Question

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.

Answer 1

(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:

$E_{ij}=\frac{ith\,row\,total\cdot jth\,row\,total}{grand\,total}$

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:

$\chi^{2}=\sum \left ( \frac{(O-E)^{2}}{E} \right )=54.97$

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)"