Select the method or formula of your choice.

The measurement error model is:

In orthogonal regression, the best fitting line is the one that minimizes the weighted orthogonal distances from the plotted points to the line. If the error variance ratio is 1, the weighted distances are Euclidean distances.

Term | Description |
---|---|

Y_{t} | observed response |

β_{0} | intercept |

β_{1} | slope |

X_{t} | observed predictor |

x_{t} | true and unobserved value of predictor |

e, _{t}u_{t} | measurement errors; e_{t}, u_{t} are independent with mean 0 and error variances of δ_{e}^{2} and δ_{u}^{2} |

Let the sample mean be (, ) and the sample covariance matrix be:
*m*_{ZZ} is a 2X2 symmetric matrix:

Term | Description |
---|---|

Z_{t} | (Y)_{t}, X_{t} |

n | sample size |

The sample covariance matrix is a 2 × 2 matrix:

If the element m_{XY} of the sample covariance matrix does not equal 0, then:

If *m _{XY}* = 0 and

Term | Description |
---|---|

estimate of error variance for X | |

estimate of error variance for Y | |

δ | ratio of error variances |

m_{XY} | element of sample covariance matrix |

m_{YY} | element of sample covariance matrix |

m_{XX} | element of sample covariance matrix |

If the element *m*_{XY} of the sample covariance matrix does not equal 0, then:

If *m _{xy}* = 0 and

If *m _{xy}* = 0 and

Term | Description |
---|---|

estimate of slope | |

estimate of intercept | |

m_{xy} | element of sample covariance matrix |

m_{yy} | element of sample covariance matrix |

δ | ratio of error variances |

mean of response values | |

mean of predictor values |

An estimate of the covariance matrix of the approximate distribution of the intercept and slope:

where:

and

If *m _{XY}* does not equal 0:

If *m _{XY}* equals 0 and

Term | Description |
---|---|

estimate of slope | |

estimate of intercept | |

m_{XY} | element of sample covariance matrix |

m_{YY} | element of sample covariance matrix |

m_{XX} | element of sample covariance matrix |

δ | ratio of error variances |

mean of response values | |

mean of predictor values |

The 100(1 - *α*)% confidence interval for *β*_{0} is:
where:

Z (1 - *α* / 2) is the 100 * (1 - *α* / 2 ) percentile for the standard normal distribution

and

, which is an element in the covariance matrix of the approximate distributionTerm | Description |
---|---|

estimate of slope | |

estimate of intercept | |

α | level of significance |

The 100(1 - *α*)% confidence interval for *β*_{1} is:

where:

Z(1 - *α* / 2) is the 100 * (1 - *α* / 2) percentile for the standard normal distribution

and

Term | Description |
---|---|

estimate of slope | |

estimate of intercept | |

α | level of significance |

The fitted value for the predictor *x* in orthogonal regression is:

Term | Description |
---|---|

δ | ratio of error variances |

Y_{t} | t^{th} response value |

intercept estimate | |

slope estimate |

The fitted value for the response *y* in orthogonal regression is:

Term | Description |
---|---|

intercept estimate | |

slope estimate | |

t^{th} fitted value for x |

The residual of an observation in orthogonal regression is:

Term | Description |
---|---|

Y_{t} | t^{th} response value |

intercept | |

X_{t} | t^{th} predictor value |

slope |

The standardized residual is helpful in identifying outliers. It is calculated as:

where

Term | Description |
---|---|

residual | |

standard deviation of residual | |

δ | error variance ratio |

estimate of slope | |

estimate of error variance for X |

The predictor of *Y*_{n + 1} is:

where:

and

Term | Description |
---|---|

X_{t} | t^{th} predictor value |

mean of predictor values | |

Y_{t} | t^{th} response value |

mean of response values |

where:

Term | Description |
---|---|

m_{yy} | sample variance of Y |

m_{xy} | sample covariance between X and Y random variables |