Fixed Effects:
The term "fixed effects" (as contrasted with "random effects") is related to how particular coefficients in a model are treated - as fixed or random values. Which approach to choose depends on both the nature of the data and the objective of the study. A fixed effect approach can be used for both random and non-random samples. Random effect models are usually applied only to random samples.
Suppose the data at hand are values of the annual income of 100 school teachers - N1=30 males and N2=70 females. The following model is chosen:
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where Yij is the income of the j-th individual belonging to the i-th sex-group (say, i=1 means "male", i=2 - "female"), Ti is the unknown mean income for the i-th level of "sex" in the population, Eij are values of the deviation from Ti.
Suppose the 100 individuals has been drawn randomly from a population, for example, from all school teachers of New York.
If the question of interest is the average income of New York school teachers, then the random effects approach is reasonable. We treat Ti as values of a random variable taking on two values - T1 and T2 . For example, we simply use the mean of the 100 values Yij as an estimate of the average income of New York teachers.
If the question of interest is the average income of female and male teachers separately, then we treat T1 and T2 as two fixed values. For example, we use the mean value {Y1j, j=1, ... , 30} as an estimate of the average income T1 of male teachers, and we use the mean value of {Y2j, j=1, ... ,70} as an estimate of T2 - the average income of female teachers.
Suppose a researcher decided to pick up N1=30 male teachers randomly from all male teachers of New York, and N2=70 female teachers from all female teachers. In this case, only the fixed effect approach is reasonable - because the N1 values T1 and the N2 values of T2 in the sample of 100 have not been drawn randomly from the population of interest.
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