(This is a draft and truncated version - for final and full version, see
Concise Encyclopedia of Biostatistics for Medical Professionals)
curvilinear regression, see also regression (types of)
This is a regression of y on x (or set of xs) that graphically yields a curve instead of the usual straight line. A curvilinear regression is obtained when quadratic (square), cubic, logarithmic, exponential type of terms are added among the regressors. Another term for this is polynomial regression.
For simplicity we explain this with one dependent and one independent variable. Linear regression in this case is y = a + bx. It becomes curvilinear when
Curvilinear regression of power K: y = a0 + a1x + a2x2 + a3x3 + … + aKxK.
Depending upon the maximum degree of x, this is called polynomial regression of degree K. When K =2, this is called quadratic or parabolic curve because of the shape it gets, for K= 3 is called cubic, K= 4 quartic, etc. Each power of x is considered as a new variable in this regression although this may raise the question of multicollinearity. Figure C.35 shows a linear regression by a line, a quadratic regression by a thick curve and a cubic regression by the dotted curve. A linear regression is a straight line with no turn, quadratic will have one turn, cubic will have two turns, etc. A linear regression is appropriate when y is increasing at a constant rate for each unit increase in x such as between body temperature and pulse rate, quadratic is appropriate when y show increase for initial value of x and then leveling off of or decrease (or the vice versa), and cubic is appropriate when y shows increase then decrease and then increase again. Regression of total glomerular filtration rate plasma on creatinine level is quadratic  and of chronological age on dental age is cubic . Polynomials can be fitted sequentially – first try linear then examine quadratic, then go to cubic, etc. However, the regression coefficients will also change: the regression coefficient of x in quadratic equation will not be the same as in linear equation.
Let us extend this concept to two regressors. A linear regression with two regressors will graphically give a plane (Figure C.36a). When quadratic terms are introduced, the plane bends as in Figure C.36b. Because of the complexity, this kind of regression is rarely used in health and medicine.
Figure C.35: Linear and two curvilinear regressions with one regressor (x)
Figure C.36: Regressions with two regressors (a) linear; (b) curvilinear
For final and full version, see
Concise Encyclopedia of Biostatistics for Medical Professionals