Aptitude Treatment Interaction

ATI Analysis

• Combines both categorical and continuous independent variables.
• Hierarchical analysis to determine appropriate model to represent data.
• Continuous independent variable is a meaningful variable, not a nuisance variable.

Sample Size

• Cronbach & Snow, 1977 recommend 200 Ss.
• With at least 100 Ss per group.

Notation

• a, X1, T -> Coded categorical variable.
• b, X2, X -> Continuous variable.
• c, X3, TX -> Interaction vector (axb interaction).

ATI Flow Chart

Step 1: Test Proportion of Variance

n.s. -> Stop
sig. -> Go to Step 2

Step 2: Test Interaction

sig. -> Go to Step 7
n.s. -> Go to Step 3

Step 3: Test Common Regression Coefficient

sig. -> Go to Step 5
n.s. -> Go to Step 4

Step 4: Test for Treatment

sig. -> Significant Treatment Effect

n.s. -> No Significant Effects, Stop

Step 5: Test Intercepts

sig. -> Separate Intercepts

n.s. -> Go to Step 6

Step 6: Compute Single Regression Using the Continuous Variable

Stop

Step 7: Compute Separate Regression Equations. Establish Regions of Significance

Stop

Numerical Example

• X1 -> Treatment: Incentive (n=12) vs No Incentive (n=12)
• X2 -> Continuous: Study Time
• X3 -> Interaction
• R²_y.123 = .827
• R²_y.12 = .823
• R²_y.1 = .580
• R²_y.2 = .243

Step 1: Test Proportion of Variance

sig. -> Go to Step 2

Step 2: Test Interaction

n.s. -> Go to Step 3

Step 3: Test Common Slope

sig. -> Go to Step 5

Step 5: Test Intercepts

sig. -> Calculate Separate Intercepts

Y' = 4.92 - 2.04X1 + .24X2

4.92 + (-2.04) = 2.88
4.92 - (-2.04) = 6.97

Y' = 2.88 + .24X2
Y' = 6.96 + .24X2

Stop

Categorizing Continuous Variables

• Giving up information concerning variables.
• Produces illusion of experimental control.
• Results in less sensitive analysis.

Determining the Point of Intersection

• Consider two regressions:
  Y1' = 7 + .3X
  Y2' = 2 + .8X

  p = (a₁-a₂)/(b₂-b₁) = (7-2)/(.8-.3) = 10

Johnson-Neyman Regions