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I have some questions on the application of statistical methodology to a
tumor efficacy study involving measurement of tumor volumes as a function
of time (days) following administration of specific drug treatment groups
(solid tumor growth inhibition measurements). If somebody in this group
could render their views, it would be awesome!
Experimental design: Studies involved solid tumor bearing mice. Once
these solid tumors reached about 100 mg (measurable size), treatment was
initiated with the anticancer drug injected on days 1, 5, and 9. Tumor
size was measured daily until conclusion of study at 15 days. We
performed statistical comparisons using ANOVA (between various treatment
groups). Now, the specific questions we have are:
 Is it appropriate to perform ANOVA on each's days data? If not, then
would performing ANOVA on the final times for statistical inference with
just description of the curves prior to that time point is considered
 What are the accepted procedures for analysis using methods for
 What does non-monotonicity of data and generalized estimating
equations (GEE) mean? How do they apply in such toxicological and
 It is likely that in such toxicological studies, one would have to
appropriately deal with missing data. What patterns of missingness
dictate the appropriate statistical interpretation of such results?
Any response on the above will be welcome and appreciated.
Department of Advanced Therapeutics
BC Cancer Agency
600 West 10 Avenue
V5Z 4E6 Canada
604/877 6010 ext. 3051
604/877 6011 (Fax)
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rajesh krishna wrote:
> I have some questions on the application of statistical methodology to
> tumor efficacy study involving measurement of tumor volumes as a
> of time (days) following administration of specific drug treatment
> (solid tumor growth inhibition measurements).
>  Is it appropriate to perform ANOVA on each's days data? If not,
> would performing ANOVA on the final times for statistical inference
> just description of the curves prior to that time point is considered
I would suggest you consider a different question. You have data that
describe the time course of drug effect so why not use a PKPD model to
describe it? ANOVA is the last resort of intellectually challenged
statisticians who have no understanding of biology and no interest in
using methods which are more revealing than a P value.
>  What are the accepted procedures for analysis using methods for
> longitudinal data?
NONMEM is a method which can describe PKPD and take into account
differences between individuals. There are other methods but NONMEM is
the most readily available.
>  What does non-monotonicity of data and generalized estimating
> equations (GEE) mean? How do they apply in such toxicological and
> efficacy studies?
Non-monotonicity means the time course of the observations can go up and
down. This means to me that you need to describe your data with a
non-linear model. GEE are still only used by statisticians (as far as I
know) and I think they are restricted to linear models. NONMEM has no
such restriction and has lots of people who know how to apply it to
>  It is likely that in such toxicological studies, one would have
> appropriately deal with missing data. What patterns of missingness
> dictate the appropriate statistical interpretation of such results?
There is no simple answer to this. Broadly, missingness can be
considered uninformative (eg data are missing at random of for reasons
quite unrelated to the drug effect) or informative (eg data are missing
because a patient dropped out because of treatment related side
effects). The former can essentially be ignored. The latter is harder to
deal with and is the subject of state of the art investigation by
workers such as Lew Sheiner and Don Rubin.
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford.-a-.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
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Per Dr. Krishna's request I have posted my response to him below.
Dear Dr. Krishna:
I am not the right person to answer your statistical questions, since I
am an engineer and only a recent convert to pharmacology. But I thought
I would provide my thoughts which you are certainly welcome to dismiss
as nonsense if you so choose.
question 1: performing ANOVA on each day's data.... Certainly you could
do this. Presumably the reason would be to estimate the variability in
the rate of efficacy (and toxicity perhaps) across your population. The
first question I would have is "is your sample population large enough
to provide you a reasonable confidence interval on this estimate of rate
of efficacy (toxicity)?".... I would at least do it if the data are
Just using the endpoint is probably more meaningful since this is what
will likely be the most important clinical endpoint when you take the
drug into humans....i.e. did my tumor get smaller (or larger) and if so
by how much and what were the side effects? I would perform both
analyses but concentrate my efforts on the later.
question 2: I will leave this for the pure statisticians in the group.
My only insight is that I use the sampled data you have (it is
essentially discrete samples) to build a model of the time-continuous
function of tumor growth. I use a maximum likelihood estimation method
to calculate the curve parameters using a Gompertz model for tumor
growth. I then use this model (basically the differential equations for
the PK and PD models and clinical endpoint models) to simulate the
results of a planned clinical trial for the new drug. Last Fall I
presented a poster session on this process at a conference in Geneva for
a "psuedo" cancer drug (to show people how to do it). I would be glad
to send you a copy of the paper if you are interested.
question 3: I will leave this for the statistics professionals.
question 4: The issue of missing data (or incorrect data) is always a
difficult thing to address. Perhaps this is where the science of
statistics meets the art of applied math.
First you may want to consider a filter on the data you have collected
(which may not sit to well with the statisticians in the audience). I
have used a windowed median approach in the past as a way to filter
sampled data of time continuous functions (such as tumor growth) but
have not applied it to cancer modeling as yet. Once you have filtered
out bad data, the question becomes what do we do with the gaps. If the
data are extremely sparse there are some algorithms for bridging them,
but they all make some assumptions about the behavior of the underlying
function you are trying to model, so you will have to choose wisely.
Essentially, your sparse data will mostly impact your confidence
intervals when you try to estimate the underlying function. At some
point, and it depends on the function, the confidence interval becomes
so large (because of sparse data) that your conclusions become
unwarranted. This will become apparent during your ANOVA.
Like I said, I am not a statistician. I just use statistics in my
work. Hopefully, these few words shed some light on the issues you
Please feel free to contact me directly if I can be of any assistance.
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