# Measurement Errors in Regression Analysis

General data includes Measurement Errors . For example, "True is 5.2. But measured value is 5.4. So data is 5.4"

In this page, I use Simulation of Dispersion Model to consider the effect of Measurement Errors in Regression Analysis . We can analyze "What is happend usinf true values?"

## Regression Analysis by True Values

Histogram is the data of 1000 samples. Average is 0. Standard deviation is 1. And normal distribution.

If "Y = X", plots are on the straight line in the 2-Dimension Scatter Plot . The formulation,
---(1)
is calculated by Single Regression Analysis .

Data table is the examples of 3 samples in the 1000 sample.

## With Measurement Errors

Make data with measurement errors. The measurement errors are that average is 0 and standard deviation is 1.

There are "X and Y" and "True and Measured". There are 4 types of combination of them.

Common of those 4 is that intercepts are near 0.
There are 2 types of slopes. One of them is almost 1. It depends on X is TRUE. The other is almost 0.81. It depends on X is Measured.

TRUE slope is 1. But if X is Measured, the output of regression analysis is not good.

General data includes measurement errors. It means that "Regression analysis finds the lower slope formulation generally."

### Other Example

The example above is using measurement errors.

There is simpler example that regression analysis does not go well.

In this example,
---(1)
is the best output. But there is the case that lower slope is the output.

## Model of Regression Analysis

---(2)
is used as the model of Single Regression Analysis . The part, "E" is needed to include the dispersion of the data from the strait line.

In this formulation, measurement errors of X is not considered. So if the data of X includes measurement errors, wrong output is calculated.