Comparison of surveillancebased metrics for the assessment and monitoring of disease detection: simulation study about type 2 diabetes
 Ralph Brinks^{1, 2}Email author,
 Annika Hoyer^{1},
 Deborah B. Rolka^{3},
 Oliver Kuss^{1, 4} and
 Edward W. Gregg^{3}
DOI: 10.1186/s1287401703282
© The Author(s) 2017
Received: 19 July 2016
Accepted: 23 March 2017
Published: 11 April 2017
Abstract
Background
Screening and detection of cases are a common public health priority for treatable chronic conditions with long subclinical periods. However, the validity of commonlyused metrics from surveillance systems for rates of detection (or casefinding) have not been evaluated.
Methods
Using data from a Danish diabetes register and a recently developed illnessdeath model of chronic diseases with subclinical conditions, we simulate two scenarios of different performance of casefinding. We report different epidemiological indices to assess casefinding in both scenarios and compare the validity of the results.
Results
The commonly used ratio of detected cases over total cases may lead to misleading conclusions. Instead, the ratio of undetected cases over persons without a diagnosis is a more valid index to distinguish the quality of casefinding. However, incidencebased measures are preferable to prevalence based indicators.
Conclusion
Prevalencebased indices for assessing casefinding should be interpreted with caution. If possible, incidencebased indices should be preferred.
Keywords
Compartment model Incidence Prevalence Diabetes Chronic disease Undiagnosed disease Casefinding ScreeningBackground
Chronic conditions like coronary heart disease, type 2 diabetes, hypertension, cancer, osteoporosis, and dementia frequently have long periods wherein the condition is undiagnosed. Although the specific policies related to active population screening are sometimes controversial, the prevalence of these undiagnosed conditions can be substantial and the period of undiagnosis is sometimes a missed opportunity to implement preventive care to reduce the risk of subsequent morbidity. Because of the lack of direct data on casefinding, population surveys with information on prevalence of diagnosed and undiagnosed cases are often incorporated into indirect indices to make inferences about levels of casefinding that are occurring. For example, national monitoring of public health efforts in the U.S. includes the tracking of proportion of cases of total diabetes [1], chronic kidney disease [2], and hypertension [3], who are aware of their condition or have been diagnosed. Similar metrics are used in diverse international settings to assess the degree of awareness, treatment and diagnosis of hypertension [4], hyperlipidemia [5], and diabetes [6]. Although these metrics are intended to assess the degree of case detection that occurs in clinical and public health settings, the validity of these surrogates of the rate of casefinding has not been evaluated. In this analysis we examine common approaches to using crosssectional data and find that the choice of metric can yield vastly different conclusions about trends in detection, with some yielding misleading conclusions.
In this article, we use a recently developed multistate model and simulate two different scenarios about diabetes casefinding. Then, we apply and compare different metrics to assess the casefinding in the two scenarios.
Methods
Based on data from the national Danish diabetes register on diagnosed diabetes, we simulate two different scenarios about casefinding, in one scenario the prevalence of undetected disease increases over time, and in another scenario the prevalence of undetected disease decreases over time. For brevity, we will denote the two scenarios as PU^{+} and PU^{−}. Here, PU means prevalence of undiagnosed disease, and the + and – signs indicate the upward and downward trend in time, respectively.
For the simulation we use a previously published multistate model. We then apply different surveillancebased indices to the PU^{−} and PU^{+} scenarios and compare the results of different indices.
After a brief description of the multistate model, we describe the different indices for assessing casefinding. Then, details of the simulation are presented.
Multistate model
The transition rates λ _{ ℓ }(t,a), ℓ=0,1, and μ _{ k }(t,a), k=0,1,2, in the model and the percentages of persons in the states, the prevalences, are related by a twodimensional system of partial differential equations [7]. As in [7] let p _{ k }(t,a) denote the fraction of persons aged a at time t in state k. In epidemiological settings time t is also called period. For example, p _{2}(t,a) is the fraction of persons in the population who are aged a at time t and are in the Diagnosed state (k=2). It can be calculated as \(p_{2}(t, a) = \tfrac {N_{2}(t, a)}{N(t, a)}\) where N _{ k }(t,a) denotes the number of people in state k, k=0,1,2, aged a at time t and N(t,a)=N _{0}(t,a)+N _{1}(t,a)+N _{2}(t,a). Similary, the prevalence of the undiagnosed disease (p _{1}) is defined as \(p_{1}(t, a) = \tfrac {N_{1}(t, a)}{N(t, a)}.\)
where z=p _{1} (μ _{1}−μ _{0})+p _{2} (μ _{2}−μ _{0}).
The rate λ _{0} can be interpreted as the true incidence or total incidence of type 2 diabetes. It describes how many persons develop diabetes in the considered population – irrespective of whether diabetes is diagnosed or not. The total incidence λ _{0} is affected by the aetiologic risk profile of the population. If risk factors become more prevalent in the considered population over time, the rate λ _{0} will increase accordingly. Vice versa, if risk factors become less frequent, λ _{0} will go down.
The rate λ _{1} describes the transition from the undiagnosed state to the diagnosed state. While λ _{0} is affected by aetiological factors, λ _{1} mainly depends on societal factors, e.g., medical progress in detecting the disease, awareness of patients and physicians, reimbursement of diagnostic testing etc. Epidemiologists are interested in both rates, λ _{0} as it reflects the risk profile of a population and λ _{1} as a societal construct. However, if an epidemiologist refers to “the incidence” of a chronic disease and the undiagnosed state is ignored, this “incidence” essentially refers to λ _{1}. Hence, it allows little inference about the changes of risk profiles in the population.
Indices for assessing casefinding
For better clarity, we categorize the indices on whether they are based on prevalence or transition rates in the multistate model. The last category refers to an index, which is combined from prevalence and transition rates.
Indices based on the prevalence
The simplest index is the prevalence of the undiagnosed disease (p _{1}). A high or an increasing value of p _{1} is intuitively considered to be unfavourable. For example, if p _{1} is 5% in a specific age group at a specific point in time, and is 10% two years later, then there is an (unfavourable) accumulation of undetected cases during these two years.
The reciprocal of ω _{1} describes the factor the diagnosed cases have to be multiplied with to obtain the number of all cases of the chronic disease. If ω _{1} equals 0.5, for instance, this means that for each detected case there is one undetected case. Obviously, it holds 0≤ω _{1}≤1. A high value in ω _{1} is usually interpreted as advantageous with respect to casefinding [1].
This proportion relates the number of persons in the undiagnosed state to all persons who do not have a diagnosis, i.e., the healthy and the undiagnosed. The idea behind the measure ω _{2} is that casefinding can be thought of the task of distinguishing persons from a pool consisting of healthy and undiagnosed persons. This pool of healthy and undiagnosed persons may be seen as the search space. The search space is subject to the activities of casefinding. Once an undiagnosed person is identified as a case, this person gets a diagnosis and is removed from the search space henceforth. As the disease under consideration is chronic, there is no way back into the search space, i.e., no remission can occur. In contrast to ω _{1}, the figure ω _{2} just refers to the persons who are at risk for a possible diagnosis. Thus, the fraction of persons with a diagnosis does not play a role for ω _{2}. The reciprocal of ω _{2} is the average number of persons without a diagnosis a physician must see to meet one undiagnosed case.
Again, it holds 0≤ω _{2}≤1. Ideally, ω _{2} is 0, i.e., all undetected cases become diagnosed and are removed from the search space. The closer ω _{2} approaches 1, the more the search space is dominated by the undiagnosed persons. Thus, a lower value of ω _{2} is advantageous in assessing casefinding. This is consistent with the interpretation of ω _{2} being the reciprocal of the average number of persons without diagnosis a physician must meet to have one undiagnosed case. The lower ω _{2} is, the greater the reciprocal is. Hence, a lower ω _{2} implies a higher average number a physician must meet to find an undiagnosed case. This means, casefinding worked well in the time before the physician met these persons.
Indices based on the transition rates
is equivalent with increasing p _{1} in (t,a), i.e., p _{1}(t,a)<p _{1}(t+δ,a+δ) for small δ>0.
Then, Φ _{ γ }(t,a) is the number of persons aged a at time t, who become incident undiagnosed cases at t and die within γ time units without diagnosis. These originally healthy persons never had the chance of obtaining a treatment.
Composite indices
A proof for this relation can be found in the Appendix. As the CDR includes the transition rate ratio DR_{0} and the prevalencebased index ω _{2}, we call CDR a composite index for assessing casefinding.
Summary of all indices for assessing casefinding
Index  Formula  Remark 

ω _{1}  \(\tfrac {p_{2}}{p_{1} + p_{2}}\)  Percentage of diagnosed over total cases 
ω _{2}  \(\tfrac {p_{1}}{p_{0} + p_{1}}\)  Inverse of the average number of persons without a diagnosis whom a physician must see in order to meet one undiagnosed case 
DR_{ γ }(t,a)  \(\tfrac {\lambda _{1}(t + \gamma, a + \gamma)}{\lambda _{0}(t, a)}\)  Rate ratio of diagnosing a person exactly γ years after contracting the disease 
CDR  DR_{0} ω _{2}  Case detection ratio 
Φ _{ γ }(t,a)  λ _{0}(t,a) P γ(dead)(t,a)  Number of healthy persons aged a at t who die with at most γ years of undiagnosed disease 
Simulation
Carstensen and coworkers have described an increase of the incidence of diagnosed diabetes in Denmark during the second half of the 1990ies [9]. The data this estimate was based upon stem from a nationwide diabetes register with a catchment population of more than 5 million people. The data about diagnosed diabetes from Denmark are very comprehensive and very detailed. Unfortunately, there are no data of similar high quality and details about undiagnosed diabetes for Denmark during that time. So, we use the rates reported in [9] and additional assumptions to include the state of undiagnosed diabetes in our simulation. The scenarios are hypothetical, because the Danish register does not report data about undiagnosed diabetes. The aim of this work is not to state that the situation of undiagnosed diabetes in Denmark has been as we describe, we are only interested in simulating somewhat realistic situations that could have been. Our aim is the exploration and comparison of how the indices of casefinding perform in these scenarios. We confine ourselves to the data from the male Danish population.
Together with an initial condition, the simulation uses the rates λ _{ ℓ }, ℓ=0,1, and μ _{ k }, k=0,1,2 as described below and integrates Eqs. (1) and (2) to obtain the functions p _{ k }, k=1,2. We will numerically integrate the system (1)–(2) using the Method of Characteristics [10] and RungeKutta numerical integration [11]. After numerically integrating system (1)–(2), we apply the epidemiological indices that have been described above.
Initial condition
The agespecific prevalence of diagnosed diabetes in the male Danish population in the year 1995 serves as the initial condition for p _{2}. Additionally, we assume that the prevalence of undiagnosed diabetes (p _{1}) is half of p _{2}.
Mortality
The mortality rate μ _{0} in our simulation is chosen to be the mortality of the nondiabetic population in Denmark. Following [9], a yearly decrease of 2.5% for all ages is assumed. Because there is evidence that the allcause mortality of men with undiagnosed diabetes is elevated by 25% as compared to normoglycemic men [12], we assume μ _{1}=1.25 μ _{0}. The mortality of men with diagnosed diabetes (μ _{2}) is taken from the Danish register.
Transition rate λ _{0}
Transition rate λ _{1}
The rate λ _{1} is chosen as a sigmoid function (right part of Fig. 2). It mimics an awareness for diabetes, which increases from age 15 until some saturation level at about 40 years of age. After an age of 40 years, the level remains constant. From year 1995 to year 2000, we assume that there is no annual trend in λ _{1}. During 2000–2005, for λ _{1} we assume an annual change of −5% and +15% in the PU^{+} and the PU^{−} scenarios, respectively. These choices are merely assumptions because we do not have any empirical data about λ _{1}.
Based on the compartment model shown in Fig. 1 and the data from the Danish register, the indices summarized in Table 1 are applied to the two scenarios PU^{+} and PU^{−}. Then, the results are compared. All calculations for this work have been performed with the statistical software R (The R Foundation for Statistical Computing). The R source files for running the simulation and applying the indices for assessing the casefinding are provided as an Additional file 1 to this manuscript.
Results
Scenario of increasing prevalence of undiagnosed disease
The index ω _{2} (right part of Fig. 4) finds that casefinding worsens during the five years from 2000 to 2005 for all ages older than 40 years. For lower ages, the index does not show a difference between the years 2000 and 2005.
Scenario of decreasing prevalence of undiagnosed disease
For all ages above about 45, there is a decrease of the prevalence of undiagnosed diabetes during the period from 2000 to 2005. The agespecific prevalences of undiagnosed and diagnosed diabetes in year 2000 agree with the corresponding curves in the PU^{+} scenario (see Fig. 3). For virtually all ages, there is a decrease of the prevalence of undiagnosed diabetes during the period 2000–05. During the same period the prevalence of the diagnosed disease increases for all ages.
For comparison with the PU^{+} scenario, consider 100,000 healthy persons aged 90 in year 2000. Again, 1587 develop diabetes during that year. During the next five years, 713 (45%) of these 1587 die without obtaining a diagnosis. In 2005, from those 2055 persons who become diabetic in that year 566 (28%) die without a diagnosis during the next five years. This is a substantial reduction, which indicates an improvement in casefinding.
Summary of the different indices of assessing casefinding
Prevalence of undiagnosed disease  

Index  Increasing (PU^{+})  Decreasing (PU^{−}) 
ω _{1}  (+)  + 
ω _{2}  –  + 
DR_{0}  –  + 
DR_{5}  –  + 
CDR  –  + 
Φ _{5}  (–)  (+) 
Discussion
Based on a recently developed multistate model, we simulated two hypothetical scenarios about trends in undiagnosed type 2 diabetes. In one scenario the incidence rate of diagnoses decreased over time reflecting a worsened casefinding. In the other scenario, the performance of casefinding improved. Several indices for assessing casefinding have been applied to the two scenarios.
We found that the measure ω _{1} leads to an inconsistent assessment of the casefinding performance in our scenarios. In the scenario of increasing prevalence of undiagnosed disease (PU^{+}), ω _{1} indicates an improvement whereas all other indices indicate a worsening of the quality of casefinding.
Thus, we can see that ω _{2} plays a direct role in determining if the prevalence of the undiagnosed disease is increasing or decreasing.
Similarly, the detection ratios DR_{ γ } assess the different simulation scenarios for γ=0 and γ=5 consistently with the other indices (except for ω _{1}). The figure Φ _{5} is an important measure, which refers to a cohort of healthy persons who contract the disease but never get the chance of being treated.
To our knowledge, the disease model in Fig. 1 has only been reported in [7]. In contrast to existing state models with a compartment preceding the diagnosis, typically called preclinical state [15, 16], our model includes the possibility of dying from the undiagnosed (preclinical) state. In diabetes, there is a considerably increased mortality from this state [12].
In the literature, there is another index to assess casefinding, the mean sojourn time (MST) in the preclinical phase (see [17] for a review). Usually, a low MST is considered advantageous. However, the MST may be low if the mortality from the undiagnosed state is high. Thus, the MST is not an appropriate figure for evaluating casefinding if there is considerable mortality from the undiagnosed (or preclinical) state. Hence, the MST has not been considered in this work.
Apart from the systematic comparison of the different indices, we introduced two new indices ω _{2} and Φ _{ γ }. The later quantifies the number of healthy persons who become incident undiagnosed cases and die before ever obtaining a diagnosis. In populations with a high incidence (i.e., large λ _{0}) and a poor casefinding (low λ _{1}), the number Φ _{ γ } will be high. This can be observed in the two simulation scenarios. While in the PU^{+} scenario Φ _{5} increases from 2000 to 2005, there is a large decrease of Φ _{5} in the PU^{−} scenario. In such cases, it would be interesting to compare this index for different strata of people in the considered population, e.g., low vs. high age, or low versus high socioeconomic status.
Compared to the other figures, the measure \(\omega _{2} = \tfrac {p_{1}}{p_{0}+p_{1}}\) has the advantage that it requires prevalence data only. Prevalences can be obtained from crosssectional studies. Those measures that include the incidence rates either require costly followup data or the application of specialized estimation techniques [7].
The question arises, how the different indices may be surveyed. The prevalencebased metrics ω _{1} and ω _{2} can be estimated from crosssectional surveys that comprise estimation of diagnosed and undiagnosed disease. A variety of crosssectional studies ask participants about prior diagnoses, for instance, in case of diabetes [18–20] or hypertension [21].
The indices based on the transition rates can be estimated by studies with followup information such as cohort studies. A practical demonstration using data from the Health and Retirement Study has been described in a recent manuscript [7], which also illustrates how to deal with statistical uncertainty.
Although the data and examples of this manuscript are given primarily for diabetes, the presented concepts can be applied in other chronic diseases like those mentioned in the introduction. For example, the estimation of ω _{1} and ω _{2} in a nationally representative study [21] about hypertension is straight forward.
Conclusion
This article compiles and compares several indices for casefinding in the field of chronic diseases. To our knowledge, this is the first systematic comparison of indirect surveillancebased metrics for the monitoring of case finding. We found that in assessing casefinding, incidencebased metrics should be preferred, because they gain a deeper insight into the dynamics of the compartment system than prevalencebased indices. Eqs. (1)–(2) show that the prevalences p _{ k }, k=1,2, depend on all transition rates in the multistate model, on the incidence rates λ _{ ℓ }, ℓ=0,1, and also on the mortality rates μ _{ k }(t,a), k=0,1,2. In this sense, the prevalences are the result of a complex interplay of the transition rates, both incidence and mortality rates.
Thus, prevalencebased indices of casefinding should be interpreted with caution, because they may lead to inconsistent findings (ω _{1}). If prevalencebased metrics have to be used, the new index ω _{2} is preferable over ω _{1}, because ω _{2} has shown to be the better index to distinguish the simulated scenarios of increasing or decreasing prevalence of undiagnosed disease.
Indices based on transition rates provide more insights into the system. The detection ratio (DR) provides a necessary and sufficient condition whether the prevalence of undiagnosed disease is increasing or decreasing. Moreover, only the true incidence rate λ _{0} indicates if the distributions of the underlying risk factors of the considered chronic disease changes. Thus, if the incidence of a chronic disease is used for surveillance, it is important to always consider the Undiagnosed state. If this state is ignored, and instead the combined state of Normal and Undiagnosed is considered, the rate λ ^{′} surveyed (see Appendix). Changes in the rate λ _{1} reflect societal processes like medical progress and changes in patients’ or physicians’ awareness.
In summary, these simulation analyses identify potential pitfalls of commonlyused indirect measures of case detection. These findings suggest that the ω _{2} term (proportion of all persons without a diagnosis who have the disease) is a preferable index to the ω _{1} term (proportion of all cases who have been diagnosed). However, these findings also underscore the need to use more direct estimates of incidence itself and to incorporate more direct estimates of detection into ongoing chronic disease surveillance systems.
Appendix
Abbreviations
 CDR:

Case detection rate
 DR:

Detection rate ratio
 PU^{−} :

Improving casefinding
 PU^{+} :

Worsening casefinding
Declarations
Acknowledgements
Not applicable.
Funding
The authors have not recieved any funding with respect to this work.
Availability of data and materials
The data used in this article were completely published in [9]. No further collection of individual persons’ data has been accomplished. All results can be reproduced from the source files provided as Additional file 1. The source files can be run with the freely available statistical software R (The R Foundation for Statistical Computing).
Authors’ contributions
EWG had the initial idea for this project. RB suggested the indices ω _{2} and Φ _{ γ }. RB developed and analysed the simulation and drafted the manuscript. RB, AH, DBR, OK and EWG critically revised the text, gave important intellectual contributions and final approval of the version to be published.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable as no collection of individual persons’ data has been accomplished.
Ethics approval and consent to participate
Not applicable as no collection of individual persons’ data has been accomplished.
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Authors’ Affiliations
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