### Simple Effects

Suppose you use the following statements to fit a full factorial model to a two-way design:

data twoway;
input A B Y @@;
datalines;
1 1 10.6 1 1 11.0 1 1 10.6 1 1 11.3
1 2 -0.2 1 2 1.3 1 2 -0.2 1 2 0.2
1 3 0.1 1 3 0.4 1 3 -0.4 1 3 1.0
2 1 19.7 2 1 19.3 2 1 18.5 2 1 20.4
2 2 -0.2 2 2 0.5 2 2 0.8 2 2 -0.4
2 3 -0.9 2 3 -0.1 2 3 -0.2 2 3 -1.7
3 1 29.7 3 1 29.6 3 1 29.0 3 1 30.2
3 2 1.5 3 2 0.2 3 2 -1.5 3 2 1.3
3 3 0.2 3 3 0.4 3 3 -0.4 3 3 -2.2
;

proc glm data=twoway;
class A B;
model Y = A B A*B;
run;

Partial results for the analysis of variance are shown in Figure 39.21. The Type I and Type III results are the same because this is a balanced design.

**
Figure 39.21
Two-Way Design with Significant Interaction**

The GLM Procedure

Dependent Variable: Y

2 |
219.905000 |
109.952500 |
165.11 |
<.0001 |

2 |
3206.101667 |
1603.050833 |
2407.25 |
<.0001 |

4 |
487.103333 |
121.775833 |
182.87 |
<.0001 |

2 |
219.905000 |
109.952500 |
165.11 |
<.0001 |

2 |
3206.101667 |
1603.050833 |
2407.25 |
<.0001 |

4 |
487.103333 |
121.775833 |
182.87 |
<.0001 |

The interaction A*B is significant, indicating that the effect of A depends on the level of B. In some cases, you might be interested in looking at the differences between predicted values across A for different levels of B. Winer (1971) calls this the *simple effects* of A. You can compute simple effects with the LSMEANS statement by specifying the SLICE= option. In this case, since the GLM procedure is interactive, you can compute the simple effects of A by submitting the following statements after the preceding statements.

lsmeans A*B / slice=B;
run;

The results are shown Figure 39.22. Note that A has a significant effect for B=1 but not for B=2 and B=3.

**
Figure 39.22
Interaction LS-Means and Simple Effects**

The GLM Procedure

Least Squares Means

10.8750000 |

0.2750000 |

0.2750000 |

19.4750000 |

0.1750000 |

-0.7250000 |

29.6250000 |

0.3750000 |

-0.5000000 |

The GLM Procedure

Least Squares Means

2 |
704.726667 |
352.363333 |
529.13 |
<.0001 |

2 |
0.080000 |
0.040000 |
0.06 |
0.9418 |

2 |
2.201667 |
1.100833 |
1.65 |
0.2103 |

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