Practice problems for these concepts can be found at:
Completely Randomized Design
A completely randomized design for a study involves three essential elements: random allocation of subjects to treatment and control groups; administration of different treatments to each randomized group (in this sense we are calling a control group a "treatment"); and some sort of comparison of the outcomes from the various groups. A standard diagram of this situation is the following:
There may be several different treatment groups (different levels of a new drug, for example) in which case the diagram could be modified. The control group can either be an older treatment (like a medication currently on the market) or a placebo, a dummy treatment. A diagram for an experiment with more than two treatment groups might look something like this:
Remember that each group must have enough subjects so that the replication condition is met. The purpose of the placebo is to separate genuine treatment effects from possible subject responses due to simply being part of an experiment. Placebos are not necessary if a new treatment is being compared to a treatment whose effects have been previously experimentally established. In that case, the old treatment can serve as the control. A new cream to reduce acne (the treatment), for example, might be compared to an already-on-the-market cream (the control) whose effectiveness has long been established.
example: Three hundred graduate students in psychology (by the way, a huge percentage of subjects in published studies are psychology graduate students) volunteer to be subjects in an experiment whose purpose is to determine what dosage level of a new drug has the most positive effect on a performance test. There are three levels of the drug to be tested: 200 mg, 500 mg, and 750 mg. Design a completely randomized study to test the effectiveness of the various drug levels.
solution: There are three levels of the drug to be tested: 200 mg, 500 mg, and 750 mg. A placebo control can be included although, strictly speaking, it isn't necessary is our purpose is to compare the three dosage levels. We need to randomly allocate the 300 students to each of four groups of 75 each: one group will receive the 200 mg dosage; one will receive the 500 mg dosage; one will receive the 750 mg dosage; and one will receive a placebo (if included). No group will know which treatment its members are receiving (all the pills look the same), nor will the test personnel who come in contact with them know which subjects received which pill (see the definition of "double-blind" given below). Each group will complete the performance test and the results of the various groups will be compared. This design can be diagramed as follows:
Double-Blind Experiments
In the example above, it was explained that neither the subjects nor the researchers knew who was receiving which dosage, or the placebo. A study is said to be double-blind when neither the subjects (or experimental units) nor the researchers know which group(s) is/are receiving each treatment or control. The reason for this is that, on the part of subjects, simply knowing that they are part of a study may affect the way they respond, and, on the part of the researchers, knowing which group is receiving which treatment can influence the way in which they evaluate the outcomes. Our worry is that the individual treatment and control groups will differ by something other than the treatment unless the study is double-blind. A double-blind study further controls for the placebo effect.
Randomization
There are two main procedures for performing a randomization. They are:
- Tables of random digits. Most textbooks contain tables of random digits. These are usually tables where the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 appear in random order (well, as random as most things get, anyhow). That means that, as you move through the table, each digit should appear with probability 1/10, and each entry is independent of the others (knowing what came before doesn't help you make predictions about what comes next).
- Calculator "rand" functions. The TI-83/84 calculator has several random functions: rand, randInt (which will generate a list of random integers in a specified range), randNorm (which will generate random values from a normal distribution with mean μ and standard deviation s), and randBin (which will generate random values from a binomial distribution with n trials and fixed probability p—see Chapter 10). If you wanted to generate a list of 50 random digits similar to the random digit table described above, you could enter randInt(0,9) and press ENTER 50 times. A more efficient way would be to enter randInt(0,9,50). If you wanted these 50 random integers stored in a list (say L1), you would enter andInt(0,9,10) →L1 (remembering that the → is obtained by pressing STO).
We will use tables of random digits and/or the calculator in Chapter 9 when we discuss simulation.
Block Design
Earlier we discussed the need for control in a study and identified randomization as the main method to control for lurking variables—variables that might influence the outcomes in some way but are not considered in the design of the study (usually because we aren't aware of them). Another type of control involves variables we think might influence the outcome of a study. Suppose we suspect, as in our previous example, that the performance test varies by gender as well as by dosage level of the test drug. That is, we suspect that gender is a confounding variable (its effects cannot be separated from the effects of the drug). To control for the effects of gender on the performance test, we utilize what is known as a block design. A block design involves doing a completely randomized experiment within each block. In this case, that means that each level of the drug would be tested within the group of females and within the group of males. To simplify the example, suppose that we were only testing one level (say 500 mg) of the drug versus a placebo. The experimental design, blocked by gender, could then be diagramed as follows.
Randomization and block designs each serve a purpose. It's been said that you block to control for the variables you know about and randomize to control for the ones you don't. Note that your interest here is in studying the effect of the treatment within the population of males and within the population of females, not to compare the effects on men and women so that there would be no additional comparison between the blocks—that's a different study.
Matched Pairs Design
A particular block design of interest is the matched pairs design. One possible matched pairs design involves before and after measurements on the same subjects. In this case, each subject becomes a block in which the experiment is conducted. Another type of matched pairs involves pairing the subjects in some way (matching on, say, height, race, age, etc.).
example: A study is instituted to determine the effectiveness of training teachers to teach AP Statistics. A pretest is administered to each of 23 prospective teachers who subsequently undergo a training program. When the program is finished, the teachers are given a post-test. A score for each teacher is arrived at by subtracting their pretest score from their post-test score. This is a matched pairs design because two scores are paired for each teacher.
example: One of the questions on the 1997 AP Exam in Statistics asked students to design a study to compare the effects of differing formulations of fish food on fish growth. Students were given a room with eight fish tanks. The room had a heater at the back of the room, a door at the front center of the room, and windows at the front sides of the room. The most correct design involved blocking so that the two tanks nearest the heater in the back of the room were in a single block, the two away from the heater in a second block, the two in the front near the door in a third, and the two in the front near the windows in a fourth. This matching had the effect of controlling for known environmental variations in the room caused by the heater, the door, and the windows. Within each block, one tank was randomly assigned to receive one type of fish food and the other tank received the other. The blocking controlled for the known effects of the environment in which the experiment was conducted. The randomization controlled for unknown influences that might be present in the various tank locations.
You will need to recognize paired-data, as distinct from two independent sets of data, later on when we study inference. Even though two sets of data are generated in a matched-pairs study, it is the differences between the matched values that form the one-sample data used for statistical analysis.
Practice problems for these concepts can be found at:
View Full Article
From 5 Steps to a 5 AP Statistics. Copyright © 2010 by The McGraw-Hill Companies. All Rights Reserved.
Celebrate Memorial Day! Worksheets and Activities About American History 
May Workbooks are Here!
Get Outside! 10 Playful Activities
7 Parenting Tips to Take the Pressure Off 
Add your own comment