosing the smoothness parameter in the generalized spline smoothing problem, Ann. Statist. 13 (1985) 1378–1402. [33] A.C. Davison, D.V. Hinkley, Bootstrap Methods and their Application, Cambridge University Press, Cambridge, 1997. [34] A.W. Bowman, A. Azzalini, Applied Smoothing Techniques for Data Analysis: the Kernel Approach with S-Plus Illustrations, Oxford University Press, Oxford, 1997. Dr. John Newell is a lecturer in Statistics in the Department of Mathematics, National University of Ireland, Galway. Dr. Newell is a Fulbright Scholar, holds a B.Sc. (Hons.) in Mathematics (National University of Ireland, Galway) an M.Sc. in Statistics (National University of Ireland, Cork) and a Ph.D. in Statistics (University of Glasgow, Scotland). His main area of research involves Applied Statistics including survival analysis, computational inference, functional data analysis and applications in sports science. He has co-authored over 25 peer-revived publications. He is the consultant statistician for the Sports Performance Unit, Glasgow Celtic Football Club. Kenny McMillan is the Head Sports Physiologist with Glasgow Celtic FC, having held this position since 1999. He received his B.Sc. in Physiology and Sports Science, M.Sc. in 1997, and also completed an M.Sc. (by research) in Physiology and Sports Science in 2002, both degrees being from the University of Glasgow. He is currently finishing his Ph.D. at NTNU, Trondheim, in Norway under the supervision of Professors Hoff and Helgerud. Kenny has co-authored several publications, and his research interests include physiological assessment techniques in exercise science and strength and endurance training interventions of soccer players and other intermittent team sport players. Dr. Stan Grant is a senior lecturer in the Institute of Biomedical and Life Sciences at the University of Glasgow. His research interests involve the development of techniques for monitoring habitual physical activity in pre-school children and profiling of physical activity in a range of groups of young children. Other interests include the physiological and genetic determinants of success at high altitude and evaluation of endurance in climbing specific tasks. The assessment of clothing using human responses to heat and cold and the physiology of soccer are other research areas. A recently funded study involves functional electrical stimulation and aerobic training in spinal cord injury patients. Dr. Grant has been a member of the advisory panel to the Scottish Institute of Sports Medicine and Sports Science, and is an exercise consultant to Celtic Football Club. In addition, he has provided guidance on strategies to combat heat and humidity to the Scottish Commonwealth Games Squads. George P. McCabe is a Professor of Statistics and Associate Dean for Academic Affairs in the School of Science at Purdue University. Until 2004 he was head of the Statistical Consulting Service. In 1966 he received a B.S. degree in mathematics from Providence College and in 1970 a Ph.D. in mathematical statistics from Columbia University. His entire professional career has been spent at Purdue with sabbaticals at Princeton, the Commonwealth Scientific and Industrial Research Organization (CSIRO) in Melbourne (Australia), the University of Berne (Switzerland), the National Institute of Standards and Technology (NIST) in Boulder, CO and the National University of Ireland, Galway. Professor McCabe is an elected fellow of the American Statistical Association and was 1998 Chair of its section on Statistical Consulting. He has served on the editorial boards of several statistics journals. He has consulted with many major corporations and has testified as an expert witness on the use of statistics in several cases.

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He was recently a member of an Institute of Medicine Committee on the Use of Dietary Reference Intakes in Nutrition Labeling. He is a coauthor with David Moore of the elementary text Introduction to the Practice of Statistics, 4th ed. and with Moore, Duckworth and Sclove, The Practice of Business Statistics. Professor McCabe’s research interests have focused on applications of statistics. He is author or coauthor of over 130 publications in many different journals.