1. RDPStatistical Methods in Scientific Research - Lecture 21 Lecture 2 Regression relationships 2.1 The influence of actual widths of the anorexics 2.2 Testing the importance of each influence 2.3 Comments on the anorexia study

2. RDPStatistical Methods in Scientific Research - Lecture 22 2.1 The influence of actual widths of the anorexics AnorexicsControls BPIActual widthBPIActual width 13022.520218.2 19419.214024.2 16019.316816.0 12023.316021.3 15221.314721.3 14422.813324.9 12028.222917.2 14121.917219.9 13022.0 20619.2 15322.1

3. RDPStatistical Methods in Scientific Research - Lecture 23 Scatter plot

4. RDPStatistical Methods in Scientific Research - Lecture 24 Observations  BPI decreases with actual width  The controls have smaller waists than the anorexics!  Actual width appears to be a stronger determinant of BPI than anorexic status

5. RDPStatistical Methods in Scientific Research - Lecture 25 Five models for the data 1 INTERCEPT Neither anorexia nor actual width affect BPI 2 INTERCEPT + GROUP Anorexia affects BPI, but actual width does not

6. RDPStatistical Methods in Scientific Research - Lecture 26 3 INTERCEPT + AW Actual width affects BPI, but anorexia does not 4 INTERCEPT + GROUP + AW Anorexia and actual width affect BPI additively

7. RDPStatistical Methods in Scientific Research - Lecture 27 5 INTERCEPT + GROUP + AW + INTERACTION Anorexia and actual width affect BPI non-additively Which model fits the data best? How can we judge? How should we play off goodness-of-fit against complexity?

8. RDPStatistical Methods in Scientific Research - Lecture 28 Residuals The residuals are the vertical distances between the observed points and the fitted models: residual = BPI observed – BPI fitted For example, for Model 4 we have:

9. RDPStatistical Methods in Scientific Research - Lecture 29

10. RDPStatistical Methods in Scientific Research - Lecture 210 Showing only the residuals, we have:

11. RDPStatistical Methods in Scientific Research - Lecture 211 Moving them all down to 0 gives: The goodness-of-fit of a models is assessed in terms of the residual sum of squares, RSS, (the smaller, the better):

12. RDPStatistical Methods in Scientific Research - Lecture 212 Model fits degrees-of-freedom (df) = n  # parameters Goodness-of-fit improves as terms are added into the model, although model complexity (number of parameters) increases (which is a bad thing) AnorexicsControls ModelinterceptslopeinterceptslopeRSSdf 1157.90 016952.9518 2145.10167.3014681.0617 3338.5  8.47 338.5  8.47 6368.0517 4324.2  8.03 332.4  8.03 6087.2416 5296.2  6.77 346.0  8.93 5936.1215

13. RDPStatistical Methods in Scientific Research - Lecture 213 Interaction  We start with the most complex model (Model 5), and see whether it can be simplified  That is, we test H 0 : there is no aw  group interaction (Model 4 is valid)  If the observations are normally distributed, then if Model 4 is true, then F int follows an F-distribution with (1, 15) degrees-of-freedom: that is F int ~ F 1,15, where 2.2 Testing the importance of each influence

14. RDPStatistical Methods in Scientific Research - Lecture 214 Interaction Large values of F int indicate that H 0 is false Here, we have This value is too small to suggest that interaction is important The p-value is p = P(F  0.38) where F ~ F 1,15, and p = 0.5459

15. RDPStatistical Methods in Scientific Research - Lecture 215 Actual width  Take the model in which BPI depends on actual width only (Model 3), and see whether the effect of actual width is necessary  That is, we test H 0 : actual width does not effect BPI, which means that Model 1 is valid  If the observations are normally distributed, then if Model 1 is true, then F aw ~ F 1,17, where

16. RDPStatistical Methods in Scientific Research - Lecture 216 Actual width We have This value is too large to come from the F 1,17 distribution The p-value is p = P(F  28.26) where F ~ F 1,17, and p < 0.0001 H 0 : actual width does not effect BPI is rejected

17. RDPStatistical Methods in Scientific Research - Lecture 217 Group  Accepting that actual width is needed in the model, now take Model 4, and see whether it can be simplified by removing the effect of anorexia  That is, we test H 0 : anorexia does not effect BPI (once aw is allowed for), which means that Model 3 is valid  If the observations are normally distributed, then if Model 3 is true, then F group  aw ~ F 1,16, where

18. RDPStatistical Methods in Scientific Research - Lecture 218 Group We have This value is too small to suggest that group is important The p-value is p = P(F  0.74) where F ~ F 1,16, and p = 0.4030

19. RDPStatistical Methods in Scientific Research - Lecture 219 Final model This is Model 3, which states that BPI has mean =  8.47 + 338.5 aw standard deviation =  395.74 = 19.89 and that being anorexic has no significant effect on body perception index

20. RDPStatistical Methods in Scientific Research - Lecture 220 Order of fitting is important  Test interaction first: if this is significant, then the two main effects should not be tested: Model 5 is needed to describe the data  Then determine whether actual width is needed in the model  As actual width is needed, test the effect of group (the factor that is of interest), by comparing Model 3 with Model 4  If actual width were not needed, test the effect of group by comparing Model 1 with Model 2

21. RDPStatistical Methods in Scientific Research - Lecture 221 Order of fitting is important To compare Model 1 with Model 2, find which is The p-value is p = P(F  2.63) where F ~ F 1,16, and p = 0.1232

22. RDPStatistical Methods in Scientific Research - Lecture 222 Order of fitting is important The t-statistic for testing the effect of anorexia shown on Slide 1.11 was equal to  1.622 The square of  1.622 is 2.631, which is equal to F group This is no coincidence: these two tests are in fact identical BUT, in this case, due to the important influence of actual width, any analysis that fails to account for aw is invalid

23. RDPStatistical Methods in Scientific Research - Lecture 223 Choice of subjects  The anorexics were consecutive unmarried female patients at St George’s Hospital, London  The controls were volunteer fifth form pupils from Putney Girls’ High School, with normal dietary habits Ages: Anorexics mean = 19.7, sd = 3.6 Controls mean = 15.4, sd = 0.5 This was not a suitable control group for this study 2.3 Comments on the anorexia study