### Abstract

In a regression setting, the partial correlation coefficient is often used as a measure of 'standardized' partial association between the outcome y and each of the covariates in x′ = (X_{1}, . . . , X_{K}). In a linear regression model estimated by using ordinary least squares, with y as the response, the estimated partial correlation coefficient between y and x_{k} can be shown to be a monotone function of the z-statistic for testing whether the regression coefficient of x_{k} is 0; it can also be shown to be a monotone function of the likelihood ratio statistic. When y is normal but the data are clustered so that y and x are obtained from each member of a cluster, maximum likelihood using a multivariate normal distribution is often used to estimate the regression parameters of the model for y given x. For clustered data, we propose two measures of partial association: the exact same monotone function of the repeated measures' z-statistic that is used to form the partial correlation coefficient in the normal linear regression setting and the exact same monotone function of the likelihood ratio statistic that is used to form the partial correlation coefficient in the normal linear regression setting. We also propose an R^{2}-like measure which can be used to measure the overall predictive ability of the model. To illustrate the method, we use a longitudinal study concerning the stress on the wall of the heart from chemotherapy in children with leukaemia.

Original language | English |
---|---|

Pages (from-to) | 87-95 |

Number of pages | 9 |

Journal | Journal of the Royal Statistical Society Series D: The Statistician |

Volume | 50 |

Issue number | 1 |

State | Published - Dec 1 2001 |

Externally published | Yes |

### Fingerprint

### Keywords

- Clustered data
- Coefficient of determination
- Likelihood ratio statistic
- Longitudinal data
- Wald statistic

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Journal of the Royal Statistical Society Series D: The Statistician*,

*50*(1), 87-95.

**A partial correlation coefficient and coefficient of determination for multivariate normal repeated measures data.** / Lipsitz, Stuart R.; Leong, Traci; Ibrahim, Joseph; Lipshultz, Steven E.

Research output: Contribution to journal › Article

*Journal of the Royal Statistical Society Series D: The Statistician*, vol. 50, no. 1, pp. 87-95.

}

TY - JOUR

T1 - A partial correlation coefficient and coefficient of determination for multivariate normal repeated measures data

AU - Lipsitz, Stuart R.

AU - Leong, Traci

AU - Ibrahim, Joseph

AU - Lipshultz, Steven E

PY - 2001/12/1

Y1 - 2001/12/1

N2 - In a regression setting, the partial correlation coefficient is often used as a measure of 'standardized' partial association between the outcome y and each of the covariates in x′ = (X1, . . . , XK). In a linear regression model estimated by using ordinary least squares, with y as the response, the estimated partial correlation coefficient between y and xk can be shown to be a monotone function of the z-statistic for testing whether the regression coefficient of xk is 0; it can also be shown to be a monotone function of the likelihood ratio statistic. When y is normal but the data are clustered so that y and x are obtained from each member of a cluster, maximum likelihood using a multivariate normal distribution is often used to estimate the regression parameters of the model for y given x. For clustered data, we propose two measures of partial association: the exact same monotone function of the repeated measures' z-statistic that is used to form the partial correlation coefficient in the normal linear regression setting and the exact same monotone function of the likelihood ratio statistic that is used to form the partial correlation coefficient in the normal linear regression setting. We also propose an R2-like measure which can be used to measure the overall predictive ability of the model. To illustrate the method, we use a longitudinal study concerning the stress on the wall of the heart from chemotherapy in children with leukaemia.

AB - In a regression setting, the partial correlation coefficient is often used as a measure of 'standardized' partial association between the outcome y and each of the covariates in x′ = (X1, . . . , XK). In a linear regression model estimated by using ordinary least squares, with y as the response, the estimated partial correlation coefficient between y and xk can be shown to be a monotone function of the z-statistic for testing whether the regression coefficient of xk is 0; it can also be shown to be a monotone function of the likelihood ratio statistic. When y is normal but the data are clustered so that y and x are obtained from each member of a cluster, maximum likelihood using a multivariate normal distribution is often used to estimate the regression parameters of the model for y given x. For clustered data, we propose two measures of partial association: the exact same monotone function of the repeated measures' z-statistic that is used to form the partial correlation coefficient in the normal linear regression setting and the exact same monotone function of the likelihood ratio statistic that is used to form the partial correlation coefficient in the normal linear regression setting. We also propose an R2-like measure which can be used to measure the overall predictive ability of the model. To illustrate the method, we use a longitudinal study concerning the stress on the wall of the heart from chemotherapy in children with leukaemia.

KW - Clustered data

KW - Coefficient of determination

KW - Likelihood ratio statistic

KW - Longitudinal data

KW - Wald statistic

UR - http://www.scopus.com/inward/record.url?scp=0041075396&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041075396&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0041075396

VL - 50

SP - 87

EP - 95

JO - Journal of the Royal Statistical Society. Series A: Statistics in Society

JF - Journal of the Royal Statistical Society. Series A: Statistics in Society

SN - 0964-1998

IS - 1

ER -