Validity of an error term regarding individual variability (R)

Question

Here is my data:

_ I measured the response of a variable "Y" to 3 different treatments over time. "Treatment" is hence a 3 levels factor and "Time" is a continuous and numeric variable (unit=hours, from 1 to 6, time step=1).

_ In each experimentation, 1 and only 1 treatment was applied on the same individual over time (6h). Each treatment was replicated 10 times, but each replicate was done on different individuals.

My question is: Do I have to consider the error term due to the variability between individuals?

Like this--> model=lm(Y ~ Treatment * time + Error(Individual/Time)) ... or is it unnecessary because each experimentation is done on a different individual? I don't know yet how to add this error term if needed, just wondering about it.

It is not clear for me, and I didn't found any clear answer in the different questions asked on "Cross Validated". Thanks for your help!

Answer 1

If I understand you correctly, you applied each treatment on 10 different patients, and no patients received more than one treatment (so you have 30 different individuals in total). You then measured the dependent variable Y once per patient, 1-6 hours after the treatment was applied (or is it that the treatment was applied during 1-6 hours?), so you have one measurement of Y per patient. If all of this is correct, then you do not need to use any random effects, specify a correlational structure for the error terms or such.

Provided that I have understood the design of your experiment correctly, your model should be fine:

lm (Y ~ Treatment * time)

However, if you only have 30 observations, then you might not have any luck in using the interaction terms, since you will then have 6 parameters to estimate compared to 4 parameters without the interaction term:

lm (Y ~ Treatment + time)

Hope this helps!

EDIT: From the commentary I now understand that you have 6 measurements per patient. You then need a linear mixed model (perhaps a repeated measures ANOVA will do, but I'm not very familiar with it) to take the repeated measurements into account. This is because there will probably be correlations between the measurements of each individual.

library(lme4)
lmer (Y ~ Treatment * time + (1|ID))

Where ID is unique for each patient. But perhaps different patients metabolizes the drug att different speeds or something similar, so you could add a random effect on the slope of time for each individual:

lmer (Y ~ Treatment * time + (1 + time|ID))

Of course, all of this assumes that the effect of time on Y is linear, which may not be the case..