It is often overlooked that the results of a statistical test depend not only on the observed data but also on the choice of statistical model. The statistician doing analysis of the data has a choice between several tests which are based on different models and assumptions. Unfortunately, many research workers who know little about statistics leave the statistical analysis to statisticians who know little about medicine; and the end result may well be a series of meaningless calculations.
Nothing can be further from the truth. The present paper endeavors to explain the meaning of probability, its role in everyday clinical practice and the concepts behind hypothesis testing. Probability is a recurring theme in medical practice.
No doctor who returns home from a busy day at the hospital is spared the nagging feeling that some of his diagnoses may turn out to be wrong, or some of his treatments may not lead to the expected cure. Encountering the unexpected is an occupational hazard in clinical practice. Doctors after some experience in their profession reconcile to the fact that diagnosis and prognosis always have varying degrees of uncertainty and at best can be stated as probable in a particular case.
Critical appraisal of medical journals also leads to the same gut feeling. One is bombarded with new research results, but experience dictates that well-established facts of today may be refuted in some other scientific publication in the following weeks or months.
When a practicing clinician reads that some new treatment is superior to the conventional one, he will assess the evidence critically, and at best he will conclude that probably it is true. The statistical probability concept is so widely prevalent that almost everyone believes that probability is a frequency. It is not, of course, an ordinary frequency which can be estimated by simple observations, but it is the ideal or truth in the universe , which is reflected by the observed frequency.
For example, when we want to determine the probability of obtaining an ace from a pack of cards which, let us assume has been tampered with by a dishonest gambler , we proceed by drawing a card from the pack a large number of times, as we know in the long run, the observed frequency will approach the true probability or truth in the universe. Mathematicians often state that a probability is a long-run frequency, and a probability that is defined in this way is called a frequential probability.
The exact magnitude of a frequential probability will remain elusive as we cannot make an infinite number of observations; but when we have made a decent number of observations adequate sample size , we can calculate the confidence intervals, which are likely to include the true frequential probability.
The width of the confidence interval depends on the number of observations sample size. The frequential probability concept is so prevalent that we tend to overlook terms like chance, risk and odds, in which the term probability implies a different meaning. Few hypothetical examples will make this clear. A probabilistic statement incorporates some amount of uncertainty, which may be quantified as follows: A politician may state that there is a fifty-fifty chance of winning the next election, a bookie may say that the odds of India winning the next one-day cricket game is four to one, and so on.
At first glance, such probabilities may appear frequential ones, but a little reflection will reveal the contrary. We are concerned with unique events, i. It follows from the above deliberations that we have 2 types of probability concepts. In the jargon of statistics, a probability is ideal or truth in the universe which lies beneath an observed frequency — such probabilities may be called frequential probabilities.
In literary language, a probability is a measure of our subjective belief in the occurrence of a particular event or truth of a hypothesis. Such probabilities, which may be quantified that they look like frequential ones, are called subjective probabilities. Bayesian statistical theory also takes into account subjective probabilities Lindley, ; Winkler, The following examples will try to illustrate these rather confusing concepts.
A young man is brought to the psychiatry OPD with history of withdrawal. He also gives history of talking to himself and giggling without cause. There is also a positive family history of schizophrenia. We ask the psychiatrist what makes him make such a statement. The statement therefore may not be based on observed frequency. Instead, the psychiatrist states his probability based on his knowledge of the natural history of disease and the available literature regarding signs and symptoms in schizophrenia and positive family history.
From this knowledge, the psychiatrist concludes that his belief in the diagnosis of schizophrenia in that particular patient is as strong as his belief in picking a black ball from a box containing 10 white and 90 black balls.
The probability in this case is certainly subjective probability. Let us consider another example: A year-old married female patient who suffered from severe abdominal pain is referred to a hospital. She is also having amenorrhea for the past 4 months. The pain is located in the left lower abdomen. As before, we ask the gynecologist to explain on what basis the diagnosis of ectopic pregnancy is suspected.
So in this case also, the probability is a subjective probability which was based on an observed frequency. One might also argue that even this probability is not good enough. We might ask the gynecologist to base his belief on a group of patients who also had the same age, height, color of hair and social background; and in the end, the reference group would be so restrictive that even the experience from a very large study would not provide the necessary information.
If we went even further and required that he must base his belief on patients who in all respects resembled this particular patient, the probabilistic problem would vanish as we will be dealing with a certainty rather than a probability.
Recorded experience is never the sole basis of clinical decision making. The two situations described above are relatively straightforward. The physician observed a patient with a particular set of signs and symptoms and assessed the subjective probability about the diagnosis in each case.
Such probabilities have been termed diagnostic probabilities Wulff, Pedersen and Rosenberg, In practice, however, clinicians make diagnosis in a more complex manner which they themselves may be unable to analyze logically. First a formal analysis may be attempted, and then we can return to everyday clinical thinking.
The frequential probability which the doctor found in the literature may be written in the statistical notation as follows:. However, such probabilities are of little clinical relevance. We of course do not suggest that clinicians should always make calculations of this sort when confronted with a diagnostic dilemma, but they must in an intuitive way think along these lines.
Clinical knowledge is to a large extent based on textbook knowledge, and ordinary textbooks do not tell the reader much about the probabilities of different diseases given different symptoms. The practical significance of this point is illustrated by the European doctor who accepted a position at a hospital in tropical Africa. In order to prepare himself for the new job, he bought himself a large textbook of tropical medicine and studied in great detail the clinical pictures of a large number of exotic diseases.
However, for several months after his arrival at the tropical hospital, his diagnostic performance was very poor, as he knew nothing about the relative frequency of all these diseases. The same thing happens on a smaller scale when a doctor trained at a university hospital establishes himself in general practice.
At the beginning, he will suspect his patients of all sorts of rare diseases which are common at the university hospital , but after a while he will learn to assess correctly the frequency of different diseases in the general population. Besides predictions on individual patients, the doctor is also concerned in generalizations to the population at large or the target population.
We may say that probably there may have been life at Mars. These probabilities are again subjective probabilities rather than frequential probabilities. It simply means that our belief in the truth of the statement is the same as our belief in picking up a red ball from a box containing 95 red balls and 5 white balls.
We can, I repeat, thus hypothesize so long as we see no impossibility. Consequently, we hypothesize the independent development of these dermal changes in S. Still, we can hypothesize , even if we cannot prove and establish. New Word List Word List. Save This Word! We could talk until we're blue in the face about this quiz on words for the color "blue," but we think you should take the quiz and find out if you're a whiz at these colorful terms.
Words nearby hypothesize hypothenuse , hypothermal , hypothermia , hypothesis , hypothesis testing , hypothesize , hypothetical , hypothetical imperative , hypothetically , hypothetico-deductive , hypothetico-deductive method. The two groups could not have exactly the same mean age if measured precisely enough such as in days. Perhaps a physician's age affects how long physicians see patients. There are innumerable differences between the groups that could affect how long they view patients.
With this in mind, is it plausible that these chance differences are responsible for the difference in times? Since this is such a low probability, we have confidence that the difference in times is due to the patient's weight and is not due to chance. It is very important to understand precisely what the probability values mean. This is not at all what it means.
It is not the probability that a state of the world is true. Although this might seem like a distinction without a difference, consider the following example. On each trial, a number is displayed on a screen and the bird pecks at one of two keys to indicate its choice. As a scientist, you would be very skeptical that the bird had this ability.
Certainly not! In statistics, it is conventional to refer to possible states of the world as hypotheses since they are hypothesized states of the world. Using this terminology, the probability value is the probability of an outcome given the hypothesis.
It is not the probability of the hypothesis given the outcome. This is not to say that we ignore the probability of the hypothesis. If the probability of the outcome given the hypothesis is sufficiently low, we have evidence that the hypothesis is false. However, we do not compute the probability that the hypothesis is false.
In the James Bond example, the hypothesis is that he cannot tell the difference between shaken and stirred martinis. However, we have not computed the probability that he can tell the difference. A branch of statistics called Bayesian statistics provides methods for computing the probabilities of hypotheses. These computations require that one specify the probability of the hypothesis before the data are considered and, therefore, are difficult to apply in some contexts.
The hypothesis that an apparent effect is due to chance is called the null hypothesis.
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