# 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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