CURVILINEAR EFFECTS

•  Several types of data transformations are appropriate for linearizing a curvilinear relationship.
• Direct approaches involve modifying the values through some arithmetic transformation (e.g., taking the square root or logarithm of the variable). However, such transformations are subject to the following limitations:
• They are applicable only in a simple curvilinear relationship (a relationship with only one turning or inflection point).
• They do not provide any statistical means for assessing whether the curvilinear or linear model is more appropriate.
• They accommodate only univariate relationships and not the interaction between variables when more than one independent variable is involved.

Specifying a Curvilinear Effect.

(Nguồn: https://www.semanticscholar.org/paper/)

• Power transformations of an independent variable that add a nonlinear component for each additional power of the independent variable are known as polynomials. 
• The power of 1 (X1) represents the linear component.
• The power of 2, the variable squared (X2), represents the quadratic component. In graphical terms, X2 represents the first inflection point of a curvilinear relationship. 
• A cubic component, represented by the variable cubed (X3), adds a second inflection point. 
• With these variables, and even higher powers, we can accommodate more complex relationships than are possible with only transformations.
• Although any number of nonlinear components may be added, the cubic term is usually the highest power used. 
• Multivariate polynomials are created when the regression equation contains two or more independent variables. We follow the same procedure for creating the polynomial terms as before, but must also create an additional term, the interaction term (X1X2), which is needed for each variable combination to represent fully the multivariate effects. 

Interpreting a Curvilinear Effect

• As each new variable is entered into the regression equation, we can also perform a direct statistical test of the nonlinear components, which we cannot do with data transformations.
• However, multicollinearity can create problems in assessing the statistical significance of the individual coefficients to the extent that the researcher should assess incremental effects as a measure of any polynomial terms in a three-step process:
1. Estimate the original regression equation.
2. Estimate the curvilinear relationship (original equation plus polynomial term).
3. Assess the change in R2. If it is statistically significant, then a significant curvilinear effect is present. The focus is on the incremental effect, not the significance of individual variables.

    For interpretation purposes, 

• The positive quadratic term indicates a U -shaped curve, whereas a negative coefficient indicates a inverse U-shaped curve. 
• The use of a cubic term can represent such forms as the S-shaped or growth curve quite easily, but it is generally best to plot the values to interpret the actual shape.

     How many terms should be added? 

• Common practice is to start with the linear component and then sequentially add higher-order polynomials until nonsignificance is achieved. 

Potential problems of using of polynomials

• First, each additional term requires a degree of freedom, which may be particularly restrictive with small sample sizes.  This limitation does not occur with data transformations. 
• Also, multicollinearity is introduced by the additional terms and makes statistical significance testing of the polynomial terms inappropriate. Instead, the researcher must compare the R2 values from the equation model with linear terms to the R2 for the equation with the polynomial terms.

Nguồn:

• Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2013). Multivariate data analysis (8th ed.). Boston: Cengage.