Article type
Year
Abstract
Background: primary studies and systematic reviews of prognosis commonly report relative measures of association between prognostic factors and outcomes of interest. For decision making, however, evidence users need the absolute risk of the outcomes in those with and without the prognostic factors of interest.
Objectives: to develop a framework for calculating the absolute risk of the outcomes in those with and without the prognostic factors of interest.
Methods: we developed a mathematical approach to calculate the absolute risk of events from the relative measure of association, the total number of events and patients at risk, and the prevalence of the prognostic factor, all of which are usually reported in cohort studies assessing prognostic factors. We present our framework using the simplest case, in which the measure of association is a relative risk, and provide extensions of the formula to odds ratios and hazard ratios.
Results: for relative measures of association reported as relative risk, one could apply the following formula:
(overall risk) / π1 = (p1× RR) + p2) / RR
π1 = risk in patients with prognostic factor
π2 = risk in patients without prognostic factor
p1 = prevalence of patients with prognostic factor
p2 = prevalence of patients without prognostic factor
RR = relative risk = π1/π2
The equality statement above works because the risk of an event in the entire cohort (i.e. those with and without the prognostic factor of interest) can be re-written as π1 × p1 + π2 × p2. Subsequently, division of the numerator and denominator by the risk in those without prognostic factors results in the right side of the equality statement presented above.
(overall risk) / π1 = (π1 × p1 + π2 × p2) / π1
= (π1/π2 × p1 + π2/ π2 × p2) / (π1 / π2)
= (RR × p1 + p2) / RR
Expansion of the formula for instances using odds ratios and hazards ratio requires the utilization of odds or hazard instead of risk. The derivation of odds and hazard requires the transformation of probability. To transform probability to odds, we divide the probability of having the outcome by the probability of not having the outcome:
odds = p / (1-p)
To transform probability to hazard, we need to obtain the probability of not having the outcome (1 – the probability of having the outcome; often referred to as survival probability). By taking the natural logarithm of the probability of not having the outcome, dividing by −1 and the time, we obtain the hazard:
hazard (h) = ln(survival) / (−1 × time).
Conclusions: our proposed formulas facilitate accurate calculation of measures of absolute risk in those with and without prognostic factors of interest for studies reporting the total number of events and patients at risk, the prevalence of the prognostic factor and relative risk, odds ratio or hazard ratio.
Objectives: to develop a framework for calculating the absolute risk of the outcomes in those with and without the prognostic factors of interest.
Methods: we developed a mathematical approach to calculate the absolute risk of events from the relative measure of association, the total number of events and patients at risk, and the prevalence of the prognostic factor, all of which are usually reported in cohort studies assessing prognostic factors. We present our framework using the simplest case, in which the measure of association is a relative risk, and provide extensions of the formula to odds ratios and hazard ratios.
Results: for relative measures of association reported as relative risk, one could apply the following formula:
(overall risk) / π1 = (p1× RR) + p2) / RR
π1 = risk in patients with prognostic factor
π2 = risk in patients without prognostic factor
p1 = prevalence of patients with prognostic factor
p2 = prevalence of patients without prognostic factor
RR = relative risk = π1/π2
The equality statement above works because the risk of an event in the entire cohort (i.e. those with and without the prognostic factor of interest) can be re-written as π1 × p1 + π2 × p2. Subsequently, division of the numerator and denominator by the risk in those without prognostic factors results in the right side of the equality statement presented above.
(overall risk) / π1 = (π1 × p1 + π2 × p2) / π1
= (π1/π2 × p1 + π2/ π2 × p2) / (π1 / π2)
= (RR × p1 + p2) / RR
Expansion of the formula for instances using odds ratios and hazards ratio requires the utilization of odds or hazard instead of risk. The derivation of odds and hazard requires the transformation of probability. To transform probability to odds, we divide the probability of having the outcome by the probability of not having the outcome:
odds = p / (1-p)
To transform probability to hazard, we need to obtain the probability of not having the outcome (1 – the probability of having the outcome; often referred to as survival probability). By taking the natural logarithm of the probability of not having the outcome, dividing by −1 and the time, we obtain the hazard:
hazard (h) = ln(survival) / (−1 × time).
Conclusions: our proposed formulas facilitate accurate calculation of measures of absolute risk in those with and without prognostic factors of interest for studies reporting the total number of events and patients at risk, the prevalence of the prognostic factor and relative risk, odds ratio or hazard ratio.