The impact of the use of multiple risk indicators for fracture on case-finding strategies: a mathematical approach

Abstract

The value of bone mineral density (BMD) measurements to stratify fracture probability can be enhanced in a case-finding strategy that combines BMD measurement with independent clinical risk indicators. Putative risk indicators include age and gender, BMI or weight, prior fracture, the use of corticosteroids, and possibly others. The aim of the present study was to develop a mathematical framework to quantify the impact of using combinations of risk indicators with BMD in case finding. Fracture probability can be expressed as a risk gradient, i.e. a relative risk (RR) of fracture per standard deviation (SD) change in BMD. With the addition of other continuous or categorical risk indicators a continuous distribution of risk indicators is obtained that approaches a normal distribution. It is then possible to calculate the risk of individuals compared with the average risk in the population, stratified by age and gender. A risk indicator with a gradient of fracture risk of 2 per SD identified 36% of the population as having a higher than average fracture risk. In individuals so selected, the risk was on average 1.7 times that of the general population. Where, through the combination of several risk indicators, the gradient of risk of the test increased to 4 per SD, a smaller proportion (24%) was identified as having a higher than average risk, but the average risk in this group was 3.1 times that of the population, which is a much better performance. At higher thresholds of risk, similar phenomena were found. We conclude that, whereas the change of the proportion of the population detected to be at high risk is small, the performance of a test is improved when the RR per SD is higher, indicated by the higher average risk in those identified to be at risk. Case-finding strategies that combine clinical risk indicators with BMD have increased efficiency, while having a modest impact on the number of individuals requiring treatment. Therefore, the cost-effectiveness is enhanced.

Appendix

In this appendix we detail the derivation of the formulae.

Let X denote a risk variable for fracture or a combination of risk variables, which we assume to have a normal distribution. Without loss of generality we can assume that the mean is zero and the standard deviation is 1. The hazard function of fracture when X=x is given by:

$$ \exp {\left( {\beta _{0} + \beta _{1} x} \right)} \cdot {\text{(average}}\;{\text{fracture}}\;{\text{risk}}\;{\text{in}}\;{\text{the}}\;{\text{population).}} $$

The expected value E[exp(β₀+β₁X)] has to be equal to 1 in order to give an average risk equal to that of the population. We use the following general relationship: if Y has a normal distribution with mean μ and standard deviation σ, then E[exp(Y)]=exp(μ+σ²/2). The variable β₀+β₁X has a normal distribution with mean β₀ and standard deviation β₁.

Therefore, E[exp(β₀+β₁X)]=exp(β₀+β₁²/2) and the condition that the mean is 1 implies that β₀=−β₁²/2.

The definition of a gradient of risk (GR) for the variable X implies that an increase of 1 standard deviation corresponds to a risk ratio of GR. Moreover, the standard deviation of X is 1. Thus, exp(β₀+β₁(x+1))/exp(β₀+β₁x)=GR. This implies that β₁=ln(GR), and we find that β₀=−ln(GR)²/2. If the value x of the risk variable X exceeds a specific risk threshold level (RT), then, exp(β₀+β₁x)>RT, and thus, x>(ln(RT)−β₀)/β₁. The probability of this event is therefore given by

$$P = \Phi {\left[ z \right]}$$

(4)

where Φ is the cumulative distribution function of the standardized normal distribution, and z is given by

$$z{\text{ = }}{\left[ {\frac{{{\text{ - }}{\left[ {{\text{ln}}{\left( {RT} \right)}{\text{ + }}{\left( {{\text{ln}}{\left( {GR} \right)}} \right)}^{{\text{2}}} {\text{/2}}} \right]}}}{{{\text{ln}}{\left( {GR} \right)}}}} \right]}$$

(5)

Let z=−(ln(RT)−β₀)/β₁ The average risk (AR) in the group above the chosen risk threshold relative to the average risk in the general population is then given by

$$\begin{aligned} & AR = \frac{{{\int\limits_{ - z}^\infty {\exp {\left( {\beta _{0} + \beta _{1} x} \right)} \cdot \exp {\left( { - x^{2} /2} \right)}{\sqrt {2\pi } }dx} }}}{{\Phi {\left( z \right)}}} = \\ & \frac{{\exp {\left( {\beta _{0} + \beta ^{2}_{1} /2} \right)} \cdot {\int\limits_{ - z}^\infty {\exp {\left( { - {\left( {x^{2} - 2\beta _{1} x + \beta ^{2}_{1} } \right)}/2} \right)}{\sqrt {2\pi } }dx} }}}{{\Phi {\left( z \right)}}} = \\ & \exp {\left( {\beta _{0} + \beta ^{2}_{1} /2} \right)} \cdot \Phi {\left( {z + \beta _{1} } \right)}/\Phi {\left( z \right)} \\ \end{aligned}$$

(6)

Since exp(β₀+β₁²/2)=1, this becomes

$$AR = \frac{{\Phi {\left( {z + {\mathbf{ln}}{\left( {GR} \right)}} \right)}}}{{\Phi {\left( z \right)}}}$$

(7)