Question

In 2004, the FDA released a report suggesting that a particular class of antidepressants (selective serotonin reuptake inhibitors or SSRIS) may increase the risk of "suicidality" in adolescents. The concern led to a black box warning on SSRIS and other antidepressants for users younger than 25. The issue spurred a debate in the psychiatric research community concerning the core issue of whether suicidality, a measure of suicidal tendencies, accurately represents how likely an adolescent is to commit suicide. The reason suicidality was at the core of the debate was that the number of suicides among teenagers is low. So, to be able to detect a difference in the number of suicides among those taking SSRIS and those taking a placebo, for example, requires a huge sample size. Thus, using the number of actual suicides was not possible, and researchers had to look at a measure of the possibility of suicide instead, or "suicidality." Since the black box warning was added, the number of prescriptions for SSRIS for adolescentss has decreased, while the number of suicides among adolescents has actually increased, suggesting that the antidepressants were actually preventing suicides and the original concern was unfounded. (Source: Karen L. Spittler, "SSRIS and Suicide-What Effect Has the Black Box Warning Had?" NeuroPsychiatry Review 9, no. 9 [2008] http://www.neuropsychiatryreviews.com/08sept/sSRISuicide.html) Suppose the rate of suicide among all people aged 15-24 in 2002 is one in 10,000. Then suppose a researcher wants to use the binomial test to test whether the suicide rate among a sample of people aged 15-24 who take SSRIS is different than the rate among all people aged 15-24 in 2002. The null hypothesis would be: Ho: For those aged 15-24 who take SSRIs, p = p(suicide) 0.0001 Ho: For those aged 15-24 who take SSRIS, p = p(suicide) = 0.01 Ho: For those aged 15-24 who take SSRIS, p p(suicide) 0.5 Ho: For those aged 15-24, p = p(suicide) = 0.0001 To use the normal approximation to the binomial test,, the researcher would need a sample size of O O O O

Answer 1

Hypothesized proportion, p = 1 /10000 = 0.0001
H0: For those aged 15-24 who take SSRIs, p = p(suicide) = 0.0001

For normal approximation, np > 10
n * 0.0001 > 10
n > 10 / 0.0001 = 100000
Sample size needed = 100000

Hypothesized proportion, p = 1 /10000 = 0.0001
H0: For those aged 15-24 in 2006, p = p(suicide) = 0.0001

np = 110000 * 0.0001 = 11
n(1-p) = 110000 * (1 - 0.0001) = 109989
Since, np and n(1-p) are both greater than 10,
The normal approximation to the binomial distribution is appropriate to use in this case.

Sample proportion, $\hat{p}$ = 18 / 110000 = 0.0001636364

Standard error of proportion, SE = = 0.00003014963

P(1 - p/n = 0.0001 + (1 - 0,0001)/110000

The z-score = ($\hat{p}$ - p) / SE = (0.0001636364 - 0.0001) / 0.00003014963 = 2.110686

With $\alpha$ = 0.05, the critical region is z-scores greater than 1.96 or less than -1.96. The z-score for the sample data is inside the critical region (as Z = 2.11 > 1.96)

Since z-score for the sample data is inside the critical region,

Reject the null hypothesis.