A recent study was conducted on a sample of 97 rheumatoid arthritis patients being treated with Etanercept and Etanercept with methotrexate. Patients were allocated to one of two treatment groups, based on the use of Etanercept (N=47) or Etanercept with methotrexate(N=50). The goal of the study was to compare the treatments, determining the effect of the treatment on the number of swollen joints for each patient in the treatment group. The samples taken were independent samples as there were no attempts to pair the patients or to give both treatments to those sampled.
First, summary statistics were calculated for each of the two treatments. From Result 1, we see that the mean number of swollen joints for the 40 patients treated with Etanercept is 5.25 swollen joints with a standard deviation of 5.90 swollen joints. For those being treated with Etanercept alone, there was a minimum of no joints swollen and a maximum of 31 swollent joints. Additionally, 57 patients had 4.14 swollen joints with a standard deviation of 3.74 when treated with Etanercept with methotrexate. In the group treated with Etanercept with methotrexate, there was a minimum of no joints swollen to a maximum of 15 joints swollen. Using the rule of thumb to test standard deviations, on can assume that the variability in number of swollen joints is the same for the two treatment groups because twice the smaller standard deviation (2*3.74 = 7.48) is greater than the largest standard deviation (5.90).
Summary statistics:
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We can also consider the distribution of the swollen joints for each treatment group by the histograms created in Result 2. From these histograms, we see that the number of swollen joints for both treatments are right-skewed, which does not meet the assumption of normally distributed data. The sample sizes taken were moderate and close in number (N1=40, N2=57). Polit argues that the t-test is robust to non-normality for sample sizes of at least 40.
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Sample data is used to draw an inference about the mean number of swollen joints for all rheumatoid arthritic patients given the two treatments. The statistical procedure used to address the research question is an independent samples t-test with equal variances. The null hypothesis to be tested is that the treatments have the same mean number of swollen joints against the alternative hypothesis of different mean number of swollen joints. The test is conducted with a significance level of 0.01. StatCrunch output is given in Result 3:
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Hypothesis test results:
μ1 : mean of Etanercept μ2 : mean of Etanercept plus Methotrexate μ1 - μ2 : mean difference H0 : μ1 - μ2 = 0 HA : μ1 - μ2 ≠ 0 (with pooled variances)
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At the 0.01 level (t= 1.13, p-value 0.26), there is no statistical evidence of a difference in the mean number of swollen joints for the group of Rheumatoid Arthritis patients treated with Etanercept and those treated with Etanercept with methotrexate. The study does not give support to using Etanercept alone as more or less effective than the use of Etanercept with methotrexate. Further studies would need to be conducted in order to make this determination.
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Hypothesis test results:
μ1 : mean of Cortisol where Group = 1 μ2 : mean of Cortisol where Group = 2 μ1 - μ2 : mean difference H0 : μ1 - μ2 = 0 HA : μ1 - μ2 < 0 (with pooled variances)
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Hypothesis test results:
μ1 : mean of Etanercept μ2 : mean of Etanercept plus Methotrexate μ1 - μ2 : mean difference H0 : μ1 - μ2 = 0 HA : μ1 - μ2 > 0 (with pooled variances)
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Sorry, should be sample of N1 = 40 and N2=57
The asumption of independent random samples of interval/ratio data are met is this study.
The interval/ratio data is not normally distributed; however, Polit argues that t-test is robust for non-normality when the sample size is of at least 40. In these sample the sizes are N1=47 and N2=50, meeting Polit's argument.
The need for the population variances to be assumed to be equal is also met. The mean number of swollen joints in the Etanercept groups is 5.25 and in the Etanercept with methotrexate group the mean number of swollen joints is 4.14. The two sample standard deviations (S1=5.90 and S2=3.74) are relatively close because if we double the smaller standard deviation (3.74 * 2 =7.48), it exceeds the larger sample standard deviation of 5.90. thus we can assume that population variance are equal.
Because these 3 assumptions are met, the t-test is appropriate for this data.