What are biostatistics?

The Methodology of Statistics

The aim of every study in the field of biostatistics is to discover something about a population—all patients with a particular disease, for example. Populations are usually much too large to be studied in their entirety: It would be impractical to round up all the patients with diabetes in the world for a study, and the researcher would still miss those who lived in the past or who have yet to be born. For this reason, most research focuses on a sample drawn from the population of interest—for example, those diabetics who were studied at a particular hospital over a two-year period. Ideally, the sample would be a random sample, one in which each member of the population has an equal chance of being included. In practice, a random sample is hard to achieve: The patients with diabetes at a hospital in Chicago, for example, will differ in various ways from those at hospitals in London, Hong Kong, or rural Mexico.

Descriptive statistics are statistics that describe samples. The most commonly used statistics are mean, median, mode, and standard deviation. The mean of n values is simply the sum of all the values divided by n, or, symbolically, x̄=Σx/n, where x̄ (pronounced “x-bar”) stands for the mean, n stands for the number of observations, and Σx (pronounced “sum of x”) stands for the sum obtained when all the different values of x (n of them) are added together. The median of a series of values is the middle value; it must always be the case that half of the values are above the median and half are below. The mode is simply the most common value, the value that occurs most often.

The mean, median, and mode are all “typical” values that characterize the center of distribution of all the values. The standard deviation, which is always positive, is a measure of variation that describes how close all the values cluster about a central value.

The numerator is calculated by subtracting the mean (x̄) from each value, squaring the difference, and adding together all these squared differences. After the sum is divided by n -1, the square root of the entire quantity is taken to determine the standard deviation. A small standard deviation indicates that the values differ very little from one another; a large standard deviation indicates greater variability among the values.

When two quantities vary, such as height and weight, correlation and regression coefficients are also calculated. A correlation coefficient is a number between -1 and +1. A correlation near +1 shows a very strong relationship between the two variables: When either one increases, the other also increases. A correlation near -1 is also strong, but when either variable increases, the other decreases. A correlation of 0 shows independence, or no relationship, between the variables: An increase in one has no average effect on the value of the other. Correlations midway between 0 and 1 show that an increase in one variable corresponds only to an average increase in the other, but not a dependable increase in each value. For example, taller people are generally heavier, but this is not true in every case.

Regression is a statistical technique for finding an equation that allows one variable to be predicted from the value of the other. For example, a regression of weight on height allows a researcher to predict the average weight for persons of a given height.

Inferential statistics, the statistical study of populations, begins with the study of probability. A probability is a number between 0 and 1 that indicates the certainty with which a particular event will occur, where 0 indicates an impossible occurrence and 1 indicates a certain occurrence. If a certain disease affects 1 percent of the population, then each random sampling will be subject to a .01 probability that the next person sampled will have the disease. For a larger sample than only one individual, the so-called binomial distribution describes the probability that the sample will include no one with the disease, one person with the disease, two people with the disease, and so on.

A variable such as height is subject to so many influences, both genetic and environmental, that it can be treated mathematically as if it were the sum of thousands of small, random variations. Characteristics such as height usually follow a bell-shaped curve, or normal distribution (see figure), at least approximately. This means that very few people are unusually tall or unusually short; most have heights near the middle of the distribution.

Each study using inferential statistics includes four major steps. First, certain assumptions are made about the populations under study. A common assumption, seldom tested, is that the variable in question is normally distributed within the population. Second, one or more samples are then drawn from the population, and each person or animal in the population is measured and tested in some way. Most studies assume that the samples are randomly drawn from the populations in question, even though true randomness is extremely difficult to achieve. Third, a particular assumption, called the null hypothesis, is chosen for testing. The null hypothesis is always an assumption that can be expressed numerically and that can be tested by known statistical tests. For example, one might assume that the mean height of the population of all diabetics is 170 centimeters (5.5 feet). Fourth, each statistical procedure allows the calculation of a theoretical probability, often a calculated value from a table. If the calculated probability is moderate or high, the null hypothesis is consistent with the observed results. If the calculated probability is very small, the observed results are very unlikely to occur if the null hypothesis is true; in these cases, the null hypothesis is rejected.

Some common types of inferential statistics are t-tests, chi-square tests, and the analysis of variance (also called ANOVA).

The Application of Statistics to Medicine

Nearly all medical research studies include biostatistics. Descriptive statistics are often presented in tables of data or in graphic form.

In the following example, height and weight data were gathered from a sample of six diabetic patients, and height was also measured in a sample of six nondiabetic patients (see table 1). The mean height of these diabetic patients is (166 + 171 + 157 + 161 + 179 + 186) ÷ 6 = 1,020 ÷ 6 = 170 centimeters. In order to calculate the standard deviation, this mean is subtracted from each of the six values, to yield differences of -4, 1, -13, -9, 9, and 16. Squaring these differences and adding them up gives a numerator of 16 + 1 + 169 + 81 + 81 + 256 = 604. Dividing this by n - 1 = 5 (n is 6) and taking the square root reveals that the standard deviation is 11.0 centimeters (rounded to the nearest tenth). Similar calculations show that the mean height of the nondiabetic sample is 177 centimeters, with a standard deviation of 8.0 centimeters.

In this example, the nondiabetic sample averaged 7 centimeters taller than the diabetic sample. Inferential statistics can be used to find out if this difference is a meaningful one. In this case, a technique known as a t-test can compare the means of two samples to determine whether they might have come from the same population (or from populations with the same mean value). The assumptions of this test (not always stated explicitly) are that the six diabetic patients were randomly drawn from a normally distributed population and that the six nondiabetic patients were randomly drawn from another normally distributed population. The null hypothesis in this case is that the populations from which the two samples were drawn have the same mean height. The value of t calculated in this test is 1.141; this value is looked up in a table to reveal that the probability is larger than .1 (or 10 percent). In other words, if the null hypothesis is true, then this value of t (or a larger value) is expected to arise by chance alone more than 10 percent of the time. Under the usual criterion of a test at the 5 percent level of significance, one would keep or accept the null hypothesis. This means that the difference between the above sample means is not large enough to demonstrate a difference
between the two populations from which these samples are derived. There may in fact be an average difference in height between diabetic and nondiabetic populations, but samples this small cannot detect such a difference reliably. In general, when differences between populations are small, larger sample sizes are required to demonstrate their existence.

A t-test similar to the one above can also be used in
drug testing. A drug is given to one set of patients, and a placebo (a fake medicine lacking the essential drug ingredient being tested) is given to a second group. Some relevant measurement (such as the drug’s level of an important chemical that the patients normally lacked) is then compared between the two groups. The null hypothesis would be that the groups are the same and that the drug makes no difference. Rejection of the null hypothesis would be the same as demonstrating that the drug is effective.

For the diabetic patients in the above sample, a correlation coefficient of .60 can be calculated. This moderate level of correlation shows that, on the average, the taller among these patients are also heavier and thus the shorter patients are also lighter. Despite this average effect, however, individual exceptions are likely to occur. The square of the correlation coefficient, .36, indicates that about 36 percent of the variation in weight can be predicted from height. The equation for making the prediction in this case is “weight = 13.99 + .404 (height),” where the numbers 13.99 and .404 are called regression coefficients. The variable being predicted (weight in this example) is sometimes called the dependent variable; the variable used to make the prediction (in this case, height) is called the independent variable. A high positive correlation signifies that the regression equation offers a very reliable prediction of the dependent variable; a correlation near zero signifies that the regression equation is hardly better than assigning the mean value of the dependent variable to every prediction.

To illustrate another type of inferential statistics, consider the following data on the diseases present among elderly patients who own pets, compared to a comparable group who do not. This type of table is called a contingency table.

As illustrated in table 2, 90 � 240, or three-eighths of the patients sampled, are pet owners. Thus, if there were no relationship between the diseases and pet ownership (which is the null hypothesis), one would expect three-eighths of the eighty arthritis patients (or 30) to be pet owners and five-eighths of the eighty (or 50) to be nonowners. Calculating all the expected frequencies in this way, one can compare them with the actual observations shown above.

The result is a statistic called chi-square, which in this problem has a value of 18.88. A table of chi-square values shows that, for a 2 4 contingency table, a chi-square value this high occurs by chance alone much less than 1 percent of the time. Thus, something has been observed (a chi-square value of 18.88) that is extremely unlikely under the null hypothesis of no relationship between disease and pet ownership, so the null hypothesis is rejected and one must conclude that there is a relationship. This conclusion applies only to the population from which the sample was drawn, however, and it does not reveal the nature of the relationship. Further investigation would be needed to discover whether pet ownership protected people from arthritis, whether people who already had arthritis were less inclined to take on the responsibility of pet ownership, or whether people who had pets gave them up when they became arthritic. All these possibilities (and more) are consistent with the findings.

The analysis of variance (also called ANOVA) is a very powerful technique for comparing many samples at once. Suppose that overweight patients were put on three or four different diets; one diet might be better than another, but there is also much individual variation in the amount of weight lost. Analysis of variance is a statistical technique that allows researchers to compare the variation between diets (or other treatments) with the individual variation among the people following each diet. The null hypothesis would be that all the diets are the same and that individual variation can account for all the observed differences. Rejecting the null hypothesis would demonstrate that a consistent difference existed and that at least one diet was better than another. Further tests would be required to pinpoint which diet was best and why.

Perspective and Prospects

The historical foundations of biostatistics go back as far as the development of probability theory by Blaise Pascal (1623–62). Karl Friedrich Gauss (1777–1855) first outlined the characteristics of the normal (also called Gaussian) distribution. The chi-square test was introduced by Karl Pearson (1857–1936). The greatest statistician of the twentieth century was Ronald A. Fisher (1890–1962), who clearly distinguished descriptive from inferential statistics and who developed many important statistical techniques, including the analysis of variance.

Although medicine existed long before statistics, it has become a nearly universal practice for every research study to use statistics in some important way. New medical procedures and new drugs are constantly being evaluated by using statistical techniques to compare them to other procedures or older drugs.

Bibliography

Brase, Charles Henry, and Corrinne Pellillo Brase. Understandable Statistics. 10th ed. Boston: Brooks/Cole, 2012.

Daniel, Wayne W. Biostatistics: A Foundation for Analysis in the Health Sciences. 9th ed. Hoboken, N.J.: John Wiley & Sons, 2009.

Glaser, Anthony N. High-Yield Biostatistics, Epidemiology, and Public Health. Philadelphia: Lippincott, Williams and Wilkins, 2013.

Hebel, J. Richard, and Robert J. McCarter. A Study Guide to Epidemiology and Biostatistics. 7th ed. Burlington, Mass.: Jones and Bartlett, 2012.

Le, Chap T. Health and Numbers: Problems-Based Introduction to Biostatistics. Malden, Mass.: Blackwell, 2009.

Merrill, Ray M. Fundamentals of Epidemiology and Biostatistics: Combining the Basics. Burlington, Mass.: Jones and Bartlett, 2013.

Phillips, John L. How to Think About Statistics. 6th ed. New York: Henry Holt, 2002.

Sokal, Robert R., and F. James Rohlf. Biometry. 4th ed. San Francisco: W. H. Freeman, 2012.

Walpole, Ronald E. Elementary Statistical Concepts. New York: Macmillan, 1976.

Zar, Jerrold H. Biostatistical Analysis. 5th ed. Englewood Cliffs, N.J.: Pearson, 2010.

What is a character sketch of Martha Washington in the book The Story of My Life?

Martha Washington was a child, a little older and taller than Helen, "ebony" of skin with black hair tamed by multiple braids. She was both compliant (or yielding) and mischievous, getting into trouble right along with Helen. She disliked conflict while liking play. Her mother was Helen's family's cook and gave them sweet treats, especially at Christmas.

The making ready for Christmas was always a delight to me. .... the pleasant odours that filled the house and the tidbits that were given to Martha Washington and me to keep us quiet. We were sadly in the way, but that did not interfere with our pleasure in the least.

Martha was valued by Helen particularly because Martha understood Helen's signs, so Helen was able to communicate. Martha Washington appears only briefly for a short time at the beginning of Helen's recollections.

In those days ... Martha Washington, the child of our cook, and Belle, an old setter and a great hunter in her day, were my constant companions. Martha Washington understood my signs, and I seldom had any difficulty in making her do just as I wished. It pleased me to domineer over her, and she generally submitted to my tyranny....

Martha Washington and an old and lazy setter named Belle were the two companions Helen had. Belle of course could not understand Helen's signs (not for want of trying to teach her). Martha of course could understand Helen's signs, such as when the hen laid her eggs out and about the grounds and Helen wanted to hunt them. Helen would make fists and put them on the grass. Martha immediately understood the fists meant "eggs" and the grass meant "let's hunt!"

Very different in appearance, Martha and Helen were much alike in interests. Both were curious and didn't mind crossing the line between curiousness and trouble, such as when they crossed from cutting paper dolls to cutting their shoelaces to cutting each other's hair. Helen cut Martha's black braids, then Martha cut Helen's golden curls (or at least one before Helen's mother stopped their play).

Martha loved the out-of-doors, as did Helen. Martha and she explored in the corn storage sheds, the horses stables and the cow yard. Together they pestered the milkman and the cows being milked. The cows defended themselves against the two small nature lovers when they "switched" their tails at the two curious girls.

The sheds where the corn was stored, the stable where the horses were kept, and the yard where the cows were milked morning and evening were unfailing sources of interest to Martha and me.

In Shakespeare's The Merchant of Venice, what are Portia's comments on the different suitors (besides Bassanio)? What does this reveal about her...

In Act 1, Scene 2 of Shakespeare's The Merchant of Venice, Nerissa asks Portia what she thinks about all of the different suitors who have come to try their luck to win her hand in marriage. Portia has her lady-in-waiting name off the different men and she comments on each. The list of suitors is as follows: Neapolitan prince, County Palatine, Monsieur le Bon, Falconbridge of England, a Scottish lord and a German, the Duke of Saxony's nephew. Portia has something sarcastic to say about each man's annoying habits and hopes none of them win her hand in marriage.

First, the Neapolitan annoys her because he only talks about his horse and the fact that he can shoe the animal himself. Next, the County Palatine never smiles. Portia feels as though he will become a "weeping philosopher when he grows old," which does not excite her. Then, the French Lord is bad because he has the bad habits that both of the first two suitors have: he frowns more than the County and he speaks of his horse as well. Not only that, but the Frenchman is also crazy enough, or cowardly enough, to fence with his own shadow, so of this Portia has no respect.

The Englishman only speaks English, which Portia does not, though she speaks Latin, French and Italian; so, marrying him would be a quiet venture. The Englishman also has a bad sense of fashion because his clothes aren't properly fitted. His clothes are so bad that she feels he bought each piece of clothing from a different country. 

Then the Scottish lord threatens the Englishman when they come in sight of each other, but he made friends with the Frenchman on this matter. Finally, there is the German who is constantly drunk, who Portia says "is a little worse than a man, and when he is worst he is little better than a beast" (I.iii.73-75).

From Portia's descriptions, one might say that she is picky, or selfish, and should be willing to marry any of these men. However, she is young and intelligent as well and values her happiness. Rather than picky or selfish, Portia can be considered wise because she knows with whom she will be compatible or not. She also knows that she wants to marry for love, not for status or money. Portia is witty because she cleverly mocks each suitor, but she is also strong-willed and knows what or whom will make her happy. She is much distressed over the fact that her father's game must be played in order to determine her lifelong mate, and wishes that she could choose him for herself.

In Lee's To Kill a Mockingbird, why does Aunt Alexandra come to visit? How does Scout feel about her arrival?

One day, Scout and Jem come home to find Aunt Alexandra sitting on their front porch.  This is unusual, because she rarely visits.  She says she is planning to stay for a while, and Scout feels uneasy.  She does not know how long a while will be.  Aunt Alexandra tells Scout that she came "to have some feminine influence" on her (To Kill a Mockingbird, Chapter 14).  She wants to prepare Scout as she becomes a young lady.

Atticus asks Scout if she would like for her aunt to live with them.  Scout lies and answers that she would.  She truthfully does not enjoy spending time with her aunt.  Aunt Alexandra is often negative toward Scout, and she shows her disapproval.  On top of that, her aunt is strict and meddling.  Aunt Alexandra wants Scout to behave like a lady.  Scout would rather play outside with Jem and Dill and wear denim overalls all day.

What is the climax of "A Bird came down the Walk" by Emily Dickinson?

It's hard to say what the climax of Emily Dickinson's "A Bird came down the Walk" is; indeed, it's difficult to say if there even is a climax in the poem, as the piece (like all of Dickinson's poems) is spare, radically unique, and difficult. For my part, however, I'd like to make a guess that the climax of the poem (if there is one) could be found in the lines "He stirred his Velvet Head / Like one in danger, Cautious, / I offered him a Crumb" (12-14). In these lines, it seems as though the bird becomes aware of his observer, and there is a tense moment as both parties are apparently "Cautious" of one another. Indeed, the offering of the crumb appears to be a dangerous transaction between a human being and the natural world, and neither parties are quite sure what to make of this interaction. As such, since one could argue these lines encapsulate the tensest action in the poem, one could also argue they constitute the climax. That said, it really is quite difficult to discern where the climax in this poem is, as the ambiguity of Dickinson's prose makes the work difficult to decipher.

Darcy's housekeeper tells Elizabeth that she has never had a cross word from him. How does this affect Elizabeth? A- She is worried that Jane will...

The best option here is the last one, D. When Elizabeth hears from Darcy's housekeeper, who has known him since his early childhood, that he has never spoken an unkind word to her, Elizabeth realizes that she may have misjudged Darcy as someone stuck-up and rude, when in fact Darcy may actually be very kind in his private life, kind even to those below him in social status, like his housekeeper.

Let's take a close look at that passage:

"I say no more than the truth, and everybody will say that knows him," replied the other. Elizabeth thought this was going pretty far; and she listened with increasing astonishment as the housekeeper added, "I have never known a cross word from him in my life, and I have known him ever since he was four years old."

This was praise, of all others most extraordinary, most opposite to her ideas. That he was not a good-tempered man had been her firmest opinion. Her keenest attention was awakened; she longed to hear more...

As you can see, the narrator states that the housekeeper's statements are in "praise" of Darcy, and that this praise is "opposite to" Elizabeth's understanding of him. She had thought Darcy had a bad temper, yet that was only her opinion that she'd formed on her own. She doesn't actually know him very well at all, and she's realizing now that she wants to--which we can tell, again from the narration, by the statement that she's very interested and hopes to hear more of what the housekeeper says about Darcy. "In what an amiable light does this place him!" she thinks, a moment later.

This question is an important one to consider as you think about the message of the novel as well as its title: Elizabeth believes that Mr. Darcy is full of pride and prejudice, but really, it's Elizabeth herself who exhibits those unattractive qualities as she misjudges Mr. Darcy. She had observed his handsome stature and reserved manners and wrongly assumed that he was an unkind man unworthy of her time. Their ensuing romance can only occur after she lets go of these false assumptions about him as well as her own pride that she once took in avoiding associating with him, and the match is a blessing for Elizabeth and her entire family. Mr. Darcy's housekeeper helps Elizabeth make this important discovery that radically changes her life, and her way of thinking, for the better.

What conditions are necessary for industrialization to occur?

An industrial revolution is a change in economic and social conditions marked by a significant increase in manufacturing and urbanization. There are a number of conditions necessary to spawn an industrial revolution. First, advances must be made in agriculture that makes crop production more efficient. These conditions could be climactic, scientific, or mechanization of machinery. With a more efficient system of farming, a surplus of laborers is created that may used in the factories.

Another factor that can help the labor supply is an increase in the population. This can naturally occur when the food supply increases, but other factors can contribute to population increases. Medical advances that improve infant mortality or allow people to live longer can increase the population. A sustained period of peace can also allow population growth.  With a larger population, you also create a larger market for manufactured goods. 

An interest in developing new technologies in a culture can also allow for industrialization. New machinery to make production more efficient is usually the first step. Improvements in transportation are also very important and capital must be available to develop a transportation network. These innovations can be enhanced with a stable and democratic government system. Another factor for industrialization is that there are raw materials available to be used for production. When all of these factors are met, a nation may become a successful industrial power.