Curvilinear Regression

Polynomial analysis is an extension of simple linear regression, where a model is used to allow for the existence of a systematic dependence of the dependent y variable (blood pressure) on the independent x variable (time) different from a linear dependence. Polynomial extension from the basic model can be done as follows:

y = a + bx (first order) linear relationship

y = a + bx + cx² (second order) parabolic relationship

y = a + bx + cx ²+ dx³ (third order) hyperbolic relationship

y = a + bx + cx² + dx ³ + ex ⁴ (fourth order) sinusoidal relationship

where a is the intercept and b, c, d, and e are the partial regression coefficients. Statistical software can be used to calculate for the data the regression line that provides the best fit for the data. In addition, regression lines of higher than 4 orders can be calculated. Fourier analysis is a more traditional way of analyzing these type of data, and is given by the function f(x) = p + q₁ cos (x) + ..+q_n cos n (x) + r₁ sin (x) +..+ r_n sin n (x) with p, q₁…q _n, and r₁…r_n = constants for the best fit of the given data.

As an example, ambulatory blood pressure monitoring (ABPM) using light weight automated portable equipment is given. ABPM has greatly contributed to our understanding of the circadian patterns of blood pressures in individual patients¹ as well as to the study of effects of antihypertensive drugs in groups of patients.² However, a problem is that ABPM data using mean values of arbitrarily separated daytime hours are poorly reproducible^3,4, undermining the validity of this diagnostic tool. Previous studies have demonstrated that both in normo-⁵ and in hypertensive groups⁶ time is a more powerful source of variation in 24 hour ABPM data than were other sources of variation (between P<0.01 and <0.001 versus between not significant and <0.01). This reflects the importance of the circadian rhythm in the interpretation of ABPM data, and the need for an assessment that accounts for this very rhythm more adequately than does the means of separated daytime hours. We also demonstrated that polynomial curves can be produced of ABPM data from both normo-⁵ and hypertensive⁶ groups, and that these polynomial curves are within the 95% confidence intervals of the sample means. However, intra-individual reproducibility of this approach has not been assessed, and is a prerequisite for further implementing this approach.