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What is the difference between a fixed effect and a random effect? Are fixed and random effects different from fixed and random factors?

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Here is the basic idea (simplified a bit).

For simplicity let's define as our 'effect' (can also be called 'factor') as day of the month

A fixed effect is usually something where

  1. You have made an intentional choice of which values to collect data on. For example, the first 10 days of the month
  2. You only want to make inference (i.e., estimate stuff) to the values you have collected data on (so only on the first 10 days of the month).

A random effect is usually something where

  1. You have randomly selected the values to collect data on. For example, you randomly choose 10 days of the month
  2. You want to make inference (i.e., estimate stuff) for all of the values that you sampled from (not just those in your sample, but the whole population of things you sampled from). For example for all days of the month (typically that would be something you would estimate the average of)

If I want to estimate money in a month spent buying milk I would randomly select 10 days (any nonrandom selection of 10 days could give a biased estimate).

If I want to estimate money spent on only the first 10 days of the month, then obviously I need data on the first 10 days.

Although the term 'random effect' is usually used only for certain types of factors, almost all studies actually have at least one random effect. We generally sample a set of subjects from a population (the randomly sampled part of the definition) and we generally want to make inference to the whole population we sampled from. So actually, here subject is a random effect.

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