The classical analysis of variance (ANOVA) technique that is based on least squares assumes that the underlying experimental errors are normally distributed. However, data often contain outliers as a result of recording or other errors. In other cases, extreme responses occur when control variables in the experiments are set to extremes. It is important to distinguish among these extreme points and determine whether they are outliers or important extreme cases. You can use the ROBUSTREG procedure for robust analysis of variance based on M estimation. Usually there are no high-leverage points in a well-designed experiment, so M estimation is appropriate.

This example shows how to use the ROBUSTREG procedure for robust ANOVA.

An experiment studied the effects of two successive treatments (`T1`

, `T2`

) on the recovery time of mice that had certain diseases. Sixteen mice were randomly assigned to four groups for the four
different combinations of the treatments. The recovery times (`time`

) were recorded (in hours) as shown in the following data set:

data recover; input T1 $ T2 $ time @@; datalines; 0 0 20.2 0 0 23.9 0 0 21.9 0 0 42.4 1 0 27.2 1 0 34.0 1 0 27.4 1 0 28.5 0 1 25.9 0 1 34.5 0 1 25.1 0 1 34.2 1 1 35.0 1 1 33.9 1 1 38.3 1 1 39.9 ;

The following statements invoke the GLM procedure (see Chapter 44: The GLM Procedure) for a standard ANOVA:

proc glm data=recover; class T1 T2; model time = T1 T2 T1*T2; run;

Output 84.2.1 indicates that the overall model effect is not significant at the 10% level, and Output 84.2.2 indicates that neither treatment is significant at the 10% level.

Output 84.2.1: Overall ANOVA

The GLM Procedure

Dependent Variable: time

Source | DF | Sum of Squares | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|

Model | 3 | 209.9118750 | 69.9706250 | 1.86 | 0.1905 |

Error | 12 | 451.9225000 | 37.6602083 | ||

Corrected Total | 15 | 661.8343750 |

R-Square | Coeff Var | Root MSE | time Mean |
---|---|---|---|

0.317167 | 19.94488 | 6.136791 | 30.76875 |

Output 84.2.2: Model ANOVA

Source | DF | Type I SS | Mean Square | F Value | Pr > F |
---|---|---|---|---|---|

T1 | 1 | 81.4506250 | 81.4506250 | 2.16 | 0.1671 |

T2 | 1 | 106.6056250 | 106.6056250 | 2.83 | 0.1183 |

T1*T2 | 1 | 21.8556250 | 21.8556250 | 0.58 | 0.4609 |

The following statements invoke the ROBUSTREG procedure and use the same model:

proc robustreg data=recover; class T1 T2; model time = T1 T2 T1*T2 / diagnostics; T1_T2: test T1*T2; output out=robout r=resid sr=stdres; run;

Output 84.2.3 shows some basic information about the model and the response variable `time`

.

Output 84.2.3: Model-Fitting Information and Summary Statistics

The ROBUSTREG Procedure

Model Information | |
---|---|

Data Set | WORK.RECOVER |

Dependent Variable | time |

Number of Independent Variables | 2 |

Number of Continuous Independent Variables | 0 |

Number of CLASS Independent Variables | 2 |

Number of Observations | 16 |

Method | M Estimation |

Summary Statistics | ||||||
---|---|---|---|---|---|---|

Variable | Q1 | Median | Q3 | Mean | Standard Deviation |
MAD |

time | 25.5000 | 31.2000 | 34.7500 | 30.7688 | 6.6425 | 6.8941 |

The “Parameter Estimates” table in Output 84.2.4 indicates that the main effects of both treatments are significant at the 5% level.

Output 84.2.4: Model Parameter Estimates

Parameter Estimates | |||||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | DF | Estimate | Standard Error | 95% Confidence Limits | Chi-Square | Pr > ChiSq | |||

Intercept | 1 | 36.7655 | 2.0489 | 32.7497 | 40.7814 | 321.98 | <.0001 | ||

T1 | 0 | 1 | -6.8307 | 2.8976 | -12.5100 | -1.1514 | 5.56 | 0.0184 | |

T1 | 1 | 0 | 0.0000 | . | . | . | . | . | |

T2 | 0 | 1 | -7.6755 | 2.8976 | -13.3548 | -1.9962 | 7.02 | 0.0081 | |

T2 | 1 | 0 | 0.0000 | . | . | . | . | . | |

T1*T2 | 0 | 0 | 1 | -0.2619 | 4.0979 | -8.2936 | 7.7698 | 0.00 | 0.9490 |

T1*T2 | 0 | 1 | 0 | 0.0000 | . | . | . | . | . |

T1*T2 | 1 | 0 | 0 | 0.0000 | . | . | . | . | . |

T1*T2 | 1 | 1 | 0 | 0.0000 | . | . | . | . | . |

Scale | 1 | 3.5346 |

The reason for the difference between the traditional ANOVA and the robust ANOVA is explained by Output 84.2.5, which shows that the fourth observation is an outlier. Further investigation shows that the original value of 24.4 for the fourth observation was recorded incorrectly.

Output 84.2.6 displays the robust test results. The interaction between the two treatments is not significant. Output 84.2.7 displays the robust residuals and standardized robust residuals.

Output 84.2.6: Test of Significance

Robust Linear Test T1_T2 | |||||
---|---|---|---|---|---|

Test | Test Statistic | Lambda | DF | Chi-Square | Pr > ChiSq |

Rho | 0.0041 | 0.7977 | 1 | 0.01 | 0.9431 |

Rn2 | 0.0041 | 1 | 0.00 | 0.9490 |

Output 84.2.7: PROC ROBUSTREG Output

Obs | T1 | T2 | time | resid | stdres |
---|---|---|---|---|---|

1 | 0 | 0 | 20.2 | -1.7974 | -0.50851 |

2 | 0 | 0 | 23.9 | 1.9026 | 0.53827 |

3 | 0 | 0 | 21.9 | -0.0974 | -0.02756 |

4 | 0 | 0 | 42.4 | 20.4026 | 5.77222 |

5 | 1 | 0 | 27.2 | -1.8900 | -0.53472 |

6 | 1 | 0 | 34.0 | 4.9100 | 1.38911 |

7 | 1 | 0 | 27.4 | -1.6900 | -0.47813 |

8 | 1 | 0 | 28.5 | -0.5900 | -0.16693 |

9 | 0 | 1 | 25.9 | -4.0348 | -1.14152 |

10 | 0 | 1 | 34.5 | 4.5652 | 1.29156 |

11 | 0 | 1 | 25.1 | -4.8348 | -1.36785 |

12 | 0 | 1 | 34.2 | 4.2652 | 1.20668 |

13 | 1 | 1 | 35.0 | -1.7655 | -0.49950 |

14 | 1 | 1 | 33.9 | -2.8655 | -0.81070 |

15 | 1 | 1 | 38.3 | 1.5345 | 0.43413 |

16 | 1 | 1 | 39.9 | 3.1345 | 0.88679 |