Hi everyone,
my question is the following. I have to perform an experiment with the aim of finding a correlation between a quantity X (e. g. temperature) and a quantity Y (depending on X). Which is the best statistical approach to it? I mean, the correlation between the log of Y and X is linear, hence how do I design the experiment according to the a priori power analysis? Moreover, once the different points have been experimentally determined, is it needed to repeat each test a certain number of times?
Thanks
Hello Luigi,
If the scenario is that you'll be manipulating the temperature (X) and taking a subsequent measurement of the target outcome (Y), then some of the salient questions are:
1. Are you more interested in whether a correlation (different from zero) exists, or are you more interested in the nature of that relationship (e.g., is it linear, quadratic, geometric, etc.)? The first is a test of r = 0; the second is curve-fitting.
2. Over what range of temperatures do you wish to know the result?
3. What degree of temperature change is ignorable (e.g., 1 degree or less; 0.25 degrees or less)?
4. How much precision do you want (if it's curve-fitting), or how much power do you want (if it's testing the hypothesis that r = 0)?
In the case of curve fitting, I'd suggest collecting data at all temperatures that are in the desired range and vary by your threshold for non-ignorable differences. A minimum of 3 data points per temperature chosen would allow for at least some within-temperature variation estimate; more would be better.
In the case of r= 0, then you'd want to cover the full range of target temperatures (as above); the multiple measures per temperature, while helpful, are not as essential. Here, the key point is to identify what degree of departure from r = 0 is worth detecting, should it exist in the population. That's your target effect size (some would use r-squared instead, or a related statistic, depending on the power calculation tool being used), and will help govern the derivation of suitable sample size given your desired power, and alpha level.
Good luck with your work.
amoung the most used technics to determin the correlation between two variable the PCA (Proncipal component analysis) after the statistical pretraitement you want.
Good luck
I would not consider looking for r or R² as some "best statistical approach".
You have the idea (assumption) that the relationship of X and logY is linear. So eech Kelvin increase in Temperature related to an y% change in Y. To my opinion, estimating y% (with a simple linear regression) would then be the "best approach".
If you rely on the linearity assumption, the best design would be to sample half of the values at the lowest tempretature and the other half at the highest tempreature in the range considered. If the linearity is in question, better distribute the measurements equally over the entire temp. range (you don't need replicates at a particular temperature to get a within-temperature variation). Even then, the average change is still best be estimated using a linear regression, but you may use other functional relationships (either theoretically founded functions or scatterplot smoothers like splines, as over-parametrized models).
@Jochen I agree with your recommendation if the continuity of y is assured. If not I would suggest a more appropriate method than OLS depending on the properties of the regression DV. This may be contained in your closing paragraph since you mentioned things like scatterplot smoothers. If so I agree completely with the suggestion. Best, David Booth
Dear David Morse and Jochen Wilhelm
thank you for your answers. The dependence is linear, I just have to perform again a certain well-known experiment with better instrumentation. The linearity can be reasonably assumed, according also to other literature rsults. The slope of the curve is not well-defined: for certain authors is positive, for other ones is negative, for other ones is null. The range is between 300 and 500 degC, effects on Delta T smaller than 5 degC are negligible. Concerning the linear regression, how do I estimate the minimum number of measurements (experiments) to be done? Does it have a sense?
Hello again Luigi,
So, you're really just estimating the Pearson correlation between temperature and quantity (since, in OLS regression with one IV, the standardized slope/beta = Pearson r, and when both X and Y are standardized, b = beta). Here, the operative question becomes, how precisely do you wish to estimate it, and with what level of confidence?
From your answer about temperature values, having just one observation each for the set (300, 305, 310,...,500) implies a minimum of 41 cases. How much precision does that N offer? It depends not only on N and the level of confidence, but on r(X,Y).
Assuming that both X and Y are standardized, and potential values of r(X,Y) is the set (.1, .2, ..., .9), 95% CI estimates of b would range from +/- 0.32 (for r = .10) to +/- 0.14 (for r = .90), when N = 41.
N of 410 yields CI limits for b of +/- 0.097 (for r = .10) to +/- 0.042 (for r = .90).
To get the precision to within +/- .05, an N of 1640 would yield CI limits of +/- 0.048 (for r = .10) to +/- 0.021 (for r = .90).
You'll have to decide how accurately you want to estimate r(X,Y) and the corresponding slope for your relationship between temperature and quantity.
Good luck with your work.
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