If I have to count the bacterial load of 3 different stations in a lake along time (4 seasons), how should I analyse the results? Should I use Mixed ANOVA or 2-way ANOVA? Any other suggestion and why?
Unless you had multiple subjects with each station, then Station cannot be a between-subjects factor. The way you described the design above, at each timepoint you have one measurement per station. A true factor would have multiple measurements within each level, e.g., if you had 10 samples within each station, then you could treat samples as your "subjects" and treat station as a between-subject factor. But if you only took one measurement per station per timepoint, then your design matrix looks like this, with only 1 observation per cell (see attached image):
t1 t2 t3 t4
A 1 1 1 1
B 1 1 1 1
C 1 1 1 1
In such a situation, it would be impossible to do an ANOVA (or inferential statistics at all). ANOVA is based on looking at the amount of variance within cells vs. the amount of variance between cells; if you only have one measurement per cell then you have no variance within cells. So if this was your design, then you cannot use ANOVA to compare different stations.
If you think about what kind of factors you are analyzing, that will answer your question. You have one factor: time (4 seasons), which is a repeated-measures factor (for each station, you took 4 measurements from that same station). So you know that you need a repeated measures ANOVA.
You can't do a 2-way ANOVA because you only have one factor. Likewise, you can't do a mixed ANOVA because a mixed ANOVA by definition needs at least two factors (one repeated measures factor and one between-subjects factor). You need a repeated measures ANOVA with one factor (time).
Thank you Stephen for your reply. How about different station? I would consider them as between-subject factor as I need to compare them as well. If so, I guess I need to use a mixed ANOVA. Station can be considered as between subject factor and time as within subject. Am I right?
It depends, do you have more than one measurement within each station? If not, you can not take station as a between factor, because there would be not variance within each station for a single measurement point.
Unless you had multiple subjects with each station, then Station cannot be a between-subjects factor. The way you described the design above, at each timepoint you have one measurement per station. A true factor would have multiple measurements within each level, e.g., if you had 10 samples within each station, then you could treat samples as your "subjects" and treat station as a between-subject factor. But if you only took one measurement per station per timepoint, then your design matrix looks like this, with only 1 observation per cell (see attached image):
t1 t2 t3 t4
A 1 1 1 1
B 1 1 1 1
C 1 1 1 1
In such a situation, it would be impossible to do an ANOVA (or inferential statistics at all). ANOVA is based on looking at the amount of variance within cells vs. the amount of variance between cells; if you only have one measurement per cell then you have no variance within cells. So if this was your design, then you cannot use ANOVA to compare different stations.
Thank you Stephen and Rainer. I got the point. Since I had 3 measurements per station, I guess I should treat station as between-subject factor and use a mixed-model.
Then it is possible, but I would expect that your design is under-powered.
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