From the Division of Biostatistics and Medical Informatics, University College of Medical Sciences, Dilshad Garden, Delhi 110 095, India and *Institute of Cytology and Preventive Onco-logy, Maulana Azad Medical College Campus, New Delhi 110 002, India.

Reprint requests: Dr. A. Indrayan, Professor of Biostatistics, Division of Biostatistics and Medical Informatics, University College of Medical Sciences, Dilshad Garden, Delhi 110 095, India.

We earlier defined biostatistics as the science of management of uncertainties in health and disease(1). In the previous articles of this series(2,3), we also included the numerical and graphical methods to summarize data. The sheet anchor of all this is the measure-ment because it is through measurements that quantities are obtained, and variations and uncertainties aptly studied. Thus, biostatistics can also be perceived as the quantitative aspects of health and disease. The qualitative aspects such as symptoms and relief are more important in evaluating an individual’s health but that is the domain of a clinician.

Some quantitative measures commonly used for assessing the health of a child are Apgar score, respiration rate, body temperature, and weight for age. Measurements are used not only in assessing the levels of health and its various components but also in assessing the disease severity. One example is Yale’s observation scale(4) which is used to identify serious illness in febrile children. Measurements are also used in establishing and interpreting the reference values of various medical parameters, in evaluating probailities in diagnosis and patient management, in assessing the validity of medical tools, etc.

The usual practice in medicine is to evaluate various parameters of a subject against a single or a range of reference values. The methodology generally used to delineate such reference values as well as their implications are discussed in Section 6.1. Because of variations and un-certainties, this assessment is done in terms of probabilities. These are discussed in Section 6.2. Section 6.3 is on assessment of validity of diagnostic tests.

Reference values are extensively used for decisions on managing patients. It is known that the normal body temperature in humans is 98.6° F and an Apgar score of 8 or more is considered normal. A birth weight of <2.5kg is con-ventionally considered low. A ponderal index [(weight in g/length in cm3) *100] ³2.5 is considered normal in neonates and a child with index < 2.0 is classified as low ponderal index. The anthropometric measurements are assessed by the percentile point achieved by a child relative to the healthy children of that age and gender in the same population. Median is regarded as a reference value, and 3rd and 97th percentiles as the thresholds to indicate abnormally low and abnormally high values.

Weight for age and height for age are the most commonly used indicators to assess growth but are more effective when the trend over age for the same child is studied. The interpretation and comparison of anthropometric measurements with reference values is some-times performed by computing an index called Z-score. This for weight is given by

Z–score = (Weight – Median)/SD

where Median and SD are calculated for the reference healthy population of that age or of that height. A Z–score below –2 is considered low and below –3 very low.

The other index used to assess growth is "percent of median". A measurement below 80% of median is regarded to indicate under-nutrition of Grade I; below 70%, of Grade II; below 60%, of Grade III; and below 50%, of Grade IV. For weight measurement, a velocity of less than normal for a younger age group indicates failure to thrive. Recently, a 3-in-1 weight monitoring chart for infants has been developed(5). Velocity is the rate of growth per unit of time. This is higher at the beginning of life and tapers off as age increases, with a slight upswing at 6 or 7 years and a spurt in adole-scence. Preece and Brans(6) have developed models that can be used to evaluate height parameters such as age at take-off and peak height velocity.

The above discussion indicates that the measurements are evaluated against reference or normal values. The evaluation can be less risky and more meaningful if the basic principles of establishing such normals are known.

Establishment of normals needs an understanding of the distributional aspects of the measurements. You may like to revisit a previous Article of this series(3) and refresh yourself with the frequency distribution of measurements along with their histogram, frequency polygon and frequency curve. The shape of the distribution of a measurement such as birth weight in healthy babies is nearly symmetrical. The frequencies are high in the center and they rapidly decline on either side in almost a similar fashion. The measurements such as cholesterol level, serum iron and blood pressure (BP) in healthy subjects also tend to follow a symmetric shape, called Gaussian.

This distribution is symmetric about mean and has a shape of a bell. The shape of the curve is as shown in Fig. 1 for the distribution of serum iron in healthy adults. This has the properties such as (i) mean, median and mode coincide, and (ii) the limits from (mean - 2 SD) to (mean + 2 SD) cover the measurements of nearly 95% subjects. Such a distribution is also called "normal distribution" but we avoid using this term because normal has a different meaning in medicine.

While many medical measurements in healthy subjects do indeed follow a Gaussian pattern, all do not. The distribution of blood lipids in children has a long tail on the right because higher level is more common than lower level. This is called a right-skewed distribution. On the other hand, the distribution of hemoglobin level is generally left-skewed because lower values are quite common. Fig. 2 shows the age-distribution of deaths in a population of a developing country. This has a bathtub shape and is entirely different from a Gaussian pattern.

Fig1. Distribution of serumiron in healthy subjects (smothened curve)

Normal or Reference Values

In this series, we use the two terms–reference value and normal value–inter-changeably for values generally seen in healthy subjects. The reference values could be separate for different segments of the population. A level of BP seen normally in adults would not be normal for children. A normal weight of 2-year olds in Sudan may not be the same as normal weight of 2-year olds in Sweden. Normals may also change from time to time.

The normal values or reference values are based on measurements of healthy subjects, preferably the most healthy segment of the population. Generally, not less than 200 subjects should be included in each group for which normal values are to be obtained. Normal values are obtained generally by mean but sometimes also by median and mode. See our previous Article(2) for situations where median is preferable or mode is preferable. When the inter-individual variation in healthy subjects is large, a single normal value is not sufficient and we need a range of normal values.

Even though normal is the level generally seen in healthy subjects, there would always be persons with very high or very low values yet absolutely healthy. The usual practice in such cases is to exclude 2.5% subjects on either side from the range of normality. This is arbitrary and purely statistical but has become a con-vention in the absence of any other acceptable criterion.

When the distribution is Gaussian, property-(ii) is invoked to say that (mean –2 SD, mean +2 SD) are the normal limits. They exclude 2.5% subjects with extreme measurements on either side from the range of normality. These are popularly known as ±2SD limits. Most of the normal ranges used in medical practice are obtained in this manner. In case of birth weight, if the mean in healthy babies in a population is 3.3 kg and SD 0.2 then the normal range is 2.9 kg to 3.7 kg for that population.

The use of ±2 SD limits as reference is not without risk. These limits in any case exclude those 5% of healthy subjects who have very low or very high values. In addition, many subjects with disease may have values well within a normal range. Fig. 3 illustrates this overlap. This figure incidentally also illustrates the wider dispersion generally seen among the diseased subjects compared to the healthy subjects. This overlap gives rise to false positivity and false negativity about which we discuss in Section 6.3. Thus there is always a risk of misdiagnosis and missed diagnosis in marginal cases. Such risk or uncertainty is measured as follows.

Fig2. Distribution of deaths by age at death in a developing country

An accepted measure of uncertainty is probability. The term has everyday meaning but its computation could be nerve-wrecking in some intricate cases. Mathematically speaking, an event which is impossible to occur, such as human male giving birth to a child, has a probability zero. The event which is certain to occur, such as death, has probability one. Statistical definition is based primarily on empiricism and thus is milder. If a women of age 54 years has never been seen to conceive in the history of a community, the statistical probability of occurrence of such an event in that community is zero. It does not necessarily imply that the event is an impossibility. No probability could be negative nor can it exceed one. Probabilities have extensive usage in medicine. When the first heart transplantation was done, the chances of success were rated as 80%. The probability of recovery of a patient of tetanus, after manifestation, is less than 50%. The chance of one year graft survival of children with renal transplantation is 80%(7). Thus probabilities have extensive usage.

Fig. Overlap of values in healthy and diseased subjects.

Probability measures the likelihood of an event and is complementary to uncertainty. An interpretation of probability is the relative frequency of an outcome in a large number of cases. This can be stated as

Probability of an outcome = Number of cases with the desired outcome / Total number of cases

If the records of a community show that the occurrence of Down syndrome is 1 in 700 live births, then the probability of that complaint is 1/700 = 0.0014. Similarly, probability of occurrence of one or more diseases together can also be computed. Some laws of probability are helpful in this context.

Occurrence of blindness and deafness in a child are independent events in the sense that occurrence of one does not increase or decrease the chance of occurrence of the other. For such independent events, the joint probability of the two occurring together in a child can be easily computed by the product of the individual probabilities. Thus P (blindness and deafness) = P (blindness) ´ P (deafness). This is called law of multiplication of probabilities.

After corrective surgery for residual paralysis in paralytic polio, the recovery could be full, partial or none. In such mutually exclusive categories, the probability of belong-ing to one or the other is computed by the law of addition. That is,

P (full or partial recovery) =P (full recovery) + P (partial recovery).

If the probability of full recovery is 0 . 30 and of partial recovery 0.40 then the probability of at least some recovery is 0.70.

We find diagnosis an easy portal for communicating the concept of probability. But the usage in other clinical activities is equally common.

Sometimes diseases are described in the literature in the form of complaints commonly seen in that disease. The track then is from disease to complaints. The actual diagnostic process is just the reverse, from complaints to the disease. Suppose the analysis of records show that 60% children of tuberculous meningitis (TBM) presented with complaints of fever, altered sensorium and convulsions. Then P (fever, altered sensorium, convulsions/TBM) = 0.60, where P denotes probability. Because of restriction to a specific group, which is mentioned after the slash (/) sign, such probability is called conditional probability. Note that this probability of signs and symptoms in a particular disease is of very little value to a clinician. The inverse probability, P (TBM/fever, altered sensorium, convulsions) is useful because it gives the diagnostic value of the complaints. Bayes’ rule helps to calculate the latter probability from the former.

Use of Bayes’ rule: The probabilities actually required in practice such as P (disease/complaints) can be obtained from P (complaints/disease) using Bayes’ rule. This is given below.

P (Disease/Complaints) = P (Complaints/Disease) X P (Disease) / P (Complaints)

P (Disease) is the prevalence of the disease in the subjects under investigation. This is generally available from various reports or books, or can be derived from records. The second is P (Complaints) which is the relative frequency of the complaints in the subjects. Special efforts may have to be made to compute this. Once these two are known, the required inverse probability can be calculated. For example, P (Infant death/Low birth weight) can be computed from P (Low birth weight), say 23.5%, P (Low birth weight/Infant death), say 30.4% and P (Infant death), say 5.9%. The required probability of infant death in children with low birth weight (LBW) then is given by

P (Infant death/LBW) = P (LBW/Infant death) * P (Infant death) / P (LBW)

This in this case is (0.304 ´ 0.059)/0.235 = 0.076 or 7.6%. Note that this is very different from P (LBW/Infant death).

The tools used for evaluation and management of health and disease are seldom perfect. They produce correct results in many cases but not in all the cases. The ability of a tool or of a procedure to correctly perform its assigned function is called its validity. A valid diagnostic test would correctly detect the presence as well as the absence of the disease. For more details of this concept refer to Griner et al.(8). Some tests are more valid than others though they may be more expensive also. Western Blot is considered more efficacious than ELISA in detecting HIV positivity. In contrast to blood culture, the C-reactive protein (CRP) is utilized as a rapid diagnostic test for septicemia in children. Fine needle aspiration cytology (FNAC) for tissue diagnosis is nearly as valid as tissue biopsy.

Malaria is characterized by high fever with chills and rigors, splenomegaly and a positive blood smear. How valid is this set of criteria? Can it correctly identify all the cases of malaria and can it correctly exclude all the non-malarial cases? We discuss these two aspects in the following paragraphs.

The ability of a test to give positive results in true cases of diseases is called sensitivity. Specificity is the ability to give negative results in cases not suffering from the disease. These are two components of validity of a test. The components are best illustrated with the help of an example.

Example 1: In rural areas, where measuring weight is not feasible due to logistic problems, an alternative measure is mid-arm circumference (MAC). This is considered age and sex independent for detecting malnutrition between the ages 12-60 months. MAC was measured for 453 children of preschool age(9). They were also assessed for grade of malnutrition by weight for age criteria. MAC was used as a test criteria for detecting malnutrition with weight for age as the gold standard. The results obtained are recorded in Table I. The following can be noted.

True positives (TP) = 45

False positives (FP) = 67

True negatives (TN) = 330

False negatives (FN) = 11

Sensitivity and specificity can be calculated as:

Sensitivity =TP / TP + FN,
and Specificity =TN / TN + FP

Both can be converted to percentage by multiplying by 100. Sensitivity and speficity of MAC against weight for age are 0.804 and 0.831 respectively. These are converted as percentages and shown in Table I.

The actual problem in practice is to detect the presence or absence of a suspected disease by using a test. The diagnostic value of a test is measured by the probability of presence of disease among those who are test positives, and the probability of absence of disease among those who are test negatives. These indicators are called positive predictivity and negative predictivity respectively. These are also called post-test probabilities.

In terms of notations,

Positive predictivity = TP / TP + FP

and Negative predictivity =TN / TN + FN

Predictivities are severly affected by the prevalence of disease among those tested.On the other hand, sensitivity and specificity are absolute and do not depend on prevalence. The predictivities for some specific values of sensitivity and specificity, and for different prevalences are shown in Table II. As the prevalence increases the positive predictivity also increases and this increase is more pronounced when specificity is low. Higher prevalence leads to less negative predictivity, more so when sensitivity is low. In summary, the calculation of predictivities should be done on subjects that correctly represent the propor-tion of diseased and non-diseased cases among those who are to be tested.

The dependence of predictivities on the "prevalence" is to be cautiously interpreted. This prevalence is among those who are administered the test. A diagnostic test is generally adminis-tered to those who are suspected to have the disease and in them, the proportion with disease is likely to be high. This proportion is the same as prevalence in the sense used here. When this is high, it becomes difficult to correctly identify the negatives.

Another very useful interpretation of prevalence is the extent of belief or of confi-dence that a clinician has in a particular subject for the presence of disease. On the basis of the information available on the subject before the test, if a clinician evaluates that the chance of disease in that subject is 60% then this has exactly same connotation as prevalence. Thus prevalence can also be understood as the pre-test probability. Predictivities can be assessed using this probability in place of prevalence.

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