I keep coming back to this thread, which is really interesting. As a consultant, I ask myself:
How can I explain a complex concept to a client using a quick example if I only had a limited amount of time to do it? The easiest explanation that I can think of is that
fixed effects focus on what is "typical" while
random effects focus on "deviations from what is typical".
First, I would give a quick example to explain what "typical" is. For example, imagine we select 30 students at random from a single school (all in grade 5) and record their grade on the same math test. A "typical" grade for these students would consist of the average of the 30 students' grade and would estimate the unknown "typical" grade of all grade-5 students in that school. The "typical" grade is specific to this school.
Then, I would expand on this example to explain what "deviation from what is typical" means. In the previous example, imagine we randomly select 10 schools in the same district and then, within each school, we randomly select 30 students who will take the same math test (all in grade 5). Obviously, each of the 10 schools will have a "typical" grade. School A might have a "typical" grade of 75 (out of 100), School B might have a "typical" grade of 80 (out of 100). School C might have a "typical" grade of 85 out of 100, etc.
For each grade, the "typical" grade can be conceived of in relation to the "typical" grade corresponding to a "typical" school. For example, School A's "typical" grade might be lower than the "typical" grade of the "typical" school. School B's "typical" grade might be higher than the "typical" grade of the "typical" school, etc. School C's "typical" grade might be exactly equal to the "typical" grade of the "typical" school, etc.
We need a school-specific random effect to keep track of where the school-specific "typical" grade is in relation to the "typical" grade of the "typical" school. For example, a negative school-specific random effect associated with School A would indicate that School A's "typical" grade is lower than the "typical" grade of the "typical" school. On the other hand, a positive school-specific random effect associated with School B would indicate that School B's "typical" grade is higher than the "typical" grade of the "typical" school. A zero school-specific random effect associated with School C would indicate that School C's "typical" grade is exactly equal to the "typical" grade of the "typical" school, etc.
While the sign of the school-specific random effect tells us whether a school-specific "typical" grade is smaller than (negative sign), equal to, or larger than (positive sign) the "typical" grade of the "typical" school, the magnitude of the school-specific random effect tells us the extent of the deviation of the school-specific "typical" grade from the "typical" grade of the "typical" school. The larger the magnitude of the school-specific random effect, the larger this deviation. Conversely, the smaller the magnitude of the school-specific random effect, the smaller this deviation. A school-specific random effect equal to 0 implies no deviation.
Before we fit a mixed effects models to data such as the one described above for the 10 randomly selected schools, we won't know what the "typical" grade in the "typical" school is and to what extent the school-specific "typical" grades deviate from it. Using the model, we can estimate the "typical" grade in the "typical" school and quantify the magnitude and sign of the deviations of the school-specific "typical" grades from it. This will allow us to also estimate the school-specific "typical" grades.
One way I think about random effects is like a "sponge" term which captures all influences that may affect how the school-specific "typical" grade deviates from the "typical" grade in the "typical" school. Examples of such influences may include school size, school type (e.g., private, public), etc. These influences play out in different ways for each school, which is why we see School A behaving differently from all other schools, School B behaving differently from all other schools, School C behaving differently from all other schools, etc. Of course, if two schools are subject to the same influences which interplay in the same way, their corresponding random effects will be the same.
Note that, in the above example, the random school-specific effect is a so-called random intercept effect.
Anyway, the example given above can be further expanded if one wishes to.
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Isabella Ghement
Ghement Statistical Consulting Company Ltd.
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Original Message:
Sent: 03-25-2019 09:44
From: Ikenna Nnabue
Subject: mixed model
Dear all, can someone help me with a simple clarification on what random effect and fixed effect are in a mixed model. I have been appraoched many times for explanation on that. Definitions i got online couldnt help at all. A simple explanation with example will help a great deal.
Best,
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Ikenna Nnabue
Research Officer
National Root Crops Research Institute.
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