I have a very simple data set: a parameter (concentration of a chemical) was measured at day 0, day 1 and day 2 in three subjects (there is a control group as well, but here all the values are always 0). The results are as follows:
S1 S2 S3 day 0 0 0 0 day 1 0 5 45 day 2 70 250 60
The zero hypothesis is that there is no increase in this parameter (in the untreated control group, the values are always 0 at any time point in any subject).
The alternative hypothesis is that there is a significant increase in the parameter with time.
As a biologist, I would say that no statistics is even necessary, the data are obvious: not a single observation of a non-zero concentration in the control group, and an observable increase in all of the treated subjects. However, the reviewer insists on having this thing called a "p-value".
At first, I considered using a mixed random model as follows. Treat the control group and the treatment group separately. Control group tells us only that the chemical in question is never observed without intervention, and that the H0 should be: no observation of increase in the chemical.
Now, fit a mixed random model to the treatment group and ask whether the slope is significantly greater than zero. In R:
d <- read.table( text= "day c S 0 0 S1 1 0 S1 2 70 S1 0 0 S2 1 5 S2 2 250 S2 0 0 S3 1 45 S3 2 60 S3", header= TRUE ) require( nlme ) l.m <- lme( c ~ day, random= ~ 1 + day | S, data= d, control= lmeControl( opt= 'optim' ) ) anova( l.m )
The result of this naive approach is as follows:
numDF denDF F-value p-value (Intercept) 1 5 1.451292 0.2822 day 1 5 3.753617 0.1104
I don't get it. Intuitively, this seems all wrong.
I have driven myself in utmost confusion. There must be a simple way of handling these data! (my question is: how should I handle the above data?)