Hypothesis Testing in Action

Transcription

1 Hypothesis Testing in Action Jonathan Bagley School of Mathematics, University of Manchester Jonathan Bagley, September 23, 2005 The t-tests - p. 1/23

2 Overview Today we ll examine three data sets and use hypothesis tests to explore them: the main new tool will be Student s t-test. Differences in proportions: The Boston aspirin Differences in means: the t-tests and H.H. Koh s macular pigment data Jonathan Bagley, September 23, 2005 The t-tests - p. 2/23

3 The Boston aspirin In a famous and very large during the 1980 s, several hospitals in the Boston area worked together to conduct a placebo controlled, double-blind of the efficacy of aspirin in preventing heart attacks. The results were: Group N-attacks N Patients Aspirin Placebo Is this an important difference? Jonathan Bagley, September 23, 2005 The t-tests - p. 3/23

4 Statistical setup The first question to ask is: how likely is this difference to have arisen by chance? We begin with a hypothesis test based on a z-score that addresses this question. Null Hypothesis The two proportions are the same. Alternative Hypothesis Either The two proportions differ (two-sided test) or A specific one of the proportions is larger (one-sided test). Jonathan Bagley, September 23, 2005 The t-tests - p. 4/23

5 Differences of proportions, The ingredients for this test are two experimentally observed proportions: p 1 = (r 1 /n 1 ) and p 2 = (r 2 /n 2 ). (a) As the null hypothesis is that the proportions are the same, combine the data to get a single estimate of the underlying proportion: p = (r 1 + r 2 )/(n 1 + n 2 ) (b) Estimate the standard error of the difference between the two proportions measured from similarly-sized : ( 1 SE = p(1 p) + 1 ) n 1 n 2 ( ) n1 + n 2 = p(1 p) n 1 n 2 Jonathan Bagley, September 23, 2005 The t-tests - p. 5/23

6 Differences of proportions, continued (c) Compute z = p 1 p 2 = = SE p 1 p 2 ( ) n p(1 p) 1 +n 2 n 1 n 2 (p 1 p 2 ) n 1 n 2 p(1 p)(n1 + n 2 ) (d) Consult the table for the standard normal. Jonathan Bagley, September 23, 2005 The t-tests - p. 6/23

7 Application to the aspirin (a) Under the null hypothesis our best estimate for p is p = ( )/( ) (b) The standard deviation of the difference is then ( 1 SE = p(1 p) ) (c) The z-score is z = (189/11034) (104/11037) Jonathan Bagley, September 23, 2005 The t-tests - p. 7/23

8 Application to the aspirin data, continued (d) This is a massively implausible z-score: we can reject the null hypothesis in favour of the alternative that the aspirin group has fewer heart attacks with confidence 99.99%. Jonathan Bagley, September 23, 2005 The t-tests - p. 8/23

9 Visual pigments and macular The next two examples involve measurements of Macular Pigment Optical Density (MPOD) collected from two groups: patients suffering from macular and healthy control subjects. Raw data are 10 total measurements per subject, collected in two sessions of 5 measurements with around a 30 minute break in between. The MPOD is the difference between measurements taken at central fixation and another in the periphery (5 degree visual angle). All measurements are on healthy eyes even those from the patients, each of whom had only a single d eye. These data were collected by Ms. Hui Hiang Koh (now Dr. Koh) and her advisor, Dr. Ian Murray. Jonathan Bagley, September 23, 2005 The t-tests - p. 9/23

10 Are the two groups different? Patients Controls MPOD SEM MPOD SEM m x m y s x s y Jonathan Bagley, September 23, 2005 The t-tests - p. 10/23

11 Testing for If anything, the patients seem to have more pigment than the controls. Is this apparent difference significant? Test with a new hypothesis test, the Two Sample t-test, designed for in the means of small. Null Hypothesis MPOD for Patients and Controls are drawn from the same normal distribution (same mean, same variance). Alternative Hypothesis MPOD for the two groups drawn from normal distributions with different means, but the same variance. This will involve a two-sided test based on a new statistic, t. Jonathan Bagley, September 23, 2005 The t-tests - p. 11/23

12 Aside: statistical folklore The t-test was developed by W.S. Gosset ( ), a statistician who worked for the Guiness brewing company. Employees of the firm were not allowed to publish under their own names so he wrote under the pseudonym Student. The t-statistic is: similar to a z-score, but is applicable when the sample is too small to assume that s 2 x and s 2 y provide good estimates of the variances; this advantage comes at a small cost: the t-distribution (and hence the tables one consults to use it) are less straightforward than those for the normal distribution; depends on the sizes of the when these grow large the distribution of t tends to the normal. Jonathan Bagley, September 23, 2005 The t-tests - p. 12/23

13 Computing t for two The ingredients are two of lists of numbers, say, {x 1, x 2,..., x Nx } and {y 1, y 2,..., y Ny }. (a) Computes the two sample means, m x and m y. Recall that, for example, Nx j=1 m x = x j. N x (b) Computes the two standard deviations, s x and s y. Recall that, for example, Ny s 2 j=1 y = (y j m y ) 2. (N y 1) Jonathan Bagley, September 23, 2005 The t-tests - p. 13/23

14 Two-sample t continued (c) Computes the pooled standard deviation, s, which satisfies (d) Last, one computes s 2 = (N x 1)s 2 x + (N y 1)s 2 y (N x 1) + (N y 1) t = m x m y s N x N y N x + N y. (e) Consult the t-table for ν = N x + N y 2 degrees of freedom.. Jonathan Bagley, September 23, 2005 The t-tests - p. 14/23

15 Distribution of t t 4 Student s t-distribution for ν = 2, 4 and 8.The dashed curve at the top is the standard normal distribution (µ = 0, σ = 1). Jonathan Bagley, September 23, 2005 The t-tests - p. 15/23

16 Testing the MPOD data Working through the recipe, N x = N y = 9 and: (a) Patients had m x = 0.293; Controls had m y = (b) Patients had s x = 0.135; Controls had s y = (c) The pooled variance is thus s 2 = (N x 1)s 2 x + (N y 1)s 2 y (N x 1) + (N y 1) = 8(0.135)2 + 8(0.142) Jonathan Bagley, September 23, 2005 The t-tests - p. 16/23

17 Testing the MPOD data, continued (d) Thus t = m x m y N x N y s N x + N y = (e) This is smaller than the critical value, 2.120, for a two-sided test with ν = 16 degrees of freedom at 95% confidence. We cannot reject the null hypothesis. Jonathan Bagley, September 23, 2005 The t-tests - p. 17/23

18 Paired sample design The considerable variation within the groups may make it hard to see whether there is much systematic difference between groups. Design a new type of in which Patients and Controls are matched for age, gender, eye (left or right), iris colour and smoking habits. Compare with the Paired Sample t-test. Jonathan Bagley, September 23, 2005 The t-tests - p. 18/23

19 Computing t for paired The only ingredient is a list of N pairs of numbers {(x 1, y 1 ),..., (x N, y N )}. Null hypothesis is that the two members of each pair are drawn from normal distributions having the same mean. All the distributions for the x s are assumed to share the same variance as are all the y s, but the variance shared by the x s need not equal that shared by the y s. Jonathan Bagley, September 23, 2005 The t-tests - p. 19/23

20 Computing t for paired, continued (a) Compute the δ j = (x j y j ); (b) Compute the mean of the N j=1 m = δ j N ; (c) Estimate the variance of the N s 2 j=1 = (δ j m) 2 ; N 1 Jonathan Bagley, September 23, 2005 The t-tests - p. 20/23

21 Computing t for paired, continued (d) Compute the paired-sample t-statistic t = m N s (e) Check against critical values in the t-table, here using ν = N 1 degrees of freedom.. Jonathan Bagley, September 23, 2005 The t-tests - p. 21/23

22 Paired MPOD data, MPOD Control Patient δ Jonathan Bagley, September 23, 2005 The t-tests - p. 22/23

23 Paired MPOD data, The mean difference is m = with standard deviation s = This leads to t = m N s = This far exceeds the critical value, 2.306, for a two-sided test at the 95% confidence level (α = 0.025, ν = 8). We can reject the null hypothesis and conclude that the difference is nonzero. Jonathan Bagley, September 23, 2005 The t-tests - p. 23/23