60. Polynomial Analysis of Circadian Rhythms (1 Patient with Hypertension)

Ambulatory blood pressure measurements and other circadian phenomena are traditionally analyzed using mean values of arbitrarily separated daytime hours. The poor reproducibility of these mean values undermines the validity of this diagnostic tool. In 1998 our group demonstrated that polynomial regression lines of the 4th to 7th order generally provided adequate reliability to describe the best fit circadian sinusoidal patterns of ambulatory blood pressure measurements (Van de Luit et al., Eur J Intern Med 1998; 9: 99–103 and 251–256).

We should add that the terms multinomial and polynomial are synonymous. However, in statistics terminology is notoriously confusing, and multinomial analyses are often, though not always, used to indicate logistic regression models with multiple outcome categories. In contrast, polynomial regression analyses are often used to name the extensions of simple linear regression models with multiple instead of first order relationships between the x and y values (Chap. 16, Curvilinear regression, pp 187–198, in: Statistics applied to clinical studies 5th edition, Springer Heidelberg Germany 2012, from the same authors as the current work). Underneath polynomial regression equations of the first-fourth order are given with y as dependent and x as independent variables. This chapter is to assess whether this method can readily visualize circadian patterns of blood pressure in individual patients with hypertension, and, thus, be helpful for making a precise diagnosis of the type of hypertension, like borderline, diastolic, systolic, white coat, no dipper hypertension.

Can higher order polynomes visualize longitudinal observations in clinical research.

In an untreated patient with mild hypertension ambulatory blood pressure measurement was performed using a light weight portable equipment (Space Lab Medical Inc, Redmond WA) every 30 min for 24 h. The question was, can 5th order polynomes readily visualize the ambulatory blood pressure pattern of individual patients? The first ten measurements are underneath, the entire data file is entitled “chapter60polynomes”, and is in extras.springer.com.

SPSS statistical software will be used for polynomial modeling of these data. Open the data file in SPSS.

For analysis the module General Linear Model is required. It consists of four statistical models:

We will use here Univariate. ....then click the green triangle in the upper graph row of your screen.

The underneath table is in the output sheets, and gives you the partial regression coefficients (B values) of the 5th order polynomial with blood pressure as outcome and with time as independent variable (−7,135E-6 indicates 0.000007135, which is a pretty small B value). However, in the equation it will have to be multiplied with x⁵, and a large very large term will result even so.

The entire equations can be written from the above B values: This equation is entered in the polynomial grapher of David Wees available on the internet at “davidwees.com/polygrapher/”, and the underneath graph is drawn. This graph is speculative as none of the x terms is statistically significant. Yet, the actual data have a definite patterns with higher values at daytime and lower ones at night. Sometimes even better fit curves are obtained by taking higher order polynomes like 5th order polynomes as previously tested by us (see the above section General Purpose). We should add that in spite of the insignificant p-values in the above tables the two polynomes are not meaningless. The first one suggests some white coat effect, the second one suggests normotension and a normal dipping pattern. With machine learning meaningful visualizations can sometimes be produced of your data, even if statistics are pretty meaningless.

24 h ABPM recording (30 min measures) of untreated subject with hypertension and 5th order polynome (suggesting some white coat effect)

24 h ABPM recording (30 min measures) of the above subject treated and 5th order polynome (suggesting normotension and a normal dipping pattern).

Polynomes of ambulatory blood pressure measurements can be applied for visualizing not only hypertension types but also treatment effects, see underneath graphs of circadian patterns in individual patients (upper row) and groups of patients on different treatments (Figure from Cleophas et al, Chap. 16, Curvilinear regression, pp 187–198, in: Statistics applied to clinical studies 5th edition, Springer Heidelberg Germany 2012, with permission from the editor).

Polynomes can of course be used for studying any other circadian rhythm like physical, mental and behavioral changes following a 24 hour cycle.

More background, theoretical and mathematical information of polynomes is given in Chap. 16, Curvilinear regression, pp 187–198, in: Statistics applied to clinical studies 5th edition, Springer Heidelberg Germany 2012, from the same authors.