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One of the most important areas of medical inquiry is the identification of risk factors for specific disorders. Such research is usually aimed at discovering new causes of a disease, but risk factors can also be used as screening tests. The fact that a risk factor must be very strongly associated with a disorder if it is to be a worthwhile screening test is not widely recognised. If this were better understood, fewer risk factors would be proposed unnecessarily as screening tests. Serum cholesterol measurement, for example, would probably never have been considered seriously as a screening test for ischaemic heart disease. Although a high cholesterol concentration is a strong risk factor for ischaemic heart disease in aetiological terms, the association is not sufficiently strong for it to be used as a screening test—in practice, its screening performance is poor.1
In this article we specify the quantitative relation between risk factors and screening tests and show how strongly a risk factor needs to be associated with a disease before it is likely to be a useful screening test. For simplicity, we consider only risk factors with a Gaussian distribution, though the general principles we present can be applied to all frequency distributions.
The strength of an association between a risk factor and a disorder can be quantified by a dose-response relation between the incidence of the disorder and increasing values of the risk factor. When risk (or incidence) is expressed on a log scale, a straight line relation with increasing levels of the risk factor is commonly seen. This simply means that there is a proportional, rather than absolute, change in the risk of the disease for a given change in the level of the risk factor. This is illustrated in figure figure1.1. The slope of the line summarises the dose-response relation. This relation can be expressed quantitatively as the risk of people in the highest fifth of the distribution (for unaffected individuals) of the risk factor developing the disorder compared with the risk of people in the lowest fifth of the distribution; 50/1000÷10/1000 in figure figure1.1. It can also be expressed as the relative odds (or odds ratio) for people in the highest and lowest fifth of the distribution, so in figure figure11 it would be (50:950)÷(10:990). Here this relation is described as the ROQ1-5. If the disorder is rare, the relative risk and relative odds are almost the same.
The relation between a risk factor and a disorder can also be described as two overlapping relative frequency distributions of the risk factor in people with and without the disorder. This approach, in which the risk factor is considered as a screening test, is illustrated in figure figure2.2. The same approach applies to individuals who are at risk of developing a disorder in the future—for example, people with high blood cholesterol who are at risk of developing ischaemic heart disease. In this case affected individuals are defined as those who develop the disorder over a given period of time, and unaffected individuals as those who remain free of the disorder over the same period. People with positive results have screening test values that are above a specified cut off level, and the approach yields the detection rate for a specified false positive rate. The detection rate (sensitivity) is defined as the proportion of affected individuals (or those who become affected during a given period of time) with positive results and the false positive rate (1 minus specificity) is defined similarly as the proportion of unaffected individuals with positive results. The performance of a screening test is often specified by the detection rate for a 5% false positive rate, which can be abbreviated as the DR5. In the example shown in figure figure2,2, the DR5 is 25%. A similar notation can be used for detection rates equivalent to any specified false positive rate—for example, DR1 and DR10 for false positive rates of 1% and 10% respectively.
The two methods of quantifying the relation between a variable and a disorder (illustrated by figs figs11 and and2)2) are directly related if the frequency distribution of the variable is Gaussian in both affected and unaffected individuals. Thus, for any value of the ROQ1-5 there will be an equivalent DR5 and vice versa. The relation depends on the means and standard deviations (SDs) of the variable in affected and unaffected individuals, but it does not depend on other factors such as the incidence or prevalence of the disorder.
Figure Figure33 shows how the ROQ1-5 and the DR5 are linked quantitatively. In the example shown in the figure, the distributions of a variable in affected and unaffected individuals are known (each distribution having the same SD) and the distributions are separated by 1.4 SD units. The top part of the figure shows that at a cut off level chosen to give a 5% false positive rate the detection rate is 40%. The bottom part shows the same distribution yielding an ROQ1-5 of 57. It is derived as the ratio of the shaded areas on the right (defined by the upper fifth) divided by the ratio of the shaded areas on the left (defined by the lower fifth), which is 71%/20%÷1.25%/20%. The figure therefore illustrates how an ROQ1-5 of 57 is equivalent to a detection rate of 40% for a 5% false positive rate (DR5). The formal mathematical relation between the two is given in the statistical appendix (see website).
In this paper we have assumed that screening variables are distributed in a Gaussian manner. Often an adequate transformation (such as log transformation) can be found to allow data to fit a Gaussian distribution. Even if this is not the case, the approach illustrated in figure figure33 whereby the distributions of a screening variable in affected and unaffected individuals are used to calculate both the detection rates and the relative odds is generally useful in linking quantitatively the screening performance to the strength of the risk factor.
The table shows the relation between the ROQ1-5 and the DR5 when the SD of the screening variable is the same in affected and unaffected individuals. An ROQ1-5 of 10, for example, which would be regarded as a high odds ratio in aetiological terms, corresponds to a DR5 of only 20%; even an ROQ1-5 as high as100 corresponds to a DR5 of only 48%. To achieve a DR5 of over 70% the ROQ1-5 would have to be about 800 or more.
The measurement of α fetoprotein in pregnancy (at about 16-22 weeks of gestation) is a proved screening test for open spina bifida. The upper diagram in figure figure44 shows the distribution of α fetoprotein in spina bifida and unaffected pregnancies at 17 weeks' gestation.2,3 The ROQ1-5 is 246, indicating a strong association between α fetoprotein and open spina bifida. The extent of the association is perhaps not fully recognised by many people. The figure shows that the corresponding screening performance is high—91% of affected pregnancies can be detected in this way for a false positive rate of 5%. The detection rate, given the ROQ1-5 value, is somewhat higher than is suggested by the table because the SD in open spina bifida pregnancies is greater than that in unaffected pregnancies. This is a highly effective screening test.
Because serum cholesterol is an established risk factor for ischaemic heart disease, it was believed that it would be a useful screening test. According to this view, individuals with high cholesterol concentrations—that is, those regarded as being “screen positive”—would be offered cholesterol lowering drugs to reduce the risk of a myocardial infarction. This belief was unfounded, as illustrated by figure figure4.4. The lower diagram in figure figure44 shows the distribution of serum cholesterol concentrations in men aged 35-65 in the United Kingdom who did or did not subsequently die from ischaemic heart disease over a period of about 10 years. The ROQ1-5 value is 2.7, indicating that people with a high serum cholesterol concentration (in the highest fifth) are nearly three times more likely to die from ischaemic heart disease than those with a low serum cholesterol concentration (in the lowest fifth). This is a moderately strong association, but when it is assessed as a screening test the performance is poor. For a false positive rate of 5%, only 15% of those who would later die of ischaemic heart disease would be identified.1 Again, this screening performance is slightly better than predicted by the table because of the somewhat higher SD of serum cholesterol concentrations in affected individuals.
If the ratio of the SDs in affected and unaffected individuals is less than 1, the detection rate is less than those shown in the table for the same ROQ1-5. For example, if the ratio is 0.8, the DR5 values are 2%, 6%, 9%, 20%, and 27% for ROQ1-5 values of 1, 5, 10, 50, and 100 respectively. If the ratio is greater than 1, the detection rates would be greater than those shown in the table. For example, if the ratio is 1.2, the DR5 values are 9%, 23%, 31%, 53%, and 63% for ROQ1-5 values of 1, 5, 10, 50, and 100 respectively.
The boxes present two practical examples illustrating the above concepts. In each example, the risk factor is known to be associated with the particular disorder, but only in the first example is the screening performance acceptable.
A risk factor has to be extremely strongly associated with a disease within a population before it can be considered to be a potentially useful screening test. Even a relative odds of 200 between the highest and lowest fifths will yield a detection rate of no more than about 56% for a 5% false positive rate, provided, as is commonly the case, that the distribution of the screening variable is approximately Gaussian (or log Gaussian) and shows a similar SD in affected and unaffected people.
It is not unusual for a strong risk factor of aetiological importance to be proposed as a screening test for the disorder, perhaps in the belief that strong risk factors make good screening tests. We show that this is not necessarily the case, giving as an example serum cholesterol concentration in relation to ischaemic heart disease. A risk factor with an ROQ1-5 of 10 or greater is unusual. Few risk factors in epidemiology are as strongly associated with a disease as this. A risk factor with an ROQ1-5 would, however, perform poorly as a screening test. It would have a detection rate of about 20% for a false positive rate of 5% if the SDs were similar for affected and unaffected individuals. Even if the SD in affected individuals was 20% greater, the detection rate would only be 30% for a 5% false positive rate.
The fact that a strong risk factor can be a poor screening test may seem counterintuitive. The paradox is explained when it is recognised that the relative odds (or relative risk), usually used to evaluate risk factors as possible causes of a disease, is usually assessed by comparing the risk of disease at each end of the distribution of the risk factor. In this way the effect of being highly “exposed” to the factor is compared with being slightly “exposed.” The groups being compared are mutually exclusive and most people in the middle of the distribution are ignored. When the risk factor is examined as a screening test, the likelihood of having (or developing) a disease given a positive result (say, 95th centile) is estimated relative to the average risk in the entire population, which not only includes all those below the cut off but also those above it. The aim in screening is to identify a group with a high risk relative to everyone.
Another reason why strong risk factors may make poor screening tests is that there may be little variation in exposure within populations. For example, we know that smoking cigarettes is a risk factor for lung cancer. However, if everyone in a certain population smoked 20 cigarettes a day, asking about cigarette consumption would not distinguish those who are more likely to develop lung cancer from those who are not.
Failure to recognise the above considerations may explain why serum cholesterol determination was proposed as a screening test for ischaemic heart disease even though it performed poorly as a screening test when cut off levels corresponding to the 95th centile were used.1 Before a risk factor is considered as a screening test it would be worth determining the ROQ1-5 and then examining the table. This should help to determine which tests are potentially useful in medical screening.
We thank Tiesheng Wu for his help in preparing the figures and Malcolm Law for comments.
Competing interests: None declared.
website extra: A mathematical appendix appears on the BMJ's website www.bmj.com