I am interested in calculating oxidation reaction kinetics parameters of a material at a particular temperature. Essentially I have a curve of data (mass change) vs. time for the particular temperature. I have repeated the experiment three times so I have three individual curves. Technically each of these curves should be the exact same, and the calculations of their curve's parameters (such as slope) should result in the same values. However, experimental variance causes the curves and their respective parameters to have slightly different values.

I am wondering if there is any mathematical difference between averaging the curves of data and calculating a single parameter from the single averaged curve (that parameter would have a C.I. determined from regression of the parameter), vs. calculating individual parameters for each individual curve and then finding the average and C.I. of the set of those three parameters/curves?

I have been trying to obtain some insight into this, but I cannot seem to figure out the proper phrasing to search google effectively.

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    $\begingroup$ Neither of the options you suggest would be my first thought. It seems you'd be better asking for help with the original problem rather than help with choosing among your proposed solutions to it. $\endgroup$ – Glen_b Jun 23 '19 at 5:47
  • $\begingroup$ @Glen_b. I don't completely see what you mean. As far as I am aware, I am going about doing the analysis correctly, but in literature studies there seems to only be sample size 1. In my experiment I have a sample size of 3, but I am unsure if there is a fundamental intricacy with averaging data vs. averaging calculations from multiple data. $\endgroup$ – D. Hallatt Jun 23 '19 at 18:33

As I understand your question, you have done 3 separate experiments under the same conditions, each time fitting a model of the change of sample weight over time at fixed temperature to obtain estimates of the model parameter values. Your question is whether you should average the parameter values among the 3 separate analyses or to average the curves first and then determine the parameter values from the averaged curve.

The correct answer, as implied by the comment from @Glen_b, is neither of those. The best approach will generally be to use all the data points individually, without prior averaging, together in a combined analysis, while taking into account the experiment from which each data point was derived.

The errors of estimates of parameter values depend on the residual degrees of freedom in the analysis, typically the number of data points minus the number of estimated parameter values. As a rule of thumb, you might expect the precision (reciprocal of error) to be proportional to the square root of the number of degrees of freedom.

So if you do 3 experiments each with $N$ observations and estimate $P$ parameter values from all $3N$ data points, then the precision is proportional to $\sqrt{3N-P}$.

For each experiment analyzed separately the precision would be lower, proportional to $\sqrt{N-P}$. Averaging the values over 3 experiments would somewhat improve the precision by a factor of $\sqrt{3}$, with a net result for precision proportional to $\sqrt{3N-3P}$. That is lower precision than for the pooled analysis.

Working simply with the averaged curves would improve the precision at each individual time point by a factor of $\sqrt{3}$, but you are now using only $N$ (averaged) data values to estimate $P$ parameter values, so you are have roughly the same (poor) precision as for averaging the 3 parameter-value estimates.

The above only holds, however, if there were no substantial systematic differences among the 3 experiments. So you should let your data tell you how to proceed.

The trick is to include an experiment identifier along with each separate data point, and then do an analysis of variance to determine whether there is a significant difference among the experiments. There is an example of how to do this with nonlinear curve fitting in R on this page; see under the heading of "Comparing fits to different data sets" and the associated R code farther down the page.

If there are no significant differences among experiments (or the differences are too small to be of practical importance even if they are "significantly different" due to very high inherent precision or large numbers of observations), then the estimates from pooled analysis will be best. If there are substantial differences among the 3 experiments, then you perhaps shouldn't be averaging in any way and you should explore why nominally identical experiments gave different results.

  • $\begingroup$ (+1) In the case of 'significant differences' between experiments, wouldn't it still be best to proceed with all the data in the same model but with a random effect for experiment? Or a fixed effect, perhaps, since 3 groups is not great for random effect estimation. $\endgroup$ – mkt - Reinstate Monica Jun 24 '19 at 8:23
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    $\begingroup$ @mkt 3 experiments are too few for random effects, as you say. The analysis of variance approach documented in the linked page treats experiments as fixed effects and tests whether there is a significant difference among experiments. Then the analyst can use that information to decide how best to proceed with reporting results or evaluating why experiments might have differed. $\endgroup$ – EdM Jun 24 '19 at 14:06
  • $\begingroup$ Agreed, thanks. $\endgroup$ – mkt - Reinstate Monica Jun 24 '19 at 14:13
  • $\begingroup$ If I follow correctly, the whole discussion on 'significant differences' is simply to see if combining the data together would be appropriate? If I deem so I can perform a pooled regression analysis of the data? My data is certainly different even to my naked eye, with a clear systematic offset of all three curves. However, my aim is to calculate the slope of the linear section of these curves, so I could just normalize/zero each curve to the beginning of each of their linear sections and cancel out the systemic error I believe? Then acquire an estimate from a pooled analysis of all data. $\endgroup$ – D. Hallatt Jun 24 '19 at 22:54
  • $\begingroup$ @DanHallatt if there is essentially a constant offset between each of the experiments, then the best way to proceed is to include the experiment as a fixed factor in a (potentially nonlinear) regression of all the data. Or do a linear regression on the linear sections of the curves with experiment as a fixed factor. If you just make all curves coincide at the beginning of their linear sections, you are putting a lot of weight on those 3 individual data points while discounting information about inter-experiment differences from all the other data points in the linear range. $\endgroup$ – EdM Jun 25 '19 at 2:15

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