Anova¶
Anova is capable of performing multiple factor between, within, and
mixed analyses of variance. Within and mixed analyses of variance provide
corrections for violations of sphericity (Huynh-Feldt, Greenhouse-Geisser, Box).
The Anova objects also have a convenience method that aids in creating interaction
plots once an analysis has been run.
If you would like to run multiple factor ANOVAs but don’t want the hassle of coding your analyses you might want to check out .. _GUANO: http://sourceforge.net/projects/guano/ GUANO. It uses pyvttbl and Anova behind the scenes but provides a point-and-click interface.
Running an ANOVA¶
The recommended way to run an ANOVA is to first load data into a DataFrame and use DataFrame.anova to run the analysis. Calling DataFrame.anova will return an Anova object.
Example Within-Subjects ANOVA¶
>>> from __future__ import print_function
>>>
>>> df = DataFrame()
>>> fname = 'error~subjectXtimeofdayXcourseXmodel.csv'
>>> df.read_tbl(fname)
>>> aov = df.anova('ERROR',
wfactors=['TIMEOFDAY','COURSE','MODEL'])
>>> print(type(aov))
<class 'anova.Anova'>
Example Between-Subjects ANOVA¶
>>> df=DataFrame()
>>> fname='words~ageXcondition.csv'
>>> df.read_tbl(fname)
>>> aov=Anova()
>>> aov.run(df, 'WORDS', bfactors=['AGE','CONDITION'])
Notice in this case we are not using DataFrame.anova. We are initializing an Anova object and passing a DataFrame instance to it.
Example Mixed-Subjects ANOVA¶
To run a mixed ANOVA we just have to specify both wfactors and bfactors.
Note
Both wfactors and bfactors must be iterable (lists of labels).
>>> df = DataFrame()
>>> fname = 'suppression~subjectXgroupXcycleXphase.csv'
>>> df.read_tbl(fname)
>>> aov = df.anova('SUPPRESSION',
wfactors=['CYCLE','PHASE'],
bfactors=['GROUP'])
Specifying a Data Transformation¶
The analysis is also capable of applying log10, square-root, arc-sine, reciprocal, an Windsoring data transforms. These are specified with the transform keyword.
transform keyword
Transform
Comments
‘’
X
default
‘log’ or ‘log10’
numpy.log10(X)
‘reciprocal’ or ‘inverse’
1/X
‘square-root’ or ‘sqrt’
numpy.sqrt(X)
‘arcsine’ or ‘arcsin’
numpy.arcsin(X)
‘windsor01’
anova.windsor(X, 1)
1% trim
‘windsor05’
anova.windsor(X, 5)
5% trim
‘windsor10’
anova.windsor(X, 10)
10% trim
Printing a Summary¶
Once an analysis is ran the results can be viewed by printing the object. By default estimated marginal means are also provided.
>>> from __future__ import print_function
>>>
>>> df = DataFrame()
>>> fname = 'error~subjectXtimeofdayXcourseXmodel.csv'
>>> df.read_tbl(fname)
>>> aov = df.anova('ERROR',
wfactors=['TIMEOFDAY','COURSE','MODEL'])
>>> print(aov)
ERROR ~ TIMEOFDAY * COURSE * MODEL
TESTS OF WITHIN SUBJECTS EFFECTS
Measure: ERROR
Source Type III eps df MS F Sig. et2_G Obs. SE 95% CI lambda Obs.
SS Power
=====================================================================================================================================================
TIMEOFDAY Sphericity Assumed 140.167 - 1 140.167 120.143 0.008 3.391 27 0.456 0.894 1621.929 1
Greenhouse-Geisser 140.167 1 1 140.167 120.143 0.008 3.391 27 0.456 0.894 1621.929 1
Huynh-Feldt 140.167 1 1 140.167 120.143 0.008 3.391 27 0.456 0.894 1621.929 1
Box 140.167 1 1 140.167 120.143 0.008 3.391 27 0.456 0.894 1621.929 1
-----------------------------------------------------------------------------------------------------------------------------------------------------
Error(TIMEOFDAY) Sphericity Assumed 2.333 - 2 1.167
Greenhouse-Geisser 2.333 1 2 1.167
Huynh-Feldt 2.333 1 2 1.167
Box 2.333 1 2 1.167
-----------------------------------------------------------------------------------------------------------------------------------------------------
COURSE Sphericity Assumed 56.778 - 2 28.389 1022.000 3.815e-06 1.374 18 0.056 0.109 9198.000 1
Greenhouse-Geisser 56.778 0.501 1.002 56.667 1022.000 9.664e-04 1.374 18 0.056 0.109 9198.000 1
Huynh-Feldt 56.778 0.504 1.008 56.336 1022.000 9.349e-04 1.374 18 0.056 0.109 9198.000 1
Box 56.778 0.500 1 56.778 1022.000 9.770e-04 1.374 18 0.056 0.109 9198.000 1
-----------------------------------------------------------------------------------------------------------------------------------------------------
Error(COURSE) Sphericity Assumed 0.111 - 4 0.028
Greenhouse-Geisser 0.111 0.501 2.004 0.055
Huynh-Feldt 0.111 0.504 2.016 0.055
Box 0.111 0.500 2 0.056
-----------------------------------------------------------------------------------------------------------------------------------------------------
MODEL Sphericity Assumed 51.444 - 2 25.722 92.600 4.470e-04 1.245 18 0.176 0.345 833.400 1
Greenhouse-Geisser 51.444 0.507 1.013 50.770 92.600 0.010 1.245 18 0.176 0.345 833.400 1.000
Huynh-Feldt 51.444 0.527 1.054 48.817 92.600 0.009 1.245 18 0.176 0.345 833.400 1.000
Box 51.444 0.500 1 51.444 92.600 0.011 1.245 18 0.176 0.345 833.400 1.000
-----------------------------------------------------------------------------------------------------------------------------------------------------
Error(MODEL) Sphericity Assumed 1.111 - 4 0.278
Greenhouse-Geisser 1.111 0.507 2.027 0.548
Huynh-Feldt 1.111 0.527 2.108 0.527
Box 1.111 0.500 2 0.556
-----------------------------------------------------------------------------------------------------------------------------------------------------
TIMEOFDAY * Sphericity Assumed 5.444 - 2 2.722 2.085 0.240 0.132 9 0.540 1.057 9.383 0.446
COURSE Greenhouse-Geisser 5.444 0.814 1.628 3.345 2.085 0.255 0.132 9 0.540 1.057 9.383 0.373
Huynh-Feldt 5.444 1 2 2.722 2.085 0.240 0.132 9 0.540 1.057 9.383 0.446
Box 5.444 0.500 1 5.444 2.085 0.286 0.132 9 0.540 1.057 9.383 0.244
-----------------------------------------------------------------------------------------------------------------------------------------------------
Error(TIMEOFDAY * Sphericity Assumed 5.222 - 4 1.306
COURSE) Greenhouse-Geisser 5.222 0.814 3.255 1.604
Huynh-Feldt 5.222 1 4 1.306
Box 5.222 0.500 2 2.611
-----------------------------------------------------------------------------------------------------------------------------------------------------
TIMEOFDAY * Sphericity Assumed 16.778 - 2 8.389 37.750 0.003 0.406 9 0.223 0.436 169.875 1.000
MODEL Greenhouse-Geisser 16.778 0.540 1.079 15.545 37.750 0.021 0.406 9 0.223 0.436 169.875 0.993
Huynh-Feldt 16.778 0.571 1.142 14.697 37.750 0.018 0.406 9 0.223 0.436 169.875 0.996
Box 16.778 0.500 1 16.778 37.750 0.025 0.406 9 0.223 0.436 169.875 0.985
-----------------------------------------------------------------------------------------------------------------------------------------------------
Error(TIMEOFDAY * Sphericity Assumed 0.889 - 4 0.222
MODEL) Greenhouse-Geisser 0.889 0.540 2.159 0.412
Huynh-Feldt 0.889 0.571 2.283 0.389
Box 0.889 0.500 2 0.444
-----------------------------------------------------------------------------------------------------------------------------------------------------
COURSE * Sphericity Assumed 8.778 - 4 2.194 3.762 0.052 0.212 6 0.367 0.719 11.286 0.504
MODEL Greenhouse-Geisser 8.778 0.354 1.415 6.204 3.762 0.157 0.212 6 0.367 0.719 11.286 0.223
Huynh-Feldt 8.778 0.354 1.415 6.204 3.762 0.157 0.212 6 0.367 0.719 11.286 0.223
Box 8.778 0.500 2 4.389 3.762 0.120 0.212 6 0.367 0.719 11.286 0.292
-----------------------------------------------------------------------------------------------------------------------------------------------------
Error(COURSE * Sphericity Assumed 4.667 - 8 0.583
MODEL) Greenhouse-Geisser 4.667 0.354 2.830 1.649
Huynh-Feldt 4.667 0.354 2.830 1.649
Box 4.667 0.500 4 1.167
-----------------------------------------------------------------------------------------------------------------------------------------------------
TIMEOFDAY * Sphericity Assumed 2.778 - 4 0.694 1.923 0.200 0.067 3 0.408 0.800 2.885 0.152
COURSE * Greenhouse-Geisser 2.778 0.290 1.159 2.397 1.923 0.293 0.067 3 0.408 0.800 2.885 0.087
MODEL Huynh-Feldt 2.778 0.290 1.159 2.397 1.923 0.293 0.067 3 0.408 0.800 2.885 0.087
Box 2.778 0.500 2 1.389 1.923 0.260 0.067 3 0.408 0.800 2.885 0.109
-----------------------------------------------------------------------------------------------------------------------------------------------------
Error(TIMEOFDAY * Sphericity Assumed 2.889 - 8 0.361
COURSE * Greenhouse-Geisser 2.889 0.290 2.318 1.246
MODEL) Huynh-Feldt 2.889 0.290 2.318 1.246
Box 2.889 0.500 4 0.722
TABLES OF ESTIMATED MARGINAL MEANS
Estimated Marginal Means for TIMEOFDAY
TIMEOFDAY Mean Std. Error 95% Lower Bound 95% Upper Bound
==================================================================
T1 5.778 0.457 4.882 6.674
T2 2.556 0.229 2.108 3.003
Estimated Marginal Means for COURSE
COURSE Mean Std. Error 95% Lower Bound 95% Upper Bound
===============================================================
C1 5.222 0.608 4.031 6.414
C2 4.500 0.562 3.399 5.601
C3 2.778 0.432 1.931 3.625
Estimated Marginal Means for MODEL
MODEL Mean Std. Error 95% Lower Bound 95% Upper Bound
==============================================================
M1 5.333 0.686 3.989 6.678
M2 4.222 0.558 3.129 5.315
M3 2.944 0.328 2.301 3.588
Estimated Marginal Means for TIMEOFDAY * COURSE
TIMEOFDAY COURSE Mean Std. Error 95% Lower Bound 95% Upper Bound
===========================================================================
T1 C1 7.222 0.641 5.966 8.478
T1 C2 6.111 0.790 4.564 7.659
T1 C3 4 0.577 2.868 5.132
T2 C1 3.222 0.401 2.437 4.007
T2 C2 2.889 0.261 2.378 3.400
T2 C3 1.556 0.294 0.979 2.132
Estimated Marginal Means for TIMEOFDAY * MODEL
TIMEOFDAY MODEL Mean Std. Error 95% Lower Bound 95% Upper Bound
==========================================================================
T1 M1 7.444 0.835 5.807 9.081
T1 M2 6.111 0.512 5.107 7.115
T1 M3 3.778 0.465 2.867 4.689
T2 M1 3.222 0.434 2.372 4.073
T2 M2 2.333 0.408 1.533 3.133
T2 M3 2.111 0.261 1.600 2.622
Estimated Marginal Means for COURSE * MODEL
COURSE MODEL Mean Std. Error 95% Lower Bound 95% Upper Bound
=======================================================================
C1 M1 6.667 1.085 4.540 8.794
C1 M2 5.167 1.195 2.825 7.509
C1 M3 3.833 0.601 2.656 5.011
C2 M1 6.167 1.195 3.825 8.509
C2 M2 4.167 0.792 2.614 5.720
C2 M3 3.167 0.477 2.231 4.102
C3 M1 3.167 0.872 1.457 4.877
C3 M2 3.333 0.882 1.605 5.062
C3 M3 1.833 0.307 1.231 2.436
Estimated Marginal Means for TIMEOFDAY * COURSE * MODEL
TIMEOFDAY COURSE MODEL Mean Std. Error 95% Lower Bound 95% Upper Bound
===================================================================================
T1 C1 M1 9 0.577 7.868 10.132
T1 C1 M2 7.667 0.333 7.013 8.320
T1 C1 M3 5 0.577 3.868 6.132
T1 C2 M1 8.667 0.882 6.938 10.395
T1 C2 M2 5.667 0.882 3.938 7.395
T1 C2 M3 4 0.577 2.868 5.132
T1 C3 M1 4.667 1.202 2.311 7.022
T1 C3 M2 5 0.577 3.868 6.132
T1 C3 M3 2.333 0.333 1.680 2.987
T2 C1 M1 4.333 0.333 3.680 4.987
T2 C1 M2 2.667 0.882 0.938 4.395
T2 C1 M3 2.667 0.333 2.013 3.320
T2 C2 M1 3.667 0.333 3.013 4.320
T2 C2 M2 2.667 0.333 2.013 3.320
T2 C2 M3 2.333 0.333 1.680 2.987
T2 C3 M1 1.667 0.333 1.013 2.320
T2 C3 M2 1.667 0.882 -0.062 3.395
T2 C3 M3 1.333 0.333 0.680 1.987
}}}
Writing Summary to File¶
If you are familiar with Python it should be obvious that the class essential
has a __str__() method that generates the above output. If you are not as familiar
with Python all you need to know is that turning the object into a string (via str(aov))
yields the summary as a big string. This means that writing the summary to a file is pretty
straightforward.
>>> with open('output.txt','w') as f:
f.write(str(aov))
>>>
Working with Anova Objects (Advanced)¶
- If you wish perform additional operations with the results the data from the main effects and
interactions can be accessed directly. The Anova are dictionaries whose keys coorespond to the main effects and interactions.
>>> for d in aov: print(d) ('TIMEOFDAY',) ('COURSE',) ('MODEL',) ('TIMEOFDAY', 'COURSE') ('TIMEOFDAY', 'MODEL') ('COURSE', 'MODEL') ('TIMEOFDAY', 'COURSE', 'MODEL')
The values are dictionaries of the various values pertaining to the effect.
>>> from pprint import pprint as pp
>>> pp(aov[('TIMEOFDAY', 'COURSE', 'MODEL')])
{'F': 1.9230769230769222,
'F_gg': 1.923076923076922,
'F_hf': 1.923076923076922,
'F_lb': 1.9230769230769222,
'ci': 0.80005506708787966,
'ci_gg': 0.80005506708787966,
'ci_hf': 0.80005506708787966,
'ci_lb': 0.80005506708787966,
'critT': 2.3060041350333709,
'critT_gg': 2.3060041350333709,
'critT_hf': 2.3060041350333709,
'critT_lb': 2.3060041350333709,
'df': 4,
'df_gg': 1.1590909090909087,
'df_hf': 1.1590909090909087,
'df_lb': 2.0,
'dfe': 8.0,
'dfe_gg': 2.3181818181818175,
'dfe_hf': 2.3181818181818175,
'dfe_lb': 4.0,
'eps_gg': 0.28977272727272718,
'eps_hf': 0.28977272727272718,
'eps_lb': 0.5,
'eta': 0.067204301075268813,
'lambda': 2.8846153846153828,
'lambda_gg': 2.8846153846153828,
'lambda_hf': 2.8846153846153828,
'lambda_lb': 2.8846153846153828,
'mse': 0.36111111111111122,
'mse_gg': 1.2461873638344234,
'mse_hf': 1.2461873638344234,
'mse_lb': 0.72222222222222243,
'mss': 0.69444444444444431,
'mss_gg': 2.3965141612200438,
'mss_hf': 2.3965141612200438,
'mss_lb': 1.3888888888888886,
'obs': 3.0,
'obs_gg': 3.0,
'obs_hf': 3.0,
'obs_lb': 3.0,
'p': 0.19999514760153031,
'p_gg': 0.2930377602206829,
'p_hf': 0.2930377602206829,
'p_lb': 0.2599000384467513,
'power': 0.15191672432754222,
'power_gg': 0.087331953378021465,
'power_hf': 0.087331953378021465,
'power_lb': 0.10875612151993008,
'se': 0.40819136075912227,
'se_gg': 0.40819136075912227,
'se_hf': 0.40819136075912227,
'se_lb': 0.40819136075912227,
'ss': 2.7777777777777772,
'sse': 2.8888888888888897,
'y2': array([ 9. , 7.66666667, 5. , 8.66666667, 5.66666667,
4. , 4.66666667, 5. , 2.33333333, 4.33333333,
2.66666667, 2.66666667, 3.66666667, 2.66666667, 2.33333333,
1.66666667, 1.66666667, 1.33333333])}
Hopefully that is enough to get you started.