- Balanced and unbalanced designs
- Missing cells
- Factorial, nested, and mixed designs
- Repeated measures
- Box, Greenhouse–Geisser, and Huynh–Feldt corrections

Afifi and Azen (1979) fitted a model of the change in systolic blood pressure for 58 patients, each suffering from one of three diseases, who were randomly assigned one of four different drug treatments:

Source | Partial SS df MS F Prob > F | ||||||

Model | 4259.33851 11 387.212591 3.51 0.0013 | ||||||

drug | 2997.47186 3 999.157287 9.05 0.0001 | ||||||

disease | 415.873046 2 207.936523 1.88 0.1637 | ||||||

drug#disease | 707.266259 6 117.87771 1.07 0.3958 | ||||||

Residual | 5080.81667 46 110.452536 | ||||||

Total | 9340.15517 57 163.862371 |

An important feature of Stata is that it does not have modes or modules.
You do not enter the ANOVA module to fit an ANOVA model, but you simply type
the command. The advantage in this is that Stata’s other commands can be
interspersed to help you better understand these data. For instance, the
data here are almost balanced, as revealed by Stata's **table** command:

Patient's Disease | ||||||

Drug Used | 1 2 3 Total | |||||

1 | 6 4 5 15 | |||||

2 | 5 4 6 15 | |||||

3 | 3 5 4 12 | |||||

4 | 5 6 5 16 | |||||

Total | 19 19 20 58 | |||||

**table** can also be used to help you better understand the relationship
of the increase in blood pressure by drug and disease:

Patient's Disease | ||||||

Drug Used | 1 2 3 Total | |||||

1 | 29.33 28.25 20.40 26.07 | |||||

2 | 28.00 33.50 18.17 25.53 | |||||

3 | 16.33 4.40 8.50 8.75 | |||||

4 | 13.60 12.83 14.20 13.50 | |||||

Total | 22.79 18.21 15.80 18.88 | |||||

The **test** command allows you to perform tests directly on the
coefficients of the underlying regression model. For instance, we can test
if the coefficient on the third drug is equal to the coefficient on the
fourth.

We find that the two coefficients are not significantly different, at least at any significance level smaller than 73%.

For more complex tests, the **contrast** command often provides
a more concise way to specify the test we are interested in and prevents
us from having to write tests in terms of the regression coefficients.
With **contrast**, we instead specify our tests in terms of differences
in the marginal means for the levels of a particular factor. For
example, if we want to compare the third and fourth drugs, we can test the
difference in the mean impact on systolic blood pressure separately
for each disease using the **@** operator. We also use the reverse
adjacent operator, **ar.**, to compare the fourth level of the drug
with the previous level.

df F P>F | ||||

drug@disease | ||||

(4 vs 3) 1 | 1 0.13 0.7234 | |||

(4 vs 3) 2 | 1 1.76 0.1917 | |||

(4 vs 3) 3 | 1 0.65 0.4230 | |||

Joint | 3 0.85 0.4761 | |||

Denominator | 46 | |||

Contrast Std. Err. [95% Conf. Interval] | |||||

drug@disease | |||||

(4 vs 3) 1 | -2.733333 7.675156 -18.18262 12.71595 | ||||

(4 vs 3) 2 | 8.433333 6.363903 -4.376539 21.24321 | ||||

(4 vs 3) 3 | 5.7 7.050081 -8.491077 19.89108 | ||||

**test** and **contrast** can still access the estimates, even though
two tabulations have intervened. Similarly, **anova** is integrated with
Stata’s **regress** command for estimating linear regressions. We
can review the underlying regression estimates by typing **regress**
without arguments:

Source | SS df MS | Number of obs = 58 | |

F( 11, 46) = 3.51 | |||

Model | 4259.33851 11 387.212591 | Prob > F = 0.0013 | |

Residual | 5080.81667 46 110.452536 | R-squared = 0.4560 | |

Adj R-squared = 0.3259 | |||

Total | 9340.15517 57 163.862371 | Root MSE = 10.51 |

systolic | Coef. Std. Err. t P>t| [95% Conf. Interval] | |||||

drug | ||||||

2 | -1.333333 6.363903 -0.21 0.835 -14.14321 11.47654 | |||||

3 | -13 7.431438 -1.75 0.087 -27.95871 1.958708 | |||||

4 | -15.73333 6.363903 -2.47 0.017 -28.54321 -2.923461 | |||||

disease | ||||||

2 | -1.083333 6.783944 -0.16 0.874 -14.7387 12.57204 | |||||

3 | -8.933333 6.363903 -1.40 0.167 -21.74321 3.876539 | |||||

drug#disease | ||||||

2 2 | 6.583333 9.783943 0.67 0.504 -13.11072 26.27739 | |||||

2 3 | -.9 8.999918 -0.10 0.921 -19.0159 17.2159 | |||||

3 2 | -10.85 10.24353 -1.06 0.295 -31.46916 9.769157 | |||||

3 3 | 1.1 10.24353 0.11 0.915 -19.51916 21.71916 | |||||

4 2 | .3166667 9.301675 0.03 0.973 -18.40663 19.03997 | |||||

4 3 | 9.533333 9.202189 1.04 0.306 -8.989712 28.05638 | |||||

_cons | 29.33333 4.290543 6.84 0.000 20.69692 37.96975 | |||||

In our original estimation, the direct effect of disease was found to be
insignificant, as was the **drug#disease** interaction. We might now
compare our two-way factorial model with a simpler, one-way layout:

Source | Partial SS df MS F Prob > F | |

disease drug#disease | 1126.1 8 140.7625 1.27 0.2801 | |

Residual | 5080.81667 46 110.452536 |

With the **test** command above, we found that a one-way model
fits these data well. We could use either Stata's **anova** command or
Stata’s **oneway** command to fit a one-way model.

Analysis of Variance | |||||

Source SS df MS F Prob > F | |||||

Between groups 3133.23851 3 1044.41284 9.09 0.0001 | |||||

Within groups 6206.91667 54 114.942901 | |||||

Total 9340.15517 57 163.862371 |

Bartlett's test for equal variances: chi2(3) = 1.0063 Prob>chi2 = 0.800 |

Comparison of Increment in Systolic B.P. by Drug Used |

(Bonferroni) |

Row Mean- | ||||

Col Mean | 1 2 3 | |||

2 | -.533333 | |||

1.000 | ||||

3 | -17.3167 -16.7833 | |||

0.001 0.001 | ||||

4 | -12.5667 -12.0333 4.75 | |||

0.012 0.017 1.000 |

Table 7.7 of Winer, Brown, and Michels (1991) provides a repeated-measures ANOVA example involving both nested and crossed terms. There are four dial shapes and two methods for calibrating dials. Subjects are nested within the calibration method, and an accuracy score is obtained.

Here is the
Stata **anova** command for this problem.

Source | Partial SS df MS F Prob > F | ||||||

Model | 123.125 11 11.1931818 9.06 0.0003 | ||||||

calib | 51.0416667 1 51.0416667 11.89 0.0261 | ||||||

subject|calib | 17.1666667 4 4.29166667 | ||||||

shape | 47.4583333 3 15.8194444 12.80 0.0005 | ||||||

calib#shape | 7.45833333 3 2.48611111 2.01 0.1662 | ||||||

Residual | 14.8333333 12 1.23611111 | ||||||

Total | 137.958333 23 5.99818841 |

------------ | Prob > F | ------------ |

Source | df F Regular H-F G-G Box | ||||||

shape | 3 12.80 0.0005 0.0011 0.0099 0.0232 | ||||||

calib#shape | 3 2.01 0.1662 0.1791 0.2152 0.2291 | ||||||

Residual | 12 | ||||||

- Afifi, A. A., and S. P. Azen. 1979.
*Statistical Analysis: A computer-oriented approach*. 2nd ed. New York: Academic Press.

- Winer, B. J., R. Brown, and K. M. Michels. 1991.
*Statistical Principles in Experimental Design*. 3rd ed. New York: McGraw–Hill.