ABSTRACT

Selecting and Interpreting Inferential Statistics To understand the information in this chapter, it will be necessary to remember or to review the sections in Chapter 1 about variables and levels of measurement (nominal, dichotomous, ordinal, and approximately normal/scale). It is also necessary to remember the distinction we made between difference and associational research questions and between descriptive and inferential statistics. This chapter focuses on inferential statistics, which as the name implies refers to statistics that make inferences about population values based on the sample data that you have collected and analyzed. What we call difference inferential statistics lead to inferences about the differences (usually mean differences) between groups in the populations from which the samples were drawn. Associational inferential statistics lead to inferences about the association or relationship between variables in the population. Thus, the purpose of inferential statistics is to enable the researcher to make generalizations beyond the specific sample data. Before we describe how to select and interpret inferential statistics, we will introduce design classifications. General Design Classifications for Difference Questions Many research questions focus on whether there is a statistically significant difference between two or more groups or conditions. The designs in this section all regard this type of design. Labeling difference question designs. Brief descriptive labels identify the design for other researchers and also guide us toward appropriate statistics to use. We do not have design classifications for the descriptive or associational research questions, so this section applies only to difference questions. Designs are usually labeled in terms of (a) the overall type of design (between-groups or within-subjects), (b) the number of independent variables, and (c) the number of levels within each independent variable. When a group comparison or difference question is asked, the independent variable and design can be classified as between-groups or within-subjects. Understanding this distinction is one essential aspect of determining the proper statistical analysis for this type of question. Between-groups designs. These are designs where each participant in the research is in one and only one condition or group. For example, there may be three groups (or levels or values) of the independent variable, treatment type. If the investigator wished to have 20 participants in each group, then 60 participants would be needed to carry out the research. Within-subjects or repeated-measures designs. These designs are conceptually the opposite of between-groups designs. In within-subjects (sometimes called dependent) designs, each participant in the research receives or experiences all of the conditions or levels of the independent variable. These designs also include examples where the participants are matched by the experimenter or in some natural way (e.g., twins, husband and wife, or mother and child). When each participant is assessed more than once, these designs are also referred to as repeated-measures designs. Repeated-measures designs are common in longitudinal research and intervention research. Comparing performance on the same dependent variable assessed before and after intervention (pretest and posttest) is a common example of a repeatedmeasures design. We might call the independent variable in such a study “time of measurement” or “change over time.” Our HSB data did not really have a within-subjects aspect to the design. However, one of the variables is repeated (visualization with two levels: visualization test and visualization retest)

and one is within (education, each student has both a mother’s education and father’s education). To demonstrate a within-subjects design and the use of repeated-measured ANOVA, we will use another data set, called Product Data, which is found on the companion website. This small data set has withinsubjects data, a rating by each participant for each of four different products (e.g., DVDs, but they could be any four stimuli). The same types of analysis could be done if, instead of each participant rating four different products in the same session, the ratings were done for satisfaction with the same product at four times. In that case, the data would be repeated-measures data. In addition, to demonstrate a doubly multivariate design, in which there are repeated assessments of several measures, we will use the data set called mixedMANOVAdata. Single-factor designs. If the design has only one independent variable (in either a between-groups design or a within-subjects design), then it should be described as a basic or single-factor or one-way design. Factor and way are other names for difference independent variables. Note that the number of factors or “ways” refers to the number of independent variables not the number of levels of an independent variable. For example, a between-groups design with one independent variable that has four levels is a single-factor or one-way between-groups design with four levels. If the design is a withinsubjects design with four levels, then it would be described as a single-factor, repeated-measures design with four levels (e.g., the same test being given four times). Between-groups factorial designs. When there is more than one group difference independent variable, and each level of each variable (factor) is possible in combination with each level of each of the other variable, the design is called factorial. For example, a factorial design could have two independent variables (i.e., factors) gender and ethnicity, allowing for male and female members of each ethnic group. In these cases, the number of levels of each variable (factor) becomes important in the description of the design. If gender had two levels (i.e., males and females) and ethnicity had three levels (e.g., EuropeanAmerican, Hispanic-American, and African-American), then this design is a 2 × 3 between-groups factorial design. In this 2 × 3 notation, then, the number of numbers is the number of factors or ways, and the numbers themselves refer to the number of levels of each of those factors. This design could also be called a two-way or two-factor design because there are two independent variables. Mixed factorial designs. If the design has a between-groups variable and a within-subjects independent variable, it is called a mixed design. For example, if the independent variables are gender (a betweengroups variable) and time of measurement (with pretest and posttest as within-subjects levels); this is a 2 × 2 mixed factorial design with repeated measures on the second factor. The mixed design is common in experimental studies with a pretest and posttest. Remember, when describing a design, that each independent variable is described using one number, which is the number of levels for that variable. Thus a design description with two numbers (e.g., 3 × 4) has two independent variables or factors, which have three and four levels, respectively. The dependent variable is not part of the design description, so it was not considered in this section.