Methods and formulas for Orthogonal Regression

Select the method or formula of your choice.

Regression equation

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

TermDescription
Ytobserved response
β0intercept
β1slope
Xtobserved predictor
xttrue and unobserved value of predictor
et, utmeasurement errors; et, ut are independent with mean 0 and error variances of δe2 and δu2

Sample covariance matrix

Let the sample mean be (, ) and the sample covariance matrix be:
mZZ is a 2X2 symmetric matrix:

Notation

TermDescription
Zt(Yt, Xt)
nsample size

Error variances

The sample covariance matrix is a 2 × 2 matrix:

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

If mXY = 0 and mYY < δmXX,

If mXY = 0 and mYY > δmXX, the remaining parameter estimates are undefined.

Notation

TermDescription
estimate of error variance for X
estimate of error variance for Y
δratio of error variances
mXYelement of sample covariance matrix
mYYelement of sample covariance matrix
mXXelement of sample covariance matrix

Coefficients

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

If mxy = 0 and myy < δm xx','

If mxy = 0 and myy > δmxx, the remaining parameter estimates are undefined.

Notation

TermDescription
estimate of slope
estimate of intercept
mxyelement of sample covariance matrix
myyelement of sample covariance matrix
δratio of error variances
mean of response values
mean of predictor values

Covariance matrix of approximate distribution

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

where:

and

If mXY does not equal 0:

If mXY equals 0 and mYY < δmXX:

Notation

TermDescription
estimate of slope
estimate of intercept
mXYelement of sample covariance matrix
mYYelement of sample covariance matrix
mXXelement of sample covariance matrix
δratio of error variances
mean of response values
mean of predictor values

Confidence interval for intercept

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

TermDescription
estimate of slope
estimate of intercept
αlevel of significance

Confidence interval for slope

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

TermDescription
estimate of slope
estimate of intercept
αlevel of significance

Fitted values for x

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

Notation

TermDescription
δratio of error variances
Yttth response value
intercept estimate
slope estimate

Fitted values for y

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

Notation

TermDescription
intercept estimate
slope estimate
tth fitted value for x

Residuals

The residual of an observation in orthogonal regression is:

Notation

TermDescription
Yttth response value
intercept
Xttth predictor value
slope

Standardized residuals

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

where

Notation

TermDescription
residual
standard deviation of residual
δerror variance ratio
estimate of slope
estimate of error variance for X

Predictor of Y

The predictor of Yn + 1 is:

where:

and

Notation

TermDescription
Xttth predictor value
mean of predictor values
Yttth response value
mean of response values

Standard deviation for the prediction error

where:

Notation

TermDescription
myysample variance of Y
mxysample covariance between X and Y random variables
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