A small pilot study is run to compare a new drug for chronic pain to one that is currently available. Participants are randomly assigned to receive either the new drug or the currently available drug and report improvement in pain on a 5-point ordinal scale: 1 = Pain is much worse, 2 = Pain is slightly worse, 3 = No change, 4 = Pain improved slightly, 5 = Pain much improved. Is there a significant difference in self-reported improvement in pain? Use the Mann-Whitney U test with a 5% level of significance.

New Drug | 4 | 5 | 3 | 3 | 4 | 2 |
---|---|---|---|---|---|---|

Standard Drug | 2 | 3 | 4 | 1 | 2 | 3 |

### Solution

Let $X$ denote improvement in chronic pain due to new drug and $Y$ denote the improvement in chronic pain due to standard drug.

Hypothesis testing problem is

$H_0:$ There is no significant difference in self-reported improvement in pain.

$H_1:$ There is a significant difference in self-reported improvement in pain

First Combine the two groups and arrange them in order of magnitude. Assign ranks to each.

X Y rx ry

4 2 10.0 3.0

5 3 12.0 6.5

3 4 6.5 10.0

3 1 6.5 1.0

4 2 10.0 3.0

2 3 3.0 6.5

The Ranks of New Drugs and standard drugs after combining the two samples are 10, 12, 6.5, 6.5, 10, 3 and 3, 6.5, 10, 1, 3, 6.5

Sum of ranks of new drugs $R_1 = 48$, sum of ranks of standard drug $R_2 = 30$.

Define

`$$ \begin{aligned} U_1 & = n_1n_2 + \frac{n_1(n_1+1)}{2}-R_1\\ &= 6* 6 + \frac{6*(6 +1)}{2}- 48\\ & = 9\\ U_2 & = n_1n_2 + \frac{n_2(n_2+1)}{2}-R_2\\ &= 6* 6 + \frac{6*(6 +1)}{2}- 30\\ & = 27\\ \end{aligned} $$`

The test statistics is $U = min$ { $U_1, U_2$} $= 9$.

The critical value of $U$ at 0.05 level of significance is $U_{\alpha, n1,n2} = 5$.

As the observed value of $U= 9$ is not less than $U_{crit}=5$, we do not reject the null hypothesis at $0.05$ level of significance, i.e. we fail to reject the null hypothesis.

We conclude that there is a significant difference in self-reported improvement in pain.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators