Circular covariance patterns arise naturally from many important biological and physical processes. Modelling these patterns can be immensely important for proper analyses. In this article, we propose a circular linear exponent autoregressive (LEAR) correlation structure for cyclical longitudinal data. Special cases of this parsimonious correlation model include the equal correlation and first-order moving average (MA(1)) correlation structures and a circular analogue of the continuous-time AR(1) model. We discuss properties and estimation of the circular LEAR model in the context of cyclical longitudinal data concerning diet and hypertension (the DASH study). Analysis of these data exemplifies the benefits of the circular LEAR correlation structure.
HartleyAMNaikDN. Estimation of familial correlations under autoregressive circular covariance. Commun Stat–Theory Methods2001; 30: 1811–1828.
2.
MullerKEEdwardsLJSimpsonSLTaylorDJ. Statistical tests with accurate size and power for balanced linear mixed models. Stat Med2007; 26: 3639–3660.
3.
SimpsonSLEdwardsLJMullerKESenPKStynerMA. A linear exponent AR(1) family of correlation structures. Stat Med2010; 29: 1825–1838.
4.
MartinRJ. A note on the asymptotic eigenvalues and eigenvectors of the dispersion matrix of a second-order stationary process on a d-dimensional lattice. J Appl Probab1986; 23: 529–535.
5.
FullerWA. Introduction to statistical time series, 2nd edn. New York: John Wiley & Sons, 1996.
6.
NiuXTiaoGC. Modeling satellite ozone data. J Am Stat Assoc1995; 90: 969–983.
7.
KhattreeRMultivariate statistical inference involving circulant matrices: a review. In: GuptaAKGirkoV (eds). Multidimensional statistical analysis and theory of random matrices: proceedings of the sixth eugene lukacs symposium1996, pp. 101–110.
8.
ArtesR. Hypothesis tests for covariance analysis models for circular data. Commun Stat–Theory Methods2008; 37: 1632–1640.
9.
SchmidCH. An EM algorithm fitting first-order conditional autoregressive models to longitudinal data. J Am Stat Assoc1996; 91: 1322–1330.
10.
AppelLJMooreTJObarzanekE. A clinical trial of the effects of dietary patterns on blood pressure. DASH collaborative research group. N Eng J Med1997; 336: 1117–1124.
11.
MooreTJVollmerWMAppelLJ. Effect of dietary patterns on ambulatory blood pressure: results from the dietary approaches to stop hypertension (DASH) trial. Hypertension1999; 34: 472–477.
12.
StreitbergBMayer-SabellekWBaumgartP. Statistical analysis of circadian blood pressure recordings in controlled clinical trials. J Hypertension1989; 7(suppl 3): S11–S17.
13.
StreitbergBMayer-SabellekW. Smoothing twenty-four-hour ambulatory blood pressure profiles: a comparison of alternative methods. J Hypertension1990; 8(suppl 6): S21–S37.
14.
DicksonDHasfordJ. 24-Hour blood pressure measurement in antihypertensive drug trials: data requirements and methods of analysis. Stat Med1992; 11: 2147–2158.
15.
SelwynMRDifrancoDM. The application of large gaussian mixed models to the analysis of 24 hour ambulatory blood pressure monitoring data in clinical trials. Stat Med1993; 12: 1665–1682.
16.
JaccardJWanCK. Statistical analysis of temporal data with many observations: issues for behavioral medicine data. Ann Behav Med1993; 15: 41–50.
17.
LambertPCAbramsKRJonesDRHalliganAWFShennanA. Analysis of ambulatory blood pressure monitor data using a hierarchical model incorporating restricted cubic splines and heterogeneous within-subject variances. Stat Med2001; 20: 3789–3805.
18.
EdwardsLJStewartPJMacDougallJEHelmsRW. A method for fitting regression splines with varying polynomial order in the linear mixed model. Stat Med2006; 25: 513–527.
19.
FanJHuangTRunzeL. Analysis of longitudinal data with semiparametric estimation of covariance function. J Am Stat Assoc2007; 102: 632–641.
20.
SchottJR. Matrix analysis for statistics, New York: John Wiley & Sons, 1997.
21.
BasilevskyA. Applied matrix algebra in the statistical sciences, New York: North Holland, 1983.
22.
DigglePJ. An approach to the analysis of repeated measures. Biometrics1988; 44: 959–971.
23.
LouisTA. General methods for analysing repeated measures. Stat Med1988; 7: 29–45.
24.
DemidenkoE. Mixed models theory and applications, New York: John Wiley & Sons, 2004.
25.
SAS Institute. SAS/IML, Version 9. Cary, NC: SAS Institute, Inc., 2002.
26.
Edwards LJ and Simpson SL. Analysis of 24-hour ambulatory blood pressure monitoring data using orthonormal polynomials in the linear mixed model. arXiv:1010.1224v1 [stat.AP].
27.
AkaikeH. A new look at the statistical model identification. IEEE Trans Automatic Control1974; AC-19: 716–723.
28.
SchwarzSR. Estimating the dimension of a model. Ann Stat1978; 6: 461–464.
29.
FitzmauriceGMLairdNMWareJH. Applied longitudinal analysis, New York: John Wiley & Sons, 2004.
30.
BurnhamKPAndersonDR. Model selection and inference: a practical information-theoretical approach, New York: Springer-Verlag, 1998.
