Dear Alan,
This is a standard problem in bioequivalence, where the usual solution is
to calculate area under the concentration time curves and compare them.
This presupposes that you have replicated samples for each of the two
treatments (that is to say that you can calculate several AUCs for each).
Failing that you could fit a linear model
Here is a crude example
> #Input data
day <- c(rep(c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10), 2))
> conc <- c(0, 1, 5, 8, 10, 15, 20, 40, 60, 62, 0, 5, 6, 9, 10, 14, 21, 38,
55, 60)
> treat <- factor(c(rep(1, 10), rep(2, 10)))
> #Set tuning parameter par for transformation
par <- 4
> trans <- log(100 - conc + par) - log(conc + par)
> results.frame <- data.frame(day, treat, conc, trans)
> results.frame
day treat conc trans
1 1 1 0 3.2580965
2 2 1 1 3.0252911
3 3 1 5 2.3978953
4 4 1 8 2.0794415
5 5 1 10 1.9042375
6 6 1 15 1.5441974
7 7 1 20 1.2527630
8 8 1 40 0.3746934
9 9 1 60 -0.3746934
10 10 1 62 -0.4519851
11 1 2 0 3.2580965
12 2 2 5 2.3978953
13 3 2 6 2.2823824
14 4 2 9 1.9889275
15 5 2 10 1.9042375
16 6 2 14 1.6094379
17 7 2 21 1.1999648
18 8 2 38 0.4519851
19 9 2 55 -0.1857171
20 10 2 60 -0.3746934
> plot(day, trans)
> fit1 <- lm(trans ~ day + treat)
> summary(fit1)
Call: lm(formula = trans ~ day + treat)
Residuals:
Min 1Q Median 3Q Max
-0.4678 -0.1262 -0.02138 0.2127 0.3573
Coefficients:
Value Std. Error t value Pr(>|t|)
(Intercept) 3.6895 0.1243 29.6771 0.0000
day -0.4022 0.0200 -20.0760 0.0000
treat -0.0239 0.0575 -0.4148 0.6835
Residual standard error: 0.2574 on 17 degrees of freedom
Multiple R-Squared: 0.9595
F-statistic: 201.6 on 2 and 17 degrees of freedom, the p-value is 1.443e-012
Correlation of Coefficients:
(Intercept) day
day -0.8864
treat 0.0000 0.0000
Here I have fiddled about to find a transformation that is approximately
linear but you could use one of the non-linear fitting procedures in Splus
or GenStat or SAS.
However, without replication the assumptions involved in any analysis are
very strong.
Regards
Stephen
At 14:57 26/08/2004 +0100, Alan Sharpe wrote:
>I am examining data which consist of % decay of a chemical in a sample.
>
>For 2 different treatments, I have a % decay value for days 1 upto 10,
>
>e.g.
>day 1,2,3,4, 5, 6, 7, 8, 9,10
>Treatment 1 - 0,1,5,8,10,15,20,40,60,62
>Treatment 2 - 0,5,6,9,10,14,21,38,55,60
>
>These are mean % values and I have a varying number of reps for each day
>and each treatment. I would not like to make any assumptions about the
>distribution of the data.
>
>I want to find out if the time vs % curves produced by this data are
>significantly different between treatments. Is a Kolmogorov 2-sample test
>ok for this type of data, or maybe Wilcoxons paired sample test.
>
>Can anyone suggest any other more appropriate/powerful tests
>
>Apologies if this is a simple one, but I am unsure whether the fact that
>the data are increasing day by day is going to influence the choice of test
>that is most appropriate
>
>Thanks in advance
>Alan
Stephen Senn
Professor of Statistics
Department of Statistics
15 University Gardens
<http://www.gla.ac.uk>University of Glasgow
G12 8QQ
Tel: +44 (0)141 330 5141
Fax: +44(0)141 330 4814
email [log in to unmask]
Private webpage: http://www.senns.demon.co.uk/home.html
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