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 |
---|---|
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 |
Term | Description |
---|---|
Zt | (Yt, Xt) |
n | sample size |
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.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 mxy = 0 and myy < δm xx','
If mxy = 0 and myy > δmxx, the remaining parameter estimates are undefined.
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 |
where:
and
If mXY does not equal 0:
If mXY equals 0 and mYY < δmXX:
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 |
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 |
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 |
Term | Description |
---|---|
δ | ratio of error variances |
Yt | tth response value |
intercept estimate | |
slope estimate |
Term | Description |
---|---|
intercept estimate | |
slope estimate | |
tth fitted value for x |
Term | Description |
---|---|
Yt | tth response value |
intercept | |
Xt | tth predictor value |
slope |
where
Term | Description |
---|---|
residual | |
standard deviation of residual | |
δ | error variance ratio |
estimate of slope | |
estimate of error variance for X |
where:
and
Term | Description |
---|---|
Xt | tth predictor value |
mean of predictor values | |
Yt | tth response value |
mean of response values |
where:
Term | Description |
---|---|
myy | sample variance of Y |
mxy | sample covariance between X and Y random variables |