Model selection

Models that contravene coefficient polarity expectations cannot be considered valid models, as a plausible power output cadence relationship will not follow from the model. In the linear models, in order for an optimum cadence to exist, the coefficient for cadence (β) must be negative and the coefficient for the square of cadence (γ) must be positive. Also, the optimum cadence ought to take a sensible value. For non-linear models, coefficients relating to cadence (both α and β) must be positive. When non-linear models are developed further, the coefficient relating to lagged heart rate (γ) should be positve so that increasing power output requires an increase in heart rate, whilst the coefficient relating to athlete effect (µ) should be negative in the additive model and positive in the multiplicative model.

Furthermore, the fitted models can be compared statistically. A standard criterion for model choice is the Akaike Information Criterion (AIC), which is defined in the following way: AIC 2ln(L)2p, where L is the log-likelihood value for the model and p is the number of parameters in the model (Stuart et al., 1999, p.748). The minimum AIC model is then sought. Comparisons of AIC between different linear models can be made as each model uses the same data and therefore all linear models have the same sample size. An alternative is the Bayesian Information Criterion (BIC), BIC 2ln(L) plog(n), which places a higher penalty on the number of parameters in the model, and leads to more parsimonious models.

As the AIC and BIC are largely affected by sample size, when developing the initial non- linear models it is difficult to compare different models as each data set (each heart rate subset for each athlete) has a different sample size. If the non-linear gamma models produce a lower AIC compared to the linear models, we can conclude that the non-linear models are a better fit of the data than the linear models. We would expect such a finding, but it is difficult to predict how great the difference will be.

We also calulate explanatory power R2 – this ostensibly represents the amount of variation in the response variable that is explained by the predictor variable. For the linear models this means the amount of variation in heart rate that is explained by power output. For the grand non-linear models, the explanatory power refers to the amount of variation in power output that is accounted for by variation in cadence and heart rate. The larger the R2, the more variation is explained, and we may consider a larger R2 therefore to indicate better the model. We would expect the explanatory power to increase with the addition of each additional covariate. However, if some covariates offer only negligible increase in explanatory power, we may consider using a more parsimonious model that does not necessarily include all covariates.

As R2 is not affected by sample size, comparisons can be made more easily between different linear and non-linear models in terms of R2 rather than AIC. We fit models in statistical packages SPSS and R. For models fitted in SPSS the R2 is given in the output –

linear models are fitted in SPSS so R2 is imediately displayed for these models. We fit non- linear models of power output and cadence in R. In R, for non-linear models, R2 is not given, as it is not a very reliable indicator of model fit for non-linear models. This is because, in non- linear models, the total sum of squares is not equal to the regression sum of squares plus the residual sum of squares. However it can be calculated nonetheless. The residual standard error is provided from the output in R for each model (each heart rate group and each athlete). In order to calculate the explanatory power (R2) of the model the residual error from a null model is also required – the null model is simply a model with just one parameter applied to the same data set: P  for each athlete. The explanatory power (R2) is then calculated in the following way:

error) standard tal error / to standard residual ( 1 2 _{ } R ,

where total standard error is the addition of the residual standard error of the non-linear model and that of the equivalent null model.

We can then check that R2 (along with coefficient estimates and standard errors) calculated from the R program is the same as that found from the same model fitted in SPSS. For non-linear models (where we take natural logs of each element of the model to transform it), we can fit models in both R and SPSS, hence R2 is provided in SPSS output.

Although R2 indicates the amount of variation in the response variable that is accounted for by variation in the predictor variables, we cannot solely use it (or indeed other statistical measures AIC and BIC) to determine whether or not we are satisfied with a given model. R2 indicates how closely a fitted curve from a model fits the real data, or how small are the differences between points on the fitted curve and the real data points. For a regression model to provide an accurate, reliable indication of optimum cadence, the fitted curve from the regression model must accurately reflect the true relationship between the variables involved – for this the coefficient estimates must have certain polarities (as discussed at the beginning of this section 4.1.9.). Nonethless we can use statistical indicators such as R2 to compare similar models (with similarly shaped fitted curves), for example in exploring the extent to which additional variables are worth including in regression models. Therefore overall we use a combination of statistical measures (R2, AIC and BIC, and coefficient estimates) and check that fitted curves from regression models are a realistic shape when assessing different models.

The power output cadence relationship of a model can be assessed through plotting the power output calculated from a regression model, along with the power output measurements in our sampled data, against cadence. We observe the extent to which the fitted power output from the model overlaps with power output measurements in our sampled data, and the plot provdes a guide as to whether the model is a valid fit of the data. Should the estimates of the coefficients be of the expected polarity the fitted power output cadence curve should be a valid shape, but the plot allows us to observe the fit in more detail, and provides a visual demonstration as to the power output candence relationship that the model produces. For fitted power output / cadence curves for non-linear models of power output, heart rate and cadence, heart rate must be fixed to an appropriate value for the curve to be fitted (illustrating the cadence that maximises power output for a given heart rate). In some more complex non- linear models (such as those featuring an interaction between cadence and additional

variables) it may not be immediately obvious from observing coefficient polairities whether the model represents the true relationship between power output and cadence. In such cases however the extent to which the model represents the true power output / cadence relationship found in the data becomes clear when the fitted curve has been plotted.