Tests whether two tests of the same athlete differ only by a horizontal shift of the lactate curve, or also in its
shape. Both tests are fitted with the exponential plus constant, lactate = a + b * exp(c * intensity) (Hughson et
al., 1987), in which a pure horizontal shift by d changes only b:
Usage
curve_shape(before, after, level = 0.95)
# S3 method for class 'lactate_shape'
autoplot(object, ...)Arguments
- before, after
The results of
lactate_threshold()for the two tests, with the same sport and at least five stages each.- level
The confidence level of the shift. Defaults to
0.95.- object
The result of
curve_shape().- ...
Not used.
Value
a lactate_shape list with shift (intensity units, positive when the curve moved to the right),
shift_low and shift_high, f, df1, df2, p_value, aicc_shift and aicc_free, verdict
("shape changed" when the F test and AICc both prefer the free shape, "shift" otherwise: none
was detected), the peak lactate of both tests and the two
exponential curves. Print it for a summary, and plot it with autoplot(): the two fitted curves, and the first one
moved by the shift (dashed).
Details
Shift only: one shape for both tests (shared
a,bandc), with the second test moved byd.Free shape: each test with its own
a,bandc.
The two fits are compared with an extra-sum-of-squares F test (Motulsky & Ransnas, 1987) and with AICc, the
Akaike criterion corrected for small samples (Hurvich & Tsai, 1989); the verdict is "shape changed" only when
both prefer the free shape. The shift d comes with a profile confidence interval.
Read the result with care:
The test finds a change of shape only when it is large and the tests are long. In simulated pairs of tests of a trained cyclist (analyzer error of 0.1 mmol/L plus 4%), a baseline 0.5 mmol/L higher was found in 96% of pairs with 8 stages per test and in 20% with 6; a curve steep enough to double the peak lactate in 55% and 10%. So "no shape change detected" is not proof of the same shape, least of all with 6 stages.
The p-value is approximate: the F test is exact only for models that are linear in their parameters, and a low p-value from one pair of curves needs a repeat before it counts (Motulsky & Ransnas, 1987). In the same simulation it fell below 0.05 in 8% to 10% of pairs with no change of shape; with AICc beside it, the verdict was wrong in 1% to 6%.
In this model, a shift to the right is the same as a proportional fall of lactate above baseline. Better endurance, a lower glycolytic rate (VLamax) at the same VO2max, low glycogen stores and overreaching can all produce it, so check the peak lactate, the diet before the test and the performance too (Maassen & Busse, 1989; Faude et al., 2009; Wackerhage et al., 2025). A shift to the left can likewise come from a higher VLamax, not only from lost endurance.
The exponential cannot follow an early lactate dip; check both tests with
lactate_diagnostics()first.
References
Faude O, Kindermann W, Meyer T. Lactate threshold concepts: how valid are they? Sports Medicine. 2009;39(6):469-490.
Hurvich CM, Tsai CL. Regression and time series model selection in small samples. Biometrika. 1989;76(2):297-307. doi:10.1093/biomet/76.2.297
Hughson RL, Weisiger KH, Swanson GD. Blood lactate concentration increases as a continuous function in progressive exercise. Journal of Applied Physiology. 1987;62(5):1975-1981.
Maassen N, Busse MW. The relationship between lactic acid and work load: a measure for endurance capacity or an indicator of carbohydrate deficiency? European Journal of Applied Physiology and Occupational Physiology. 1989;58(7):728-737.
Motulsky HJ, Ransnas LA. Fitting curves to data using nonlinear regression: a practical and nonmathematical review. FASEB Journal. 1987;1(5):365-374.
Wackerhage H, Kabasakalis A, Seiler S, Heck H. Is the vLamax for glycolysis what the VO2max is for oxidative phosphorylation? Sports Medicine. 2025;55(8):1853-1866.
Examples
series <- subset(lactate_tests, athlete == "cyclist_series")
before <- lactate_threshold(subset(series, day == 0), intensity, lactate, plot = FALSE)
after <- lactate_threshold(subset(series, day == 58), intensity, lactate, plot = FALSE)
curve_shape(before, after)
#> Shift only vs free shape: F(2, 9) = 2.88, approximate p = 0.108; AICc -49.7 vs -43.8.
#> No shape change detected: one curve shifted by +24 W (95% CI 21 to 28) describes both tests, but with fewer than 8 stages the test finds few changes of shape.
#> Peak lactate 6.4 -> 7.9 mmol/L. A shift can also come from low glycogen or overreaching, not only fitness.