# Random and fixed factors

In my masters work I want analyse whether a treatment with a specific drug changes a specific physiological response (dependent variable) in rats. However, it needs be stimulated so I can compare the magnitude of the response between the groups treated and non-treated. For the stimulation I have to administer another drug. This stimulater drug is given in 3 specific doses for each rat.

So thats my question! I control the levels of this factor (stimulater drug), I intentionally select these 3 doses, but my intention with this drug is just to analyse the physiological response and with the results conclude the effect of the treatment with my specific drug upon the physiological response in a general way.

I'm using rat ID as a random factor because each rat has received the 3 doses and the three responses are not independent. But I don't really know whether I should consider this stimulater drug as a random or a fixed factor!

• Have I understood this correctly? There are two groups of rats - a control group and a treatment group. The rats in the treatment group are given drug A. Then each rat in the treatment group is given dose 1 of drug B. You measure the physiological response. Then you give dose 2 to the treatment group and measure the response. And then dose 3. How many measurements are you making on the rats in the control group? Are the doses (1, 2 & 3) of drug B the same for all rats? Apr 14, 2017 at 6:36
• Yes! I have 2 groups, control and treatment (drug A). each rat from control group has received doses(1, 2 and 3) of drug "B" and for each dose I have a physiological response. The other group Treated (which has received drug A 24 hours before the running test) also received the doses (1, 2 and 3) of drug "B" and also has 3 physiological responses. Apr 14, 2017 at 14:34

You have two fixed factors and one random factor. 'Drug A' (with levels "control" and "treatment") and 'Drug B' (with levels "dose 1", "dose 2" and "dose 3") are fixed factors. 'Rat' is random factor, nested within Drug A.

It is not always easy to decide whether a factor is fixed or random. If a factor is something that the experimentor has deliberately applied, is under their control, and the levels can be reproduced by someone else, then this is a fixed factor.

You have deliberately applied drug A, or not (the control group), and also deliberately given 3 specific doses of drug B to the rats. Someone else could follow your methodology and the meaning of drug A, drug B, and the timing and amounts of drug B would be the same in their experiment as it is in yours.

In contrast, each rat has an individual ID and is a level of the 'Rat' factor. Let's take the first rat in the treatment group, for example. No-one else can replicate that rat directly in their experiments, they would have to use a different rat.

Your conclusions from your experiment are about the physiological effects of drug A in combination with drug B in those specific conditions (fixed factors) on the population of rats in general (from which your test subjects are a sample of, and hence a random factor).

Sokal and Rohlf (1995, pp198-205) give a more detailed account of the difference between fixed and random factors with respect to Model I and Model II ANOVAs and with the consequences for interpretation and the formulas for expected Mean Squares.

The Minitab support pages http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/anova/anova-models/fixed-and-random-factors/ and The Analysis Factor website http://www.theanalysisfactor.com/specifying-fixed-and-random-factors-in-mixed-models/ also have some useful information regarding when to consider a factor as fixed or random. And, there are some more thoughts on this highly viewed CrossValidated question What is the difference between fixed effect, random effect and mixed effect models?.

Reference: Sokal RR and Rohlf FJ (1995) Biometry. The principles and practice of statistics in biological research. 3rd edition. WH Freeman & Company, New York.

• If you think this is a useful answer, could you show this to other users by "accepting" the answer. Thanks. Apr 18, 2017 at 8:12
• Groovy, You said that Rat is nested within Drug A. So, I've looked in SPSS and there I have to choose a type of covariance. I've read about and it's not clear which one I should choose. could you help me? mixed models are quite new for me! Apr 24, 2017 at 20:11
• Sorry, I don't have SPSS so I can't answer that. Apr 26, 2017 at 9:31
• It seems to be a variance-covariance structure for the RAT ID. I found this Guideline www2.sas.com/proceedings/sugi30/198-30.pdf since the rats come from a controled vivarium and belong to a specific lineage, would be reasonable to think in a variance component or compound symmetry as a CS ? Apr 26, 2017 at 21:12