Question

Critical Thinking: Is the pain medicine Duragesic effective in reducing pain? Listed below are measures of pain intensity before and after using the drug Duragesic (fentanyl) (based on data from Janssen Pharmaceutical Products, L.P.). The data are listed in order by row, and corresponding measures are from the same subject before and after treatment. For example, the first subject had a measure of 1.2 before treatment and a measure of 0.4 after treatment. Each pair of measurements is from one subject, and the intensity of pain was measured using the standard visual analog score. A higher score corresponds to higher pain intensity.

Pain Intensity Before Duragesic Treatment

1.2	1.3	1.5	1.6	8	3.4	3.5	2.8	2.6	2.2
3	7.1	2.3	2.1	3.4	6.4	5	4.2	2.8	3.9
5.2	6.9	6.9	5	5.5	6	5.5	8.6	9.4	10
7.6

Pain Intensity After Duragesic Treatment

0.4	1.4	1.8	2.9	6	1.4	0.7	3.9	0.9	1.8
0.9	9.3	8	6.8	2.3	0.4	0.7	1.2	4.5	2
1.6	2	2	6.8	6.6	4.1	4.6	2.9	5.4	4.8
4.1

Correlation Use the given data to construct a scatterplot, then use the methods of Section 10-1 to test for a linear correlation between the pain intensity before and after treatment. If there does appear to be a linear correlation, can we conclude that the drug treatment is effective?

Short answer

The scatterplot between the two pain intensity value is constructed below:

The value of the linear correlation is equal to 0.304.

The linear correlation between the pain intensities before and after the treatment is not significant.

No, it cannot be concluded that the drug is effective in reducing pain even if there was a significant linear correlation between the two pain intensities.

Step-by-step solution

Given information

The pain intensities of a group of subjects are recorded before using the drug Duragesic and after using the drug Duragesic.

Construction of scatterplot

Follow the given steps to construct a scatterplot between the pain intensities before and after the treatment:

• Mark the values 0, 2, 4, ……., 12 on the horizontal scale and label the axis as “Pain Intensity Before Treatment”.
• Mark the values 0, 1, 2, ……., 10 on the vertical scale and label the axis as “Pain Intensity After Treatment”.
• Plot points on the graph for each value of the pain intensity after treatment for the corresponding value on the pain intensity before treatment.

The following scatterplot is constructed:

The points on the plot appear to be scattered randomly, indicating that there is not a significant linear correlation between the pain intensities before and after the treatment.

Correlation coefficient

Let x denote the values of the pain intensity before treatment.

Let y denote the values of the pain intensity after treatment.

The formula for computing the correlation coefficient (r) between the values of x and y is as follows:

\(r = \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\)

The following calculations are done to compute the value of r:

x	y	xy	\({x^2}\)	\({y^2}\)
1.2	0.4	0.48	1.44	0.16
1.3	1.4	1.82	1.69	1.96
1.5	1.8	2.7	2.25	3.24
1.6	2.9	4.64	2.56	8.41
8	6	48	64	36
3.4	1.4	4.76	11.56	1.96
3.5	0.7	2.45	12.25	0.49
2.8	3.9	10.92	7.84	15.21
2.6	0.9	2.34	6.76	0.81
2.2	1.8	3.96	4.84	3.24
3	0.9	2.7	9	0.81
7.1	9.3	66.03	50.41	86.49
2.3	8	18.4	5.29	64
2.1	6.8	14.28	4.41	46.24
3.4	2.3	7.82	11.56	5.29
6.4	0.4	2.56	40.96	0.16
5	0.7	3.5	25	0.49
4.2	1.2	5.04	17.64	1.44
2.8	4.5	12.6	7.84	20.25
3.9	2	7.8	15.21	4
5.2	1.6	8.32	27.04	2.56
6.9	2	13.8	47.61	4
6.9	2	13.8	47.61	4
5	6.8	34	25	46.24
5.5	6.6	36.3	30.25	43.56
6	4.1	24.6	36	16.81
5.5	4.6	25.3	30.25	21.16
8.6	2.9	24.94	73.96	8.41
9.4	5.4	50.76	88.36	29.16
10	4.8	48	100	23.04
7.6	4.1	31.16	57.76	16.81
\(\sum x \)=144.9	\(\sum y \)=102.2	\(\sum {xy} \)=533.78	\(\sum {{x^2}} \)=866.35	\(\sum {{y^2}} \)=516.4

Substituting the above values, the following value of r is obtained:

\(\begin{aligned} r &= \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\\ &= \frac{{31\left( {533.78} \right) - \left( {144.9} \right)\left( {102.2} \right)}}{{\sqrt {31\left( {866.35} \right) - {{\left( {144.9} \right)}^2}} \sqrt {31\left( {516.4} \right) - {{\left( {102.2} \right)}^2}} }}\\ &= 0.304\end{aligned}\)

Therefore, the value of r is equal to 0.304.

Significance of r

Here, n=31.

If the value of the correlation coefficient lies between the critical values, then the correlation between the two variables is considered insignificant else, it is considered significant.

The critical values of r for n=31 and\(\alpha = 0.05\)are -0.355 and 0.355.

The corresponding p-value of r is equal to 0.0964.

Since the computed value of r equal to 0.304 lies between the values -0.355 and 0.355, and the p-value is greater than 0.05, it can be said that the correlation between the two variables is not significant.

Therefore, the linear correlation between the pain intensities before and after the treatment is not significant.

Effectiveness of the drug

Even if there is a significant linear correlation between the pain intensities before and after the treatment, it could not be concluded that the drug is effective in reducing pain. The linear correlation only tells us about the association between the two variables and not about the difference in the values of the two variables and whether the overall value of one variable is lower than the other variable.

A separate kind of hypothesis test needs to be conducted to conclude the effectiveness of the drug.