Experimental Design B/C

[binary010101]

Is there any way to easily make a log scale?

[andrewwski]

Actually, I think you'd take the log of the x values if you were looking for a logarithmic function, not a square root function (in which case, you'd just take the square root of the Xs!). You'd take the log of the Ys if you were looking for an exponential function, and both sides (as you suggested) if you were looking for power data. Also extremely common is inverse data, in which you'll just raise all of the X values to the negative first. You might also get inverse square, inverse root, etc.

Getting non-linear data is definitely a possibility. We also try to choose an experiment that'll give us linear data, but sometimes the experiment that clearly would be the best option involves a nonlinear relationship, in which case, you've gotta go for it. The transformations shouldn't be too bad, even without a graphing calculator.

Oh my, I really messed up my post (I was doing homework). The square root function is a power function. So you log the X and Y. And if it is exponential, you log the Ys.

Correct. Nejanimb's explanation is very good.

You can then use this linearized data to find the Least-Squares Regression Line, or line of best fit. This would be your best fit line for the linearized data. If you do essentially a reverse-linearization it will give you the best fit curve.

I've never done this event so I don't really know what data you'll run across, but it can often be difficult to distinguish between curves, especially exponential and power. To see which is the appropriate one, you need to calculate the r-values (correlation coefficient) for each. To do that by hand, you need to do a residual plot. That's going to be a lot of time consuming work. If you have a calculator that will do it for you, that's your best bet, as it can run the computations including the correlation coefficient very quickly. I'm not sure what the capability of the scientific calculators is in that respect though; the TI-89 (or 83/84) does it really easily, but they're not allowed.

[nejanimb]

Paradox, if you're hypothesizing a square root function (or for whatever reason, you have the inkling that it's a square root function), it's better to actually just take the square root of the Xs than to log both sides and call it a power regression. Yes, the square root is a power function (just ^.5) but if you take the power regression you might get a b value (for y = ax^b) that isn't exactly 0.5 (in fact, you almost definitely will). This means you cannot evaluate your hypothesis of a square root function properly. Similarly, you could call an inverse function a power function (just ^-1), but that also is not going to be as powerful for a regression. Basically, when you hypothesize a certain relationship, *that's* the transformation you should apply. If you hypothesize a power regression, log both sides; if you hypothesize inverse square, raise all of your X values to the -2; if you hypothesize exponential, log the Y values.

Andrewwski, I'd agree that determining the difference between exponential and power is tough when you're just looking at the graphed curves, but it's pretty easy to do a quick ratio-test do determine if you're looking at exponential data. What I've always found very tough is looking at inverse vs. inverse square. Does anyone know an easy test to figure out between the two of those without doing the full regression?