Blood Pressure Monitor Hardware

Analysis of 24-Hour Ambulatory Blood Pressure Monitoring Data using Orthonormal Polynomials in the Linear Mixed Model

Winston-Salem, Nc 27157-1063

SUMMARY. The use of 24-hour ambulatory blood pressure monitoring (ABPM) in clinical practice and observational epidemiological studies has grown considerably in the past 25 years. ABPM is a very effective technique for assessing biological, environmental, and drug effects on blood pressure. In order to enhance the effectiveness of ABPM for clinical and observational research studies via analytical and graphical results, developing alternative data analysis

Approaches Are

. The linear mixed model for the analysis of longitudinal data is

About

population and subject-specific trajectories for ABPM data. Subject-specific trajectories are of great importance in ABPM studies, especially in clinical research, but little emphasis has been placed on this dimension of the problem in the statistical analyses of the data. We propose using a linear mixed model with orthonormal polynomials across time in both the fixed and random

Rthonormal Polynomials In The Linear Mixed Model

to develop model-based, subject-specific 24-hour ABPM correlates of cardiovascular disease outcomes. e demonstrate the proposed analysis technique using data from the Dietary

W

Approaches to Stop Hypertension (DASH) study, a multicenter, randomized, parallel arm feeding study that tested the effects of dietary patterns on blood pressure. KEY WORDS: Cubic Spline, DASH Study, Graphical Display, Hypertension, Longitudinal,

1. Introduction - Motivation And Background

Ambulatory blood pressure monitoring (ABPM) is a powerful research tool for examining blood pressure (BP) variability and the physiologic and environmental factors that affect BP.1 The evidence that ABPM gives information over and above conventional blood pressure measurement (CBPM) has been growing steadily over the past 25 years and the technique is now accepted as being indispensable to good clinical practice.

2-3

in which a standard cuff, attached to a lightweight, portable data recording unit, is placed around the upper arm and inflated at regular preset intervals during a 24 hour time period. Traditionally, clinicians have used office blood pressure measurements as the preferred method of monitoring blood pressure and consequently the method of diagnosing hypertension.

Unfortunately, physician's office blood pressure measurements often can give rise to what is termed "white-coat hypertension", that is, an artificially high blood pressure reading. ABPM

4

provides a profile of blood pressure away from the medical environment, thereby allowing identification of individuals with a white coat response.

5 It Also Provides Several Other Important

advantages discussed by O'Brien5: ABPM can demonstrate the efficacy of antihypertensive medication over a 24 hour period rather than making a decision based on one or a few CBPMs confined to a short period of the diurnal cycle; ABPM can identify patients whose blood pressure does not reduce at night-time (the non-dippers) who are probably at high risk for a number of conditions; and the technique can demonstrate a number of patterns of blood pressure behavior that may be relevant to clinical management (isolated systolic hypertension, hypotension, dipping and non-dipping, etc.).

That Abpm Is A Much Stronger

predictor of cardiovascular morbidity and mortality than CBPM. The importance of the technique is further evidenced by the fact that the Centers for Medicare and Medicaid Services

2,5

should be available to all hypertensive patients given its

Indispensable

patients with established and suspected hypertension Th

Tool For

.

Is

recommendation has important implications for clinical practice. Among the questions that O'Brien5 posed were: 1. how should the data be presented?; 2. how best can doctors and nurses unfamiliar with the technique be educated in its use and interpretation of the data? Lambert et al. noted that there has been relatively little work

Done

analysis of 24-hour ABPM data. Though some authors have attempted to address the issue, there is not a generally accepted 'standard' method of analyzing 24-hour ABPM. Some analyses of ABPM data have used means and/or medians, either obtaining means over the entire 24-hour period or obtaining means over defined intervals of the 24-hour period, for example daytime and nighttime intervals.

Other analyses have stressed the use of traditional simple or extended

7-9

cosinor models. Some authors have proposed smoothing splines or other smoothing

10-12

techniques for the analysis of 24-hour ABPM data .

16

cross-sectional pooled time series designs for the analysis of 24-hour ABPM data. Schwartz et al. proposed a very limited variance-components model which is equivalent to a mixed model

17

using very simple explanatory factors (time is not explicitly used as a factor). Selwyn and DiFranco demonstrated the use of the general linear mixed model as an approach to analyzing

18

24-hour ABPM data. Typically the emphasis of 24-hour ABPM data analyses are on between- subject effects (fixed effects) and less emphasis is placed on within-subject effects (random

Effects).6

We propose using a linear mixed model with orthonormal polynomials in both the fixed and random effects as an effective approach to analyzing 24-hour ABPM data, particularly when subject-specific trajectories are of primary emphasis The general linear mixed model

Has

.

19-20

become a very powerful statistical tool in the analysis of longitudinal data with continuous outcomes in both clinical and non-clinical studies.

21-23

of longitudinal data is particularly well-suited for

About

interpretation of both population and subject-specific trajectories for ABPM data. Subject-

3

specific trajectories are of great importance in ABPM studies but little emphasis has been placed on this dimension of the problem in data analyses. Using orthonormal polynomials provides the

Nonlinear Trajectories Represented By

the same complexity in which the mean model (fixed effects) are addressed. Graphical presentations are proposed that help to address questions posed by O'Brien regarding how the

5

data should be presented. We demonstrate the proposed technique using data from the Dietary Approaches to Stop Hypertension (DASH) study, a multicenter, randomized, parallel arm feeding study that tested the effects of dietary patterns on blood pressure.

In Section 2, a description of the DASH trial is given In Section 3, the basic linear mixed . model formulation and the details for the orthonormal polynomial statistical methods are presented. We discuss the results of applying the models to the DASH data in Section 4. Section

And A

Section 6. 2. Description of Dietary Approaches to Stop Hypertension (DASH) Clinical Trial The Dietary Approaches to Stop Hypertension (DASH) trial was a multicenter, randomized, parallel arm feeding study that tested the effects of dietary patterns on blood pressure (Appel et al 1997 - 9). The three diets were a control diet (low in fruits, vegetables, and dairy products, with a fat content typical of the average diet in the United States), a diet rich in fruits and vegetables (a diet similar to the control except it provided more fruits and vegetables and fewer snacks and sweets), and a combination diet rich in fruits, vegetables, and low-fat dairy foods and reduced in saturated fat, total fat, and cholesterol. The combination diet will be subsequently referred to as the DASH diet.

Participants were healthy, community-dwelling adults 22 years of age or older who were not taking antihypertensive medication. Each subject had an average systolic blood pressure of less than 160 mm Hg and a diastolic blood pressure of 80 to 95 mm Hg (mean of six measurements across three screening visits). Study subjects were enrolled sequentially in groups; the first group

4

began the run-in phase of the trial in September 1994, and the fifth and last group started in January 1996. For each group, data were collected during three phases (screening, run-in, and intervention).

Run-in was a three-week period in which all participants were fed the control diet. Toward the end of run-in, 24-hour ambulatory blood pressure monitoring was obtained once. This constituted the "baseline" ABPM reading. During the third week, participants were randomized to one of three diets. Intervention was an eight-week period in which participants were fed their assigned diets. During the last two weeks, one 24-hour ambulatory blood pressure monitoring was obtained (end-of-intervention ABPM).

ABPM was attempted on the 362 participants enrolled in groups 2-5. We use 357 subjects for baseline 24-hour ABPM data analysis, including incomplete data. We also use the 288 subjects who had complete data at baseline and the 287 subjects who had complete data during the intervention for our AUC example.

3. The Linear Mixed Model and Orthonormal Polynomials As discussed in Section 1.2, several statistical methods have been proposed for the analysis of 24-hour ABPM data. In this paper, we consider the linear mixed model with orthonormal polynomials with degree of polynomial ranging from 4 to 9.

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The advantage of reducing bias in covariance estimation has made restricted maximum likelihood (REML) estimation very popular for the linear mixed model.

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variance estimates all parameter estimates in this paper are done using REML. However, the

,

formulations also apply to computations based on maximum likelihood estimates.

3.1 Why Orthonormal Polynomials?

Polynomial models play a prominent role in mathematics. In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed

24

interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The Extendend Mean Value Theorem is a remarkable theorem

24

usually associated with Brook Taylor (1685-1731) that enables us to approximate various complicated functions by much simpler natural polynomial functions. Although a power series

Expansion Cannot

be obtained, most of the familiar functions in calculus can be

Always

represented as a sum of a convergent power series. For any individual, blood pressure as a

6

function of time on the interval can be assumed to be a continuous function that meets the regularity conditions posed by the Weierstrass approximation theorem and the Extendend Mean Value Theorem.

Polynomials are commonly used to describe curved relationships in statistical models where a model must be developed empirically. Due to the heterogeniety of 24-hour ABPM trajectories

Empirically

, polynomials can be very effective statistical tools for

Developing

models for subject-specific trajectories. The goal of polynomial regression is to model a non- linear relationship between the independent and dependent variables (technically, between the independent variable and the conditional mean of the dependent variable). An advantage of traditional polynomial regression is that the inferential framework of the linear mixed model can be used.

The use of natural polynomials for the analysis of 24-hour ABPM recordings poses problems, particularly when modeling subject-specific trajectories. The problems include the existence of

;

model software convergence problems when used in random effects. The lack of convergence in

Cases

is caused by the multicol inearity and large values

Ir

resulting ffect on the estimation of the random effects covariance

.3

.

, But This

subject-specific trajectories. The use of orthonormal polynomials helps alleviate this problem and hence allows for more accurate modeling of the heterogeneity in subject-specific 24-hour ABPM trajectories than natural polynomials.

Orthonormal polynomial predictors in linear regression are transformations of the natural polynomials that provide a new set of predictors that meet the following criteria: 1. The new predictors contain the same information as the original set; 2. The new predictors are linear

7

combinations of the original natural polynomials; 3. The new columns of predictors all have mean zero (except for the constant term); 4. The new columns of predictors are all mutually orthogonal. Combining (3) and (4) implies that the new predictors

(Except The Constant)

mutually uncorrelated. he first new column captures the information in the intercept, the second

T

captures all of the linear term information adjusted for the intercept, the third captures quadratic

Linear And Intercept

and so forth.

Information

In the case of 24-hour ABPM trajectories, the main interest is in the shape of the trajectories that the orthonormal polynomials produce collectively and not in any one single polynomial component. Two big advantages of using orthonormal polynomials are

That

multicollinearity is greatly reduced; and b) the orthonormal polynomials are "normalized" and

Bounded In The Interval (

1, 1). Thus, orthonormal polynomials have few problems with



rounding error or very large (small) regression coefficients. Using the linear mixed model with orthonormal polynomials, subject-specific 24-hour ABPM trajectories can be represented and parameterized, within- and between- treatment comparisons based on individual trajectories can be made, and discrimination between different trajectories can be obtained. The linear mixed model can account for circadian rhythms, subject effects, and treatment effects.

3.2 The Case of Complete, Balanced Data with Equally Spaced Observations For the analysis of 24-hour ABPM data, we advocate the use of a linear mixed model with orthonormal polynomials used in both the fixed and random effects, particularly when an emphasis is placed on accurate modeling of subject-specific parameters. Here, we assume that the degree of the polynomials used in the fixed effects is equal that used in the random effects.

To

For demonstration purposes, we first start with a linear mixed model with only time effects and intercept, but no additional covariates. We assume that time is equally spaced 1-hour intervals ranging from 1 to 24 and that each subject has all 24 observations (complete and balanced). We

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design matrix of orthonormal polynomials for person ;

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orthonormal polynomials with rank for person . Gaussian

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mean and variance given in equation 2.

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Because the polynomials are orthogonal, mathematically we can model

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vector of variances for each element of the random effects vector

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. This greatly simplifies the complexity of the random

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the number of diagonal elements. However, in actual data analysis, the best fitting

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Natural Polynomials

. present in the orthonormal fits due to the lack of fit of the fixed effects,

3 "S,

address this issue in the example data. The maximum degree orthonormal polynomial that we consider for the DASH data is a 9th degree. Our goal of modeling the nonlinear ABPM trajectories without overfitting the data along with our prior work and experience analyzing DASH data guided us to begin with a maximum model with a polynomial degree between 6 and 9.

25-26

degree polynomial was significant in the SBP model (though not in the DBP model). Simpson and Edwards used a fixed-effects only model for the DASH data and the order of the

26

orthonormal polynomial was chosen to be 6 when considering additional covariates and aiming for parsimony. However, for the example analysis in this paper, we also consider a range of polynomials between 4 and 9 with a focus on the 9th degree model since it is the largest polynomial considered. We do not consider additional covariates but we discuss how the covariates can be included in the model.

; Œ

polynomials), 10 random effects covariance parameters and

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And

) for a total of 11 covariance parameters. We also compare the fit

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Using An Unstructured Covariance For

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However, With

conventional mixed effects software like SAS Proc Mixed, it would be impossible to use an

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

 with natural polynomials for the subject-specific models for the reasons discussed in section 3.1.

3.3 The Case Of Incomplete, Unbalanced Data

In the DASH Study, about 81% (288/357) of the subjects had complete data at baseline. The additional 19% of subjects had very little missing data. We assumed that the data were missing competely at random for these analyses, i.e., an “intent-to-treat” treatment of missing data by including every observation of the dependent measure (no observations are discarded and no data are imputed).

One of the main advantages of the linear mixed model for longitudinal data is that it uses all available data, rather than excluding cases with missing observations, and accommodates unbalanced designs. This feature of the linear mixed model is of particular relevance to ABPM data since it is common for a significant proportion of participants to be missing at least one

Observation.27

In 24-hour ABPM studies, typically there are criteria for controlling the quality of the monitoring data. For example, in the DASH study, if fewer than 14 acceptable readings were obtained, the subject was asked to repeat the monitoring. Among participants with acceptable ABPM measurements, more than 90% of the possible waking and sleeping readings were obtained. Sometimes more rigid criteria are used, e.g., for Hermida et al. , BP series were not

29

considered valid for analysis if 30% of the measurements were lacking, if data were missing for >2-hour spans, if data were collected from subjects while experiencing an irregular rest–activity schedule, or if the nighttime sleep span was <6 hours or >12 hours during monitoring.

10

4. DASH Baseline Data Analysis Results - Estimation, Inference and Goodness-of-fit Table 1 provides estimates, standard errors (SE), and p-values for

The

polynomial fit to the DASH baseline 24-hour ABPM data. Table 2 provides model comparisons using ornothormal polynomial models of degree ranging 4-9. All computations were done using

Iml

time spacings when generating orthonormal polynomials. Restricted maximum likelihood estimation (REML) was used for estimation and the Kenward-Roger F and adjusted denominator degrees of freedom were used for all fixed effect inference. However, maximum likelihood estimation results were almost identical (results not shown). From Table 1, we can see that the sizes of the absolute values of the orthonormal polynomial regression coefficients indicate that orthonormal polynomials 1-6 have the largest effect on BP, changing from being greater than 4 to less than 2.

30

and Bayesian Information Criterion for orthonormal polynomial models using 4-9th degree

31

polynomials in both the fixed and random effects. AIC and BIC were computed for both

.3

. .

.3

.

) And

Unstructured (UN). Morrell et al. commented that the best

32

way to select among linear mixed-effects models based on various information criteria is still not clearly determined. The conclusion from both the AIC and BIC is that the 9th degree

.3

.

Œ

yields the smallest AIC and BIC values when compared to all other orthonormal polynomial

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covariance fit, but the lack of fit in the fixed effects is carried over into the variance (the

Estimated

was not diagonal). There were no model convergence problems using

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œ UN. To check the model distribution assumptions for the random effects and residual error, we used an important result from Gurka et al. which states that if



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And

are multivariate normal given the model assumption that

And

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3

3

3

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are independent. Using this result, we used standard residual analysis for the estimated "stacked" %3 and concluded that the errors were approximately normally distributed for each of the orthonormal polynomial models considered.

The linear mixed model easily accommodates additional explanatory variables. For example, we can do a simple check for baseline homogeneity of average 24-hour blood pressure for the 3 diet groups by including intercept effects for diet (two indicator variables with control diet as the reference group) and interaction effects in time for diet. Let

(Both Main

effects and diet x time interaction effects) in the linear mixed model as follows

C

^ .

#

.

, And Are

vectors of fixed effect population parameters where

;‚"

effect of the control diet (intercept and time) on blood pressure,

Differential

relative to the control diet Note that the structure of random

Effect Of The Dash Diet

. effects and within-subject errors remain the same as in equation (4) as does the corresponding covariance structure. We can rewrite the model as a standard linear mixed model

\

^ .

"

.

Is A

fixed effects design matrix for person and is the corresponding

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vector of fixed effect parameters. Tests of interaction

And Intercept Effects

not signifcant and hence we concluded that there was no evidence of a difference in 24-hour BP profiles at baseline. Using the same model for the intervention period, there was no evidence of interaction effects but there was evidence of diet main effects (again, results not shown here). An

12

example of including several explanatory variables in the analysis of the DASH data can be found in Simpson and Edwards , which features a general linear model for longitudinal data

26

with the circular LEAR correlation structure and uses diet groups, race, age, and orthonormal polynomials as fixed effects. In order to incorporate both baseline and intervention into the model, we use a shared

/

.

‚ 7

for baseline and intervention periods. The shared parameter model is appropriate since the response is the same for baseline and intervention. The scalar representation of the model is

Where

,1 represents baseline and intervention BP for subject ,

D

orthonormal time effects. The model assumes that within-subject variation is the same at baseline and intervention and the between-subject effect of intervention is modeled using the

&

provides the results for the change in SBP and DBP for each diet main effect. Both the DASH and fruit/vegetable diets lowered SBP and DBP significantly compared with the control diet

(P

0.0001 for SBP and DBP on both diets: control diet,









similar to the results Moore et al. (1999 - 28) found using mean ABP where both the

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