Assumption Of A Linear Relationship Between The
Independent And Dependent Variable(s).
Standard multiple regression can only accurately
estimate the relationship between dependent and
independent variables if the relationships are linear in
nature. As there are many instances in the social
sciences where non-linear relationships occur (e.g.,
anxiety), it is essential to examine analyses for nonlinearity.
If the relationship between independent
variables (IV) and the dependent variable (DV) is no
linear, the results of the regression analysis will
under-estimate the true relationship. This underestimation
carries two risks: increased chance of a
Type II error for that IV, and in the case of multiple
regression, an increased risk of Type I errors (overestimation)
for other IVs that share variance with that
IV.
Authors such as Pedhazur (1997), Cohen and
Cohen (1983), and Berry and Feldman (1985)
suggest three primary ways to detect non-linearity.
The first method is the use of theory or previous
research to inform current analyses. However, as
many prior researchers have probably overlooked the
possibility of non-linear relationships, this method is
not foolproof. A preferable method of detection is
examination of residual plots (plots of the
standardized residuals as a function of standardized
predicted values, readily available in most statistical
software). Figure 1 shows scatterplots of residuals
that indicate curvilinear and linear relationships.The
third method of detecting curvilinearity is to routinely
run regression analyses that incorporate curvilinear
components (squared and cubic terms; see Goldfeld
and Quandt, 1976 or most regression texts for details
on how to do this) or utilizing the nonlinear
regression option available in many statistical
packages. It is important that the nonlinear aspects of
the relationship be accounted for in order to best
assess the relationship between variables.