independent t-Test is used to determine significant changes between treatment and control(each group 3 replicates).
Three replicates, not good.
Of course you should consider error propagation, but with three replicates the accuracy in your estimate of the error (from any source) is no better than a random number generator.
Try this: run your experiment twice more (at least). Analyze each data set separately. Do you get the same result each time? Of course, this is the bare minimum, and it would be better if you could run your experiment with three replicates a few dozen times.
Alternatively, use the observed means and standard deviations as input to a random number generator. Assuming that your three replicates were an accurate approximation to the population values, what is the probability of another experiment with three replicates drawing the same conclusion?
Thanks Dr.Timothy for your guide, I will run with more replicate and will consider the error propagation in the calculations.
Where do you want the error to propagate to? I don't see where error propagation would really apply to in your case.
The t-test is about the expected difference between groups, and the error of this difference is calculated from the Fisher information. That's estimating the error of a statistic, not propagating. If you do have an estimate of this error, you might propagate it to some derived property that can be expresses as some algebraic relation (I don't see any meaningful example here).
A nice example for error propagation is following:
you want to estimate the area of a rectangular field. However, you can only measure the lengths of the sides a and b from which you will have to calculate the area as a*b. Now both your measurements are estimates, associated with some uncertainty or "error" (ua and ub, respectively). Error propagation can now be used to propagate these erorrs to the estimate of the area. Assuming the errors are not correlated, on gets sqrt( (b*ua)² + (a*ub)² ) for the error of the area a*b. See: https://en.wikipedia.org/wiki/Propagation_of_uncertainty#Example_formulas (f = AB).
@Jochen,
That I can easily see as error propagation.
In this case I assumed that we are using division rather than multiplication. There is a treatment effect on the chlorophyll/carotenoid ratio before versus the chlorophyll/carotenoid ratio after. There is error in both chlorophyll and carotenoid measures. How you assess this error with three replicates is the real mystery, and this is likely the wrong experimental design for addressing this issue.
I still don't see why one should propagate errors here. If the ratios per plant and time-point are calculated, their variance will already include the errors from both components (Chl.a and Carot.), so these errors are propagated "by design".
The estimate of the error is surely very vague for n=3, but it may be good enough if the difference between the time-points is large.
If there are several measurements of the ratios, these ratios vary, and this variation depends on the variations of the measurements for Chl.a and for Carot. If I am only interested in comparing such ratios, all I need is the variance of such ratios. If the variance of Chl.a or Carot. measurements was higher, it would impact the variance of the ratios.
The error in the ratio is a function of the error in the numerator and the error in the denominator and any correlations between the errors. So if Bob is taking both measurements then the care with which Bob takes the measurements could be a correlation in the errors. This only becomes apparent if you run an experiment designed to examine how any given protocol influences experimental outcome. You would start by hiring Alex, Rhonda, Joe and Alice.
@Jochen
Yes. The variance of three replicates does not require propagation of error. As you noted, n = 3 could be a problem.
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