Probability without Equations: Concepts for Clinicians

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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.

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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.

Probability, clinical decision making and hypothesis testing

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. It means that we are, however, almost not totally convinced that the average recovery time during treatment with a particular antidepressant is shorter than during placebo treatment.

The purpose of hypothesis testing is to aid the clinician in reaching a conclusion concerning the universe by examining a sample from that universe.

WHAT IS PROBABILITY?

A hypothesis may be defined as a presumption or statement about the truth in the universe. It is frequently concerned about the parameters in the population about which the presumption or statement is made.

It is the basis for motivating the research project. There are two types of hypotheses, research hypothesis and statistical hypothesis Daniel, ; Guyatt et al. Hypothesis may be generated by deduction from anatomical, physiological facts or from clinical observations. Statistical hypotheses are hypotheses that are stated in such a way that they may be evaluated by appropriate statistical techniques.

The types of data that form the basis of hypothesis testing procedures must be understood, since these dictate the choice of statistical test.


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These presumptions are the normality of the population distribution, equality of the standard deviations, random samples. There are 2 statistical hypotheses involved in hypothesis testing. These should be stated a priori and explicitly. The null hypothesis is the hypothesis to be tested. It is denoted by the symbol H 0. It is also known as the hypothesis of no difference.

The null hypothesis is set up with the sole purpose of efforts to knock it down. In the testing of hypothesis, the null hypothesis is either rejected knocked down or not rejected upheld.

If the null hypothesis is not rejected, the interpretation is that the data is not sufficient evidence to cause rejection. If the testing process rejects the null hypothesis, the inference is that the data available to us is not compatible with the null hypothesis and by default we accept the alternative hypothesis , which in most cases is the research hypothesis. The alternative hypothesis is designated with the symbol H A. Neither hypothesis testing nor statistical tests lead to proof.

It merely indicates whether the hypothesis is supported or not supported by the available data. When we reject a null hypothesis, we do not mean it is not true but that it may be true. Share your thoughts with other customers. Write a customer review. There was a problem filtering reviews right now. Please try again later. I think this is the only book I have come across which explains the meaning of "alpha levels" so clearly.

Probability, clinical decision making and hypothesis testing

The author's example of flipping a two-headed coin without revealing to his students that it is two-headed , observing their reactions as "the number of heads keep going up" and finally explaining that "different students had different alpha levels" could'nt have brought out the meaning of this term in a more better way!

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