Linear vs. Quadratic in Comparisons - Why?

#1
Hello,

I have a set of data for the level of antioxidants in tea, with two variables (water temperature of either 70 or 90 degrees, and steeping time of either 2, 3 or 4 minutes). There are 4 replications each of 6 treatment combinations.

I did a trend comparison in SAS, and I got the table below:
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Contrast .....................DF.....Contrast SS.....Mean Square.....F Value.....Pr > F
Temp............................1.....0.39526667.....0.39526667.......19.70.....0.0003
Steep linear ..................1.....1.14490000.....1.14490000.......57.06.....<.0001
Steep quadratic..............1.....0.08167500.....0.08167500.......4.07.....0.0588
Temp * Steep linear........1.....0.00302500.....0.00302500.......0.15.....0.7024
Temp * Steep quadratic...1.....0.10083333.....0.10083333.......5.03.....0.0378
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So I know this is telling me that there is a significant response in temperature (because the p value is .0003) and there is a significant linear response in steeping (p value <.0001) but not in the quadratic response (p value .0588). It looks like there is a significant interaction between temperature and steep quadratic, but not temperature and steep linear.

I think I'm reading it correctly (I THINK), and I can tell you all of the above, but I don't know what it all means! I'm struggling to understand it. What does it mean to say that there is a significant linear response, or a significant quadratic response? I know we are somehow comparing the treatments, to see what changes happen during each treatment, and if it's significant. But I don't know the difference between linear and quadratic (and cubic and quartic, for that matter).

I greatly appreciate any enlightenment you can give me, so that when I run this stuff in SAS, I actually understand what I'm doing! If you need more information than what I gave above, let me know.

Thank you!
Jungle Rat
 
#2
Well the regression function you have might look something like this:

aox lovel = a + b(Temp) + c(Steep) + d(Steep^2) + e(Temp*Steep) + f(Temp*Steep^2)

Where Temp, Steep, Steep^2, Temp*Steep, Temp*Steep^2 are predictor variables, and "aox level" is the outcome variable. We say that this function is linear in Temp since it only has a power of 1. Likewise, this function is quadratic in Steep because it has a Steep^2 term. There is also a linear Steep term. Each of the b, c, d, e, f are an indicator of the relationship between the corresponding predictor variable and aox level. If one or more of the b, c, d, e, f are equal to 0 then there is not a relationship between the corresponding predictor variable and the outcome variable. The p-values in the table above are the result of a test to see if these b, c, d, e, f are zero. If the p-value for Steep linear corresponds to a test for c = 0. Since the p-value is less than 0.05, we conclude that c is not zero. We say that there is a "significant linear response in Steep" because c, which corresponds to the linear Steep term is significantly greater than 0.

The same type of argument can be given to squared terms like Steep^2. We say that there is/is not a significant quadratic response in Steep. If there is a Steep^3 or Steep^4 term in the model, we would say something about the cubic or quartic response in Steep.

Hope this makes it a little clearer.

~Matt