Methods In statistics, particularly in analysis of variance and linear regression, a contrast is a linear combination of variables (parameters or statistics) whose coefficients add up to zero, allowing comparison of different treatments (Casella, 2008). The classical methods of linear contrasts are those developed for post hoc analysis of ANOVA. In this workshop, we will first review the basic ideas of various linear-contrast methods used in ANOVA (i.e., the Tukey HSD test, Scheffe's test, and Tukey-Kramer procedure). Locating which pairs of groups that lead to the rejection of equal overall means is the key task of post hoc procedure following ANOVA, and perhaps is the oldest approach of linear contrasts. Next, we will present a specific method that can be employed in most post hoc analyses to make presentations of results produced by complicated statistical models more substantively attractive and meaningful. The method is the test of null hypothesis Qβ = r, where Q is a matrix containing linear constraints based on research questions and prior studies, β is the parameter vector being tested, and r is a vector of constants. We next will use three examples to illustrate how to perform the linear contrasts. They are: (1) the test of a hypothesis stating that the Cox regression coefficient of a young group equals to the average coefficients of two older groups; (2) a test of regression coefficients from a Probit model about equal wife's and husband's education impacts on an outcome variable; and (3) a test of logistic regression coefficients about a zero effect of wife's and husband's education and the test is simultaneous. We will employ the Stata program to demonstrate these analyses. We will show that in practice, these examples, albeit simple and basic, are extremely useful and can be applied in most social work research projects.
Conclusion and Implication Performing appropriate linear contrasts not only makes interpretation of statistical results interesting and attractive, but also is an important method helping address crucial research questions. We should follow the trend observed in biostatistics, health, and mental health research to promote the use of linear contrasts in social work research, and therefore, to enhance the level of quantitative research in our field.