Interpreting the confidence intervals of model parameters of breath‐by‐breath pulmonary O2 uptake

Keir et al. (2014) have recently compared different techniques used to assemble breath-by-breath pulmonary O2 uptake data from repeated step transitions in the moderate-intensity exercise domain. In particular, the authors evaluated the quality of the assembling methods by comparing what they called the ‘model confidence CI95’, where ‘CI95 is equal to the SE (derived from the sum of squared residuals from the model parameter estimates) multiplied by the t-distribution value for the 2.5% two-tailed dimensions’. The authors conclude that the data-processing technique had no effect on parameter estimates. However, ‘the narrowest interval for CI95 occurred when individual trials were linearly interpolated on a second-by-second basis and ensemble averaged’ before running the non-linear regression procedure. It can be easily demonstrated, and Keir et al. (2014) can do it using their experimental data, that a linear interpolation on a 0.5 s-by-0.5 s basis or 0.1 s-by-0.1 s basis results in a decrease of the SE, better called asymptotic standard error or ASE, which is calculated from the variance–covariance matrix of parameter estimates on the basis of the number of data points used by the linear regression procedure. As a consequence, the corresponding CI95 will also be narrower, in comparison to the values obtained by the linear interpolation on a second-by-second basis, roughly by the factor √ 1 s 0.5 s = √2 or √ 1s 0.1s = √10 , respectively (Francescato et al. 2014b). Following this reasoning, the narrowest CI95 would be obtained by linearly interpolating the data on an infinitesimal-by-infinitesimal basis. As discussed in detail by Francescato et al. (2014b), this phenomenon is due to the fact that the linear interpolation increases the number of data points used by the non-linear regression procedure without adding new information (‘cloning’ effect). Indeed, it can easily be shown that the same phenomenon takes place even for the average (and corresponding standard error) of a simple series of data, analysed as raw data or taking them twice. In the above conditions, the CI95 no longer meets the definition of confidence interval, i.e. the range of values that, over a set of notional repeated samples, would contain the true parameter of interest with a prespecified probability, usually set at 95%. The simulation performed by Francescato et al. (2014a) highlighted that when the individual trials were processed to obtain regular 1 s bins and then ensemble averaged before running the non-linear regression procedure, the ‘true’ (known) phase II time constant was included within the estimated CI95 in only 70% of cases. This means that, for this data-processing technique, the estimated CI95 does not satisfy the statistical definition of confidence interval at 95% level. The same simulation showed that when the individual trials are simply combined into a single data set, the ‘true’ (known) phase II time constant was included within the estimated CI95 in more than 94% of cases, thus being very close to the definition of the confidence interval at 95% level. In conclusion, the non-linear regression procedure applied following interpolation of individual trials on a second-by-second basis and ensemble averaging results in misleading CI95. As a consequence, in contrast to Keir et al. (2014), we believe that the CI95 estimated in this manner cannot be used as figure of merit to evaluate the quality of different data-processing techniques. Narrower confidence intervals should be obtained by increasing the information used (e.g. increasing the number of repeats).