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There has been a growing interest in screening programs designed to detect chronic progressive cancers in the asymptomatic stage, with the expectation that early detection will result in a better prognosis. One key element of early detection programs is a screening test. An accurate screening test is more effective in finding cases with early-stage diseases. Sensitivity, the conditional probability of getting a positive test result when one truly has a disease, represents one measure of accuracy for a screening test. Since the true disease status is unknown, it is not straightforward to estimate the sensitivity directly from observed data. Furthermore, the sensitivity is associated with other parameters related to the disease progression. This feature introduces additional numerical complexity and limitations, especially when the sensitivity depends on age. In this paper, we propose a new approach that, through combinatorial manipulation of probability statements, formulates the age-dependent sensitivity. This formulation has an exact and simple expression and can be estimated based on directly observable probabilities. This approach also helps evaluating other parameters associated with the natural history of disease more accurately. The proposed method was applied to estimate the mammography sensitivity for breast cancer using the data from the Health Insurance Plan trial.

Screening of asymptomatic individuals for chronic diseases is a rapidly growing public health initiative. Early detection programs are aimed at detecting the disease in the stage when a disease is present without symptoms. For example, in breast cancer, there had been eight randomized screening trials demonstrating that mammography screening is beneficial in finding breast cancer at an earlier stage and consequently leads to a decrease in mortality among women of 50 - 65 years of age ([

One key element of these programs is a screening test. An important measure for the effectiveness of a screening test is sensitivity (b), the conditional probability of getting a positive screening test result given that one has the disease. Ideally it should be evaluated in the setting of natural history of disease model ([

These parameters are not directly observable and add to the complexity of the problem formulation and numerical computation of the sensitivity. Furthermore, the sensitivity can be age-dependent. For example in breast cancer the mammogram sensitivity in younger women is lower than that in older women [

In this paper, we derive a formula that expresses age-dependent sensitivity in terms of probabilities that are directly observable in screening trials/programs and discuss the characteristics and generalization of this formula. We apply the formula to breast cancer screening trial (Health Insurance Plan) data to estimate the mammography sensitivity [

Consider the following health states in the natural history of disease progression:

·

·

·

A progressive disease model assumes that the disease progresses in the direction

·

·

·

·

Note that

We now introduce two important probabilities that all the subsequent derivations are based upon. Suppose there are no screening examinations before age a, and a series of examinations are scheduled (but not necessarily taken) at times

·

·

The importance of

Note that it is implied in the definition of T and X that by the time of the last examination, one would either be in

We derive a formula for

· before time 0 (or age a), go undetected at time 0, and stay in

· between time 0 and

Therefore,

Or,

Similarly, for

· before age a and go undetected at times 0 and

· between ages a and

· between ages

Therefore,

From (1), we have

which gives

Observing (1) and (2), we arrive at our main result in this section:

Theorem 2.1.

Therefore by writing out a few recursions, we are able to eliminate all

Let

Corollary 2.2. When at least three repeated tests with the same sensitivity

The result is achieved by simply taking limits

Theorem 2.1 is far from being intuitive, but corollary is easily verified, when we realize that when

Advantage of Theorem 2.1. One main advantage of Theorem 2.1 is that it enables us to ignore the consideration for transition rate

where

Thus in just this one term, all the

However Theorem 2.1 allows us to bypass

Another important advantage of Theorem 2.1 is that once (age-dependent)

One limitation of the Theorem 2.1 is that it does not provide a way to apply the “interval cases”, or the cases that enter clinical stages between two scheduled screening exams. These cases, however, can be used later on to determine w and

An interesting mathematical result involves certain symmetry between X and T. We use the same setup as in the last section: suppose there are no examinations before age a, and a series of exams are scheduled (but not necessarily taken) at times

Under the same set up, we introduce the new notation:

where

We also recall how we have defined

Recall we derived the following expressions in the last section (notice that

or, after some reorganizing of the terms, and recognizing that

Therefore it is clear that there is a symmetric relationship between the T terms and the corresponding Y terms, in the case of 1, 2 and 3 examinations. The pattern of these expressions becomes more obvious in case of 4 examinations: (which will be proved along with the general case) to make it more obvious:

In the expression of

Theorem 3.1. For any

and

where S goes over all subsets of

Proof. Because we will be applying the inclusion-exclusion principle later, to make the corresponding arguments, we revise the definition of the

Let

·

·

·

Therefore we have

and

_{i} are mutually exclusive events. We also have

By inclusion-exclusion principle [

But

Or,

The second expression can be proved in a similar manner. With the same set up as the above, let

·

·

Recall that

and

Again by inclusion-exclusion principle,

However

Or,

Corollary 3.2. There are infinitely many expressions of age-specificity in terms of

Proof. This is because each

where

we saw in the last section that

Each

Given expressions like Theorem 3.1, the proof for Theorem 2.1 is not as straightforward. However the purpose of that proof was to relate some of the traditional considerations and reasoning’s to our new notions. This will be useful when evaluating important parameters such as age-dependent disease transition probabilities and sojourn time distribution.

We illustrate the proposed method using the data from the Health Insurance Plan Project, or HIP data [

To calculate, saying

Note that this estimate is essentially identical to Shen and Zelen’s published estimate of 0.7 ([

This work was supported by grants 1RC CA 146496-01 and 1R01 CA16430 from the National Cancer Institute/National Institute of Health.