1. RDPStatistical Methods in Scientific Research - Lecture 11 Lecture 1 Interpretation of data 1.1 A study in anorexia nervosa 1.2 Testing the difference between the samples 1.3 Confidence intervals for treatment effects

2. RDPStatistical Methods in Scientific Research - Lecture 12 1.1 A study in anorexia nervosa Ben-Tovim, Whitehead and Crisp (1979) “Sufferers from anorexia nervosa, even those whose bodies have become severely emaciated, often maintain that their bodily dimensions are quite normal”  Are anorexics able to judge their own bodily dimensions?  Are they worse at doing so than healthy controls? 8 anorexics and 11 controls participated in a study of these questions

3. RDPStatistical Methods in Scientific Research - Lecture 13 The apparatus Two lights on a horizontal beam Move them together: “Say stop when the distance apart is the same as the width of your waist” Repeat while they move apart, and then average the two measurements to give the perceived width

4. RDPStatistical Methods in Scientific Research - Lecture 14 Body perception index Let  A  = mean BPI for anorexics  C  = mean BPI for controls Null hypothesis is H 0 :  A  =  C

5. RDPStatistical Methods in Scientific Research - Lecture 15 The data Anorexics: 130, 194, 160, 120, 152, 144, 120, 141 Controls: 202, 140, 168, 160, 147, 133, 229, 172, 130, 206, 153 Summary: Overall (n = 19): mean = 157.95, standard deviation = 30.689 Anorexics (n A = 8): mean = 145.13, standard deviation = 24.398 Controls (n C = 11): mean = 167.27, standard deviation = 32.426

6. RDPStatistical Methods in Scientific Research - Lecture 16 Formulae mean: standard deviation (a measure of the spread of the data):

7. RDPStatistical Methods in Scientific Research - Lecture 17 Notes  Here we have means for anorexics and for controls and standard deviations S A for anorexics and S C for controls  These are sample means and sample standard deviations: they vary from sample to sample  The population means are  A for anorexics and  C for controls and the population standard deviations are  A for anorexics and  C for controls: these are fixed truths that will never be known precisely  and are estimates of  A and  C S A and S C are estimates of  A and  C

8. RDPStatistical Methods in Scientific Research - Lecture 18 1.2 Testing the difference between the samples The two group means are different from one another Are they significantly different? Or might the difference just be due to chance? We will use a t-test to find out

9. RDPStatistical Methods in Scientific Research - Lecture 19 The t-statistic where

10. RDPStatistical Methods in Scientific Research - Lecture 110 Notes  We begin with an estimate of the difference between the means:  This is standardised by dividing by S: S 2 is a weighted average of  Standardisation ensures that t is unit-free  Division by is a matter of convention, but it does ensure that values are not too greatly affected by sample sizes

11. RDPStatistical Methods in Scientific Research - Lecture 111 Calculation so that

12. RDPStatistical Methods in Scientific Research - Lecture 112 Theory Suppose that  the BPIs of anorexics follow the normal distribution with mean  A and standard deviation   the BPIs of controls follow the normal distribution with mean  C and the same standard deviation  Then, if  A =  C, the statistic t follows Student’s t-distribution on 17 degrees of freedom (17 = 19 – 2 = n  # parameters)  a similar shape (slightly fatter) centred on 0

13. RDPStatistical Methods in Scientific Research - Lecture 113 The t-distribution The probability that a random variable following Student’s t-distribution on 17 degrees of freedom is   1.627 is 0.061

14. RDPStatistical Methods in Scientific Research - Lecture 114 Interpretation If the null hypothesis H 0 :  A  =  C is true (and the populations have the same standard deviation), then t is unusually negative The chance of it being so negative (or even more so) is 0.061 This is the p-value against the one-sided alternative H 1 :  A  <  C The value 0.061 is not so small that one would wish to reject H 0 and conclude that there is a significant difference – it shows a trend, but does not constitute strong evidence

15. RDPStatistical Methods in Scientific Research - Lecture 115 Caution! The investigators sought evidence that anorexics had a poorer perception of their bodily dimensions than controls – that  A  >  C The trend is in the opposite direction! “So maybe the anorexics have a better perception, being so obsessed by their bodies” Investigators are going to wish to interpret the data, whichever direction the difference, so use a two-sided p-value

16. RDPStatistical Methods in Scientific Research - Lecture 116 Two-sided p-value Double the one-sided p-value to give the two-sided p-value: p = 0.122

17. RDPStatistical Methods in Scientific Research - Lecture 117 Convention  A two-sided p-value  0.05 is usually taken to represent strong evidence of an effect  This goes back to Fisher in the 1930s  It is rather arbitrary, but it is a useful yardstick  A one-sided p-value  0.025 is usually taken to represent strong evidence of an effect – this avoids “cheating” by choosing the direction of the difference once the data have been observed

18. RDPStatistical Methods in Scientific Research - Lecture 118 1.3 Confidence intervals for treatment effects We have used  A to denote the population mean of the BPIs for anorexics and  A to denote their population standard deviations These are estimated by the sample mean = 145.13 and by the sample standard deviation S A = 24.398 respectively How good an estimate of  A is 145.13? How big or small might  A actually be? A confidence interval will answer this question

19. RDPStatistical Methods in Scientific Research - Lecture 119 Another t-distributed random variable Let Note that you cannot calculate t A as it depends on the unknown  A If the BPI observations are normally distributed, then t A follows Student’s t-distribution with (n A – 1) df Now, a t 7 random variable lies between  2.365 and 2.635 with probability 0.95

20. RDPStatistical Methods in Scientific Research - Lecture 120 A confidence interval for  A It follows that, with probability 0.95, which is

21. RDPStatistical Methods in Scientific Research - Lecture 121 A confidence interval for  A So, with probability 0.95, We say that (124.73, 165.53) is a 95% confidence interval for  A The upper and lower limits are random, while  A is fixed The limits capture the true value of  A with probability 0.95 It could well be that the true mean BPI for anorexics is as low as 124.73, it could also be as high as 165.53

22. RDPStatistical Methods in Scientific Research - Lecture 122 A confidence interval for  C For the controls, n C = 11, = 167.27 and S C = 32.426 The 97.5% point of the t distribution on 10 df is 2.228 Hence, the 95% confidence interval for  C is (167.27  2.228  32.426/  11)  (145.49, 189.05) Note that the confidence intervals for  A and  C overlap What about a 95% confidence interval for  =  A  C ?

23. RDPStatistical Methods in Scientific Research - Lecture 123 A confidence interval for  =  A  C Now follows Student’s t-distribution with (n – 2) df The 97.5% point of the t distribution on 17 df is 2.110

24. RDPStatistical Methods in Scientific Research - Lecture 124 A confidence interval for  =  A  C It follows that, with probability 0.95 so that the 95% confidence interval for  is

25. RDPStatistical Methods in Scientific Research - Lecture 125 Interpretation 0 lies within the confidence interval  consistent with lack of significant evidence against H 0 :  = 0 at the 5% level (2-sided), as found from the t-test The mean BPI for anorexics could be substantially lower that that for controls (by more than 50), or slightly higher Larger sample sizes would reduce the width of the confidence intervals, and make it easier to determine whether there really is a difference