Dear all,
            It is a blessing to have this service to ask any tricky
questions. I will appreciate any guidance. Questions 2 and 3 please.
Thanking you in advance and best wishes
Ian


The Competitive index

Overall Aims
The method is used to compare the ratio of viable mutant (mut) and wildtype
(wt) bacteria in an inoculum (the input), with the mut:wt ratio obtained a
number of days after infection of a mouse (the output).  If the mutant is
attenuated, during the infection, the mut:wt ratio will be less in the
output than the input.  In the past, the method has been used successfully
to identify very attenuated mutants without much consideration given to
estimating errors.  I want to use the method to identify much more mildly
attenuated mutants, and therefore need to devise an experimental design
which reduces random errors to a tolerable level.

The method - Estimating bacterial numbers
Measuring input and output numbers of bacteria is done by plating on agar
and counting bacterial colonies which grow on the plates.  We count 'total'
bacteria on non-selective media, and mutant bacteria on media containing
antibiotic, since the mutants have an antibiotic resistance gene engineered
into them.  Thus the number of wt is not measured directly, but can be
estimated in either of two ways as follows.
        (a)     wtI = totI-mutI         This has been the standard method.
        (b)     wtI = 2xtotI - mutI      I have used this as I think it may
reduce systematic bias.  By diluting bacteria for the 'total' plate by
2-fold compared with the antibiotic plate, I count roughly the same number
of colonies on 'total' and mut plates.

Likewise for the output:
        (a)     wtO = totO-mutO
        (b)     wtI = 2xtotO - mutO

I want to be able to quantify the random error in my input and output mt:wt
ratios, and hence determine how many replicants at each stage of the process
are appropriate.

Since var(x+-y)=var(x) + var(y), I plan to use this to work out the variance
of my wt estimates.   For method (a), can I assume, var(wt)=var(tot) +
var(mut) since total and mutant are measured independently?  Of course, the
actual value of total will depend on the contribution made be the mutant, so
the variables are not really independent.  How should I work out the
variance for method (b)?


The method - Estimating the competitive index (CI)

CI=(mutO/wtO)/(mutI/wtI)

Since I am dealing with ratios, I think the best thing is to convert to
logs. Using log(a/b)=log a - log b, I obtain the following:

logCI = logmutO - log wtO - log mutI + log wtI

If I am right, the variance should be:

Var(logCI) = var(logmutO) + var(logwtO) + var(logmutI) + var(logwtI)

Ultimately, I want to work out the confidence intervals for log(CI), then
transform to obtain the confidence intervals for CI

Problems
There are a number of difficulties for me with my proposed method:
1.      As mentioned, working out the variances of my wt estimates.

        2.      At what stage should I convert to logs?  I have used the raw
data to estimate wtI and wtO.  I can convert the mean to log mean, but how
do I convert the variance to a logged variance?

        3.      How should I work out the degrees of freedom to construct
confidence intervals?  If I have n samples to estimate tot (d.f.=n-1) and m
samples to estimate mut (d.f.=m-1), what are the corresponding values for
number of samples and d.f. for my wt estimate?  And for the final analysis,
what value for n and d.f. for log mutO - log wtO - log mutI + log wtI

Additional comments
Ultimately, I would like to be able to do the analysis myself, either using
the method outlined above, or a better method of your recommendation.  I
have been using Excel but would consider acquiring more specialist packages
if it was your recommendation.  When I have some good error estimates, it
may be worthwhile doing an analysis of variance (within mouse output vs
between mouse output), and would appreciate your help for this.