Likelihood ratio slide rule (October 24, 2002) Category: Diagnostic testing

The use of likelihood ratios requires a bit of tedious calculations. I have developed a simple slide rule that will do likelihood ratio calculations for you.

Note: I am developing a special handout (PDF format) that explains the mathematics behind diagnostic testing and which illustrates many of the important points using the likelihood ratio slide rule. I distributed this handout in a talk for the American College of Allergy, Asthma & Immunology on Sunday, November 11, but ran out very quickly.

Assembly instructions

Please print out this this PDF file. Cut out the bottom piece (the sleeve) and the top piece (the insert). Also cut out the two rectangles in the middle of the sleeve. Fold the left and right portions of the sleeve behind and tape them together. Double sided tape works very well for this. Slip the insert into the sleeve. You may need to trim a tiny amount off the left and right sides of the insert to get it to fit well. You want the insert to fit not too snugly and not too loosely inside the sleeve.

For a more durable slide rule

If you print this to a regular sheet of paper, the slide rule will be okay but a bit flimsy and easy to bend. For a more durable slide rule, print out the image on a thick piece of paper or tape/glue the image to a thin piece of cardboard. You can also print the image on a full sheet adhesive label (like Avery 5165) and then attach the label to a thick piece of paper or a thin piece of cardboard.

How to use the slide rule

Slide the insert up or down until the pre-test probability in the left window lines up with the likelihood ratio. Read the post-test probability in the right window.

Examples

In Watkins et al 2001, a single question diagnostic test (the Yale-Brown obsessive-compulsive scale) was compared to a "gold standard" measure of depression, the Montgomery Asberg depression rating scale (MADRS).

On the MADRS 43 (54%) were classified as clinically depressed; 37 answered "yes" to the Yale single question and six answered "no." Of the 36 classified as not depressed, eight answered "yes" and 28 "no." The values (95% confidence intervals) for the Yale test were sensitivity 86% (75% to 97%), specificity 78% (65% to 91%), positive predictive value 82% (71% to 93%), negative predictive value 82% (69% to 95%); 82% (73% to 91%) of cases were classified correctly.

The prevalence of depression in this population was unusually high, so the authors presented additional positive predictive values (PPV) and negative predictive values (NPV) for prevalence values ranging from 10% to 90%. An abridged version of their table appears below.

Prevalence PPV NPV
90% 97% 38%
80% 94% 58%
70% 90% 70%
60% 85% 79%
50% 80% 85%
40% 72% 89%
30% 63% 93%
20% 49% 96%
10% 30% 98%

Since the PPV is simply the post-test probability after a positive test, we can use the likelihood ratio slide rule to re-create their calculations. First, we need to compute the likelihood ratio for a positive test (LR+). The formula is

LR+ = Sn / (1-Sp) = 0.86 / (1-0.78) = 3.9

where Sn and Sp are sensitivity and specificity, respectively. We will round this value to 4.

To compute the positive predictive value when the prevalence of the disease is 10%, line up the 10% pre-test probability with the likelihood ratio of 4 (the unlabelled tick mark between 3 and 5). In the right side window, the post-test probability should be slightly more than 30%, which matches the value computed by Watkins.

Slide the insert up so the 20% pre-test probability lines up with the likelihood ratio of 4. The post-test probability should be around 50% which also matches the value in Watkins.

Now slide the insert up so the 30% pre-test probability lines up with the likelihood ratio of 4. The post-test probability should be slightly more than 60%.

Repeat this for 40%, through 90% and see if you can estimate the remaining PPV values.

To compute NPV, we need to calculate the likelihood ratio for a negative test (LR-). The formula is

LR- = (1-Sp) / Sn = (1-0.86) / 0.78 = 0.18.

There is no tick mark for 0.18, so we will use a point about halfway between the 0.15 and 0.2 tick marks. Line up the prevalence of 10% with the likelihood ratio of 0.18 and read off the post-test probability of 2% in the right side window. Since there is only a 2% chance of having the disease, there is a 98% of being healthy, which matches the NPV computed by Watkins.

Line up a prevalence of 20% with the likelihood ratio of 0.18 to get a post-test probability of 4% and an NPV of 96%.

Now line up a prevalence of 30% with the likelihood ratio of 0.18 to get a post-test probability of 7% and an NPV of 93%.

Repeat this for 40% through 90% and estimate the remaining NPV values.

Second example

A letter to the editor in BMJ commented on how the use of likelihood ratios could have simplified the interpretation of results of a rapid whole blood test for diagnosing Helicobacter pylori infection.

In that study the likelihood ratio for a positive test result was 9.8. The advantage of knowing this is that it can be applied to similar patients in other populations to estimate the predictive value of the test, provided that the pre-test probability of disease can be estimated. For example, H pylori is found in 48% of dyspeptic patients in the community (the pre-test probability), so therefore a positive rapid blood test with a likelihood ratio of 9.8 applied to this population would give a post-test probability (or predictive value) of 90% (this can be estimated using a simple calculation or a nomogram). --BMJ 1997; 314: 1688.

We have to round a bit here. Line up a pre-test probability of 50% with a likelihood ratio of 10. Read the post-test probability of slightly more than 90% in the upper window.

Third example

Buschbaum et al examined the sensitivity, specificity, and likelihood ratio for the CAGE score, a series of yes/no answers to four questions (Ann Intern Med 1991; 115(10): 774-777). The four item scale was very good at detecting alcohol abuse or dependence.

Score Abuse or
Dependence
No abuse or
dependence
Likelihood
ratio
0 33 428 0.14
1 45 54 1.5
2 86 34 4.5
3 74 10 13
4 56 1 100

In this paper, the authors noted a prevalence of alcohol abuse and dependence of 36%. Find this value in the pre-test probability and line it up successively with each of the likelihood ratios listed above. You should get a post-test probability of 7%, 45%, 70%, 90% and 98% for the scores of 0 through 4, which matches up nicely with the values given in the paper. The likelihood ratio slide rule computations are shown below for the first three of these cases.

Grant et al tabulated the prevalence of alcohol abuse or dependence for demographic groups. This rate varies by age (higher among younger people), by gender (higher among males) and race (higher among non-blacks). Among non-black males, for example, the prevalence is  23%, 11%, 6%, and 1% for 18-29, 30-44, 45-64, and 65+ years of age, respectively (Alcohol Health & Research World 1994; 18(3):243-248, as quoted in alcoholism.about.com/library/nabdep4.htm).

The prevalence would be roughly twice as high among ambulatory patients than the general population and four times as high for hospitalized patients than the general population (Postgraduate Medicine Online 1996; 100(1),  www.postgradmed.com/issues/1996/07_96/blondell.htm).

Suppose you apply the CAGE score to a 70 year old hospitalized white male. This person scores 3 on CAGE. Line up a pre-test probability of 4% with a likelihood ratio of 13. The post test probability is slightly more than 30%.

Suppose you give the same test to a 35 year old white male who visits your clinic and he scores 0 on CAGE. Line up a pre-test probability of 22% with a likelihood ratio of 0.14. The post-test probability is 4%.

How does it work?

The likelihood ratio slide rule works on the same principle as a regular slide rule. The logarithms on a slide rule allow you to multiply simply by adding. It uses the simple formula

log (a*b) = log (a) + log (b).

There's an old joke well known among mathematicians about logarithms. After the flood waters receded, Noah commanded the animals to go forth and multiply. The snakes went up to Noah and told him they couldn't multiply because they were adders. So Noah built them a piece of wooden furniture with a flat top and four legs. The adders could now multiply because they had a log table.

The formula for computing post-test odds is

post-test odds = likelihood ratio * pre-test odds.

By taking logarithms of both sides of the equation, we get

log (post-test odds) = log (likelihood ratio) + log (pre-test odds)

Sliding the insert up or down will add a pre-test log odds value to a log likelihood ratio to get a post-test log odds value. The tick marks are labeled using probability rather than odds to simplify things further.

The likelihood ratio slide rule that I developed was inspired by the Fagan nomogram which also uses logarithms. In the Fagan nomogram, you draw a line connecting the pre-test probability with the likelihood ratio. Extend the line further to the right to compute the post-test probability.

Summary

The likelihood ratio slide rule allows you to compute the post-test probability of a disease given the pre-test probability and the likelihood ratio of a diagnostic test. Simply line up the pre-test probability in the left side window with the likelihood ratio. Then read the post-test probability in the right side window.

Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 United States License. It was written by Steve Simon.