Chapter 6: Non-Parametric Comparison of Means.Paired Samples and BlocksChapter 6: Non-Parametric Comparison of Means. Wilcoxon Sum Rank Test Say you believe that students who go home to their families for Thanksgiving Weekend actually do better on their exams because they need to decompress more than they need to study. Say you took a random sample of 8 students who went home for Thanksgiving and 8 who stayed in Missoula and studied, and then obtained their final exam scores. Here are the resulting data Went Home Studied 113.25 95.94 90.04 104.44 119.21 106.88 94.99 131.09 137.9134 142.4956 129.6706 115.4934 183.4077 123.5596 94.7618 102.0240 Let’s conduct a Wilcoxon Sum Rank Test on these data. Wilcoxon Rank Sum Test Home Ranked Studied Ranked Ranks 90.0400 94.9900 95.9400 104.4400 106.8800 113.2500 119.2100 131.0900 94.7618 102.0240 115.4934 123.5596 129.6706 137.9134 142.4956 183.4077 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 We now sum the ranks in for the ‘home’ data. This yields W=1+3+4+6+7+8+10+12=51. Under the null hypothesis of no difference, we have ()()()1111,var21WnN nnNWµ22++==2Our two samples are just large enough (each >7) so we can perform this test by calculating a z-score. ()wWzSD Wµ−= . For this example, 51 681.7859.52z−==−, with p-value < 0.01. So, we reject the null hypothesis of no difference between the two groups. Paired Samples and Blocks Example: Differences of electric potential occur naturally from point to point on a body’s skin. Is the natural electric field strength best for helping wounds to heal? If so, changing the field will slow healing. An experiment was conducted to investigate this question. The subjects were 14 anesthetized newts. A razor cut was made in both hind limbs. One limb was allowed to heal naturally (the control). In the other limb, an electrode was used to change the electric field to half its normal value. After two hours, the healing rate was measured in each limb in micrometers per hour. The results were: Newt Experimental limb Control limb Difference 1 24 25 -1 2 23 13 10 3 47 44 3 4 42 45 -3 5 26 57 -31 6 46 42 4 7 38 50 -12 8 33 36 -3 9 28 35 -7 10 28 38 -10 11 21 43 -22 12 27 31 -4 13 25 26 -1 14 45 48 -3 • What kind of experimental design is this?3 • If you were reviewing a paper about this experiment, is there anything else you would like to know about how this experiment was conducted? • Can we analyze these data with two-sample t procedures? Why not? • How could the experiment have been conducted to make two-sample procedures appropriate? • What are the advantages of conducting this experiment as matched pairs rather than as a completely randomized design? The analysis of matched pairs data is conducted by analyzing the differences between the two responses. The analysis proceeds just as for a single variable. The one-sample t-procedures are used on the differences. The assumptions of these procedures now apply to the differences.4Difference (experimental - control)1050-5-10-15-20-25-306543210 • Randomization condition: the data are from a simple random sample from the population. Since this is an experiment, this condition is satisfied if the treatments are assigned randomly within each pair. The inference then, is to the true difference for these 14 newts, averaging over all possible assignments of the treatments within each pair. The inference carries beyond these particular newts if they can be viewed as a random sample of a larger population of newts. • Near normal condition: since n < 15: the population distribution of the differences should be pretty normal; at least, mound shaped and symmetric with no outliers. Generally, we can only judge this by the distribution of the sample differences. In this case, we can see that there is an outlier or two on the low end. That is a concern. We will proceed with the t-procedures to illustrate their calculation, but we should be cautious about the validity of the inferences. Notation: to make it clear that we are dealing with the differences, we let dµ = true mean difference (experimental – control) between the treatments d = sample mean difference = sample standard deviation of the differences dsWe must also remember that the sample size n is the number of differences (the number of pairs). • A confidence interval for the true mean difference is )SE(*1dtdn−± where SE(d) = nsd/. • To test the hypothesis of no difference between the treatments, we test 0:0=dHµ5 0: ≠dAHµ (or 0: >dAHµ or 0:<dAHµ) The test statistic is )SE(ddt = which is compared to a t distribution with n-1 df. Descriptive Statistics14 -31 10 -5.71 10.564Difference (exp-control)N Minimum Maximum Mean Std. Deviation Compute a 95% confidence interval for the true mean difference between the healing rates of the experimental and control treatments. One-Sample Test-2.024 13 .064 -5.71429 -11.8139 .3854Difference (exp - control)t df Sig. (2-tailed)MeanDifferenceLower Upper95% ConfidenceInterval of theDifferenceTest Value = 0 What conclusions can we reach from these data about whether the electric potential affects the healing rate and by how much? Wilcoxon Signed-Rank Test for Nonnormal small, paired Sample Data – Compute Differences di (as in the paired t-test) and obtain their absolute values (ignoring 0s). n = number of non-zero differences. – Rank the observations by |di| (smallest =1), averaging ranks for ties. – Compute T+ and T- , the rank sums for the positive and negative differences, respectively. – 1-sided tests:Conclude HA: M1 > M2 if T=T- ≤ T0 – 2-sided tests:Conclude HA: M1 ≠ M2 if T=min(T+ , T- ) ≤ T0 – Values of T0 are given in Table 7, pp 684-685 for various sample sizes and significance levels. Exact p-values can be found in various statistical software packages.6Example: ECMO A study investigating the reduction of forced vital capacity (FVC) for a sample of patients with cystic fibrosis after being given an experimental drug, amiloride, compared to the same time period after being given a placebo was reported in Principles of Biostatistics, M. Pagano and K. Gauvreau, Duxbury Press, 1993. The data and their ranks are shown below. Reduction in FVC (ml) Subject Placebo Drug Difference Rank Signed Rank 1 224 213 11 1 1 2
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