One sided and two sided hypothesis testsBMJ 2010; 340 doi: https://doi.org/10.1136/bmj.c2458 (Published 12 May 2010) Cite this as: BMJ 2010;340:c2458
- Philip Sedgwick, senior lecturer in medical statistics
Researchers investigated whether vaccination against human papillomavirus (HPV) increased the risk of miscarriage in young women.1 The results of two double blind, multicentre, phase three randomised controlled trials were pooled. Participants had been randomly assigned to receive three doses of bivalent HPV 16/18 VLP vaccine with AS04 adjuvant or hepatitis A vaccine (as control) over six months. The primary outcome was rate of miscarriage after vaccination. Given that a lower miscarriage rate with the HPV vaccine was not a safety concern, a one sided test of statistical significance was performed (P=0.16).
Which of the following statements, if any, are true?
a) Null hypothesis: In the total population, the rate of miscarriage for HPV vaccine is equal to that for control
b) Null hypothesis: In the total population, the rate of miscarriage for HPV vaccine is equal to, or less than, that for control
c) Alternative hypothesis: In the total population, the rate of miscarriage for HPV vaccine is less, or greater than, that for control
d) Alternative hypothesis: In the total population, the rate of miscarriage for HPV vaccine is greater than that for control
Statements b and d are true; a and c are false.
Classic hypothesis testing starts at the position of equipoise. The null hypothesis would specify that there is no difference between HPV vaccine and control vaccine in rates of miscarriage in the total population. The aim would be to establish if the sample data support the null hypothesis or lend support to a two sided alternative. The alternative hypothesis would state that there is a difference in miscarriage rates in the entire population: that the rates for the HPV vaccine could be less, or greater, than those for the control vaccine. Statistical hypothesis testing has been described in a previous question.2
The researchers wished to establish whether there was evidence that the HPV vaccine increased the rate of miscarriage, and hence whether there was a safety concern associated with the vaccine. Given that a lower miscarriage rate with the HPV vaccine was not a safety concern, the vaccine would be considered safe if the miscarriage rate was equal to, or less than, that of the control vaccine. Therefore, the alternative hypothesis specifies that the miscarriage rate for HPV vaccine is greater than that for control vaccine in the total population (d is true; c is false). The alternative hypothesis in cases like this is termed “one sided.”
The above one sided alternative hypothesis does not account for the possibility that HPV vaccine has a lower miscarriage rate than does control. This possibility is still part of the test, and is, therefore, incorporated into the null hypothesis. The null hypothesis specifies that the rate of miscarriage for HPV vaccine is equal to, or less than, that for control in the population where the samples were obtained (b is true; a is false).
One sided tests must be specified and justified in advance of statistical analysis. Generally, one sided tests are only appropriate when a difference between treatment groups in one direction would lead to the same action as no difference at all. If the HPV vaccine had a miscarriage rate that was equal to, or less than, that of control, it would be considered safe and, therefore, recommended. The critical level of significance is usually maintained at 5% for a one sided test, although some researchers advocate 2.5%.
In the above example, there was no evidence to reject the null hypothesis in favour of the alternative hypothesis (P=0.16). We conclude that there is no evidence of a safety concern associated with the HPV vaccine, and the vaccine has a miscarriage rate that is equivalent to or less than that of the control vaccine.
Cite this as: BMJ 2010;340:c2458
Competing interests: None declared.