# Inverse Regression Analysis

In most regression problems we have to determine the value of Y corresponding to a given value of X. We will consider the inverse problem, which is called inverse regression or calibration.

Assume we have known values of X and their corresponding Y values, which both form a simple linear regression model and we have also an unknown value of X, such as X0, which cannot be measured and we can observe the corresponding value of Y, say Y0. Then, X0 can be estimated and a confidence interval for X0 can be obtained.

In regression analysis we want to investigate the relationship between variables. Regression has many applications, which occur in many fields: engineering, economics, the physical and chemical sciences, management, biological sciences and social sciences. We only consider the simple linear regression model, which is a model with one regressor X that has a linear relationship with a response Y. It is not always easy to measure the regressor X or the response Y.

We now consider a typical example for this problem. If X is the concentration of glucose in certain substances, then a spectrophotometric method is used to measure the absorbance. This absorbance depends on the concentration X. The response Y is easy to measure with the spectrophotometric method, but the concentration on the other hand is not easy to measure. If we have n known concentrations, then the absorbance can be measured. If there is a linear relation between Y and Y, then a simple linear regression model can be made with these data. Suppose we have an unknown concentration, which is difficult to measure, but we can measure the absorbance of this concentration. Is it possible to estimate this concentration with the measured absorbance? This is called the calibration problem.

Suppose we have a linear model Y = β0 + β1 X + ε and we have an observed value of the response Y, but we do not have the corresponding value of X. How can we estimate this value of X? The two most important methods to estimate X are the classical method and the inverse method.

The classical method is based on the simple linear regression model

Y = b0 + b1 X + ε    where  ε~N(0,σ2)

where the parameters b0 and b1 are estimated by Least Squares as β0 and β1 . At least two of the n values of X have to be distinct, otherwise we cannot ﬁt a reliable regression line. For a given value of X, say X0 (unknown), a Y value, say Y0(or random sample of k values of Y) is observed at the X0value. The problem is to estimate X0. The classical method uses a Y0value (or the mean of k values of Y0) to estimate X0, which is then estimated by $\hat{x_0}=\frac{\hat{Y_0}-\hat{\beta_0}} {\hat{\beta_1}}$.

The inverse estimator is the simple linear regression of X on Y. In this case, we have to ﬁt the model

X=α0+ α1Y+e           where  ε~N(0,σ2)

to obtain the estimator. Then the inverse estimator of X0 is

X0=α0+ α1Y+e