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# ANOVA, ANCOVA, MANOVA and MANCOVA

## ANOVA, ANCOVA

These are robust methods to compare two or more sets of data with respect to their averages, medians or the shape of their distributions. If the data is sparse, heavily tied, or skewed, exact non-parametric methods will be used.

### Example

The torque outputs for four pneumatic actuators at three different pressures are given in the table below:

Actuator type | Torque (N.m) | ||
---|---|---|---|

400 kPa | 600 kPa | 800 Kpa | |

A B C D |
23.5 23.8 25.2 23.2 |
24.3 24.3 26.1 23.7 |
24.5 24.6 25.9 23.8 |

### The two initial questions usually are

- Do the different actuator types affect the torque output?
- Do the different pressure settings affect the torque output?

### Summary of statistical findings

There is strong evidence that the type of actuator as well as the pressure setting have a significant effect on the torque. The p-values for both tests are well below 0.001.

Thus, we know that at least two actuator types differ significantly in average output. Using the Tukey HSD test for post-hoc tests reveal that actuator types A and B produce similar results (p=0.67) but that these two actuators differ significantly in output from actuators C and D. (All p-values below 0.001.)

A means plot supports this conclusion:

The vertical bars in the plot represent 95% confidence intervals for the mean torque output.

Note: If two or more repetitions for each setting in the experimental layout would have been made, a test for the existence of interaction between actuator type and pressure setting could also be done.

## ANCOVA

If one ore more covariates are known and measured, they can be incorporated in the analysis. By doing so, the amount of error variation will be reduced considerably and this will increase the precision of the estimates of the effects.

## MANOVA and MANCOVA

**MANOVA** and **MANCOVA **are generalizations of ANOVA and ANCOVA. These techniques are used when we want to analyze the effect of predictor variables simultaneously on more than one response variable.