Table 4.

Formulae to estimate a sample size for each of the common observational epidemiological study designs.

Study design	Formula	Arguments
Cross-sectional	$n\ge \frac{r+1}{r{(\lambda -1)}^2{\pi }^2}{[z_{1-(\alpha /2)}\sqrt{(r+1)p_c(1-p_c)}+z_{1-\beta }\sqrt{\lambda \pi (1-\lambda \pi )+r\pi (1-\pi )}]}^2$	n = the number of subjects in the sample. r = the anticipated number of subjects in the exposed group divided by the anticipated number of subjects in the unexposed group. λ = the expected prevalence ratio. π = the expected prevalence of the outcome among the non-exposed. z1−(α/2) = value from the standard normal curve corresponding to the desired level of confidence. Use z1−(α/2) = 1.96 for 95% (two-sided) confidence. pc = the common prevalence over exposed and unexposed groups.z1−β = value from the standard normal curve corresponding to the desired study power. Use z1−β = −0.84 for 80% power.
Case-control	$p_c*=\frac{p_0}{r+1}\left(\frac{r\lambda }{1+(\lambda -1)p_0}+1\right)$ $n\ge \frac{(r+1){(1+(\lambda -1)p_0)}^2}{r{p}_0^2{(p_0-1)}^2{(\lambda -1)}^2}{\left[z_{1-(\alpha /2)}\sqrt{(r+1)p_c*(1-p_c*)}+z_{1-\beta }\sqrt{\frac{\lambda p_0(1-p_0)}{{\left[1+(\lambda -1)p_0\right]}^2}+r{p}_0(1-p_0)}\right]}^2$	n = the number of subjects in the sample.p0 = the expected prevalence of exposure among the controls. r = anticipated number of subjects in the control group divided by the anticipated number of subjects in the case group. λ = the expected odds ratio. z1−(α/2) = value from the standard normal curve corresponding to the desired level of confidence. Use z1−(α/2) = 1.96 for 95% (two-sided) confidence. z1−β = value from the standard normal curve corresponding to the desired study power. Use z1−β = 0.84 for 80% power.
Cohort, count data	$n\ge \frac{r+1}{r{(\lambda -1)}^2{\pi }^2}{[z_{1-(\alpha /2)}\sqrt{(r+1)p_c(1-p_c)}+z_{1-\beta }\sqrt{\lambda \pi (1-\lambda \pi )+r\pi (1-\pi )}]}^2$	n = the number of subjects in the sample. r = the anticipated number of subjects in the exposed group divided by the anticipated number of subjects in the unexposed group. λ = the expected incidence risk ratio. π = the expected prevalence of the outcome among the non-exposed. z1−(α/2) = value from the standard normal curve corresponding to the desired level of confidence. Use z1−(α/2) = 1.96 for 95% (two-sided) confidence. pc = the common prevalence over exposed and unexposed groups.z1−β = value from the standard normal curve corresponding to the desired study power. Use z1−β = 0.84 for 80% power.
Cohort, time at risk	${\lambda }_0^{{}^{\prime }}=\frac{{\lambda }_0^{3}FT}{{\lambda }_{0}FT-1+{exp}^{(-{\lambda }_{0}FT)}}$ ${\lambda }_1^{{}^{\prime }}=\frac{{\lambda }_{10}^{3}FT}{{\lambda }_{1}FT-1+{exp}^{(-{\lambda }_{1}FT)}}$ ${\overline{\lambda }}^{{}^{\prime }}=\frac{{\overline{\lambda }}^{3}FT}{\overline{\lambda }FT-1+{exp}^{(-\overline{\lambda }FT)}}$ nA = rnB $n_A\ge \frac{{\left(z_{1-(\alpha /2)}\sqrt{(1+r){\overline{\lambda }}^{{}^{\prime }}}+z_{1-\beta }\sqrt{(r\times {\lambda }_1^{{}^{\prime }}+{\lambda }_0^{{}^{\prime }})}\right)}^2}{{r({\lambda }_1^{{}^{\prime }}-{\lambda }_0^{{}^{\prime }})}^2}$	nA = the number of subjects in the sample. λ0 = the expected incidence rate among the unexposed.λ1 = the expected incidence rate among the exposed.$\overline{\lambda }=({\lambda }_0+{\lambda }_1)/2$ FT = the expected follow-up period for the study. r = anticipated number of subjects in the exposed group divided by the anticipated number of subjects in the unexposed group. z1−(α/2) = value from the standard normal curve corresponding to the desired level of confidence. Use z1−(α/2) = 1.96 for 95% (two-sided) confidence. z1−β = value from the standard normal curve corresponding to the desired study power. Use z1−β = 0.84 for 80% power.