EDIT - Okay, You are have 20 rats and you take two measurements from 5 of the rats at a specific time but don't measure those rats again after that. You then repeat this for the 3 remaining time points. Both measurements are the level of a protein in the brain tissue BUT they measure them at different places in the brain. If you wanted to compare the two data sets, would this count as a repeated measure?
More specifically if you were doing a statistical analysis would you enter them into a mixed model two-way ANOVA (time point and location)? or an independent two-way ANOVA (time point and location)?
When the rat samples contain different rats over time, it is not a repeated measures (dependent samples) design unless there are some other dependencies that cause the observations to be correlated across time. For example, there may be some dependencies if you use a multilevel (hierarchical) sampling design in which, for example, groups of rats are nested within the same environments (e.g., cages). The key question here is whether the observations are correlated over time. This would naturally be the case (typically) if you used the same rats repeatedly. It may or may not be the case when you use different rats.
As you have described things, Time Point is a between-subjects factor and Location is a within-subjects factor (with 2 levels). You have not stated explicitly how many time points (groups) there are, but given that you described 5 rats at time points 1 and 2, and 20 rats in total, I would guess that there are 4 time points.
A conventional mixed-design (between-within) ANOVA model assumes that each rat is independent of all other rats. But as Christian Geiser noted, if the rats within each of the 4 groups are nested within cages, there could be an intracluster correlation > 0, which would indicate a violation of that independence assumption.
I am a bit puzzled about the 20 rats, as Bruce already mentioned. It is not clear to me, why this is relevant. I would read the design as follows: n1 = 5 for time point 1 and n2 = 5 for time point 2. The chosen rats are independent and are not chosen with replacement, so no rat can be chosen 2 times. Two obersavtions are recorded within each rat, hence, these are dependent. Is this correct? Or 4 time points as Bruce suggested?
In both cases, a two way between-wthin design would be appropriate, although problematic with such small sample sizes. All other considerations mentioned about possible nesting are still valid, but even more problematic with that sample size.
I think this is the key:
"I want to compare the two sets of data to look for any differences in level of protein in the two different brain locations over the 4 time points."
From each group of 5 mice you measure the protein concentration in two different brain locations. This is where you get the "two sets of data". And, yes, if you wish to say so, these data are paired between set 1 and set 2, because these data are from the same mice.
But when your data is complete (no missing values) it would be the simplest solution to calculate the ratio of the concentrations in the two locations for each individual mouse. So you end up with a single "set" of data, 20 concentration ratios for the 20 mice in total.
These 20 ratios are organized in 4 groups (each with 5 values from one time-point). All these ratios are statistcally independent, if there is nothing in the experiment that could serve as a common source of variance*, and can be analysed as such. These ratios are not normal distributed, so if you like to get p-values make sure to use the logarithms (log-ratios).**
* In the worst case, all mice in one group were kept in one cage, so that cage and group are fully confounded. In such a case there is no independent value in any group and you would not be able to answer your research question, because any difference you see in any particular time-point could be due to something related with that particular cage rather than with that time-point.
** Another way is to start with log-concentrations and calculate the differences of the log-conc between the two locations in each mouse. Yet another option would be to use generalized linear model of the quasi-Poisson family, but this is rarely used.
It is important to take care into account the definition of repeated measure and independent measure. In independent measures / between-groups: Different participants are used in each condition of the independent variable while repeated measures /within groups: the same participants take part in each condition of the independent variable. As you can realize is independent measure because the same rat did not take part in each condition of the independent variable.
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