Transcript Lecture 1
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 RDP Statistical Methods in Scientific Research - Lecture 1 1 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 RDP Statistical Methods in Scientific Research - Lecture 1 2 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 RDP Statistical Methods in Scientific Research - Lecture 1 3 Body perception index perceived width BPI 100 actual width Let mA = mean BPI for anorexics mC = mean BPI for controls Null hypothesis is H0: mA = mC RDP Statistical Methods in Scientific Research - Lecture 1 4 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 (nA = 8): mean = 145.13, standard deviation = 24.398 Controls (nC = 11): mean = 167.27, standard deviation = 32.426 RDP Statistical Methods in Scientific Research - Lecture 1 5 Formulae mean: x1 ... x n x n standard deviation (a measure of the spread of the data): S RDP x1 x 2 ... x n x 2 n 1 Statistical Methods in Scientific Research - Lecture 1 6 Notes Here we have means x A for anorexics and x C for controls and standard deviations SA for anorexics and SC for controls These are sample means and sample standard deviations: they vary from sample to sample The population means are mA for anorexics and mC for controls and the population standard deviations are sA for anorexics and sC for controls: these are fixed truths that will never be known precisely x A and x C are estimates of mA and mC SA and SC are estimates of sA and sC RDP Statistical Methods in Scientific Research - Lecture 1 7 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 RDP Statistical Methods in Scientific Research - Lecture 1 8 The t-statistic t xA xC 1 1 S n n C A where S RDP n A 1 S2A n C 1 SC2 nA nC 2 Statistical Methods in Scientific Research - Lecture 1 9 Notes We begin with an estimate of the difference between the means: x A x C This is standardised by dividing by S: S2 is a weighted average of S2A and SC2 Standardisation ensures that t is unit-free Division by 1 1 is a matter of convention, nA nC but it does ensure that values are not too greatly affected by sample sizes RDP Statistical Methods in Scientific Research - Lecture 1 10 Calculation 7 24.3982 10 32.4262 S 29.377 17 so that t RDP 145.13 167.27 1 1 29.377 8 11 1.622 Statistical Methods in Scientific Research - Lecture 1 11 Theory Suppose that the BPIs of anorexics follow the normal distribution with mean mA and standard deviation s the BPIs of controls follow the normal distribution with mean mC and the same standard deviation s Then, if mA = mC, 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 RDP Statistical Methods in Scientific Research - Lecture 1 12 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 RDP Statistical Methods in Scientific Research - Lecture 1 13 Interpretation If the null hypothesis H0: mA = mC 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 H1: mA < mC The value 0.061 is not so small that one would wish to reject H0 and conclude that there is a significant difference – it shows a trend, but does not constitute strong evidence RDP Statistical Methods in Scientific Research - Lecture 1 14 Caution! The investigators sought evidence that anorexics had a poorer perception of their bodily dimensions than controls – that mA > mC 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 RDP Statistical Methods in Scientific Research - Lecture 1 15 Two-sided p-value Double the one-sided p-value to give the two-sided p-value: p = 0.122 RDP Statistical Methods in Scientific Research - Lecture 1 16 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 RDP Statistical Methods in Scientific Research - Lecture 1 17 1.3 Confidence intervals for treatment effects We have used mA to denote the population mean of the BPIs for anorexics and sA to denote their population standard deviations These are estimated by the sample mean x A = 145.13 and by the sample standard deviation SA = 24.398 respectively How good an estimate of mA is 145.13? How big or small might mA actually be? A confidence interval will answer this question RDP Statistical Methods in Scientific Research - Lecture 1 18 Another t-distributed random variable Let tA x A mA nA SA Note that you cannot calculate tA as it depends on the unknown mA If the BPI observations are normally distributed, then tA follows Student’s t-distribution with (nA – 1) df Now, a t7 random variable lies between 2.365 and 2.635 with probability 0.95 RDP Statistical Methods in Scientific Research - Lecture 1 19 A confidence interval for mA It follows that, with probability 0.95, x A mA 2.365 SA nA 2.365 which is 2.365 SA n A x A m A 2.365 SA nA which is x A 2.365 SA n A m A x A 2.365 SA nA which is 145.13 2.365 24.398 RDP 8 m A 145.13 2.365 24.398 Statistical Methods in Scientific Research - Lecture 1 8 20 A confidence interval for mA So, with probability 0.95, 124.73 mA 165.53 We say that (124.73, 165.53) is a 95% confidence interval for mA The upper and lower limits are random, while mA is fixed The limits capture the true value of mA 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 RDP Statistical Methods in Scientific Research - Lecture 1 21 A confidence interval for mC For the controls, nC = 11, x C = 167.27 and SC = 32.426 The 97.5% point of the t distribution on 10 df is 2.228 Hence, the 95% confidence interval for mC is (167.27 2.228 32.426/11) (145.49, 189.05) Note that the confidence intervals for mA and mC overlap What about a 95% confidence interval for d = mA mC? RDP Statistical Methods in Scientific Research - Lecture 1 22 A confidence interval for d = mA mC Now td xA xC d 1 1 S n n C A follows Student’s t-distribution with (n – 2) df The 97.5% point of the t distribution on 17 df is 2.110 RDP Statistical Methods in Scientific Research - Lecture 1 23 A confidence interval for d = mA mC It follows that, with probability 0.95 1 1 1 1 x A x C 2.110 S d x A x C 2.110 S nA nC nA nC so that the 95% confidence interval for d is 1 1 x A x C 2.110 S nA nC 1 1 22.14 2.110 29.377 8 11 50.94,6.66 RDP Statistical Methods in Scientific Research - Lecture 1 24 Interpretation 0 lies within the confidence interval consistent with lack of significant evidence against H0: d = 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 RDP Statistical Methods in Scientific Research - Lecture 1 25