Methods and formulas for Orthogonal Regression

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.

Notation

Term	Description
Yt	observed response
β0	intercept
β1	slope
Xt	observed predictor
xt	true and unobserved value of predictor
et, ut	measurement errors; et, ut are independent with mean 0 and error variances of δe2 and δu2

Let the sample mean be (, ) and the sample covariance matrix be:

m_ZZ is a 2X2 symmetric matrix:

Notation

Term	Description
Zt	(Yt, Xt)
n	sample size

If the element m_XY of the sample covariance matrix does not equal 0, then:

If m_XY = 0 and m_YY < δm_XX,

If m_XY = 0 and m_YY > δm_XX, the remaining parameter estimates are undefined.

Notation

Term	Description
	estimate of error variance for X
	estimate of error variance for Y
δ	ratio of error variances
mXY	element of sample covariance matrix
mYY	element of sample covariance matrix
mXX	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 m_yy < δm _xx','

If m_xy = 0 and m_yy > δm_xx, the remaining parameter estimates are undefined.

Notation

Term	Description
	estimate of slope
	estimate of intercept
mxy	element of sample covariance matrix
myy	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 m_YY < δm_XX:

Notation

Term	Description
	estimate of slope
	estimate of intercept
mXY	element of sample covariance matrix
mYY	element of sample covariance matrix
mXX	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 β₀ 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 distribution

Notation

Term	Description
	estimate of slope
	estimate of intercept
α	level of significance

The 100(1 - α)% confidence interval for β₁ is:

where:

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

and

Notation

Term	Description
	estimate of slope
	estimate of intercept
α	level of significance

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

Notation

Term	Description
δ	ratio of error variances
Yt	tth response value
	intercept estimate
	slope estimate

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

Notation

Term	Description
	intercept estimate
	slope estimate
	tth fitted value for x

The residual of an observation in orthogonal regression is:

Notation

Term	Description
Yt	tth response value
	intercept
Xt	tth predictor value
	slope

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

where

Notation

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

Notation

Term	Description
Xt	tth predictor value
	mean of predictor values
Yt	tth response value
	mean of response values

where:

Notation

Term	Description
myy	sample variance of Y
mxy	sample covariance between X and Y random variables