# What statistical test can I run on data from an experiment with 1 independent variable and 2 dependent variables?

To specify my academic level, I am a high school student. For the lab report, I am required to conduct statistical tests to validate my results. What I had done was vary the concentration of an inhibitor (2%, 4%, 6%, 8%, 10%) and see how that impacts the activity of the enzyme. Essentially, it's an exothermic reaction meaning with a higher concentration of inhibitor, less heat would be let out. As well, when the enzyme typically breaks down its substrate, it releases an acid which changes the solution's pH. But more inhibitor means less acid.

I had calculated the initial temp/pH and final temp/pH. Then subtracted the values to get the change in temp/pH. For every concentration, I had 6 trials. I think I can get the mean change in temp/pH which would be better.

So what statistical tests are appropriate?

Welcome to CV! There are several options you could use. One relatively simple way in your situation would be ANCOVA (Analysis of Covariance) in which you would predict the final temp/pH from the concentration level having the initial temp/pH as the covariate. This way, you would get the predictions for change for different levels of concentration.

I'd probably just run separate analyses for pH and temperature. It's perhaps more elegant to analyze them simultaneously (e.g. with a multilevel model or a structural equation model), but with only two dependents, I don't think it's necessary.

I use pH as the example dependent below, you would then do the same things with temperature.

For ANCOVA, for example in R, you'd organize your data like this (sample= sample identifier, conc=concentration, prepH=initial pH, postpH=final pH)

sample conc prepH     postpH
1    2  0.5892052  0.3658990
2    4 -0.1070532 -2.3020591
3    6  1.0498823  0.7784372
4    8  0.1777278 -0.2202755
5   10 -1.3852736  0.5106044
6    2  0.2505689  1.3162620
7    4  0.3392950  0.2606596
8    6 -0.4703248 -0.2850500
9    8 -0.8250730  0.6677811
10   10  0.3909296 -0.6779350
...etc.


Then, you can run the ANCOVA

modelPH<-aov(postpH ~ factor(conc) + prepH, data=data)


Then, you can use emmeans package to find out whether pH change differs for different levels of concentration. Because you have a linear hypothesis (i.e. higher concentration = higher change), you could specify linear or consecutive contrasts:

#For linear contrasts, you first need to extract the emmeans grid for your predictor:

gridph<-emmeans(modelPH, "conc") #creates the grid

#Then, you extract the contrasts

phlin<-emmeans(gridph, "poly")
phlin

#The "linear" row in the output will tell you whether the change increases linearly as a function of concentration increasing, and how strongly. The other rows tell you whether there's a quadratic or even mode complex relationship between concentration level and change, but you can probably ignore those for the present purposes.


Another way would be to use "consec" contrast. This will tell you whether the consecutive concentration groups differ from each other (i.e. whether 2% differs from 4%, whether 4% differs from 6% etc.):

phconsec<-emmeans(modelPH, consec ~ conc)
phconsec
#The output gives you consecutive comparisons


If you want all pairwise comparisons (2% vs. 4%, 2% vs 6%, 2%vs 8% etc), use

emall<-emmeans(modPH, pairwise ~ conc)


If you have SPSS, you can do all this by using the menus, you just need to click around a little.

EDIT: To do ANCOVA in SPSS, use Analyze...General Linear Model...Univariate. Then, put the final pH to the dependent variable box, concentration to the Fixed Factor(s) box, and initial pH to the Covariate(s) box.

Then you need to decide what do you want to know about the effects of concentration: do you want to know whether change (initial --> final) increases linearly when concentration level increases? Or do you want to know whether change differs between each "concentration pair" (i.e. whether change for the 2% samples differs from change within the 4% samples, whether change for the 2% samples differs from change within 6% samples etc...), or something else. This is something you would do using the EM Means menu or the contrast menu (contrast menu if you want to test the linear increase, em means menu if you want something else).

Again, I recommend you run separate analyses for pH and temp. Don't do MANOVA, it's almost never a good idea.

• Hey! Thanks for answering. I'm very new to this so a lot of the points you were making sounded confusing but still made quite some sense. I haven't heard of ANCOVA before and I thought that perhaps MANOVA would work but honestly, I do not have the knowledge to understand which one is more appropriate in my case. I definitely must search how to do ANCOVA and I have just downloaded SPSS. Thank you! Commented Mar 11 at 15:19
• Hi! Sorry, I get that my answer was a bit overwhelming. Forget the parts dealing with the emmeans code for now (I'll leave them there as someone else might find them useful). MANOVA is a bit outdated analysis and makes assumptions that are often problematic. (cont.) Commented Mar 12 at 7:29
• (cont.) I recommend you start with ANCOVA predicting final pH (temp) from concentration level and having initial pH as a covariate. This way, you get the change rates for different concentration levels, and can test whether change is for instance stronger for 10% than for 6% etc. In SPSS you can do this via Analyze...General Linear Model...Univariate. Commented Mar 12 at 7:30
• See my edit at the bottom of the answer Commented Mar 12 at 9:18