1.2.7: Experiments
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Studies where the researchers assign treatments to cases are called experiments . When this assignment includes randomization, e.g. using a coin ip to decide which treatment a patient receives, it is called a randomized experiment . Randomized experiments are fundamentally important when trying to show a causal connection between two variables.
Principles of experimental design
Randomized experiments are generally built on four principles.
Controlling . Researchers assign treatments to cases, and they do their best to control any other differences in the groups. For example, when patients take a drug in pill form, some patients take the pill with only a sip of water while others may have it with an entire glass of water. To control for water consumption, a doctor may ask all patients to drink a 12 ounce glass of water with the pill.
Randomization. Researchers randomize patients into treatment groups to account for variables that cannot be controlled. For example, some patients may be more susceptible to a disease than others due to their dietary habits. Randomizing patients into the treatment or control group helps even out such differences, and it also prevents accidental bias from entering the study.
Replication . The more cases researchers observe, the more accurately they can estimate the effect of the explanatory variable on the response. In a single study, we replicate by collecting a sufficiently large sample. Additionally, a group of scientists may replicate an entire study to verify an earlier nding.
Blocking . Researchers sometimes know or suspect that variables, other than the treatment, inuence the response. Under these circumstances, they may rst group individuals based on this variable into blocks and then randomize cases within each block to the treatment groups. This strategy is often referred to as blocking . For instance, if we are looking at the effect of a drug on heart attacks, we might rst split patients in the study into low-risk and high-risk blocks, then randomly assign half the patients from each block to the control group and the other half to the treatment group, as shown in Figure 1.15. This strategy ensures each treatment group has an equal number of low-risk and high-risk patients.
It is important to incorporate the rst three experimental design principles into any study, and this book describes applicable methods for analyzing data from such experiments. Blocking is a slightly more advanced technique, and statistical methods in this book may be extended to analyze data collected using blocking.
Reducing bias in human experiments
Randomized experiments are the gold standard for data collection, but they do not ensure an unbiased perspective into the cause and effect relationships in all cases. Human studies are perfect examples where bias can unintentionally arise. Here we reconsider a study where a new drug was used to treat heart attack patients.17 In particular, researchers wanted to know if the drug reduced deaths in patients.
17 Anturane Reinfarction Trial Research Group. 1980. Sul npyrazone in the prevention of sudden death after myocardial infarction. New England Journal of Medicine 302(5):250-256.
These researchers designed a randomized experiment because they wanted to draw causal conclusions about the drug's effect. Study volunteers18 were randomly placed into two study groups. One group, the treatment group , received the drug. The other group, called the control group , did not receive any drug treatment.
Put yourself in the place of a person in the study. If you are in the treatment group, you are given a fancy new drug that you anticipate will help you. On the other hand, a person in the other group doesn't receive the drug and sits idly, hoping her participation doesn't increase her risk of death. These perspectives suggest there are actually two effects: the one of interest is the effectiveness of the drug, and the second is an emotional effect that is difficult to quantify.
Researchers aren't usually interested in the emotional effect, which might bias the study. To circumvent this problem, researchers do not want patients to know which group they are in. When researchers keep the patients uninformed about their treatment, the study is said to be blind . But there is one problem: if a patient doesn't receive a treatment, she will know she is in the control group. The solution to this problem is to give fake treatments to patients in the control group. A fake treatment is called a placebo , and an effective placebo is the key to making a study truly blind. A classic example of a placebo is a sugar pill that is made to look like the actual treatment pill. Often times, a placebo results in a slight but real improvement in patients. This effect has been dubbed the placebo effect .
The patients are not the only ones who should be blinded: doctors and researchers can accidentally bias a study. When a doctor knows a patient has been given the real treatment, she might inadvertently give that patient more attention or care than a patient that she knows is on the placebo. To guard against this bias, which again has been found to have a measurable effect in some instances, most modern studies employ a double-blind setup where doctors or researchers who interact with patients are, just like the patients, unaware of who is or is not receiving the treatment. 19
Exercise 1.14 Look back to the study in Section 1.1 where researchers were testing whether stents were effective at reducing strokes in at-risk patients. Is this an experiment? Was the study blinded? Was it double-blinded? 20


18.6C: Short-Term Chemical Control
Blood pressure is controlled chemically through dilation or constriction of the blood vessels by vasodilators and vasocontrictors.
- Discuss chemical substances to control blood pressure
Key Points
- Constriction or dilation of blood vessels alters resistance, increasing or decreasing blood pressure respectively.
- Generalized vasoconstriction usually results in an increase in systemic blood pressure, but it may also occur in specific tissues, causing a localized reduction in blood flow.
- Vasoconstriction results from increased concentration of calcium (Ca2+) ions within vascular smooth muscle.
- When blood vessels dilate, the flow of blood is increased due to a decrease in vascular resistance. Therefore, dilation of arterial blood vessels (mainly the arterioles ) causes a decrease in blood pressure.
- Localized tissues increase blood flow in multiple ways, including releasing vasodilators, primarily adenosine, into the local intersitial fluid, which diffuses to capillary beds provoking local vasodilation.
Key Terms
- vasodilation : The dilation (widening) of a blood vessel.
- vasoconstriction : The constriction (narrowing) of a blood vessel.
Many physical factors that influence arterial pressure. Each may in turn be influenced by physiological factors such as diet, exercise, disease, drugs or alcohol, stress, and obesity. In practice, each individual’s autonomic nervous system responds to and regulates all of these interacting factors so that the actual arterial pressure response varies widely because of both split-second and slow-moving responses of the nervous system and end organs. These responses are very effective in changing the variables and resulting blood pressure from moment to moment.
Chemical Vasoconstriction
Vasoconstriction is the narrowing of blood vessels resulting from contraction of the muscular wall of the vessels, particularly the large arteries and small arterioles. Generalized vasoconstriction usually results in an increase in systemic blood pressure, but may also occur in specific tissues, causing a localized reduction in blood flow.
The mechanism that leads to vasoconstriction results from the increased concentration of calcium (Ca2+ ions) and phosphorylated myosin within vascular smooth muscle cells. When stimulated, a signal transduction cascade leads to increased intracellular calcium from the sarcoplasmic reticulum through IP3 mediated calcium release, as well as enhanced calcium entry across the sarcolemma through calcium channels.
The rise in intracellular calcium interacts with calmodulin, which in turn activates myosin light chain kinase. This enzyme is responsible for phosphorylating the light chain of myosin to stimulate cross-bridge cycling. Once elevated, the intracellular calcium concentration is returned to its basal level through a variety of protein pumps and calcium exchanges located on the plasma membrane and sarcoplasmic reticulum. This reduction in calcium removes the stimulus necessary for contraction allowing for a return to baseline.
Endogenous vasoconstrictors include ATP, epinephrine, and angiotensin II.
Vasoconstriction : Vasoconstriction of a microvessel by pericytes and endothelial cells that encircle an erythrocyte (E).
Chemical Vasodilation
Vasodilation is the widening of blood vessels resulting from relaxation of smooth muscle cells within the vessel walls, particularly in the large veins, large arteries, and smaller arterioles. Generalized vasodilation usually results in a decrease in systemic blood pressure, but may also occur in specific tissues causing a localized increase in blood flow.
The primary function of vasodilation is to increase blood flow in the body to tissues that need it most. This is often in response to a localized need for oxygen, but can occur when the tissue in question is not receiving enough glucose, lipids, or other nutrients. Localized tissues increase blood flow by several methods, including the release of vasodilators, primarily adenosine, into the local interstitial fluid, which diffuses to capillary beds provoking local vasodilation. Some physiologists have suggested the lack of oxygen itself causes capillary beds to vasodilate by the smooth muscle hypoxia of the vessels in the region.
As with vasoconstriction, vasodilation is modulated by calcium ion concentration and myosin phosphorylation within vascular smooth muscle cells. Dephosphorylation by myosin light-chain phosphatase and induction of calcium symporters and antiporters that pump calcium ions out of the intracellular compartment both contribute to smooth muscle cell relaxation and therefore vasodilation. This is accomplished through reuptake of ions into the sarcoplasmic reticulum via exchangers and expulsion across the plasma membrane.
Endogenous vasodilators include arginine and lactic acid.


Chapter 7: Courts
Learning Objectives
After reading this section, students will be able to:
- Differentiate between what happens at trial and what happens on appeal and identify the procedural history of a criminal case by reading appellate opinions written in the case.
- Describe how a crime/criminal case proceeds from the lowest level trial court up through the U.S. Supreme Court. (i.e., students should understand the hierarchy of the federal and state courts).
- Discuss the function and selection of state and federal trial and appellate judges in the American criminal justice system
- Discuss the function and selection of state and federal prosecutors in the American criminal justice system.
- Discuss the importance of the criminal defense attorney in the American criminal justice system.
- Identify at what stages of the criminal justice process a defendant is entitled to the assistance of a court-appointed attorney.
This section examines the structure and function of the criminal courts in America. It examines the concept of jurisdiction and describes the dual court system (the federal court system and the various state court systems). This section also examines the role and function of the various courtroom participants–the people who work in the courts.
Critical Thinking Questions
- Knowing what happens at trial and what happens on appeal, would you be more interested in being a trial judge or an appellate judge? Why?
- Why is there a different standard of review for questions of fact and questions of law?
- Do you agree that cases should be overturned only when there was a fundamental or prejudicial error that occurred during the trial?
- Do you think it is easier to be a defense attorney than a prosecutor believing the defendant is guilty but knowing that the justice system has violated the defendant’s rights?
- Should the defendant ever waive the assistance of counsel?
- Is there any position as a court staff that particularly interests you? Why?


18.4: Euler's formula
Let \(\alpha\) be a real number. The following identity is called Euler’s formula :
\[e^{i\cdot\alpha}=\cos\alpha+i\cdot\sin\alpha.\]
In particular, \(e^{i\cdot\pi}=-1\) and \(e^{i\cdot\frac\pi2}=i\) .
Geometrically, Euler’s formula means the following: Assume that \(O\) and \(E\) are the points with complex coordinates \(0\) and \(1\) respectively. Assume
\(OZ=1\quad \text{and}\quad \measuredangle EOZ \equiv \alpha,\)
then \(e^{i\cdot\alpha}\) is the complex coordinate of \(Z\) . In particular, the complex coordinate of any point on the unit circle centered at \(O\) can be uniquely expressed as \(e^{i\cdot\alpha}\) for some \(\alpha\in(-\pi,\pi]\) .
The proof of Euler’s identity depends on the way you define the exponential function. If you never had to apply the exponential function to an imaginary number, you may take the right hand side in 18.4.1 as the definition of the \(e^{i\cdot\alpha}\) .
In this case, formally nothing has to be proved, but it is better to check that \(e^{i\cdot\alpha}\) satisfies familiar identities. Mainly,
\(e^{i\cdot \alpha}\cdot e^{i\cdot \beta}= e^{i\cdot(\alpha+\beta)}.\)
The latter can be proved using 18.1.2 and the following trigonometric formulas, which we assume to be known:
\(\begin{aligned} \cos(\alpha+\beta)&=\cos\alpha\cdot\cos\beta-\sin\alpha\cdot\sin\beta, \\ \sin(\alpha+\beta)&=\sin\alpha\cdot\cos\beta+\cos\alpha\cdot\sin\beta.\end{aligned}\)
If you know the power series for the sine, cosine, and exponential function, then the following might convince that the identity 18.4.1 holds:
\(\begin{aligned} e^{i\cdot \alpha } &{}= 1 + i\cdot \alpha + \frac{(i\cdot \alpha )^2}{2!} + \frac{(i\cdot \alpha )^3}{3!} + \frac{(i\cdot \alpha )^4}{4!} + \frac{(i\cdot \alpha )^5}{5!} + \cdots = \\ &= 1 + i\cdot \alpha - \frac{\alpha ^2}{2!} - i\cdot\frac{ \alpha ^3}{3!} + \frac{\alpha ^4}{4!} + i\cdot\frac{ \alpha ^5}{5!} - \cdots = \\ &= \left( 1 - \frac{\alpha ^2}{2!} + \frac{\alpha ^4}{4!} - \cdots \right) + i\cdot\left( \alpha - \frac{\alpha ^3}{3!} + \frac{\alpha ^5}{5!} - \cdots \right) = \\ &= \cos \alpha + i\cdot\sin \alpha.\end{aligned}\)