Statistics in Applied Science and Technology Supplemental: Elaborating Crosstabs: Adding a Third Variable

Key Concepts in this Chapter Direct relationship Spurious relationship Intervening relationship Conditional (interactive) relationship Limitation of Elaborating Crosstabs

Introduction Few research questions can be answered through a statistical analysis of only two variables Elaborating crosstabs (bivariate table) will allow us to look at the possible impact hat a third variable has on the original bivariate association. Elaborating crosstabs (bivariate table) in one of the multivariate analysis technique.

How is the 3 rd variable (Z) introduced? The 3 rd variable (Z) is introduced as a control variable which decomposes the data into sub- groups based on the categories of the control variable. A separate crosstab (bivariate table) for each of the subgroups defined by the control variable (Z) is then generated. The resulting crosstabs are called partial tables and we generate as many partial tables as there are categories for the control variable (Z).

Three Possible Outcomes Three possible outcomes when a third variable (Z) is introduced. A direct relationship still exists (the 3 rd variable has no effect) Either a spurious or intervening relationship exists A conditional relationship exists

Direct Relationships The relationship between X and Y is the same in all partial tables and in the original bivariate table. This pattern is often called replication since the partial tables reproduce (or replicate) the bivariate table; the cell frequencies are the same and measure of association have the same values too. This outcome indicates that the control variable has no important impact on the bivariate relationship and may be ignored in any further analysis.

Spurious or Intervening Relationship The relationship between X and Y is much weaker in the partial tables than in the original table but the same across all partials. Measure of association for the partial tables are much lower in value than the measure computed from original bivariate table.

Spurious relationship In this situation, Z is conceptualized as being antecedent to both X and Y. Z is a common cause of both X and Y, and the original bivariate relationship is said to be spurious. Z X Y

Intervening Relationship In this situation, Z may also intervene between the two variables. X is causally linked to Z, which is in turn linked to Y. This pattern indicated that although X and Y are related, they are associated primarily through the control variable Z. X Z Y

Conditional Relationship In this pattern, also called interaction, the relationship between X and Y changes markedly, depending on the value of the control variable. The partial tables differ from each other and from the bivariate table. X Z1 Z2 Y 0.0 X Z1 Z2 Y + _

Limitation of Elaborating Crosstabs The basic limitation of this technique involves sample size. Elaboration requires dividing the sample into a serious of partial tables. The greater the number of partial tables, the more likely we are to run out of cases to fill all the cells of each partial table. Small or empty cells can create problems in terms of generalizability and confidence in our findings.

Congratulations! You are done with statistics, well, at least for 445!