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Instrument Calibration.

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Introduction

Instrument Calibration

Introduction

A new device that measures the concentration of fluorescence requires calibration. The device is calibrated using the calibration lineimage00.png, method of least square estimation can be used to estimate the parameters α and β and the validity of the estimates using measurement readings from observed data.

The following are the measurements of fluorescence y, of a substance, A, in known concentrations x in μg/m3, using the new device.

Fluorescence (y)

1.0

8.0

16.0

24.0

32.0

38.0

Concentration (x)

0.0

2.0

4.0

6.0

8.0

10.0

Description of Data

The fluorescence (y) of a substance, A and its concentration (x) has a linear relationship, where concentration (x) increases as fluorescence (y) of the substance increases.

A linear model will be fitted to the observed data. The model is as follows:

image01.png

Straight-line regression

  1. Fluorescence (y) is the dependent variable
  2. Concentration (x) is called the regressor variable.

image12.pngimage20.png 1, 2, … , n        

here image01.png = image41.png

e = Error        

ei is the error in the ith observation yi.

Assumption on the errors

  1. {ei} are mutually independent
  2. E(ei) = 0
  3. Var(ei) = σ2image51.pngimage30.png

We assume ei’s are normal i.e.        ei ~ N(0, σ2)         image20.png 1, 2, … , n        

Regression by Formula

...read more.

Middle

16.05

23.62

31.19

38.76

Total

image18.png

0.095

-0.476

-0.048

0.381

0.810

-0.762

0.000

For the model to be a good fit to the data we require the sum of the errors equal zero.

image19.png

image21.png

Variance-covariance matrix

image22.png

image23.png

image24.png

image25.png

The above variance-covariance matrix gives image26.png, image27.pngand  image28.png

Design matrix

Below is the full fitted model for the data.

image29.pngimage30.png


Analysis of Variance

Source

Sum of squares

df

Mean (SS)

F ratio

Due to regression

image31.png

image32.png

image33.png

image34.png

Error SS (residual SS)

image35.png

image36.png

image37.png

Total

image38.png

image39.png

The ratioimage40.pngis distributed as F with (1, n –2 ) df.

Here we reject the null hypothesis that β = 0 if theimage40.png> Fα, 1, n – 2

Rejecting the null hypothesis implies that the variable x influences the variable y.  That is the concentration increases as the fluorescence of substance A increases.

Analysis of Variance

Source            DF         SS         MS         F      P

Regression         1    1003.21    1003.21   2478.53  0.000

Error              4       1.62       0.40                

Total              5    1004.83                            

From the ANOVA table we see that the P-value is 0.000 < 0.01, which is significant at the 1% significance level. We can conclude there is overwhelming evidence to reject the null hypothesis in favour of the alternative hypothesis, i.e. the concentration depends on fluorescence of the substance.

Goodness of fit in regression

Having found the best straight line, the next question is how well it describes the data.

...read more.

Conclusion

image00.pngand make image15.pngthe subject of the formula.

image57.png

image58.png

image59.png

When y = 17.5

hence image60.png

Having now calculated the value of image15.pngat a given image16.png, we can calculate the associated error.

Given some image16.pngthe predictor image15.pngis just image61.png, the error is given by

image62.png

Where image63.png

image64.png

hence

image65.pngimage66.png

image67.png

image68.png

image69.png= (17.35, 17.65) with 90% confidence.


Conclusion

The regression model fitted for the data is of the form:

image70.png

where         image16.pngis the dependent variable

image71.pngis the intercept

image72.pngis the slope of the regression coefficient

image15.pngis the independent variable

image73.pngis the error term.

Where

image11.png

The regression equation is image13.png

The equation will specify the average magnitude of the expected change in Y given a change in X.

Limitations of Regression

  • An assumption is made with the regression line that it is where it should be.

  • You cannot assume that the regression line is valid outside the range of the data.

  • You can interpolate, but you cannot extrapolate.

In the model the unknown parameter α and β where calculated. From the 90% confidence interval we know we have a good estimate for the gradient β of the model however more data is required before anything very definite can be said about the intercept of the model the parameter α.

...read more.

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