Curvilinear Relationships Definition
A curvilinear relationship could be a sort of any relationship, for instance, it may be square, cubic, etc. other than a linear relationship between two variables or among a set of variables.
Overview of Curvilinear Relationships
A convex or U-shaped graph will form if one of the variables decreases and another variable increases but only up to a certain point, since after that both the variables increase.
And a concave graph or inverted U-shape graph will form if both the variables increases up to a certain point but after that one variable increases and another variable decreases. Other curvilinear graphs are cubic, then polynomial, etc.
When is a curvilinear relationship used?
Sometimes, upon analyzing data regarding correlation and linear regression, it may be observed that the relationship between an independent and a dependent variable (Let X, be the former and Y be the latter), follows a curved line. This is in contrast to the straight line that would ordinarily have been observed.
In cases like these, the line of linear regression would not be able to properly describe and predict the relationship between these variables. It would also mean that the P value cannot accurately test the null hypothesis, and the variables would appear to be un-associated. So, in such a case, options that may be considered, are-
- Data transformation
- Finding a curvilinear relationship.
Before trying curvilinear regression, we should try data transformation. Data transformation is a method where we take logarithmic or exponent of the variable and then find a linear relationship. It is bit easier and take less time compare to the curvilinear regression. But in curvilinear relationships we take the square, cubic or pth power of the independent variables p=2,3,4, 5,… and then find a relationship between pth power of independent variables and dependent variables.
Curvilinear regression hypothesis and assumptions
Hypothesis:
The null hypothesis states Ho: There is no association between the dependent variable and independent variable. It means that the independent variable, X cannot predict the dependent Y variable. Adjusted-R2 is used to measure the association between the two variables.
Where is the coefficient of determination. It tells about the total variation of the dependent variable explained by the independent variable.
Adj goes on increasing as we go on adding more parameters. For example, adj in a quadratic regression is always more than the linear regression, and a cubic regression will have higher adj than the quadratic regression. We will stop adding the parameter when we see that there is no change in the value of adj .
Assumptions:
In curvilinear regression just like linear regression there are assumptions that are need to be followed before doing regression analysis. These assumptions are:
- The dependent variable is normally distributed and the variances are homoscedastic in nature i.e. the variances are same.
- The data points of the variables are independent in nature.
- It is assumed that we are fitting the right kind of curve in our data. If the data is quadratic, then we should perform quadratic regression, and if the data is exponential then we should perform exponential regression.
Testing curvilinear relationships
