# Which test to find out best concentration (the one having maximum effect)?

Hi my apologies first, I'm a biologist and not so good in statistics. In my study I'm studying the effect of concentration of feed on growth of a certain specimen. I have with me the different concentrations and the growth data for each concentration. What do you suggest is the best statistical test to find out the best concentration (i.e the one having maximum growth). Thanks in advance.

It sounds like an ANOVA (analysis of variance) setting, except that usually ANOVA is used to ask the question whether there are differences in growth between the concentrations. If you used ANOVA and found that there is a difference, then it wouldn't indicate which is the "best" one, just that they are different. There are post-hoc tests to look at all the different pairs.

I'd take a different approach though: Can the different concentration be considered as gradually increasing? (rather than just "different" from each other). What kind of relationship do you anticipate between the feed concentration and the growth? How many concentrations to you have? how many records per concentration? If you have more than 2-3 concentrations and sufficient data on each concentration level you could fit a parametric or nonparametric growth curve (x-axis is concentration, y-axis is growth). This would give you a clearer idea of what is going on, with less assumptions.

Are you not looking after method for studying dose-response relationship? From the description you gave, I guess your growth curves are highly non-linear. Moreover, the problem of using simple ANOVA to compare two growth curve (i.e., equivalent to a time course curve) is that it would be blind to the fact that the different doses are administered in a sequential order, though it should give a reasonable first idea of some kind of differences between the two, if any. However, multiple comparisons are likely not to answer the question you really want to ask since we often expect very few differences at baseline and larger ones at intermediate concentration levels, whereas what you generally want to assess is what's the lowest concentration level that yields a significant difference between the two products.

If you are using R, the drc package (see also www.bioassay.dk) allows to fit various models to DR data, and plot the resulting DR curve like the one shown below. It was generated from the on-line help with data secalonic whose description is:

Data stem from an experiment assessing the inhibitory effect of secalonic acids on plant growth.

Gong, X. and Zeng, R. and Luo, S. and Yong, C. and Zheng, Q. (2004) Two new secalonic acids from Aspergillus Japonicus and their allelopathic effects on higher plants, Proceedings of International Symposium on Allelopathy Research and Application, 27-29 April, Shanshui, Guangdong, China (Editors: R. Zeng and S. Luo), 209-217.

Ritz, C (2009) Towards a unified approach to dose-response modeling in ecotoxicology To appear in Environ Toxicol Chem.

If you have growth curves from each feed concentration, with time on the x-axis, you'll first want to estimate the growth from each curve. % change at the end points may be sufficient for your purposes, but to use all of your data consider looking to fitting some kind of model to each growth curve with a regression. Then you can estimate the parameters of the model and do a statistical comparison on those. Your institution might have access to graph pad, (can't link b/c not enough rep but google), and it is very intuitive.

This approach has some benefits. For example, one concentration might lead to faster growth initially but not peak as high, for whatever reason--maybe it becomes slightly toxic with time. This analysis could potentially detect that.

For more help we'd need to know the type of growth you expect from your specimen. If it's bacteria there are a ton of ways to do growth curves, see e.g. the classic paper by Zwietering et al here: http://aem.asm.org/cgi/reprint/56/6/1875.pdf.