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**Didier Concordet d.concordet@envt.fr**

Ecole Nationale Vétérinaire de Toulouse Linear Regression Didier Concordet ECVPT Workshop April 2011 Can be downloaded at

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An example

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**About the straight line**

Y= a + b x Y x a b>0 b<0 b=0 a=0 a = intercept b = slope

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**Questions How to obtain the best straight line ?**

Is this straight line the best curve to use ? How to use this straight line ?

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**How to obtain the best straight line ?**

Proceed in three main steps write a (statistical) model estimate the parameters graphical inspection of data

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**A statistical model Write a model Mean model :**

functionnal relationship Variance model : Assumptions on the residuals

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Write a model Mean model = residual (error term)

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**Assumptions on the residuals**

the xi 's are not random variables they are known with a high precision the ei 's have a constant variance homoscedasticity the ei 's are independent the ei 's are normally distributed normality

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Homoscedasticity homoscedasticity heteroscedasticity

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Normality Y x

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**Estimate the parameters**

A criterion is needed to estimate parameters A statistical model A criterion

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**How to estimate the "best" a et b ?**

Intuitive criterion : minimum compensation Reasonnable criterion : minimum Linear model Homoscedasticity Normality Least squares criterion (L.S.)

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**The least squares criterion**

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**Result of optimisation**

and change with samples and are random variables

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**Balance sheet True mean straight line Estimated straight line or**

Mean predicted value for the ith observation ith residual

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**Example Estimated straight line Dep Var: HPLC N: 18**

Effect Coefficient Std Error t P(2 Tail) CONSTANT CONCENT Intercept Estimated straight line Slope

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Example

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Example

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**Residual variance by construction but**

The residual variance is defined by standard error of estimate

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**Example Dep Var: HPLC N: 18**

Multiple R: Squared multiple R: 0.991 Adjusted squared multiple R: 0.991 Standard error of estimate : 8.282 Effect Coefficient Std Error t P(2 Tail) CONSTANT CONCENT

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**Questions How to obtain the best straight line ?**

Is this straight line the best curve to use ? How to use this straight line ?

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**Is this model the best one to use ?**

Tools to check the mean model : scatterplot residuals vs fitted values test(s) Tools to check the variance model : scatterplot residuals vs fitted values Probability plot (Pplot)

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**Checking the mean model**

scatterplot residuals vs fitted values structure in the residuals change the mean model No structure in the residuals OK

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**Checking the mean model : tests**

Two cases No replication Try a polynomial model (quadratic first) Replications Test of lack of fit

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**Without replication try another mean model and test the improvement**

Example : If the test on c is significant (c 0) then keep this model Dep Var: HPLC N: 18 Multiple R: Squared multiple R: 0.991 Adjusted squared multiple R: 0.991 Standard error of estimate: 8.539 Effect Coefficient Std Error t P(2 Tail) CONSTANT CONCENT CONCENT *CONCENT

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**With replications Perform a test of lack of fit Principle : compare to**

Departure from linearity Pure error Principle : compare to if - > then change the model

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**Test of lack of fit : how to do it ?**

Three steps 1) Linear regression 2) One way ANOVA 3) if then change the model

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**Test of lack of fit : example**

Three steps 1) Linear regression 2) One way ANOVA Dep Var: HPLC N: 18 Analysis of Variance Source Sum-of-Squares df Mean-Square F-ratio P CONCENT Error 3) if We keep the straight line

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**Checking the variance model : homoscedasticity**

scatterplot residuals vs fitted values No structure in the residuals but heteroscedasticity change the model (criterion) homoscedasticity OK

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**What to do with heteroscedasticity ?**

scatterplot residuals vs fitted values : modelize the dispersion. The standard deviation of the residuals increases with : it increases with x

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**What to do with heteroscedasticity ?**

Estimate again the slope and the intercept but with weights proportionnal to the variance. with and check that the weight residuals (as defined above) are homoscedastic

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**Checking the variance model : normality**

Expected value for normal distribution Expected value for normal distribution No curvature : Normality Curvature : non normality is it so important ?

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**What to do with non normality ?**

Try to modelize the distribution of residuals In general, it is difficult with few observations If enough observations are available, the non normality does not affect too much the result.

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**An interesting indice R²**

R² = square correlation coefficient = % of dispersion of the Yi's explained by the straight line (the model) 0 R² 1 If R² = 1, all the ei = 0, the straight line explain all the variation of the Yi's If R² = 0, the slope is = 0, the straight line does not explain any variation of the Yi's

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**An interesting indice R²**

R² and R (correlation coefficient) are not designed to measure linearity ! Example : Multiple R: 0.990 Squared multiple R: 0.980 Adjusted squared multiple R: 0.980

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**Questions How to obtain the best straight line ?**

Is this straight line the best curve to use ? How to use this straight line ?

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**How to use this straight line ?**

Direct use : for a given x predict the mean Y construct a confidence interval of the mean Y construct a prediction interval of Y Reverse use calibration (approximate results): for a given Y predict the mean x construct a confidence interval of the mean x construct a prediction interval of X

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**For a given x predict the mean Y**

Example :

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**Confidence interval of the mean Y**

There is a probability 1-a that a+bx belongs to this interval

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**Confidence interval of the mean Y**

U L 30

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Example

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**Prediction interval of Y**

100(1-a)% of the measurements carried-out for this x belongs to this interval

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**Prediction interval of Y**

U L 30

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Example

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**Reverse use : for a given Y=y0 predict the mean X**

Example :

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**For a given Y=y0 a confidence interval of the mean X**

U

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**Confidence interval of the mean X**

There is a probability 1-a that the mean X belongs to [ L , U ] L and U are so that

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Example

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**What you should no longer believe**

One can fit the straight line by inverting x and Y If the correlation coefficient is high, the straight line is the best model Normality of the xi's is required to perform a regression Normality of the ei's is essential to perform a good regression

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