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 nonrandom 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  N_{1}=30 males and N_{2}=70 females. The following model is chosen:

where Y_{ij} is the income of the jth individual belonging to the ith sexgroup (say, i=1 means "male", i=2  "female"), T_{i} is the unknown mean income for the ith level of "sex" in the population, E_{ij} are values of the deviation from T_{i}.
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 T_{i} as values of a random variable taking on two values  T_{1} and T_{2} . For example, we simply use the mean of the 100 values Y_{ij} 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 T_{1} and T_{2} as two fixed values. For example, we use the mean value {Y_{1j}, j=1, ... , 30} as an estimate of the average income T_{1} of male teachers, and we use the mean value of {Y_{2j}, j=1, ... ,70} as an estimate of T_{2}  the average income of female teachers.
Suppose a researcher decided to pick up N_{1}=30 male teachers randomly from all male teachers of New York, and N_{2}=70 female teachers from all female teachers. In this case, only the fixed effect approach is reasonable  because the N_{1} values T_{1} and the N_{2} values of T_{2} in the sample of 100 have not been drawn randomly from the population of interest.
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