2.4 Analysis of cohort studies: binary exposure

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Let $p_1$ and $p_0$ denote the respective disease risks (event probabilities) for the exposed and non-exposed groups.

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  The risk difference, $p_1-p_0$, is estimated:
  

  $$RD=\frac{a}{a+b}-\frac{c}{c+d}$$

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  The relative risk, $p_1/p_0$, is estimated:

  $$RR=\frac{a/(a+b)}{c/(c+d)}.$$

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  The disease odds, $p/(1-p)$, provides the ratio of success (here death) to failure (here survival). The odds ratio comparing the exposed and non-exposed groups is estimated:

  $$OR=\frac{p_1}{1-p_1}÷\frac{p_0}{1-p_0}=\frac{ad}{bc}$$

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  Intervals are based upon the Normal approximation $(1-\alpha )\%$ CI:

  $\widehat{\theta }\pm z_{1-\alpha /2}\text{SE}(\widehat{\theta })$

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  risk difference:

  $$\text{SE}(RD)=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_0(1-p_0)}{n_0}}$$

  with $p_1={\displaystyle \frac{a}{a+b}}$, $p_0={\displaystyle \frac{c}{c+d}}$, $n_1=a+b$ and $n_0=c+d.$

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  relative risk (natural log scale):

  $$\text{SE}({\log }_e(RR))=\sqrt{\frac{1}{a}-\frac{1}{a+b}+\frac{1}{c}-\frac{1}{c+d}}$$

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  odds ratio (natural log scale):

  $$\text{SE}({\log }_e(OR))=\sqrt{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}$$

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  Study of all cause mortality at 7-years in smoking and non-smoking doctors

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  Study aim: to compare the ‘all cause’ mortality risk amongst smokers and non-smokers

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  Cohort study $\to$ prospective: we can investigate association using the either the risk difference, the relative risk or the odds ratio

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  Example: relative risk (smokers compared to non-smokers)

  $$RR=\frac{133/(133+25636)}{3/(3+5436)}=9.357$$

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  RR on the natural log scale:$\text{log}(9.357)=2.236$

In addition to estimating the relative risk we should construct a confidence interval for the true relative risk

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  first compute the standard error of the natural logarithm of the relative risk:

  $$\text{SE}({\log }_e(RR))=\sqrt{\frac{1}{133}-\frac{1}{25769}+\frac{1}{3}-\frac{1}{5439}}=0.584$$

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  then compute a $95\%$ confidence interval for the logarithm of the relative risk:

  $$2.236\pm 1.96\cdot 0.584\to [1.091;3.380]$$

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  then back transform (exponentiate) to the relative risk scale:

  $$[\text{exp}(1.091),\text{exp}(3.380)]=[2.977,29.400]$$

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  Conclusion: on average the ‘all cause’ mortality risk amongst smokers was found to be 9.36 times that of non-smokers. We are 95% confident that the true relative risk lies between 2.98 and 29.40 and since the said interval does not contain one we have significant evidence (at the 5% level) of increased mortality risk amongst those who smoke compared to those who do not smoke.

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  Note that the interval is very wide and thus we have large uncertainty in the true magnitude of the relative risk