Statistics in Epidemiology

Epidemiology

Epidemiology is the study of the occurrence of disease or other health-related characteristics in human and in animal populations. Epidemiologists study the frequency of disease and whether the frequency differs across groups of people; such as, the cause-effect relationship between exposure and illness. Diseases do not occur at random; they have causes. Many diseases could be prevented if the causes were known. The methods of epidemiology have been crucial to identifying many causative factors which, in turn, have led to health policies designed to prevent disease, injury and premature death.

Basic Terms

Epidemiology is an academic discipline which deals with the occurrence of diseases, their causes, events which lead to these diseases and their consequences. The most important terms which define epidemiology are incidence, prevalence, mortality and lethality of diseases.

Incidence is the number of instances of illness commencing, during a given period in a specified population.  The number of new health-related events in a defined population within a specified period of time.

Prevalence is the proportion of population who have specific characteristic in given time.  It is disease occurrence or other factor related to health, the total number of individuals who have the condition at a particular time divided by the population at risk of having the condition at that time or midway through the period.

Example: 400 people are tested for the common flue. 100 of them in the sample group are found to have the flue. Divide the 100 flue infected people by the total sample size, which is 400, the answer is the prevalence.

100/400= 1/4 or 1 out of 4 people or 25%.

Mortality determines how many people die in a certain time period. It can be measured with calculating the death rate: (Number of deaths during a specified period in the sample group)/(The total number of people in the sample groupe period)

Example: In the flue case 10 people died from the flue and 10 from a different source. So, divide the number of people who died by the total sample size: 20/400=1/20 or 0.5%, that is the mortality rate, notice that it is the TOTAL number of deaths, not just from the disease. For a more accurate measure use lethality rate.

Lethality of diseases is a ratio which is determined by the number of people who died in a certain time divided through the number of people who fell ill in the same time period. is a description of how a disease can cause death and harm.

Example: Let’s take the flue case as an example. 10 people have died from the flue out of the 100 that were sick. So: 10/100=1/10 or in other works there is a 10% chance to die from this disease, that’s the lethality.

Diagnostic Tests

Diagnostic tests are performed in the aim of determining the presence of a certain disease or illness in a patient. The test may be carried out through performing procedures, such as various scans or merely on the basis of symptoms. Some examples of diagnostic tests include X-rays, biopsies, pregnancy tests, medical histories, and results from physical examinations.

The results obtained from the test could be from either of the 2 distinct main categories- positive or negative, where a positive result indicates the presence of the diseases. A positive or negative result can be subdivided further into true positives and negatives, and false positives and negative results. A true positive result is one that accurately determines the presence of the illness. On the contrary a false positive result indicates the presence of the disease in the patient; however, the disease is actually not present in the patient. A similar pattern is seen in true negative and false negative results.

• True positive: the patient has the disease and the test is positive.
•  False positive: the patient does not have the disease but the test is positive.
• True negative: the patient does not have the disease and the test is negative

• False negative: the patient has the disease but the test is negative.

Table illustrating different types subcategories positive and negative test results

The Fourfold Table

The fourfold (confusion matrix) table is a type of contingency table with, which is a tabular cross-classification of data in which subcategories of one characteristic are indicated horizontally (in rows) and subcategories of another characteristic are indicated vertically (in columns) to test the characteristics between the two (the rows and the columns).

Sensitivity and Specificity of Diagnostic Test Calculated from Fourfold table

Sensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as classification test:

Sensitivity (also called the true positive rate, the recall, or probability of detection in some fields) measures the proportion of positives that are correctly identified as such (e.g., the percentage of sick people who are correctly identified as having the condition).

• Specificity (also called the true negative rate) measures the proportion of negatives that are correctly identified as such (e.g., the percentage of healthy people who are correctly identified as not having the condition).

The letters a, b, c, d symbolize the numbers in the fourfold table.

• a — stand for diseased individuals detected by the test.
• b — stand for healthy individuals detected by the test.
• c — stand for diseased individuals not detectable by the test.
• d — stand for healthy individuals negative by the test.

The formula for sensitivity — a / a+c.

The formula for specificity — d / b+d.

predictive value of a positive test result — a / a+b.

Predictive value of a negative test result — d / c+d.

Receiver Operating Characteristic (ROC) Curve

- Receiver Operating Characteristic (ROC curve) is a graphical plot that illustrates the performance of a binary classifier system

- The curve is created by plotting the true positive rate (TPR) against the false positive rate (FPR) at various threshold settings.

+) The true-positive rate (sensitivity): recall or probability of detection in machine learning.

+) The false-positive rate (fall-out or probability of false alarm): be calculated as (1 − specificity).

- The ROC curve is thus the sensitivity as a function of fall-out.

-  ROC has been used in medicine, radiology, biometrics, and other areas for many decades and is increasingly used in machine learning and data mining research.

- The ROC is also known as a relative operating characteristic curve, because it is a comparison of two operating characteristics (TPR and FPR) as the criterion changes

- A classification model (classifier or diagnosis) is a mapping of instances between certain classes.

Endometrial ultrasound example

Ultrasound can be used to detect thickening in the lining of the uterus, which may be an early sign of cancer. If abnormal, a biopsy or minor surgical procedure (dilatation and curettage or D&C) is needed. This is painful and invasive, and has some risk. So our goal is to maximize the number of true positives (correctly diagnosed cancers) with an acceptable number of false positives (false alarms requiring a biopsy).

Cutoff for abnormal wall thickness	Sensitivity (%)	Specificity (%)	1 - Specificity  (%)
> 4 mm	99	50	50
> 5 mm	97	61	39
> 10 mm	83	80	20
> 15 mm	60	90	10
> 20 mm	40	95	5
> 25 mm	20	98	2

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The example of diabetes disease prediction using the blood glucose level was used to illustrated the way these curves are generated. In this example, it is supposed that the normal levels for a healthy person of glucose in blood are 70-110 mg/dL (average value 90 mg/dl) corresponding to condition negative, and the glucose concentration in blood of a non-healthy person is 90 to 180 mg/dL (corresponding to condition positive). For this specific example, it is assumed that the sample of total population is 1000 members with 500 being non-healthy (condition positive) and 500 healthy (condition negative). The distribution of the healthy and non-healthy population is given in the tables 1 and 2, whereas their graphical presentation is given in Fig. 1.:

Based on data provided in Tables 1 and 2 (or in Fig.1), the parameters TP, FN, FP and TN, as well as the respective rates (TPR, FNR, FPR and TNR) were determined for five different cut-off (threshold) levels: 90, 92, 95, 100 and 105 mg/dL. The results for these exercises are presented tin the following Table 3.

Then, the pairs of FPR and TPR were plotted to give the ROC curve.

Table 4. Pairs of FPR, TPR for plotting

FPR	TPR
0	0
0.02	0.962
0.084	0.978
0.188	0.988
0.442	0.998
0.526	1
1	1

As seen in Table 3, the maximum ACC (accuracy) value was obtained in Case A3 examined using the value of 105 mg/dL as cut-off (threshold) value, therefore, this value should be used for the classification. Furthermore, it is easily seen that the ROC curve produced is very close to the perfect classification point (0,1), therefore, the classification is very effective.

Example of a Possible Screening and Confirmation Test in Medicine

In the screening test, the initial screening test have to be very sensitive. If anybody gets a positive results on that first screening test, we have to give them a second test which is the confirmatory test. This one would be very specific. The diagnostic of any diseases need to have both test positive. If we look at just either one, we will get high false positives or false negatives. But together both tests have relatively low probability false positive and false negative.

The high sensitive and high specific test are very important. For instance, in the case of screening donated blood from a blood bank for Bloodborne pathogens. Because the drawbacks of a false-negative are way higher than the drawbacks of a false positive. You may infect anybody easily so it has to be very sensitive. Alternatively, a disease that doesn’t have mortality or morbidity. There is a treatment for it but that treatment has a serious side effects that happen. In that case you have to be very specific with your diagnosis because the drawback of giving somebody a false positive and treating them with the potentially serious side effect of the drug would be a big drawback to consider.

In the real world, you never have a test that is fully Sensitivity and full Specificity. We are usually faced with a decision to use a test with high Sensitivity (and lower spec) or high Specificity (and lower Sensitivity). Usually a test with high sensitivity is used as the Initial Screening Test. Those that receive a positive result on the first test will be given a second test with high specificity that is used as the Confirmatory Test. In these situations, you need both tests to be positive to get a definitive diagnosis. Getting a single positive reading is not enough for a diagnosis as the individual tests have either a high chance of FP or a high chance of FN. For example, HIV is diagnosed using 2 tests. First an ELISA screening test is used and then a confirmatory Western Blot is used if the first test is positive. ,

There are also specific situations where having a high specificity or sensitivity is really important. Consider that you are trying to screen donations to a blood bank for blood borne pathogens. In this situation, you want a super high sensitivity, because the drawbacks of a false negative (spreading disease to a recipient) are way higher than the drawbacks of a false positive (throwing away 1 blood donation). Now consider you are testing a patient for the presence of a disease. This particular disease is treatable, but the treatment has very serious side effects. In this case, you want a test that has high specificity, because there are major drawbacks to a false positive. ,

Links

References

1.     Porta, M. A Dictionary of Epidemiology. Oxford University Press

2.     WikiLectures. Fourfold and Contingency Tables. 2014

3.     Setiabudhi Times. The 10 Sources of Psychological Myths: Your Mythbusting Kit. 2012

4.     Lalkhen, A. McCluskey, A. Clinical tests: sensitivity and specificity.

5.     PennState Eberly College of Science. Epidemiological Research Methods, 10.3 - Sensitivity, Specificity, Positive Predictive Value, and Negative Predictive Value

6.     4.6: Receiver Operating Characteristic Curves (author unknown) link: http://ebp.uga.edu/courses/Chapter%204%20-%20Diagnosis%20I/8%20-%20ROC%20curves.html

7.     Department of Statistics, University of California, One Shields Ave, Davis, CA 95616, USA. Estimation of diagnostic-test sensitivity and specificity through Bayesian modelling. 2005