A. J. Guarino

ANOVA AND RESEARCH DESIGN

WHAT IS ANALYSIS OF VARIANCE? Analysis of variance (ANOVA) is a statistical technique used to evaluate the size of the difference between sets of scores. For example, a group of researchers might wish to learn if the room color in which college students are asked to respond to questions assessing their mood can affect their reported mood. Students are randomly assigned to complete a mood inventory in one of two rooms. Random assignment, one of the hallmarks of experimental design, is used in an attempt to assure that there is no bias in who is placed into which group by making it equally likely that any one person could have been assigned to either group. One of the rooms is painted a soft shade of blue that was expected to exert a calming effect on the students; the other room is painted a bright red that was presumed to be more agitating than calming. Higher numerical scores on the mood inventory indicate a more relaxed mood. At the end of the study, we score the mood inventory for all participants. The research question in this example is whether mood as indexed by the score on the mood inventory was affected by room color. To answer this question, we would want to compare the mood scores of the two groups. If the mood scores obtained in the blue room were higher overall than those obtained in the red room, we might be inclined to believe that room color influenced mood.

One-Way Within-Subjects ANOVA

The last command reads a csv file that contains the data from Table 11.5 in your textbook, and stores everything in the data frame mw115. The data are from a fictitious experiment that measured cognitive ability in 12 children at 30, 36, 42, and 48 months of age. The within-subjects ANOVA consists of the following steps: 1. The dependent variables are extracted from the data frame and stored in a matrix ; 2. The lm function is used to create a multivariate linear model (i.e., mlm) object that specifies the betweensubjects aspect of the experimental design; 3. The Anova function in the car package is used to i) specify the within-subjects aspects of the design; and ii) convert the mlm object to an aov object; 4. Finally, the summary function prints the ANOVA table, Mauchly Test, and the corrected p values for the within-subject variables.

Data Analysis Using Sas Enterprise Guide : Nonlinearly Transforming Variables in Order to Meet Underlying Assumptions

Overview Most of the statistical procedures we use are based on the assumption that the data are normally distributed, that there are no outliers potentially distorting the results of the analyses, and, if there are two or more distributions involved in the analysis, that the sets of scores have comparable variances (the assumption of homogeneity of variance). If these assumptions are violated, one option available to researchers is to transform the data to force the values to come closer to meeting the assumptions. Chapter 11 discussed standardizing variables based on existing norms, which is one form of transformation. Standardizing a variable (e.g., to z or linear T scores) is an example of a linear transformation , that is, one preserving the characteristics of the distribution. Thus, a distribution whose values are skewed remains so following the raw scores being converted to z scores. In the present chapter, we discuss transformations that are performed with the intention of modifying the shape of the distribution. These types of transformations are known as nonlinear transformations . Notes on transformations To transform data is to perform certain types of mathematical operations on the scores of a variable for each case in the data set. We do this by computing a new variable in much the same way as we showed in Chapter 7 when we computed new variables and in Chapter 11 when we discussed standardizing a variable based on external norms.

Data Analysis Using Sas Enterprise Guide : Multiple Logistic Regression

Overview Logistic regression is conceptually analogous to linear regression in that a single dependent variable is predicted from either a single predictor (simple logistic regression) or multiple predictors (multiple logistic regression) based on a prediction model. It is also permissible to use both quantitatively measured and dichotomously (binary) coded variables as predictors. Our example for this chapter involves a single quantitatively measured predictor variable. Some differences between linear and logistic regression Although the two regression methods are conceptually similar, the differences between linear and logistic regression are important. Three of the most salient differences are as follows: In linear regression, the dependent variable is quantitatively measured; in logistic regression, the dependent variable is categorical. We will limit ourselves to a dichotomously coded dependent variable. In linear regression, a straight line function is fitted to the data set by using an ordinary least squares method; in logistic regression, a logistic function (an S-shaped function) is fitted to the data set by using a maximum likelihood estimation procedure. In linear regression the value of the quantitatively measured dependent variable is predicted; in logistic regression the dependent variable is categorical and what is predicted is the likelihood that a case with a certain value or values on the predictor(s) is a member of a particular group (the reference group).

Bivariate Correlation: Pearson Product–Moment and Spearman Rho Correlations

Overview Correlation in statistical terms is a way to assess the degree of relationship or association that is observed between variables. Bivariate correlation focuses on the relationship between two ( bi -) variables (- variate ). Behavioral and social research almost always is concerned about the relationship of two or more variables, and so correlation plays a central role in such ventures. Some history Sir Francis Galton, the late 19th-century geographer, meteorologist, and statistician, was perhaps best known for his study of the inheritance of both physical and intellectual characteristics. As early as 1875, he distributed packets of sweet pea seeds to seven of his friends. Each packet contained seeds of uniform weight, but the weight of the seeds varied across packets. These friends were asked to plant the seeds, raise several generations of the plants, and then send the last generation of seeds back to Galton (Stanton, 2001). Upon graphing the results of this experiment he discovered that relatively heavier- and relatively lighter-weighted parent seeds ultimately produced seeds of less extreme weight. Later, Galton, on the basis of some physical characteristics of people and their family history, determined that both taller than average and shorter than average men have family and offspring who deviate less from the mean than they do. Galton presented this latter work in 1886 as the framework for introducing the concept of regression “towards the level of mediocrity” (Galton, 1886, p. 492) – what we now call regression toward the mean.

SAS Enterprise Guide Projects

A brief history of SAS The SAS Web site provides a comprehensive history of the software and the company. Here is a synopsis of that information. SAS, an acronym for Statistical Analysis Software, is a set of statistical analysis procedures housed together within a large application. The idea for it was conceived by Anthony J. Barr, a graduate student at North Carolina State University, between 1962 and 1964. Barr collaborated with Jim Goodnight in 1968 to integrate regression and analysis of variance (ANOVA) procedures into the software. The project received a major boost in 1973 from the contribution of John P. Sall. Other participants in the early years of SAS development included Caroll G. Perkins, Jolayne W. Service, and Jane T. Helwig. The SAS Institute was established in Raleigh, NC in 1976 when the first base SAS material was released. The company moved to its present location of Cary, NC in 1980. SAS began being used on mainframe computers several decades ago. At that time, the only way to instruct the software to perform the statistical analyses was by punching holes on computer cards via a card-reader machine. Later this instruction occurred by typing in this code on an otherwise blank screen. The majority of SAS users still prefer this latter process.

MEASUREMENT, CENTRAL TENDENCY, AND VARIABILITY

SCALES OF MEASUREMENT In Chapter 1, we indicated that variables can take on different values. Here, we will deal with the process of associating values with variables and how to summarize the sets of values we obtain from that process. This will prove to be important in guiding us toward situations in which it is appropriate to use the ANOVA technique. THE PROCESS OF MEASUREMENT Measurement represents a set of rules informing us of how values are assigned to objects or events. A scale of measurement describes a specific set of rules. Psychology became intimately familiar with scales of measurement when S. S. Stevens, in 1946, wrote a response to a committee of the British Association for the Advancement of Science. That committee was dealing with the possibility of measuring sensory events, and the members could not agree on whether or not it was possible to perform such measurements. Because they had focused on the work that Stevens had done to measure loudness of sound, he felt that a response to the committee from him was appropriate. But instead of arguing with the committee, Stevens adopted a different strategy, presenting instead a theory of measurement. He elaborated on this theory in 1951 when, as the editor of the Handbook of Experimental Psychology, he gave prominent treatment to this topic in his lead chapter. Stevens identified four scales in his theory: nominal , ordinal , interval , and ratio scales in that order.

Analysis of Variance Designs: THREE-WAY BETWEEN-SUBJECTS DESIGN

COMBINING TWO INDEPENDENT VARIABLES FACTORIALLY NAMING THE DESIGN In Chapter 6, we discussed a between-subjects design that contained a single independent variable (preparation time for the SAT). However, we are not limited to studying the effects of just one independent variable in a research design. In this chapter, we will deal with the inclusion of a second independent variable (note that we still have only one dependent variable). A design containing more than one independent variable is known as a factorial design when the variables are combined in a manner described in Section 8.1.2. When those independent variables are between-subjects variables, the design is called a between-subjects design or a between-subjects factorial design . Designs containing two between-subjects independent variables that are simultaneously varied are two-way between-subjects (factorial) designs . These designs are also sometimes referred to as two-way completely randomized designs because subjects are assumed to be randomly assigned to the various treatments. COMBINING INDEPENDENT VARIABLES IN A SINGLE DESIGN Intertwining two independent variables within the same design is done by combining them in a factorial fashion in which each level of one independent variable is combined with each level of the other independent variable. If our independent variables were, for example, gender (female and male) and size of city in which participants resided (large and small), then one combination of the levels of the independent variables might be females living in large cities. This would then be one condition or group in the factorial design.

Analysis of Variance Designs: COMPLEX MIXED DESIGN: ONE BETWEEN-SUBJECTS FACTOR AND TWO WITHIN-SUBJECTS FACTORS

COMBINING BETWEEN- AND WITHIN-SUBJECTS FACTORS As discussed in Chapter 13, a mixed design is one that contains at least one between-subjects independent variable and at least one within-subjects independent variable. Simple mixed designs have only two independent variables and so, by definition, must have one of each type of variable. Complex mixed designs contain at least three independent variables. The three-way complex mixed designs presented in this chapter and in Chapter 15 must (by definition) have two of one type and one of the other type of factor. In this chapter, we will focus on the design with two between-subjects factors and one within-subjects factor; Chapter 15 will present the design with one between-subjects factor and two within-subjects factors. A NUMERICAL EXAMPLE OF A COMPLEX MIXED DESIGN College students who signed up for a research study read vignettes in which their romantic partner was described as being attracted to another person; this attraction was depicted as being either emotional attraction with no physical component or physical attraction with no emotional component. Because this type-of-attraction variable was intended to be a within-subjects variable, students read both vignettes. After reading each vignette, the students completed a short inventory evaluating their feelings of jealousy; the response to this inventory served as the dependent variable. For the purposes of this hypothetical example, assume that no effects concerning the order of reading these vignettes was obtained; thus, we will present the results without considering the vignette ordering factor.