[R] Growth rate determination using ANCOVA

Andrew Robinson A.Robinson at ms.unimelb.edu.au
Sun Nov 23 10:51:00 CET 2008

Hi David,

On Fri, Nov 21, 2008 at 12:01:52PM -0800, dschruth wrote:
> I'm a programmer in a biology lab who is starting to use R to automate
> some of our statistical analysis of growth rate determination. But I'm
> running into some problems as I re-code.
> 1) Hypotheses concerning Slope similarity/difference:
>    I'm using R's anova(lm()) methods to analyse a model which looks
> like this:
>               growth.metric ~ time * test.tube
>    I understand that testing the the interaction between time and tube
> (time:test.tube) will tell us if the growth rates (for the last three
> test tubes) are significantly different from one another (Ho=slopes
> are the same).  The purpose of doing this test is so that we can be
> certain our cultures have fully acclimated to the treatment and aren't
> going to change much if we stop measuring. This is an important cost
> saving practice too as measurements can go on for years.   Yet I'm
> worried that our null and alternative hypotheses should be swapped so
> that our test is more conservative (Ho=slopes are different ... ie
> still acclimating.)

Good thinking.

> Is there a way to specify my model that flips these hypotheses?
> Should I be using a different method?  Is this even appropriate?

You could think about equivalence tests.  See e.g. references in the
equivalence package. 
> 2) Growth Rate is confounded with Variance of Growth Rate
>    I'm also worried about the fact that rates for cultures with faster
> growth are calculated using fewer data points (assuming similar
> sampling times between treatments) .  The result is that growth ~ var
> (growth).   Not only does this put a wrinkle in my analysis between
> treatments, but it also biases the growth acclimation determining
> ANCOVA test above.  Faster growing cultures will usually pass the "no
> significant difference between slopes test" more easily because there
> are fewer points from which to be certain about rejecting Ho.
> Is there a way to control for this?
> Perhaps I could include the number of points in my model?

Depending on the model that you apply, you might be able to explicitly
model the variance to allow for this possibility.  I would guess that
it's not necessarily only the fewer data points contributing to the
greater variation.  Faster-growing cultures might also be inherently
more variable.
> 3) Statistical validity of using subsets of growth.metric measurements
> within a test tube
>    There are some lab members who insist that we can throw out the
> beginning and end of our log transformed growth.metric measurements
> because they are outliers in determining maximum growth.    I've
> proposed looping through all possible combinations of 3 or more points
> within the growth curve and using the highest or best fitting (best R-
> squared) slope.  But this idea has been rejected by our PI as not be a
> valid thing to do.
> Ideas here?

I'm feeling very cautious about giving advice on this question as I
don't know enough about the area.  Sorry.

I hope that this helps, otherwise.

Andrew Robinson  
Department of Mathematics and Statistics            Tel: +61-3-8344-6410
University of Melbourne, VIC 3010 Australia         Fax: +61-3-8344-4599

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