Check the lactate curve and how much each threshold depends on single stages

Regression diagnostics for the fitted lactate curve, adapted to tests of 6 to 10 stages, and a leave-one-out check of every threshold. Print it for a short summary, and plot it with autoplot() for six panels, each with a line on what to look for; they need a tall plot, about 9 x 10 inches, and the article Checking the analysis shows them panel by panel.

Usage

lactate_diagnostics(x)

# S3 method for class 'lactate_diagnostics'
autoplot(object, ...)

Arguments

x: The result of lactate_threshold().
object: The result of lactate_diagnostics().
...: Not used.

Value

a lactate_diagnostics object: a list with stages (one row per stage used in the fit: intensity, lactate, fitted, residual, standardized, leverage, cooks_distance, influential), leave_one_out (method, stage left out, intensity without it, change), jackknife (method, intensity, lowest, highest, jackknife_se), and fits (lactate_fit_quality(), best prediction error first).

Details

Residuals: the measured lactate minus the curve. A run of residuals of the same sign, or an arch against the fitted values, means the shape of the curve is wrong for this test.
Standardized residuals: residuals divided by their standard error. With so few stages, normality and equal-variance tests have almost no power, so a check that "passes" means "could not tell", not "fine"; analyzer error also grows with lactate (Bonaventura et al., 2015).
Leverage and Cook's distance: how much one stage pulls the curve (Cook, 1977). The first and last stages always have high leverage, by design. A stage is flagged when its Cook's distance is above the median of the F(p, n - p) distribution, where p is the number of curve terms; rules such as 2p/n for leverage cannot flag anything when p is close to n.
Leave one stage out: every threshold is computed again without each stage in turn (a jackknife; Efron, 1982). The range shows how much a threshold rests on one stage, and the jackknife standard error is sqrt((n - 1) / n * sum((theta_i - mean(theta_i))^2)). Dropping the first or last stage changes the Dmax line itself, so large changes there are expected for the Dmax family. The resting sample is left out too (column "rest"), which shows how much Bsln+ rests on that one baseline sample (with include_baseline = TRUE the curve changes too, so the curve methods move as well); it is not part of the jackknife.
Which curve predicts best: the leave-one-out prediction error of every curve type, from lactate_fit_quality(). It rewards curves that follow the data without overfitting it, which the RMSE of the fit alone cannot do.

References

Bonaventura JM, Sharpe K, Knight E, Fuller KL, Tanner RK, Gore CJ. Reliability and accuracy of six hand-held blood lactate analysers. Journal of Sports Science and Medicine. 2015;14:203-214.

Cook RD. Detection of influential observation in linear regression. Technometrics. 1977;19(1):15-18.

Efron B. The jackknife, the bootstrap and other resampling plans. Philadelphia: SIAM; 1982.

Examples

female <- subset(lactate_tests, test == "cycling_female")
results <- lactate_threshold(female, intensity, lactate, heart_rate, plot = FALSE)
diagnostics <- lactate_diagnostics(results)
diagnostics
#> Lactate curve: 3rd degree polynomial on 10 points; residual SD 0.14 mmol/L.
#> No stage has a Cook's distance above 0.94 (the median of F(4, 6)).
#> Most sensitive to single stages: Log-log (jackknife SE 12 W), Dmax (jackknife SE 12 W), LTP1 (jackknife SE 12 W).
#> Best-predicting curve: 3rd degree polynomial, which missed each stage by 0.17 mmol/L on average when fitted without it (root-mean-square leave-one-out error).
## six panels: draw them in a tall window, about 9 x 10 inches
if(interactive()) autoplot(diagnostics)