Excuse the png equations, still a MathJax newbie.
I am analyzing data I have computed:
Alcohol content and Caffeine content retention after a duration of 8 hours for each.
I had gotten the data in the last four tables by using the first two tables shown below:
In the tables above, I attempt to compute the Average Rate of Change, which I believe to be correct. Using the formula;
In these tables above, I attempt to compute the Average Percent Rate of Change.
Using the formula;
I need to construct a model that could find the caffeine content after t hours. I figured it best be accomplished using the formula for exponential decay;
and so I decided to make something like this:
Until I realized I perhaps have not gotten the correct value to make r. Which is correct, because whiles my APRC for caffeine is slightly constant, that for alcohol is completely variant.
In fact the computation results for ARC and APRC in either of the four tables do not seem to make sense to me, why Alcohol that decreases at .015 every time must have a varying percentage results. All even being so far from each other in value.
Basically what I am trying to achieve is the correct rate value to use for my exponential decay formula. How do I derive it? Is it even possible, necessary or suggested that I go along with this formula? Professional insight is welcomed and appreciated.
Thank you all ahead of time.
Best Answer
1) For the Alcohol retention, you don't even need a linear regression, as the relationship is perfectly linear (straight line), thus your model is:
$$ y(t) = 0.19 -0.015t, $$
where $y$ denotes the Alcohol amount (g/dl) and $t$ time.
2) For the Caffeine, assuming that exponential decay model is reasonable fit, you should first perform a linearization, namely $$ y(t) = y(0)e^{\beta t} \to \ln(y(t))=\ln(y(0))+\beta t, $$ thus, you can view the transformed model as $$ y^*(t) = \beta_0+\beta t+\epsilon, $$ where $y^*(t)$ is the $\ln$ of Caffeine (mg) and $t$ is the time scale. So, technically, just take the natural log of the right column of the first Caffeine table and run simple OLS regression (available even in MS Excel). Finally, just take an exponent (of both sides) of the estimated equation in order to return to the original form of an exponential model.