R-Vogt.qxd 1/25/2005 11:49 AM Page 259 R R Symbol for *Pearson’s correlation, which is a *bivariate correlation (between two variables). R (a) Symbol for a *multiple correlation, that is, between more than two variables. (b) Abbreviation for *range. R A statistical software package available to download at no cost from the www. It is modeled after *S-PLUS. r2 Symbol for a *coefficient of determination between two variables. It tells you how much of the *variability of the *dependent variable is explained by (or accounted for, associated with, or predicted by) the *independent variable. Sometimes written “r-squared.” For example, if the r2 between students’ SAT scores and their college grades were .30, that would mean that 30% of the variability in students’ college grades could be predicted by variability in their SAT scores—and that 70% could not. R2 Symbol for a *coefficient of multiple determination between a *depen- dent variable and two or more *independent variables. It is a commonly used measure of the *goodness-of-fit of a *linear model. Sometimes writ- ten “R-squared.” For example, if the R2 between average individual income (the depen- dent variable) and father’s income, education level, and IQ were .40, that would mean that the effects of father’s income, educational level, and IQ together explained (or predicted) 40% of the variance in individuals’ aver- age incomes (and that they did not explain 60% of the variance). rbis *Biserial correlation rpb *Point-biserial correlation. 259 R-Vogt.qxd 1/25/2005 11:49 AM Page 260 260 rs rs Symbol for the *Spearman correlation coefficient (rho). R rtet *Tetrachoric correlation. Rc Symbol for the *canonical correlation. Radical The *root of a number as shown by the radical sign √ . A number (the “index”) to the left of the sign shows the type of root. For example, √3 means the third (cube) root. If there is no number, the root is a square root. Radix In a *life table, the starting number (100,000 in the example at the life table entry). The number perishing at different ages is subtracted from the radix to obtain the number surviving. R and D *Research and development. Random Said of events that are unpredictable because their occurrence is unrelated to their characteristics; they are governed by chance. The oppo- site of random is determined. The chief importance of randomness in research is that by using it to select or assign subjects, researchers increase the probability that their conclusions will be *valid. *Random assignment increases *internal valid- ity. Random sampling increases *external validity. See *probability sample, *pseudo-random numbers. Random Assignment Putting subjects into *experimental and *control groups in such a way that each individual in each group is assigned entirely by chance. Otherwise put, each subject has an equal probability of being placed in each group. Using random assignment reduces the likelihood of *bias. Also called “random allocation.” Random Coefficients Models See *hierarchical linear models. Random-Effects Model An *experimental design in which the *levels of the *factors are random, in the sense that they are drawn at random from a population of levels rather than fixed by an investigator. Also called “vari- ance components model” and “Model II ANOVA design.” Random-effects models are much less widely used than *fixed-effects models in which researchers set the treatment levels. Compare *mixed-effects model, *ran- dom variable. The random-effects model is used when there is a large number of cate- gories or levels of a factor. For example, say researchers in a survey orga- nization wanted to see whether different kinds of telephone interviewers get different response rates. Because there are potentially a very large number of categories (differences in accent, quality of voice, etc.), perhaps as many as there are individual telephone interviewers, a sample is chosen randomly from the population of interviewers, which is also, in this case, a popula- tion of levels. On the other hand, if the survey organization were only R-Vogt.qxd 1/25/2005 11:49 AM Page 261 Randomized Field Trial 261 interested in, say, the difference in response rate between male and female interviewers, they would used a *fixed-effects model. R Random Error Another term for *random variation. Also called “unrelia- bility” and *disturbance. See *reliability. “Error” without qualification usually means random error. Random errors are often assumed to have a *normal distribution. Random error cannot be eliminated, but it can be esti- mated. The opposite of random error is *systematic error, or *bias, which is usually more difficult to estimate. Random Factor A *factor in an *ANOVA in which the levels of the vari- able are points along a continuum, such as 5, 10, 15, and 20 minutes. See *random-effects model. Randomized-Blocks Design A *research design in which subjects are matched on a *variable the researcher wishes to control. The subjects are put into groups (blocks) of the same size as the number of *treatments. The members of each block are assigned randomly to different treatment groups. Compare *Latin square, *repeated-measures ANOVA. For example, say we are doing a study of the effectiveness of four meth- ods of teaching statistics. We use 80 subjects and plan to divide them into 4 treatment groups of 20 students each. Using a randomized-blocks design, we give the subjects a test of their knowledge of statistics. The four who score highest on the test are the first block, the next highest four are the second block, and so on to the 20th block. The four members of each block are randomly assigned, one to each of the four treatment groups. We use the blocks to equalize the variance within each treatment group by making sure that each group has subjects with a similar prior knowledge of statistics. Randomized Clinical Trial (RCT) Experimental methods applied to stud- ies of the effectiveness of treatments, originally drug treatments in medical research. The first published results from a RCT appeared in a 1948 article; it reported the effects of streptomycin on tuberculosis. *Random assign- ment is used to form treatment and control groups, and *double-blind pro- cedures are applied when possible. Randomized clinical and control trials are sometimes used interchangeably. Randomized Control Trial (RCT) Experimental methods applied to the study of the effectiveness of treatments, especially treatments not adminis- tered in a clinical or laboratory setting, such as a *field experiment of an educational intervention. Subjects are placed in *control and *treatment groups by *random assignment. Originally developed in medical research, the methods of the RCT are increasingly demanded in other fields, such as education, and in program *evaluation research more generally. Randomized Field Trial Another term for *randomized control trial. R-Vogt.qxd 1/25/2005 11:49 AM Page 262 262 Random Numbers Random Numbers A sequence of numbers in which the occurrence of any number in the sequence is no guide to the numbers that come next. In the R long run, all numbers will appear equally often. Tables of random numbers, found in most statistics texts, are frequently used to select random samples or assign subjects randomly. Random numbers generated by a computer are *pseudo-random, because they are produced according to a formula. Random Number Generator A computer *program designed to produce a series of random numbers. These programs usually produce numbers in cycles, and thus generate numbers that are only “pseudo-random.” Random Process A means by which random numbers or events are gener- ated. Rolling a pair of fair dice is an example. Random Sampling Selecting a group of subjects (a *sample) for study from a larger group (*population) so that each individual (or other *unit of analy- sis) is chosen entirely by chance. When used without qualification (such as *stratified random sampling), random sampling means “simple random sampling.” Also sometimes called “equal probability sample,” since every member of the population has an equal *probability of being included in the sample. A random sample is not the same thing as a haphazard or *acci- dental sample. Using random sampling reduces the likelihood of *bias. Compare *probability sample, *cluster sample, *quota sample, *stratified random sample. Random Selection Another term for *random sampling. “Selection” is more often used in experimental research; “sampling” is the more common term in survey research, but the underlying principle is the same. See *ran- dom assignment, *probability sample. Random Variable A variable that varies in ways the researcher does not control; a variable whose values are randomly determined. “Random” refers to the way the events, values, or subjects are chosen or occur, not to the variable itself. Men and women are not random, but gender could be a random variable in a research study; the gender of subjects included in the study could be left to chance and not controlled by the researcher. Also called *stochastic variable. Random Variation Differences in a variable that are due to chance rather than to one of the other variables being studied. Random variations tend to cancel one another out in the long run. Also called *random error. For example, say you take two random samples of 100 workers from a local factory. You find that in the first sample 52 are women and 48 men. In the second sample 49 are women and 51 men. The differences between the sex compositions of the two samples would be due to random variation. See *sampling error. R-Vogt.qxd 1/25/2005 11:49 AM Page 263 Rasch Modeling 263 Random Walk (a) A series of steps in which the direction and length of each step is uninfluenced by the previous steps.