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:

where Y_ij is the income of the j-th individual belonging to the i-th sex-group (say, i=1 means "male", i=2 - "female"), T_i is the unknown mean income for the i-th 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 - T₁ and T₂ . 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 T₁ and T₂ as two fixed values. For example, we use the mean value {Y_1j, j=1, ... , 30} as an estimate of the average income T₁ of male teachers, and we use the mean value of {Y_2j, 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.