2DS00 Statistics 1 for Chemical Engineering Lecture 3

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

Detailed contents of week 3 Least Squares Method simple linear regression –parameter estimates –residuals –confidence intervals –significance test –influential points –lack-of-fit

Least Squares measurements of time and distance estimate speed (assuming constant speed)

Tijd (sec) Gemeten afstand Berekende afstand Gemeten – Berekende afstand Kwadraat Kwadratensom Table of measurements and squares

Visualisation of sums of squares

Types of regression analysis Linear means linear in coefficients, not linear functions! Simple linear regression Multiple linear regression Non-linear regression

Surface tension nitrobenzene measurements of temperature and surface tension temperature ranges from 40 to 200 o C scatter plot indicates linear relation

Regression analysis of nitrobenzene example

Confidence intervals parameter estimates: estimate +/- t 14-2;0,025 standard error predicted values (extrapolation is dangerous, most accurate predictions at mean of independent variable)

Extrapolation

Significance testing

Model: Y i =  0 +  1 x 1 +  i  ssumptions: the model is linear (+ enough terms) the  i 's are normally distributed with  =0 and constant variance  2 the  i 's are independent. Simple Linear regression: model assumptions

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

Residuals use studentized residuals in order to obtain universal scale e versushomogeneity of variance e versuslinearity e versus timeindependence of errors e versus x i homogeneity of variance

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

Influential points regression lines tend to go to remote points: see

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

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