Medical diagnostic research, as usually practiced, is prone to bias and even more importantly to yielding information that is not useful to patients or physicians and sometimes overstates the value of diagnostic tests. Important sources of these problems are conditioning on the wrong statistical information, reversing the flow of time, and categorization of inherently continuous test outputs and disease severity. It will be shown that sensitivity and specificity are not properties of tests in the usual sense of the word, and that they were never natural choices for describing test performance. This implies that ROC curves are unhelpful (although areas under them are sometimes useful). So is categorical thinking.

This chapter outlines the many advantages of diagnostic risk modeling, showing how pre-and post-test diagnostic models give rise to clinically useful displays of pre-test vs. post-test probabilities that themselves quantify diagnostic utility in a way that is useful to patients, physicians, and diagnostic device makers. And unlike sensitivity and specificity, post-test probabilities are immune to certain biases, including workup bias.

Case-control studies use a design where sampling is done on final disease status and patient exposures are “along for the ride.” In other words, one conditions on the outcome and considers the distribution of exposures using outcome-dependent sampling. Sensitivity and specificity are useful for proof-of-concept case-control studies because sensitivity and specificity also condition on the final diagnosis. The use of sensitivity and specificity in prospective cohort studies is the mathematical equivalent of making three left turns in order to turn right. Like the complex adjustments needed for $$P$$-values when doing sequential trials, sensitivity and specificity require complex adjustments for workup bias just because of their backward consideration of time and information. In a cohort study one can use a vanilla regression model to estimate the probability of a final diagnostic outcome given patient characteristics and diagnostic test results.

## 19.1 Problems with Traditional Indexes of Diagnostic Utility

sensitivity = Prob$$[T^{+} | D^{+}]$$

specificity = Prob$$[T^{-} | D^{-}]$$

Prob$$[D^{+} | T^{+}] = \frac{\mathrm{sens} \times \mathrm{prev}}{\mathrm{sens} \times \mathrm{prev} + (1-\mathrm{spec})\times (1-\mathrm{prev})}$$

Problems:

• Diagnosis forced to be binary
• Test force to be binary
• Sensitivity and specificity are in backwards time order
• Confuse decision making for groups vs. individuals
• Inadequate utilization of pre-test information
• Dichotomization of continuous variables in general

#### Example: BI-RADS Score in Mammography

Does Category 4 Make Any Sense?

Diagnosis Number of Criteria
0 Incomplete Your mammogram or ultrasound didn’t give the radiologist enough information to make a clear diagnosis; follow-up imaging is necessary
1 Negative There is nothing to comment on; routine screening recommended
2 Benign A definite benign finding; routine screening recommended
3 Probably Benign Findings that have a high probability of being benign ($$>98$$%); six-month short interval follow-up
4 Suspicious Abnormality Not characteristic of breast cancer, but reasonable probability of being malignant (3 to 94%); biopsy should be considered
5 Highly Suspicious of Malignancy Lesion that has a high probability of being malignant ($$\geq 95$$%); take appropriate action
6 Known Biopsy Proven Malignancy Lesions known to be malignant that are being imaged prior to definitive treatment; assure that treatment is completed

How to Reduce False Positives and Negatives?

• Do away with “positive” and “negative”
• Provide risk estimates
• Defer decision to decision maker
• Risks have self-contained error rates
• Risk of 0.2 $$\rightarrow$$ Prob[error]=.2 if don’t treat
• Risk of 0.8 $$\rightarrow$$ Prob[error]=.2 if treat

See this for a nice article on the subject.

#### Binary Diagnosis is Problematic Anyway

The act of diagnosis requires that patients be placed in a binary category of either having or not having a certain disease. Accordingly, the diseases of particular concern for industrialized countries—such as type 2 diabetes, obesity, or depression—require that a somewhat arbitrary cut-point be chosen on a continuous scale of measurement (for example, a fasting glucose level $$>6.9$$ mmol/L [$$>125$$ mg/dL] for type 2 diabetes). These cut-points do not adequately reflect disease biology, may inappropriately treat patients on either side of the cut-point as 2 homogeneous risk groups, fail to incorporate other risk factors, and are invariable to patient preference. — Vickers et al. (2008)

Newman & Kohn (2009) have a strong section about the problems with considering diagnosis to be binary.

#### Back to Sensitivity and Specificity

• Backwards time order
• Irrelevant to both physician and patient
• Improper discontinuous scoring rules1
• Are not test characteristics
• Are characteristics of the test and patients
• Not constant; vary with patient characteristics
• Sensitivity $$\uparrow$$ with any covariate related to disease severity if diagnosis is dichotomized
• Require adjustment for workup bias
• Diagnostic risk models do not; only suffer from under-representation
• Good for proof of concept of a diagnostic method in a case–control study; not useful for utility
• 1 They are optimized by not correctly estimating risk of disease.

• Hlatky et al. (1984); Moons et al. (1997); Moons & Harrell (2003); Gneiting & Raftery (2007)

#### Sensitivity of Exercise ECG for Diagnosing CAD

Age Sensitivity
$$<40$$ 0.56
40–49 0.65
50–59 0.74
$$\geq$$ 60 0.84
Sex
male 0.72
female 0.57
# Diseased Coronary Arteries
1 0.48
2 0.68
3 0.85

## 19.2 Problems with ROC Curves and Cutoffs

… statistics such as the AUC are not especially relevant to someone who must make a decision about a particular $$x_{c}$$. … ROC curves lack or obscure several quantities that are necessary for evaluating the operational effectiveness of diagnostic tests. … ROC curves were first used to check how radio receivers (like radar receivers) operated over a range of frequencies. … This is not how most ROC curves are used now, particularly in medicine. The receiver of a diagnostic measurement … wants to make a decision based on some $$x_{c}$$, and is not especially interested in how well he would have done had he used some different cutoff. — Briggs & Zaretzki (2008)

In the discussion to this paper, David Hand states “when integrating to yield the overall AUC measure, it is necessary to decide what weight to give each value in the integration. The AUC implicitly does this using a weighting derived empirically from the data. This is nonsensical. The relative importance of misclassifying a case as a non-case, compared to the reverse, cannot come from the data itself. It must come externally, from considerations of the severity one attaches to the different kinds of misclassifications.”

## 19.3 Optimum Individual Decision Making and Forward Risk

• Minimize expected loss/cost/disutility
• Uses
• utility function (e.g., inverse of cost of missing a diagnosis, cost of over-treatment if disease is absent)
• probability of disease

$$d =$$ decision, $$o =$$ outcome
Utility for outcome $$o = U(o)$$
Expected utility of decision $$d = U(d) = \int p(o | d) U(o)do$$
$$d_\mathrm{opt} = d$$ maximizing $$U(d)$$

The steps for determining the optimum decision are:

1. Predict the outcome $$o$$ for every decision $$d$$
2. Assign a utility $$U(o)$$ to every outcome $$o$$
3. Find the decision $$d$$ that maximizes the expected utility

See

See this NY Times article about decision theory.

## 19.4 Diagnostic Risk Modeling

Assuming (Atypical) Binary Disease Status

Y 1:diseased, 0:normal
$$X$$ vector of subject characteristics (e.g., demographics, risk factors, symptoms)
$$T$$ vector of test (biomarker, …) outputs
$$\alpha$$ intercept
$$\beta$$ vector of coefficients of $$X$$
$$\gamma$$ vector of coefficients of $$T$$

pre$$(X)$$ = Prob$$[Y=1 | X] = \frac{1}{1 + \exp[-(\alpha^{*} + \beta^{*} X)]}$$
post$$(X,T)$$ = Prob$$[Y=1 | X,T] = \frac{1}{1 + \exp[-(\alpha + \beta X + \gamma T)]}$$

Note: Proportional odds model extends to ordinal disease severity $$Y$$.

### 19.4.1 Example Diagnostic Models

Code
knitr::include_graphics('cad-nomo.png') Figure 19.1: A nomogram for estimating the likelihood of significant coronary artery disease (CAD) in women. ECG = electrocardiographic; MI = myocardial infarction. Pryor et al. (1983). Reprinted from American Journal of Medicine, Vol. 75, D.B. Pryor et al, “Estimating the likelihood of significant coronary artery disease,” p. 778, Copyright 1983, with permission from Excerpta Medica, Inc.
Code
knitr::include_graphics('abm-nomo.png') Figure 19.2: Nomogram for estimating probability of bacterial (ABM) versus viral (AVM) meningitis. Step 1, place ruler on reading lines for patient’s age and month of presentation and mark intersection with line A; step 2, place ruler on values for glucose ratio and total polymorphonuclear leukocyte (PMN) count in cerebrospinal fluid and mark intersection with line B; step 3, use ruler to join marks on lines A and B, then read off the probability of ABM versus AVM. From Spanos et al. (1989)
Code
knitr::include_graphics('bra91usiNomogram.png') Figure 19.3: Proportional odds ordinal logistic model for ordinal diagnostic classes from Brazer et al. (1991)

## 19.5 Assessing Diagnostic Yield

### 19.5.1 Absolute Yield

Pencina et al. (2008): Absolute incremental information in a new set of markers
Consider change in predicted risk when add new variables:

Average increase in risk of disease when disease present
+
Average decrease in risk of disease when disease absent

#### Formal Test of Added Absolute and Relative Information

Likelihood ratio $$\chi^{2}$$ test of partial association of new markers, adjusted for old markers

### 19.5.2 Assessing Relative Diagnostic Yield

• Variation in relative log odds of disease = $$T \hat{\gamma}$$, holding $$X$$ constant
• Summarize with Gini’s mean difference or inter-quartile range, then anti-log
• E.g.: the typical modification of pre-test odds of disease is by a factor of 3.4

Gini’s mean difference = mean absolute difference between any pair of values

See Figure 13.6 for a graphical depiction of the relationship between odds ratio and absolute risk difference.

## 19.6 Assessing Absolute Diagnostic Yield: Cohort Study

• Patient $$i = 1, 2, 3, \ldots, n$$
• In-sample sufficient statistics: pre$$(X_{1})$$, …, pre$$(X_{n})$$, post$$(X_{1}, T_{1})$$, …, post$$(X_{n}, T_{n})$$
Code
knitr::include_graphics('hla09criFig3.png') Figure 19.4: Pre vs. post-test probability. This may be summarized with quantile regression to estimate 0.1 and 0.9 quantiles of post as a function of pre. From Hlatky et al. (2009)

Assessments assume that risk estimates are well calibrated, using, for example a high-resolution continuous calibration curve. Out-of-sample assessment: compute pre$$(X)$$ and post$$(X,T)$$ for any $$X$$ and $$T$$ of interest

#### Summary measures

• quantile regression (Koenker & Bassett (1978)) curves as a function of pre
• overall mean $$|$$post – pre$$|$$
• quantiles of post – pre
• du$$_{50}$$: distribution of post when pre = 0.5
diagnostic utility at maximum pre-test uncertainty
• Choose $$X$$ so that pre = 0.5
• Examine distribution of post at this pre
• Summarize with quantiles, Gini’s mean difference on prob. scale
• Special case where test is binary (atypical): compute post for $$T^{+}$$ and for $$T^{-}$$

## 19.7 Assessing Diagnostic Yield: Case-Control & Other Oversampling Designs

• Intercept $$\alpha$$ is meaningless
• Choose $$X$$ and solve for $$\alpha$$ so that pre = 0.5
• Proceed as above to estimate du$$_{50}$$

## 19.8 Example: Diagnosis of Coronary Artery Disease (CAD)

Test = Total Cholesterol

Code
require(rms)
getHdata(acath)
acath <- subset(acath, !is.na(choleste))
f <- lrm(sigdz ~ rcs(age,5)*sex, data=acath)
pre <- predict(f, type='fitted')
g <- lrm(sigdz ~ rcs(age,4)*sex + rcs(choleste,4) + rcs(age,4) %ia%
rcs(choleste,4), data=acath)
ageg <- c(40, 70)
psig <- Predict(g, choleste, age=ageg)
s <- lrm(tvdlm ~ rcs(age,4)*sex + rcs(choleste,4) + rcs(age,4) %ia%
rcs(choleste,4), data=acath)
psev <- Predict(s, choleste, age=ageg)
adj.subtitle=FALSE) Figure 19.5: Relative effect of total cholesterol for age 40 and 70; Data from Duke Cardiovascular Disease Databank, $$n=2258$$
Code
post <- predict(g, type='fitted')
plot(pre, post, xlab='Pre-Test Probability (age + sex)',
ylab='Post-Test Probability (age + sex + cholesterol)', pch=46)
abline(a=0, b=1, col=gray(.8))
lo <- Rq(post ~ rcs(pre, 7), tau=0.1)  # 0.1 quantile
hi <- Rq(post ~ rcs(pre, 7), tau=0.9)  # 0.9 quantile
at <- seq(0, 1, length=200)
lines(at, Predict(lo, pre=at)$yhat, col='red', lwd=1.5) lines(at, Predict(hi, pre=at)$yhat, col='red', lwd=1.5)
abline(v=.5, col='red') Figure 19.6: Diagnostic Utility of Cholesterol for Diagnosing Significant CAD. Curves are 0.1 and 0.9 quantiles from quantile regression using restricted cubic splines

## 19.9 Summary

Diagnostic utility needs to be estimated using measures of relevance to individual decision makers. Improper accuracy scoring rules lead to sub-optimal decisions. Traditional risk modeling is a powerful tool in this setting. Cohort studies are ideal but useful measures can be obtained even with oversampling. Avoid categorization of any continuous or ordinal variables.