Method Comparison Studies of OLS-Bisector Regression and Ranged Major Axis Regression

Linear regression is a widely used technique in many disciplines of science. When both the dependent and independent variables of the model are random, there are errors associated with the measurement of x and y variables, such models are called Errors-in-Variable models. Under this situation, Model II Regression techniques should be used for parameter estimation. We focus on the performance study of OLS-Bisector Method (Isobe et al., 1990). We compared this method to Ranged Major Axis (RMA), which is proposed by Legendre and Legendre (2012) as an alternative to Reduced Major Axis. Our simulation study addresses the coverage and width of the 95% confidence intervals. We conclude that performance of the two methods is very similar but OLS-Bisector gives a more accurate estimate than Ranged Major Axis when sample size is small.

1. Abstract
Linear regression is a widely used technique in many disciplines
of science. When both the dependent and independent variables
of the model are random, there are errors associated with the
measurement of x and y variables, such models are called Errors-
in-Variable models. Under this situation, Model II Regression
techniques should be used for parameter estimation. We focus on
the performance study of OLS-Bisector Method (Isobe et al., 1990).
We compared this method to Ranged Major Axis (RMA), which is
proposed by Legendre and Legendre (2012) as an alternative to
Reduced Major Axis. Our simulation study addresses the cover-
age and width of the 95% confidence intervals. We conclude that
performance of the two methods is very similar but OLS-Bisector
gives a more accurate estimate than Ranged Major Axis when
sample size is small.


Goal
● To analyze the performance of OLS-Bisector method by investi-
gating its two components: OLS (Y|X) and OLS (X|Y), which has
not been studied from this perspective before.
● To compare the performance of RMA method and OLS-Bisector
method. To date, only Neiuwoudt (2014) has studied the perfor-
mance of RMA. To our knowledge, no one has compared its per-
formance to the increasingly popular OLS-Bisector method.


Method Comparison Studies of OLS-Bisector Regression and Ranged Major Axis Regression
ZHAOCE LIU
Advisor: Dr. Melinda Holt
Sam Houston State University
The Errors-in-Variables Model




Different Model II Regression Methods
Ranged Major Axis Method:
1
st step: Transform each pair of observations to by the equations: and
2
nd step: Apply Major Axis method on the transformed observations.
3
rd step: Transform the Major Axis regression estimators back to the original units by multiplying the ratio of the
ranges: .
Y
X
Y
X
Standard
Ordinary Least Squares Methods















Slope Estimator:
Reversed
Ordinary Least Squares Methods















Slope Estimator:

Y
X
Major Axis Methods
















Slope Estimator:

Y
X
Reduced Major Axis Methods
















Slope Estimator:

OLS-Bisector Method:
The OLS-Bisector regression line bisects the smaller of the two angles between the two ordinary least squares lines:
and
Coverage Percentage Based on the Interaction of Correlation and Minimum Ratio*
Slope Estimation of
OLS (Y|X) and OLS (X|Y)


Performance Patterns of different methods
Discussion
1 We confirmed the phenomenon of attenuation in OLS (Y|X) and overestimation
in OLS(X|Y) and the resulting impact on coverage. The OLS-Bisector merges
these and thus provides a more accurate estimation. However, we found that
OLS-Bisector will be more influenced by OLS (X|Y) because of its relatively high
instability.
2 It is interesting to find that OLS-Bisector and RMA perform similarly in most sit-
uations. This indicates that OLS-Bisector could also be an alternative method to
Reduced Major Axis method. Our study shows that when the sample size is
large, RMA provides better coverage than the Reduced Major Axis and OLS-
Bisector at the cost of greater confidence interval width. OLS-Bisector is preferred
when the sample size is small, because RMA intervals are often undefined.
Future Research
Since OLS-Bisector method merges OLS(Y|X) and OLS(X|Y), we have reason to
introduce an parameter weighing these two components instead of just a
”bisector”, thus we could present an more accurate estimation. To do this, we
could also use prior information to determine our parameter, and conduct a
Bayesian analysis.
References
1. Babu, G. J. and Feigelson, E. D. (1992), Analytical and Monte Carlo Comparisons
of Six Different Linear Least Squares Fits, Common.Statist.-Simula., 21(2), 553 -549.
2. Edland, S. (1996). Bias in Slope Estimates for the Linear Errors in Variables Mod-
el by the Variance Ratio Method. Biometrics , 52(1), 243-248.
3. Isobe, T., Feigelson, E. D., Akritas, M. D. and Babu, G. J. (1990), Linear Regres-
sion in Astronomy I. Astrophysical J., 364: 104 - 113.
4. Legendre, L. , Legendre, P. (2012) Numerical Ecology.
5. Legrendre, P. (2013) Model II Regression User’s Guide, R edition.
6. Nieuwoudt (2014), Assessing the Sensitivity of Interval Estimates in Errors-in-
Variables Regression, Practicum Paper, Sam Houston State University
7. Kendall, M. and Stuart, A. (1977) The Advanced Theory of Statistics 4th ed. New
York; MacMillan


Suppose the classical simple linear regression model could be writ-
ten as:

where we assume thatt aandd aWhen con-
fronted with measurement error, the observation can be expressed
byy aandd wheree rrepresents the difference be-
tween the true value offf and what we observed, and represents
the difference between the true value off and what we observed.
Substitutingg aandd wwe get the Errors-in-Variables
model:

In addition, we assume thatt aandd
aare linearly independent. Under the assumptions above,
Kendall and Stuart (1977) provided a general formula for estimating
the slope

,where

Coverage rate of the 95% Reduced Major Axis interval estimate for true slope based on the interaction of Correlation and Minimum Ratio
Coverage rate of the 95% Ranged Major Axis interval estimate for true slope based on the interaction of Correlation and Minimum Ratio
Coverage rate of the 95% OLS-Bisector interval estimate for true slope based on the interaction of Correlation and Minimum Ratio