# Conduct and Interpret a Sequential One-Way Discriminant Analysis

**What is the Sequential One-Way Discriminant Analysis?**

Sequential one-way discriminant analysis is similar to the one-way discriminant analysis. Discriminant analysis predicts group membership by fitting a linear regression line through the scatter plot. In the case of more than two independent variables it fits a plane through the scatter cloud thus separating all observations in one of two groups –one group to the "left" of the line and one group to the "right" of the line.

Sequential one-way discriminant analysis now assumes that the discriminating, independent variables are not equally important. This might be a suspected explanatory power of the variables, a hypothesis deducted from theory or a practical assumption, for example in customer segmentation studies.

Like the standard one-way discriminant analysis, sequential one-way discriminant analysis is useful mainly for two purposes: 1) identifying differences between groups, and 2) predicting group membership.

Firstly, sequential one-way discriminant analysis identifies the independent variables that significantly discriminate between the groups that are defined by the dependent variable. Typically, sequential one-way discriminant analysis is conducted after a cluster analysis or a decision tree analysis to identify the goodness of fit for the cluster analysis (remember that cluster analysis does not include any goodness of fit measures itself). Sequential one-way discriminant analysis tests whether each of the independent variables has discriminating power between the groups.

Secondly, sequential one-way discriminant analysis can be used to predict group membership. One output of the sequential one-way discriminant analysis is Fisher's discriminant coefficients. Originally Fisher developed this approach to identify the species to which a plant belongs. He argued that instead of going through a whole classification table, only a subset of characteristics is needed. If you then plug in the scores of respondents into these linear equations, the result predicts the group membership. This is typically used in customer segmentation, credit risk scoring, or identifying diagnostic groups.

Because sequential one-way discriminant analysis assumes that group membership is given and that the variables are split into independent and dependent variables, the sequential one-way discriminant analysis is a so called structure testing method as opposed to structure exploration methods (e.g., factor analysis, cluster analysis).

The sequential one-way discriminant analysis assumes that the dependent variable represents group membership the variable should be nominal. The independent variables represent the characteristics explaining group membership.

The independent variables need to be continuous-level (interval or ratio scale). Thus the sequential one-way discriminant analysis is similar to a MANOVA, logistic regression, multinomial and ordinal regression. Sequential one-way discriminant analysis is different than the MANOVA because it works the other way around. MANOVAs test for the difference of mean scores of dependent variables of continuous-level scale (interval or ratio). The groups are defined by the independent variable.

Sequential one-way discriminant analysis is different from logistic, ordinal and multinomial regression because it uses ordinary least squares instead of maximum likelihood; sequential one-way discriminant analysis, therefore, requires smaller samples. Also continuous variables can only be entered as covariates in the regression models; the independent variables are assumed to be ordinal in scale. Reducing the scale level of an interval or ratio variable to ordinal in order to conduct multinomial regression takes out variation from the data and reduces the statistical power of the test. Whereas sequential one-way discriminant analysis assumes continuous variables, logistic/ multinomial/ ordinal regression assume categorical data and thus use a Chi-Square like matrix structure. The disadvantage of this is that extremely large sample sizes are needed for designs with many factors or factor levels.

Moreover, sequential one-way discriminant analysis is a better predictor of group membership if the assumptions of multivariate normality, homoscedasticity, and independence are met. Thus we can prevent over-fitting of the model, that is to say we can restrict the model to the relevant independent variables and focus subsequent analyses. Also, because it is an analysis of the covariance, we can measure the discriminating power of a predictor variable when removing the effects of the other independent predictors.

*The Sequential One-Way Discriminant Analysis in SPSS*

The research question for the sequential one-way discriminant analysis is as follows:

*The students in our sample were taught with different methods and their ability in different tasks was repeatedly graded on aptitude tests and exams. At the end of the study the pupils go to choose from three computer game ‘thank you’ gifts: a sports game (Superblaster), a puzzle game (Puzzle Mania) and an action game (Polar Bear Olympics). The researchers wish to learn what guided the pupils’ choice of gift.*

The independent variables are the three test scores from the standardized mathematical, reading, writing test (viz. Test_Score, Test2_Score, and Test3_score). From previous correlation analysis we suspect that the writing and the reading score have the highest influence on the outcome. In our logistic regression we found that pupils scoring lower had higher risk ratios of preferring the action game over the sports or the puzzle game.

The sequential one way discriminant analysis is not a part of the graphical user interface of SPSS. However, if we want include our variables in a specific order into the sequential one-way discriminant model we can do so by specifying the order in the */analysis* subcommand of the Discriminant syntax.

The SPSS syntax for a sequential one-way discriminant analysis specifies the sequence of how to include the variables in the analysis by defining an inclusion level. SPSS accepts inclusion levels from 99…0, where variables with level 0 are never included in the analysis.