Converting between exposure-response parameters

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Converting between exposure-response parameters helps in using different kinds of input data for exposure-response modelling. Especially, sometimes relative risks (RR) are given, while sometimes odds ratios (OR) are used. These can be transformed from one to another if enough background data is available.

Question

How to transform different exposure-response parameters from one to another and use them in a coherent way? Especially, we are interested in relative risk, odds ratio, and unit risk.

Answer

Rationale

The definition of relative risk (or risk ratio, RR) is 

RR = PX1/PX0,

where P is the incidence of disease (or alternatively the probability of the disease during a fixed time period), X1 is the exposure in the group of interest, and X0 is the exposure in the control group (or background group).

Using the same terms, odds ratio (OR) is 

OR = ( PX1/(1-PX1) ) / ( (PX0/(1-PX0) ).

We can solve the PX1 (incidence in the exposed group) from both equations: 

PX1 = RR*PX0 and

PX1 = OR*PX0/( 1-PX0+OR*PX0 ) and therefore

RR = OR/( 1-PX0+OR*PX0 ).

Thus, we can see that OR can be translated to RR (and vice versa) only if the background incidence is known. If PX0 is small, OR is close to RR, but the difference may be substantial if PX0 and OR are large. For example models, see Excel-model and Analytica-model

We can also look at these parameters using the regression coefficient ß obtained from an epidemiological study. The interpretation and usage of ß depends on the statistical model used.

If a linear model is used, then it is assumed that 

ln(PX) = α + ß*X

for each exposure level X. Then, 

RR = exp(ß*(X1-X0)).

If a logistic model is used (which is typical in case-control studies where the background parameter α cannot be estimated), then it is assumed that 

ln(PX/( 1-PX) ) = α + ß*X.

Then, 

OR = exp(ß*(X1-X0)).

It must be noted that the parameters ß are NOT the same in linear and logistic approaches. The linear and logistic betas can be converted by the following formula 

ßRR = ßOR - ln(PX0exp(ßOR*(X1-X0))-PX0+1) / (X1-X0)

Further calculations

Let us define concepts. In equations, capital letters are used for random variables (things that are typically not known), small letters are used for constants (things that are typically known or have been measured). 

P(x|y) = probability of x happening, given that y happens
P(d|e+) = A = probability of disease d given exposure e+
P(d|e-) = B = probability of disease d given no exposure e-
P(d) = c = probability of disease in the whole population
P(e+) = 1 - P(e-) = k = probability of exposure e+ 
pop = population size
OR = o = odds ratio
RR = P(d|e+)/P(d|e-) = A/B = risk ratio

When operating with odds ratio, there are two important equations: the definition of odds ratio, and the probability of disease (with or without exposure).

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle OR = o = \frac{P(d|e+)/(1-P(d|e+))} {P(d|e-)/(1-P(d|e-))} = \frac{A/(1-A)} {B/(1-B)} = \frac{A-AB}{B-AB} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(d) = c = kA + B(1-k)}

Probability of disease that is due to exposure = probability of disease altogether - probability of seeing a case not caused by exposure. Based on the previous equation, this can be expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c - B = (A - B)k}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c - B = \frac{kAB}{B} - kB, B <> 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c - B = RR*kB - kB}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c - B = (RR - 1)kB.}

The number of extra cases of disease can be calculated from

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle extra cases = (c - B) * pop = (RR - 1)kB * pop.}

However, B is typically not known. In addition, sometimes OR - not RR - is known. These problems can be overcome by solving the first pair of equations. So, let's start by solving B from the first equation and using that in the second equation.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle oB - oAB = A - AB}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \frac{A}{o - oA + A}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = kA + \frac{A(1-k)}{o - oA + A}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c - kA = \frac{A(1-k)}{o - oA + A}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle co - coA + cA - okA + okA^2 - kA^2 = A - kA }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ok - k)A^2 + (-co + c -ok -1 + k)A + co = 0} .

We can solve A from here using the formula for second power equations.

If we know A and B, we know everything there is to know about this. However, if we know RR but not A, we can directly calculate from the second equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = \frac{ B(kA + B(1-k))}{B}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = B(k*RR + 1 - k)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \frac{c}{k * RR + 1 - k}.}

For original equations, see [1]. (Note that the original equations made a deliberate bias and for practical reasons used OR, although RR should have been used.)

It is useful to convert from RR to OR and back.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RR = \frac{o}{1 - B + oB}}

and on the other hand

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RR - RR*B + RR*oB = o}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o = OR = \frac{RR - RR*B}{1 - RR*B} = \frac{RR - A}{1 - A}.}

Proof:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o = \frac{A - AB}{B - AB}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle oB - oAB = A - AB}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \frac{oB}{1 - B + oB}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RR = \frac{A}{B} = \frac{oB}{(1 - B + oB )B} = \frac{o}{(1 - B + oB)}}

  • Prevalences for nondamp and damp homes are assumed constant, for a given iteration in a given country.

These are the equations you should use:

RR for exposure = EXP(LN(RR)*(Exposure Result - MAX(Exposure Background, Exposure-response function threshold)))

Attributable fraction in the whole population = Exposed fraction * (RR for exposure – 1) / (Exposed fraction *(RR for exposure – 1)+1)

Extra cases per year =Disease incidence * Population * attributable fraction

Burden of disease of exposure = Burden of disease of the disase * attributable fraction

Personal lifetime risk = Extra cases per year * life expectancy * population

See also

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