# 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.

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