2. [13 marks] In an early clinical trial of a new drug, researchers advertised for patients with a certain difficult-to-treat condition. Applicants with other medical complications, or extreme age (either young or old) were excluded. Those participating in the trial were randomly divided into three treatment groups. One group received a placebo, another received 20 milligrams (mg) of the drug per day, and the third group received a daily dose of 40 mg. All doses (including placebo) were identical in packaging and appearance, and neither the patients nor the clinicians knew who was in which treatment group. After a month, patients were clinically assessed and the researchers classified them according to Treatment and Change in Condition. The data are summarised below:
Change in condition
Treatment Improved No Change Worse
Placebo 35 70 95
20mg 62 76 62
20mg 88 80 32
(a) Identify the predictor and response variables in these data, and justify your choice. [2]
(b) Test the hypothesis that the different doses have no effect on patient outcomes. If you conclude that they do have an effect, describe it. Use a 1% significance level.
(a)
Since researcher want to see the effect of different treatments so the predictor variable is Treatment.
The response variable is "Change in condition"
(b)
Here we need to use chi square test of independence.
Hypotheses are:
H0: The different doses have no effect on patient outcomes.
Ha: The different doses have effect on patient outcomes.
Following table shows the row total and column total:
Improved | No change | Worse | Total | |
Placebo | 35 | 70 | 95 | 200 |
20 mg | 62 | 76 | 62 | 200 |
40 mg | 88 | 80 | 32 | 200 |
Total | 185 | 226 | 189 | 600 |
Expected frequencies will be calculated as follows:
Following table shows the expected frequencies:
Improved | No change | Worse | Total | |
Placebo | 61.667 | 75.333 | 63 | 200 |
20 mg | 61.667 | 75.333 | 63 | 200 |
40 mg | 61.667 | 75.333 | 63 | 200 |
Total | 185.001 | 225.999 | 189 | 600 |
Following table shows the calculations for chi square test statistics:
O | E | (O-E)^2/E |
35 | 61.667 | 11.53175749 |
62 | 61.667 | 0.00179819 |
88 | 61.667 | 11.24469958 |
70 | 75.333 | 0.377535595 |
76 | 75.333 | 0.005905632 |
80 | 75.333 | 0.289128124 |
95 | 63 | 16.25396825 |
62 | 63 | 0.015873016 |
32 | 63 | 15.25396825 |
Total | 54.97463413 |
Following is the test statistics:
Degree of freedom: df =( number of rows -1)*(number of columns-1) = (3-1)*(3-1)=4
The p-value is: 0.0000
Conclusion: Since p-value is less than 0.01 so we reject the null hypothesis. That is we cannot conclude that the different doses have no effect on patient outcomes.
Excel function used for p-value: "=CHIDIST(54.97, 4)"
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