2DS00

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2DS00

• Statistics 1 for Chemical Engineering
• Lecture 3

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Week schedule

• Week 1 Measurement and statistics
• Week 2 Error propagation
• Week 3 Simple linear regression analysis
• Week 4 Multiple linear regression analysis
• Week 5 Nonlinear regression analysis

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Detailed contents of week 3

• Least Squares Method
• simple linear regression
• parameter estimates
• residuals
• confidence intervals
• significance test
• influential points
• lack-of-fit

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Least Squares

• measurements of time and distance
• estimate speed (assuming constant speed)

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Table of measurements and squares
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Visualisation of sums of squares
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Types of regression analysis

• Linear means linear in coefficients, not linear
  functions!
• Simple linear regression
• Multiple linear regression
• Non-linear regression

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Surface tension nitrobenzene

• measurements of temperature and surface tension
• temperature ranges from 40 to 200 oC
• scatter plot indicates linear relation

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Regression analysis of nitrobenzene example
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Confidence intervals

• parameter estimates estimate /- t14-20,025
  standard error
• predicted values (extrapolation is dangerous,
  most accurate predictions at mean of independent
  variable)

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Extrapolation
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Significance testing
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Simple Linear regression model assumptions

• Model Yi ?0 b1x1 ei
• Assumptions
• the model is linear ( enough terms)
• the ei's are normally distributed with m0 and
  constant variance s2
• the ei's are independent.

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Normality checking independence

• check normality by considering residuals
• apply both graphical checks and Shapiro-Wilks
• check independence by using the Durbin Watson
  test
• also check residuals by plotting them against
  time

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Residuals

• use studentized residuals in order to obtain
  universal scale

e versus homogeneity of variance
e versus linearity
e versus time independence of errors
e versus xi homogeneity of variance
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Lack-of-fit test

• if multiple measurements are available, then we
  may test whether model may be improved
  significantly
• test is based on two different ways of computing
  standard deviation
• note difference with testing of model is
  significant

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Influential points

• regression lines tend to go to remote points
  see http//www.stat.sc.edu/west/javahtml/Regressi
  on.html

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Check-list

1. apply regression analysis
2. check whether regression is signficant. If
   applicable, apply lack-of-fit test
3. study residual plots for constant variance
4. check for outliers
5. check normality of residuals (graphical checks,
   Shapiro-Wilks)
6. check independence of residuals (residual plots,
   Durbin Watson)
7. check for influential points

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Causality and regression

• Significant regression results do not imply
  causal relation !
• Statistical results must be explained
  (afterwards) by chemical theory.