Student’s T Test

Primer of Biostatistics,
by Stanton Glantz.

Probability and statistics is an important topic in medicine. Russ Hobbie and I discuss probability in the 4th edition of Intermediate Physics for Medicine and Biology, but we don’t delve into statistics. Yet, basic statistics is crucial for analyzing biomedical data, such as the results of a clinical trial.

Suppose IPMB did contain statistics. What would that look like? I suspect Russ and I would summarize this topic in an appendix. The logical place seems to be right after Appendix G (The Mean and Standard Deviation). We would probably not want to go into great detail, so we would only consider the simplest case: a “student’s t-test” of two data sets. It would be something like this (but probably less wordy).

Appendix G ½  Student’s T Test

Suppose you divide a dozen patients into two groups. Six patients get a drug meant to lower their blood pressure, and six others receive a placebo. After receiving the drug for a month, their blood pressure is measured. The data is given in Table G ½.1.

Table G ½.1. Systolic Blood Pressure (in mmHg)

Drug	Placebo
115	99
90	106
99	100
108	119
107	96
96	104

Is the drug effective in lowering blood pressure? Statisticians typically phrase the question differently: they adopt the null hypothesis that the drug has no effect, and ask if the data justifies the rejection of this hypothesis.

The first step is to calculate the mean, using the methods described in Appendix G. The mean for those receiving the drug is 102.5 mmHg, and the mean for those receiving the placebo is 104.0 mmHg. So, the mean systolic blood pressure was lower with the drug. The crucial question is: could this difference arise merely from chance, or does it represent a real difference? In other words, is it likely that this difference is a coincidence caused by taking too small of a sample?

To answer this question, we need to next calculate the standard deviation σ of each data set. We calculate this using Eq. G.4, except that because we do not know the mean of the data but only estimate it from our sample, we should use the factor N/(N-1) for the best estimate of the variance, where N = 6 in this example. The standard deviation is then σ = √( Σ (x -x_mean)²/(N-1) ). The calculated standard deviation for the patients who took the drug is 9.1, whereas for the patients who took the placebo it is 8.2. 

The standard deviation describes the spread of the data within the sample, but what we really care about is how accurately we know the mean of the data. The standard deviation of the mean is calculated by dividing the standard deviation by the square root of N. This gives 3.7 for patients taking the drug, and 3.3 for patients taking the placebo.

We are primarily interested in the difference of the means, which is 104.0 – 102.5 = 1.5 mmHg. The standard deviation of the difference in the means can be found by squaring each standard deviation of the mean, adding them, and taking the square root (standard deviations add like in the Pythagorean theorem). You get √(3.7² + 3.3²) = 5.0 mmHg.

Compare the difference of the means to the standard deviation of the difference of the means by taking their ratio. Following tradition we will call this ratio T, so T = 1.5/5.0 = 0.3. If the drug has a real effect, we would expect the difference of the mean to be much larger than the standard deviation of the difference of the mean, so the absolute value of T should be much greater than 1. On the other hand, if the difference of means is much smaller than the standard deviation of the difference of the means, the result could arise easily from chance and |T| should be much less than 1. Our value is 0.3, which is less than 1, suggesting that we cannot reject the null hypothesis, and that we have not shown that the drug has any effect. 

But can we say more? Can we transform our value of T into a probability that the null hypothesis is true? We can. If the drug truly had no effect, then we could repeat the experiment many times and get a distribution of T values. We would expect the values of T to be centered about T = 0 (remember, T can be positive or negative), with small values much more common than large. We could interpret this as a probability distribution: a bell shaped curve peaked at zero and falling as T becomes large. In fact, although we will not go into the details here, we can determine the probability that |T| is greater than some critical value. By tradition, one usually requires the probability p to be larger than one twentieth (p greater than 0.05) if we want to reject the null hypothesis and claim that the drug does indeed have a real effect. The critical value of T depends on N, and values are tabulated in many places (for example, see here). In our case, the tables suggest that T would have to be greater than 2.23 in order to reject the null hypothesis and say that the drug has a true (or, in the technical language, a “significant”) effect.

If taking p greater than 0.05 seems like an arbitrary cutoff for significance, then you are right. Nothing magical happens when p reaches 0.05. All it means is that the probability that the difference of the means could have arisen by chance is less than 5%. It is always possible that you were really, really unlucky and that your results arose by chance but |T| just happened to be very large. You have to draw a line somewhere, and the accepted tradition is that p greater than 0.05 means that the probability of the results being caused by random chance is small enough to ignore. 

Problem 1 Analyze the following data and determine if X and Y are significantly different. 

X	Y
94	122
93	118
104	119
105	123
115	102
96	115

Use the table of critical values for the T distribution at http://en.wikipedia.org/wiki/Student%27s_t-distribution.

I should mention a few more things.

1. Technically, we consider above a two-tailed t-test, so we’re testing if we can reject the null hypothesis that the two means are the same, implying that either the drug had a significant effect of lowering blood pressure or the drug had a significant effect of raising blood pressure. If we wanted to test only if the drug lowered blood pressure, we should use a one-tailed test.

2. We analyzed what is known as an unpaired test. The patients who got the drug are different than the patients who did not. Suppose we gave the drug to the patients in January, let them go without the drug for a while, then gave the same patients the placebo in July (or vice versa). In that case, we have paired data. It may be that patients vary a lot among themselves, but that the drug reduced everyone’s blood pressure by the same fixed percentage, say 12%. There are special ways to generalize the t-test for paired data.

3. It’s easy to generalize these results to the case when the two samples have different numbers N.

4. Please remember, if you found 20 papers in the literature that all observed significant effects with p less than but on the order of 0.05, then on average one of those papers is going to be reporting a spurious result: the effect is reported as significant when in fact it is a statistical artifact. Given that there are thousands (millions?) of papers out there reporting the results of t-tests, there are probably hundreds (hundreds of thousands?) of such spurious results in the literature. The key is to remember what p means, and to not over-interpret or under-interpret your results.

5. Why is this called the “student’s t-test”? The inventor of the test, William Gosset, was a chemist working for Guinness, and he devised the t-test to assess the quality of stout. Guinness would not let its chemists publish, so Gosset published under the pseudonym “student.”

6. The t-test is only one of many statistical methods. As is typical of IPMB, we have just scratched the surface of an exciting and extensive topic. 

7. There are many good books on statistics. One that might be useful for readers of IPMB (focused on biological and medical examples, written in engaging and nontechnical prose) is Primer of Biostatistics, 7th edition, by Stanton Glantz.

Point/Counterpoint: Low-Dose Radiation is Beneficial, Not Harmful

I have discussed the pedagogical virtues of point/counterpoint articles published by Medical Physics before in this blog (see, for instance, here and here). They are a wonderful resource to augment any medical physics class, and serve as an excellent supplement to the 4th edition of Intermediate Physics for Medicine and Biology. Medical Physics posts all its point/counterpoint articles freely available online (open access). Each article presents a somewhat controversial proposition in the title, and two leading medical physicists then debate the issue, one pro and one con. Each makes an opening statement, and each has a chance to respond to their opponents opening statement in a rebuttal.

One example that is closely related to a topic in IPMB is addressed in the July 2014 point/counterpoint article, which debates the proposition that “low-dose radiation is beneficial, not harmful.” Mohan Doss argues for the proposition, and Mark Little argues against it. The issue is central to the “linear no threshold” model of radiation risk that Russ Hobbie and I discuss in Sec. 16.13 (The Risk of Radiation) of IPMB. Mohan Doss leads off with this claim:

When free radical production is increased, e.g., from low-dose radiation (LDR) exposure (or increased physical/mental activity), our body responds with increased defenses consisting of increased antioxidants, DNA repair enzymes, immune system response, etc. referred to as adaptive protection. With enhanced protection, there would be reduced cumulative damage in the long term and reduced diseases. The disease-preventive effects of increased physical/mental activities are well known.

Little responds:

Dr. Doss discusses the well-known involvement of the immune system in cancer, and more generally the role of adaptive response. The critical issue is whether the up-regulation of the immune system or other forms of adaptive response that may result from a radiation dose offsets the undoubted carcinogenic damage that is caused. The available evidence, summarized in my Opening Statement, is that it does not.

Both cite the literature extensively. I find it fascinating that such a basic hypothesis hasn’t, to this day, been resolved. We don’t even know the sign of the effect: is low dose radiation positive or negative for our health. Although I can’t tell you who is right, Doss or Little, I can tell you who wins: the reader. And especially the student, who gets a front-row seat at a cutting-edge scientific debate between two world-class experts.

By the way, point/counterpoint articles aren’t the only articles available free-of-charge at the Medical Physics website. You can get 50th Anniversary Papers [for its 50th anniversary, Medical Physics published several retrospective papers], Vision 20/20 papers [summaries of state-of-the-art developments in medical physics], award papers, special focus papers, and more. And it’s all free.

I love free stuff.

Physics of Phoxhounds

I don’t have any grandchildren yet, but I am fortunate to have a wonderful “granddog.” This weekend, my wife and I are taking care of Auggie, the lovable foxhound that my daughter Kathy rescued from an animal shelter in Lansing, Michigan. Auggie gets along great with our Cocker-Westie mix, “Aunt Suki,” my dog-walking partner who I’ve mentioned often in this blog (here, here, here, and here).

Do dogs and physics mix? Absolutely! If you don’t believe me, then check out the website dogphysics.com. I plan to read “How To Teach Physics To Your Dog” with Auggie and Suki. According to this tee shirt foxhounds are particularly good at physics. Once we finish “How To Teach Physics To Your Dog,” we may move on to “Physics for Dogs: A Crash Course in Catching Cats, Frisbees, and Cars.” Apparently there is even a band that sings about dog physics, but I don’t know what that is all about.

Auggie is a big fan of the 4th edition of Intermediate Physics for Medicine and Biology. His favorite part is Section 7.10 (Electrical Stimulation) because there Russ Hobbie and I discuss the “dog-bone” shaped virtual cathode that arises when you stimulate cardiac tissue using a point electrode. He thinks “Auger electrons,” discussed in Sec. 17.11, are named after him. Auggie’s favorite scientist is Godfrey Hounsfield (Auggie adds a “d” to his name: “Houndsfield”), who earned a Nobel Prize for developing the first clinical computed tomography machine. And his favorite homework problem is Problem 34 in Chapter 2, about the Lotka-Volterra equations governing the population dynamics of rabbits and foxes.

How did Auggie get his name? I’m not sure, because he had the name Auggie when Kathy adopted him. I suspect it comes from an old Hanna-Barbera cartoon about Augie Doggie and Doggie Daddy. When Auggie visits, I get to play doggie [grand]daddy, and say “Augie, my son, my son” in my best Jimmy Durante voice. I’m particularly fond of the Augie doggie theme song. What is Auggie’s favorite movie? Why, The Fox and the Hound, of course.

Me holding Suki.

Our dog Suki has some big news this week. My friend and Oakland University colleague Barb Oakley has a new book out: A Mind for Numbers: How to Excel at Math and Science (Even if You Flunked Algebra). I contributed a small sidebar to the book offering some tips for learning physics, and it includes a picture of me with Suki! Thanks to my friend Yang Xia for taking the picture. Barb is a fascinating character and author of an eclectic collection of books. I suggest Hair of the Dog: Tails from Aboard a Russian Trawler. Her amazon.com author page first gave me the idea of publishing a blog to go along with IPMB. To those of you who are interested in physics applied to medicine and biology but struggle with all the equations in IPMB, I suggest Barb's book or her MOOC Learning How to Learn.

All Creatures Great and Small,
by James Herriot.

James Herriot—the author of a series of wonderful books including All Creatures Great and Small, which will warm the heart of any dog-lover—loved beagles, which look similar to foxhounds, but are smaller. If you’re looking for an uplifting and enjoyable book to read on a late-summer vacation (and you have already finished IPMB), try Herriot’s books. But skip the chapters about cats (yuck).

Auggie may not be the brightest puppie in the pack, and he is too timid to be an effective watch dog, but he has a sweet and loving disposition. I think of him as a gentle soul (even if he did chew up his grandma’s shoe). Below is a picture of Auggie and his Aunt Suki, getting ready for their favorite activity: napping.

Suki and Auggie.

On Size and Life

I have recently been reading the fascinating book On Size and Life, by Thomas McMahon and John Tyler Bonner (Scientific American Library, 1983). In their preface, McMahon and Bonner write

This book is about the observable effects of size on animals and plants, seen and evaluated using the tools of science. It will come as no surprise that among those tools are microscopes and cameras. Ever since Antoni Van Leeuwenhoek first observed microorganisms (he called them “animalcules”) in a drop of water from Lake Berkel, the reality of miniature life has expanded our concepts of what all life could possibly be. Some other tools we shall use—equally important ones—are mathematical abstractions, including a type of relation we shall call an allometric formula. It turns out that allometric formulas reveal certain beautiful regularities in nature, describing a pattern in the comparisons of animals as different in size as the shrew and the whale, and this can be as delightful in its own way as the view through a microscope.

Their first chapter is similar to Sec. 1.1 on Distances and Sizes in the 4th edition of Intermediate Physics for Medicine and Biology, except it contains much more detail and is beautifully illustrated. They focus on larger animals; if you want to see a version of our Figs. 1.1 and 1.2 but with a scale bar of about 10 meters, take a look at McMahon and Bonner’s drawing of “the biggest living things” on Page 2 (taken from the 1932 book The Science of Life by the all-star team of H. G. Wells, J. S. Huxley, and G. P. Wells).

In their Chapter 2 (Proportions and Size) is a discussion of allometric formulas and their representation in log-log plots, similar to but more extensive than Russ Hobbie and my Section 2.10 (Log-Log Plots, Power Laws, and Scaling). McMahon and Bonner present in-depth analysis of several biomechanical explanations for many allometric relationships. For instance, below is their description of “elastic similarity” in their Chapter 4 (The Biology of Dimensions).

Let us now consider a new scaling rule as an alternative to isometry (geometric similarity [all length scales increase together, leading to a change in size but no change in shape]), which was the main rule employed for discussing the theory of models in Chapter 3. This new scaling theory, which we shall call elastic similarity, uses two length scales instead of one. Longitudinal lengths, proportional to the longitudinal length scale ℓ, will be measured along the axes of the long bones and generally along the direction in which muscle tensions act. The transverse length scale, d, will be defined at right angles to ℓ, so that bone and muscle diameters will be proportional to d…When making the transformations of shape from a small animal to a large one, all longitudinal lengths (or simply “lengths”) will be multiplied by the same factor that multiples the basic length, ℓ, and all diameters will be multiplied by the factor that multiplies the basic diameter, d. Furthermore, there will be a rule connecting ℓ and d…d ∝ ℓ^3/2.

They then show that elastic similarity can be used to derive Kleiber’s law (metabolic rate is proportional to mass to the ¾ power), and justify elastic similarity using biomechanical analysis of buckling of a leg. I must admit I am a bit skeptical that the ultimate source of Kleiber’s law is biomechanics. In IPMB, Russ and I review more recent work suggesting that Kleiber’s law arises from general considerations of models that supply nutrients through branching networks, which to me sound more plausible. Nevertheless, McMahon and Bonner’s ideas are interesting, and do suggest that biomechanics can sometimes play a significant role in scaling.

Their Chapter 5 (On Being Large) presents a succession of intriguing allometric relationships related to the motion of large animals (running, flying, swimming, etc). Let me give you one example: large animals have a harder time running uphill than smaller animals. McMahon and Bonner present a plot of oxygen consumption per unit mass versus running speed, and find that for a 30 g mouse there is almost no difference between running uphill and downhill, but for a 17.5 kg chimpanzee running uphill requires about twice as much oxygen as running downhill. In Chapter 6 (On Being Small) they examine what life is like for little organisms, and analyze some of the same issues Edward Purcell discusses in “Life at Low Reynolds Number.”

Overall, I enjoyed the book very much. I have a slight preference for Knut Schmidt-Nielsen’s book Scaling: Why Is Animal Size So Important?, although I must admit that Size and Life is the better illustrated of the two books.

Author Thomas McMahon was a major figure in biomechanics. He was a Harvard professor particularly known for his study of animal motion, and even wrote a paper about “Groucho Running”; running with bent knees like Groucho Marx. Russ and I cite his paper “Size and Shape in Biology” (Science, Volume 179, Pages 1201–1204, 1973) in IPMB. I understand that his book Muscles, Reflexes and Locomotion is also excellent, although more technical, but I have not read it. Below is the abstract from the article “Thomas McMahon: A Dedication in Memoriam” by Robert Howe and Richard Kronauer (Annual Review of Biomedical Engineering, Volume 3, Pages xv-xxxix, 2001).

Thomas A. McMahon (1943–1999) was a pioneer in the field of biomechanics. He made primary contributions to our understanding of terrestrial locomotion, allometry and scaling, cardiac assist devices, orthopedic biomechanics, and a number of other areas. His work was frequently characterized by the use of simple mathematical models to explain seemingly complex phenomena. He also validated these models through creative experimentation. McMahon was a successful inventor and also published three well-received novels. He was raised in Lexington, Massachussetts, attended Cornell University as an undergraduate, and earned a PhD at MIT. From 1970 until his death, he was a member of the faculty of Harvard University, where he taught biomedical engineering. He is fondly remembered as a warm and gentle colleague and an exemplary mentor to his students.

His New York Times obituary can be found here.

Interview with Russ Hobbie in The Biological Physicist

In 2006, just as Springer was about to publish the 4th edition of Intermediate Physics for Medicine and Biology, an interview with Russ Hobbie appeared in The Biological Physicist, a newsletter of the Division of Biological Physics of the American Physical Society. Below are some excerpts from the interview. You can read the entire thing in the December 2006 newsletter.

THE BIOLOGICAL PHYSICIST: Are there any stories you have about particular physics examples you have used in the book or in the classroom that have really awakened the interest of medical students to the importance of physics?

Russ Hobbie: I cannot speak to what has triggered a response in different students. But there is one amusing story. I was working with a pediatric cardiologist, Jim Moeller, to understand the electrocardiogram. I finally wrote up a 5-page paper explaining it with an electrostatic model. When I showed what I thought was simplicity itself to Jim, he could not understand a word of it. But he finally agreed to show it to some second- year medical students. Their response: “Thanks goodness it is rational.” I think this shows the gap between our premed course and what the student needs in medical school and also the fact that the physics we love so dearly may be helpful to a medical student during the basic science years but is not so helpful later on. It also became clear to me that what we teach about x-rays and radioactivity is the only exposure to those topic that physicians will receive, unless they go into radiology!

THE BIOLOGICAL PHYSICIST: How has the book changed over its four editions? Has the way you have presented material evolved over the years?

Russ Hobbie: It is amusing to compare my explanation of the electrocardiogram in the four editions. In the first, I was thinking in terms of an electrostatic model. By the second edition, I had realized that a current dipole model was much better and had been in the literature for a long time. This has been improved even more in the 3rd and 4th editions. I am a slow learner! But as an excuse, I was confused for a long time because the physiologists called the current dipole moment “the electric force vector.”

As I have added material (such as non-linear systems and chaos) it has been necessary to remove material. For example, the first edition had 11 pages and 3 color plates on polarized light and birefringence. This was gone to save money and to make room for biomagnetism in the second edition. I wish it was still there. I did not get around to discussing acoustics, hearing, and ultrasound until the fourth edition.

THE BIOLOGICAL PHYSICIST: How would you assess the impact of the book on the field of interdisciplinary research, and on interdisciplinary education? Do you have any information on the history of how quickly it was adopted by other departments, and how it is used in other interdisciplinary programs?

Russ Hobbie: I have always hoped that a physicist without the biological background could teach from the book, and the solutions manual was written in the hope that students could use it for an independent study course. (At the request of instructors, the solutions manual is now an Adobe Acrobat file which is password-protected. Instructors can ask me or Brad for the password and give it to students if they wish.)

Many physicists are more interested in molecular biophysics than physiology- and radiology- oriented physics and find that other books better meet their needs. However, there seems to be a growing interest in the book among biomedical engineers. One teaching technique that was very successful in the early years of the course had to be abandoned while I was serving as Associate Dean, because it took too much of my time. I required the students to find an article in the research literature that interested them and then to write a paper filling in all the missing steps. They could come to me for help as often as they needed. Then, three days after they submitted the paper, I would give them an oral exam on anything that I suspected they did not fully understand. They said this was a valuable experience; my office was packed with students the week before the papers were due; and I learned a lot myself.

THE BIOLOGICAL PHYSICIST: Have you found that there is a “cultural divide” between physicists and MDs? Some people in the Division of Biological Physics describe having difficulty communicating with medical researchers. Do you ever find that?

Russ Hobbie: Absolutely. One friend, Robert Tucker, got a PhD in biophysics with Otto Schmitt and then went to medical school. Bob said that medical school destroyed his ability to reason. This was probably an extreme statement, but it does capture the “drink from a fire hose” character of medical school. On the other hand, if I am having a myocardial infarct, I would prefer that the clinician taking care of me not start with Coulomb’s law!