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Probability in Radiology¹

¹ From the DATA Group, Department of Radiology, Massachusetts General Hospital, Zero Emerson Pl, Suite 2H, Boston, MA 02114. Received October 19, 2001; revision requested December 26; revision received March 27, 2002; accepted April 9. Address correspondence to E.F.H. (e-mail: elk@the-data-group.org).

	ABSTRACT

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
In this article, a summary of the basic rules of probability using examples of their application in radiology is presented. Those rules describe how probabilities may be combined to obtain the chance of "success" with either of two diagnostic or therapeutic procedures or with both. They define independence and relate it to the conditional probability. They describe the relationship (Bayes rule) between sensitivity, specificity, and prevalence on the one hand and the positive and negative predictive values on the other. Finally, the two distributions most commonly encountered in statistical models of radiologic data are presented: the binomial and normal distributions.

© RSNA, 2002

Index terms: Statistical analysis

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
Radiologists routinely encounter probability in many forms. For instance, the sensitivity of a diagnostic test is really just a probability. It is the chance that disease (eg, a liver tumor) will be detected in a patient who actually has the disease. In the process of determining whether a sequence of successive diagnostic tests (say, both computed tomography [CT] and positron emission tomography) is a significant improvement over CT alone, radiologists must understand how those probabilities are combined to give the sensitivity of the combination.

Similarly, the prevalence of disease such as malignant liver cancer among patients with cirrhosis is a probability. It is the fraction of patients with a history of cirrhosis who have a malignant tumor. It can be determined simply from the number of patients with cirrhosis and the number of patients with both cirrhosis and malignant tumors of the liver or from the two prevalences.

The likelihood that a patient with a cirrhotic liver and positive CT findings has a malignant tumor (the positive predictive value [PPV] of CT in this setting) is another probability. It is determined by combining the prevalence of the disease among patients with cirrhosis with the sensitivity and specificity of CT.

In all its forms, probabilities obey a single set of rules that determine how they may be combined; this article presents an outline of these rules. The rules are also found and explained in greater detail in textbooks of biostatistics, such as that by Rosner (1).

	THE EARLIEST DEFINITION OF PROBABILITY

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
Probability is a numeric expression of the concept of chance, in all its guises. As such, it obeys the rules of arithmetic and the logic of mathematics.

In its earliest manifestation, probability was a tool for gamblers. The earliest expressions of probability dealt with simple gambling games such as flipping a coin, rolling a die, or dealing a single card from the top of a deck. The essential property assumed of such games was that the possible outcomes were all equally likely. The probability of some event was proportional to the number of individual outcomes that comprised the event.

In more complicated scenarios, the fundamental outcomes might not be equally likely or there might be an infinite number of them. In these situations, the definition of a probability of an event became the fraction of times that the event would occur by chance. That is, it is the fraction of times in a sufficiently long sequence of trials where the chance of the event was the same in all trials and was not affected by the results of previous or subsequent trials.

With either definition, counting possible outcomes or measuring the frequency of occurrence, probability was susceptible to the laws and rules of simple arithmetic. The simplest rule of all was the rule of addition. Addition of the fraction of patients who have a single liver tumor that is malignant to the fraction of patients who have a single tumor that is benign must yield the fraction of patients who have a single tumor, either malignant or benign.

Algebraically, we express this "additive" rule for two events, A and B, as the following: If Prob(A AND B) = 0, then Prob(A OR B) = Prob(A) + Prob(B), where we use Prob(A) to denote the probability of A. The condition "Prob(A AND B) = 0" is a way of saying that the simultaneous occurrence of both A and B is impossible.

One consequence of this rule is that the chance that an event would not happen is immediately determined by the chance that it would happen. As "A" and "NOT A" (the event that A would not happen) cannot occur simultaneously, yet one or the other is certain, Prob(A) + Prob(NOT A) = 1 or Prob(NOT A) = 1 - Prob(A). Thus, the fraction of patients who do not have a single tumor is just the complement to the fraction of patients who do.

The limitation "Prob(A AND B) = 0" is necessary, as the rule does not apply without modification when it is possible for both events to be true. For instance, addition of the fraction of patients who have at least one malignant tumor to the fraction of patients who have at least one benign tumor may cause overestimation of the fraction of patients with tumors, either malignant or benign. The extent of the overestimation is given by the fraction of patients who have both benign and malignant tumors. These patients were included in the counts for both tumor-type specific fractions but should be counted only once when patients who have at least one tumor are counted. This may be best seen in Figure 1, where the added shaded areas of A and B yield a sum greater than the total shaded area. The area representing the overlap of A and B must be accounted for.

View larger version (55K): [in this window] [in a new window] [Download PPT slide]   Figure 1. Venn diagram represents the probability of either of two events based on the probabilities of each and of both. The area of the shape made by combining A and B is the total of the areas of A and B less the area where they overlap.

	SUBJECTIVE VERSUS OBJECTIVE PROBABILITIES

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
Before a coin is flipped, the probability that it will land heads up is 0.5. After it is flipped and is seen to have landed heads up, the probability that it landed heads up is 1 and that it landed tails up is 0. But what is the probability that it landed heads up before anyone saw which face was up? The face has been determined, and the probability is either 1 or 0 that it is heads, though which is not yet known. Yet, any gambler would be willing to bet at that point on which face had landed up in precisely the same way that he or she would have bet before the coin had been flipped. The gambler has his or her own "subjective" probability that the coin will be face up when observed, which reflects his or her beliefs concerning the probability before the coin was flipped. The subjective probabilities obey the same laws and rules as the "objective" probabilities in describing future events. Subjective probabilities may be revised as one learns more about what else may be true.

In radiology, before an examination, the probability that a patient will have an abnormality detected is a fraction between 0 and 1. After the procedure and after the image has been viewed, the probability is either 1 or 0, depending on whether an abnormality has been detected. In the interim, after the procedure but before the image has been viewed, the subjective probability is the same as the objective probability was before the procedure.

	CONDITIONAL PROBABILITIES AND INDEPENDENCE VERSUS DEPENDENCE

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
While the probability that a randomly selected woman has an undetected malignant breast cancer at least 1 cm in diameter has real meaning, the probability is not the same for all women. It certainly is higher for women who have never undergone mammographic screening than for women who have—all other things being equal. The probability that a woman who has never been screened has such a tumor is called a "conditional" probability, because it is defined as the chance that the woman has a 1 cm or greater tumor, given that the woman has never been screened. The probability that a randomly chosen woman has such a tumor is the number of women with such tumors divided by the number of all women. The conditional probability that a randomly chosen woman who has never been screened has such a tumor is defined analogously. It is the number of women who have never been screened who have such tumors out of the number of women who have never been screened. By using Prob(A|B) to denote the conditional probability of A (a woman with a 1 cm or greater breast tumor) given B (she underwent no prior screening), we have Prob(A|B) = Prob(A AND B)/Prob(B).

This forms the basis of the definition of sensitivity and specificity. If we let A stand for a positive diagnostic examination result and B to represent the actual presence of the disease, then Prob(A|B) is the sensitivity, the chance of a positive examination result among individuals with disease or of a true-positive result. The definition can also be used to calculate the chances derived from a succession of diagnostic tests. For instance, if confirmation of any positive test result, D₁+, is required by means of a second positive test result, D₂+, then the chance that we will obtain two positive test results is given by the "multiplicative" law of probabilities: Prob(D₁+ AND D₂+) = Prob(D₂+|D₁+) x Prob(D₁+).

That is, the chance that both tests have positive results is a fraction of all second tests with positive results, once the first test had a positive result times the chance that the first test had a positive result. The rule can be similarly used to calculate the chance of two successive negative findings or any other combination.

Occasionally, the chances for the second examination, D₂, are unaltered by the results of the initial examination, D₁. For instance, after the diagnostic image has been obtained, the interpretation by a blinded reader, D₂, should be unaffected by the prior interpretation by another blinded reader, D₁, of the same image. In these situations, the two interpretations are said to be independent (2).

If A and B are independent, then Prob(A) = Prob(A|B) = Prob(A|NOT B); that is, the (conditional) chance of A given that B occurs is the same as the (conditional) chance of A given that B does not occur and, as a result, also equals the (unconditional) chance of A.

A consequence is the multiplicative law of probabilities for independent events; namely, if A and B are independent, then Prob(A AND B) = Prob(A) x Prob(B).

In practice, the diagnostic tests of radiology are rarely truly independent as CT, magnetic resonance imaging, and ultrasonography all rely similarly on the lesion size and differences in tissue characteristics. Yet, quite often, the multiplicative law is used as an approximation because the exact conditional probability has never been accurately determined in clinical trials of both diagnostic modalities.

The same rules may be used to calculate the risk of disease in the presence of multiple predictive characteristics (or cofactors). If the effects of the cofactors are synergistic, the more general multiplicative rule must be used. But if the effects of the cofactors are unrelated or independent, the multiplicative rule for independent events may be used. Similarly, the rules may be used to compute the overall chance of a successful treatment of disease with a succession of treatments based on the (conditional) chance of success of each treatment.

	BAYES RULE AND POSITIVE AND NEGATIVE PREDICTIVE VALUES

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
While the sensitivity and specificity of a diagnostic test are important to the clinician when he or she determines which test to use, they do not entirely address the question of concern after the test has been performed. To the patient, the issue is not "How often does the test detect real disease?" but rather, "Now that the test results are known, what is the chance that I have the disease?" The patient wants to know a conditional probability that is the reverse of sensitivity. If we use D_x to denote a positive finding and D to denote the actual presence of disease, the patient is not as concerned with the sensitivity, Prob(D_x|D), as with the PPV, Prob(D|D_x), the chance that there is disease present given that the test result was positive.

Both sensitivity and specificity are considered to be inherent invariant test characteristics. In contrast, the PPV and the negative predictive value depend not only on the sensitivity and specificity but also on the prevalence of the disease, Prob(D). They may be combined by using Bayes rule (3), which relates the PPV, Prob(D|D_x), to the sensitivity, Prob(D_x|D), the specificity, 1 - Prob(D_x|NOT D), and the prevalence, Prob(D).

In this equation, the denominator is the total number of expected positive findings in the population, while the numerator is the number of positive findings that accompany the actual disease.

Suppose that we had a diagnostic test used for screening that had both 95% sensitivity (positive in 95% of all cases where the disease is present) and 95% specificity (negative in 95% of all cases where the disease is absent). When the prevalence of the disease is 10%, out of every 10,000 screening examinations, we can expect to see 1,400 cases that result in a positive finding with the diagnostic test (Table). Rather than go through the laborious exercise of constructing tables, Bayes rule gives us the PPV directly. For 10% prevalence, Prob(D) = 0.10, we have PPV = (0.95 x 0.10)/(0.95 x 0.10) + (0.05 x 0.90) = 0.095/(0.095 + 0.045) = 0.679.

	P VALUES, POWER, AND BAYESIAN STATISTICS

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

In a conventional analysis, the probability of significant study results, S, conditional on B being false, Prob(S|NOT B), is known as the power. It is not a factor in the conclusion drawn from the study. Rather, it is the major factor in the design before the study is conducted. It determines the number of patients in the study. The study sample size is chosen to provide the desired chance of successfully showing that B is not true.

Bayesian statistics differs from conventional statistics insofar as it depends on the probability that the hypothesis holds given the observed results of the study or studies. This probability is calculated by means of the Bayes rule. In order to be able to calculate it, the (subjective) probability reflecting the prior (before the study) chance or belief that the hypothesis was true is required. Much of the dispute regarding the use of Bayesian analysis centers around the possibility of conclusions that might be largely dictated by "opinion" before hard data are obtained.

P values, power, and Bayesian analysis will be presented in later articles in this series. Software for Bayesian probabilities may be found at the University of Sheffield Web site at www.shef.ac.uk/~st1ao/1b.html.

	PROBABILITY FOR CONTINUOUS OUTCOMES

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
The interpretation of the probability that a tumor consists of a specified number of cells differs in one essential regard from the interpretation of the probability that the tumor is a specified diameter or volume. The number of cells is "discrete" in the sense that it can only be an integer value. No fractional number of cells is possible. As such, each possible number of cells has its own probability, and if that number of cells is possible, the probability is greater than 0. But the diameter or volume of a tumor can take on all fractional values, as well as an integer. The probability that the tumor is between 2 and 3 cm in diameter could be treated in the same way as all of the probabilities that we have been discussing. However, the probability that the tumor is exactly cm to the last decimal place (or any other exact value, even, say, 3 cm to the last decimal place) has to differ in meaning.

For continuous measures such as the diameter or volume of a tumor or time of an occurrence, the probability that it exactly equals a single value is analogous to the distance traveled or the radiation received in an instant. Instead, one can express the rate at each value and calculate the probability of any interval just as one calculates the distance traveled over any time interval from the instantaneous speed or the total radiation exposure from the instantaneous rate of exposure.

Happily, all of the rules discussed for discrete outcomes apply equally well to continuous ones, both for intervals and for the rates at specified values.

	DISTRIBUTIONS

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 
These rules may be used to provide formulas for calculating the distribution, the probabilities for all possible outcomes, under a variety of circumstances. Two of the distributions most commonly encountered by radiologists are the binomial and the normal distributions.

The binomial distribution, B(i|n,p), (4) describes the probability of an event occurring i times out of n tries, where the chance, p, of the event is the same for all tries and the occurrence of the event in one try is unrelated (independent) to its occurrence in any other try. The distribution then gives the probabilities of each possible number of occurrences of the event out of n cases. The specific formula for the probability of i events out of n cases, B(i|n,p), is B(i|n,p) = [n!/i!(n - i)!]pⁱ(1 - p)^{n - i}, where ! indicates factorial, as in n! = n x (n - 1) x (n - 2) x ...x 2 x 1.

In practice, this distribution is built into most statistical and spreadsheet software packages. For instance, by using Microsoft Excel software, B(i|n,p) is calculated by the function BINOMDIST.

A binomial distribution that a radiologist might encounter is the probability of detecting i cancers with screening. In order for the binomial distribution to apply, the sensitivity would have to be identical for all of the cancers. Additionally, the detection (or nondetection) of any one cancer could not influence the chance that another was detected. If both conditions applied and there were n actual cases of cancer among those screened, B(i|n,p) would be the chance that i cancers were detected if the sensitivity of the technique was given by p. Thus, if there were actually eight cancers and the sensitivity of screening was 70%, the chance that exactly six of those eight cancer were detected is B(6|8,0.7) = [8!/6!(8 - 6)!]0.7⁶(1 - 0.7)^{8 - 6} = 0.296. The full distribution is depicted in Figure 2.

View larger version (16K): [in this window] [in a new window] [Download PPT slide]   Figure 2. Binomial distribution, B(i|n,p), for the number of cancers detected out of a total of eight, if the sensitivity of the detector is P = .70.

The distribution depends on two values or parameters: the mean, µ, and the SD, . (See the preceding article [7] in this series.) The mean determines the location of the high point of the curve. The SD gives the scale. The height of the curve at any point, x, is determined by the "z score," the difference of x and µ in units of . That is, the height depends only on z = (x - µ)/.

Again, the height is found as a function in most spreadsheets and statistical software. With Excel software, it is given by NORMDIST. The distribution is depicted in Figure 3.

View larger version (10K): [in this window] [in a new window] [Download PPT slide]   Figure 3. The normal, or Gaussian, distribution.

	FOOTNOTES

 
Abbreviation: PPV = positive predictive value

	REFERENCES

TOP ABSTRACT INTRODUCTION THE EARLIEST DEFINITION OF... SUBJECTIVE VERSUS OBJECTIVE... CONDITIONAL PROBABILITIES AND... BAYES RULE AND POSITIVE... P VALUES, POWER, AND... PROBABILITY FOR CONTINUOUS... DISTRIBUTIONS REFERENCES

 

1. Rosner B. Fundamentals of biostatistics 5th ed. Pacific Grove, Calif: Duxbury, 2000.
2. DeMoivre A. The doctrine of chance London, England: W. Pearson, 1718.
3. Bayes T. An essay toward solving a problem in the doctrine of chances. Philos Trans R Soc London 1763; 53:370-418(Reprinted in Biometrika 1958; 45:293–315.).
4. Bernoulli J. Ars conjectandi Switzerland: Basel, 1713; (Reprinted in: Die werke von Jakob Bernoulli. Vol 3. Basel, Switzerland: Birkhäuser Verlag, 1975; 106-286.
5. Laplace PS. Théorie analytique des probabilités Paris, France: Ve. Courcier, 1812.
6. Gauss CF. Theoria motus corporum coelestium in sectionibus conicis solem ambientium Hamburg, Germany: F. Perthes et I. H. Besser, 1809.
7. Applegate KE, Crewson PE. An introduction to biostatistics. Radiology 2001; 225:318-322.[Abstract/Free Full Text]

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