Question

It is well known that a placebo, a fake medication or treatment, can sometimes have a positive effect just because patients often expect the medication or treatment to be helpful. The article "Beware the Nocebo Effect" (New York Times, Aug. 12, 2012) gave examples of a less familiar phenomenon, the tendency for patients informed of possible side effects to actually experience those side effects. The article cited a study reported in The Journal of Sexual Medicine in which a group of patients diagnosed with benign prostatic hyperplasia was randomly divided into two subgroups. One subgroup of size 55 received a compound of proven efficacy along with counseling that a potential side effect of the treatment was erectile dysfunction. The other subgroup of size 52 was given the same treatment without counseling. The percentage of the no-counseling subgroup that reported one or more sexual side effects was 15.3 %, whereas 43.6 % of the counseling subgroup reported at least one sexual side effect. State and test the appropriate hypotheses at significance level .05 to decide whether the nocebo effect is operating here. (Note: The estimated expected number of "successes" in the no-counseling sample is a bit shy of 10, but not by enough to be of great concern (some sources use a less conservative cutoff of 5 rather than 10).)

Short answer

Reject null hypothesis and can conclude that there does appear to be Nocebo effect.

Step-by-step solution

To determine the large- sample test procedure

There are two subgroups. Denote with \({p_1}\)the true proportion of patients which did not receive counseling, and with \({p_2}\)the true proportion of the patients which received counseling about the possible side effect.

For the no-counseling subgroup (first subgroup) 15.3 % of 52 who received no counseling, and for the second subgroup, the subgroup of patients who received counseling 43.6% of 55 reported one or more sexual side effects.

The 15.3 % of 52 is

\(m = 52 \times \frac{{15.3}}{{100}} = 8\)

and 43.6 % of 55 is

\(n = 55 \times \frac{{43.6}}{{100}} = 24\)

This means that 8 out of 52 patients who received no counseling reported sexual side effect, and 25 out of 55 patients who received counseling reported sexual side effect.

The hypotheses of interest are \({H_0}:{p_1} - {p_2} = 0\) versus\({H_a}:{p_1} - {p_2} < 0\). Even though the samples are not large enough (and the proportions), you can use the large sample test procedure (see the hint).

A Large-Sample Test Procedure:

The two proportion z statistic - Consider test in which\({H_0}:{p_1} - {p_2} = 0\). The test statistic value for the large samples is

\(z = \frac{{{{\hat p}_1} - {{\hat p}_2}}}{{\sqrt {\hat p\hat q \times \left( {\frac{1}{m} + \frac{1}{n}} \right)} }}\)

where

\(\hat p = \frac{m}{{m + n}} \times {\hat p_1} + \frac{n}{{m + n}} \times {\hat p_2}\)

Depending on alternative hypothesis, the P value can be determined as the corresponding area under the standard normal curve. The test should be used when\(m{\hat p_1},m{\hat q_1},n{\hat p_1}\), and \(n{\hat q_1}\) are at least 10.

The estimates of the proportions are

\(\begin{array}{l}{{\hat p}_1} = \frac{8}{{52}} = 0.153\\{{\hat p}_2} = \frac{{24}}{{55}} = 0.436\end{array}\)

Compute a pooled proportion \(\widehat p\)

\(\begin{array}{c}\hat p = \frac{m}{{m + n}} \times {{\hat p}_1} + \frac{n}{{m + n}} \times {{\hat p}_2}\\ = \frac{{52}}{{52 + 55}} \times \frac{8}{{52}} + \frac{{55}}{{52 + 55}} \times \frac{{24}}{{55}}\\ = 0.299.\end{array}\)

Step 2:To find the two proportion z test value

The two proportion z test value is

\(\begin{array}{c}z = \frac{{{{\hat p}_1} - {{\hat p}_2}}}{{\sqrt {\hat p\hat q \times \frac{1}{m} + \frac{1}{n}} }}\\ = \frac{{0.153 - 0.436 - 0}}{{\sqrt {(0.299 \times 0.701) \times \frac{1}{{52}} + \frac{1}{{55}}} }}\\ = - 3.2\end{array}\)

The P value is the area under the standard normal curve to the left of z because the test is lower sided (one sided). Thus

\(P = P(Z \le - 3.2) = 0.007\)

which was obtained using the table in the appendix of the book. Since

\(P = 0.007 < 0.05 = \alpha \)

Reject null hypothesis

At given significance level. You can conclude that a higher proportion of men will experience erectile dysfunction if they received consulting about the side effect of the BPH treatment, than if there were no consulting about the side effects.

Or there does appear to be nocebo effect.