In this post, you will learn how to carry out a repeated measures ANOVA in Python. A standard method in experimental psychology is within-subjects designs. One way to analysis the data collected using within-subjects designs is using repeated measures ANOVA. I recently wrote a post on how to conduct a repeated measures ANOVA using Python and rpy2. I wrote that post since the great Python package statsmodels do not include repeated measures ANOVA. However, the approach using rpy2 requires R statistical environment installed. Recently, I found a Python library called pyvttbl whith which you can do within-subjects ANOVAs. Pyvttbl enables you to create multidimensional pivot tables, process data and carry out statistical tests. Using the method anova on pyvttbl’s DataFrame we can carry out repeated measures ANOVA using only Python.

Note, pyvttbl is no longer maintained, and you should see a new post using Pingouin for carrying out repeated Measures ANOVA: Repeated Measures ANOVA in R and Python using afex & pingouin

## Table of Contents

### Why within-subject designs?

There are at least two advantages to using a within-subjects design. First, more information is obtained from each subject in a within-subjects design compared to a between-subjects design. Each subject is measured in all conditions, whereas in the between-subjects design, each subject is typically measured in one or more but not all conditions. A within-subject design thus requires fewer subjects to obtain a certain statistical power. In situations where it is costly to find subjects, this kind of design is clearly better than a between-subjects design. Second, the variability in individual differences between subjects is removed from the error term. That is, each subject is his or her control, and extraneous error variance is reduced.

Now, if you are working with data from a study with a between-subjects design and have only two groups you can carry out two-sample t-test with Python or Mann-Whitney U test in Python.

### Repeated measures ANOVA in Python

As you may be aware of, in this tutorial we are going to use an old Python package called pyvttbl to do data analysis. First, we need to install this Python package.

#### Installing pyvttbl

pyvttbl can be installed using pip:

`pip install pyvttbl`

Code language: Bash (bash)

If you are using Linux you may need to add ‘sudo’ before the pip command. This method installs pyvttbl and, hopefully, any missing dependencies. Note, if you decide to work with pyvttbl you need also inteed dto use pip to install specific versions of the dependencies.

#### Python script

I continue with simulating a response time data set. If you have your own data set you want to do your analysis on you can use the method “read_tbl” to load your data from a CSV-file.

```
from numpy.random import normal
import pyvttbl as pt
from collections import namedtuple
N = 40
P = ["noise","quiet"]
rts = [998,511]
mus = rts*N
Sub = namedtuple('Sub', ['Sub_id', 'rt','condition'])
df = pt.DataFrame()
for subid in xrange(0,N):
for i,condition in enumerate(P):
df.insert(Sub(subid+1,
normal(mus[i], scale=112., size=1)[0],
condition)._asdict())
```

Code language: Python (python)

Conducting the repeated measures ANOVA with pyvttbl is pretty straight forward. You just take the pyvttbl DataFrame object and use the method anova. The first argument is your dependent variable (e.g. response time), and you specify the column in which the subject IDs are (e.g., sub=’Sub_id’). Finally, you add your within subject factor(s) (e.g., wfactors). wfactors take a list of column names containing your within subject factors. In my simulated data there is only one (e.g. ‘condition’). Note, if your Numpy version is greater than 1.1.x you will have to install an older version. A good way to do this is to run Pyvttbl within a virtual environment (see Step-by-step guide for solving the Pyvttbl Float and NoneType error for a detailed solution both for Linux and Windows users).

```
aov = df.anova('rt', sub='Sub_id', wfactors=['condition'])
print(aov)
```

Code language: Python (python)

#### Tests of Within-Subjects Effects

##### Measure: rt

Source | Type III Sum of Squares | ε | df | MS | F | Sig. | η^{2}_{G} | Obs. | SE of x̄ | ±95% CI | λ | Obs. Power | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

condition | Sphericity Assumed | 4209536.428 | – | 1.000 | 4209536.428 | 309.093 | 0.000 | 4.165 | 40.000 | 19.042 | 37.323 | 317.019 | 1.000 |

Greenhouse-Geisser | 4209536.428 | 1.000 | 1.000 | 4209536.428 | 309.093 | 0.000 | 4.165 | 40.000 | 19.042 | 37.323 | 317.019 | 1.000 | |

Huynh-Feldt | 4209536.428 | 1.000 | 1.000 | 4209536.428 | 309.093 | 0.000 | 4.165 | 40.000 | 19.042 | 37.323 | 317.019 | 1.000 | |

Box | 4209536.428 | 1.000 | 1.000 | 4209536.428 | 309.093 | 0.000 | 4.165 | 40.000 | 19.042 | 37.323 | 317.019 | 1.000 | |

Error(condition) | Sphericity Assumed | 531140.646 | – | 39.000 | 13618.991 | ||||||||

Greenhouse-Geisser | 531140.646 | 1.000 | 39.000 | 13618.991 | |||||||||

Huynh-Feldt | 531140.646 | 1.000 | 39.000 | 13618.991 | |||||||||

Box | 531140.646 | 1.000 | 39.000 | 13618.991 |

As can be seen in the output table the Sum of Squares used is **Type III** which is what common statistical software use when calculating ANOVA (the *F*-statistic) (e.g., SPSS or R-packages such as ‘afex’ or ‘ez’). The table further contains correction in case our data violates the assumption of Sphericity (which in the case of only 2 factors, as in the simulated data, is nothing to worry about). As you can see we also get generalized eta squared as effect size measure and 95 % Confidence Intervals. It is stated in the docstring for the class Anova that standard Errors and 95% confidence intervals are calculated according to Loftus and Masson (1994). Furthermore, generalized eta squared allows comparability across between-subjects and within-subjects designs (see, Olejnik & Algina, 2003).

Conveniently, if you ever want to transform your data you can add the argument transform. There are several options here; *log* or *log10*, *reciprocal* or *inverse*,* square-root* or *sqrt*, *arcsine* or *arcsin*, and *windsor10*. For instance, if you want to use log-transformation you just add the argument “*transform*=’log'” (either of the previously mentioned methods can be used as arguments in string form):

`aovlog = df.anova('rt', sub='Sub_id', wfactors=['condition'], transform='log')`

Code language: Python (python)

Using pyvttbl we can also analyse mixed-design/split-plot (within-between) data. Doing a split-plot is easy; just add the argument “*bfactors*=” and a list of your between-subject factors. If you are interested in one-way ANOVA for independent measures see my newer post: Four ways to conduct one-way ANOVAS with Python.

Finally, I created a function that extracts the F-statistics, Mean Square Error, generalized eta squared, and the p-value the results obtained with the anova method. It takes a factor as a string, a ANOVA object, and the values you want to extract. Keys for your different factors can be found using the key-method (e.g., *aov.keys()*).

```
def extract_for_apa(factor, aov, values = ['F', 'mse', 'eta', 'p']):
results = {}
for key,result in aov[(factor,)].iteritems():
if key in values:
results[key] = result
return results
```

Code language: Python (python)

Note, the table with the results in this post was created with the private *method _within_html*. To create an HTML table you will have to import SimpleHTML:

```
import SimpleHTML
output = SimpleHTML.SimpleHTML('Title of your HTML-table')
aov._within_html(output)
output.write('results_aov.html')
```

Code language: Python (python)

That was all. There are at least one downside with using pyvttbl for doing within-subjects analysis in Python (ANOVA). Pyvttbl is not compatible with Pandas DataFrame which is commonly used. However, this may not be a problem since pyvttbl, as we have seen, has its own DataFrame method. There are also a some ways to aggregate and visualizing data using Pyvttbl. Another downside is that it seems like Pyvttbl no longer is maintained.

### References

Loftus, G.R., & Masson, M.E. (1994). Using confidence intervals in within-subjects designs. The Psychonomic Bulletin & Review, 1(4), 476-490.

Olejnik, S., & Algina, J. (2003). Generalized eta and omega squared statistics: measures of effect size for some common research designs. Psychological Methods, 8(4), 434–47. http://doi.org/10.1037/1082-989X.8.4.434

Richard HöchenbergerDear Erik,

thanks for this excellent blog post! I would like to point out that there is also relatively new, but actively maintained RM-ANOVA support in the mne-python package, see http://martinos.org/mne/stable/generated/mne.stats.f_mway_rm.html#mne.stats.f_mway_rm The data have to be transformed into a certain shape to use this function though; but this should be straightforward using pandas. It would be great if you could cover this approach in future post, as I think it might help many people making the final switch to Python.

All the best,

–Richard

Erik MarsjaHey Richard,

Glad you liked my post.

Thank you for mentioning mne-python package. It seems like I’ve missed it.

I will have a look at it later today (travelling atm).

Again, thank you for pointing it out.

Have a nice day,

Erik

Richard HöchenbergerHi Erik,

great you like my suggestion 🙂 I actually discovered mne’s RM-ANOVA through this github issue re statsmodels: https://github.com/statsmodels/statsmodels/issues/749

And by the way, although the docstring of mne-python’s f_mway_rm() states it expects a 3D array as input, the code will handle ordinary 2D (“behavioral”) data just fine: https://github.com/mne-tools/mne-python/blob/master/mne/stats/parametric.py#L295

It would be great to have a side-by-side comparison of the results from this RM-ANOVA implementation and possibly afex or JASP (which, to my knowledge, internally uses afex). Let me know if I can help you with this in any way!

(NB: I’m by no means a stats guru, but simply a neuroscientist/psychologist who happens to be willing to analyze his data 😉 That’s why I’ve been very hesitant to move away from afex so far, because I have the impression that Henrik Singmann, the author of afex — who also happens to be a psychologist! — made very “sane” decision as to how the package should behave, e.g. it allows me to replicate existing SPSS analyses etc., so I can assume that “I’m doing it right” with great confidence.)

Thanks again,

–Richard

Richard HöchenbergerOh and, of course, have a nice day too! 🙂

Erik MarsjaGreat suggestions. I have actually been thinking on comparing R-packages and Python packages. Maybe I’ll reach out to you.

I am no stats guru either… However, I enjoy programming (may sound strange but I find it relaxing to write code) and I’ve found that I understand statistical concepts better when using R (or Python).

I tend to mess things up in GUI interfaces such as SPSS (e.g., forgetting which boxes I checked in last time). Doing stats in R (or Python) make it easier to reproduce the same analysis. In my two first studies of my thesis I use R (i.e., afex and lsmeans). Afex is really great.

John TanHi Erik,

Do you know where I could find the SimpleHTML module?

Thanks,

John

Erik MarsjaHey John,

just import it from pyvttbl:

`from pyvttbl import SimpleHTML`

.Erik

dunovankStellar, thanks for posting this.

PhilGHi,

Thanks for your post.

I’m able to run your example without an issue, but when I try to run it with a second within subject factor, I get the error below;

Exception: (‘data has %d conditions; design only %d’, 2, 4)

I’ve pasted is the script that causes the error below. As you can see, the only change that I have made to your example is to add a second within subject factor (which is simply a duplicate of your single within subject factor), such that now the data contains condition1 and condition2 instead of only condition. Have you been able to run your example with multiple within subject factors?

Here is the script;

from numpy.random import normal

import pyvttbl as pt

from collections import namedtuple

N = 40

P = [‘c1’, ‘c2’]

rts = [998, 511]

mus = rts * N

Sub = namedtuple(‘Sub’, [‘Sub_id’, ‘rt’, ‘condition1’, ‘condition2’])

df = pt.DataFrame()

for subid in xrange(0, N):

for i, condition in enumerate(P):

df.insert(Sub(subid + 1, normal(mus[i], scale=112., size=1)[0], condition, condition)._asdict())

aov = df.anova(‘rt’, sub=’Sub_id’, wfactors=[‘condition1’, ‘condition2’])

print(aov)

Thanks, Phil

Erik MarsjaHey Phil,

For simulating data for a 2-way ANOVA for repeated measures look at this example: http://wp.me/p5bFFs-oV

Regards,

Erik

LucaHello Erik,

thank you for your article!

I am trying to use your guide for running a repeated measures analyisis for my experiment. I have the data in a CSV file which I imported using the method read_table. However, I am getting issues when running the anova method.

Here my script:

df = pt.DataFrame()

df.read_tbl(‘GEQ_DATA_ANALYSIS_CORE_CSV_SUMMARY_p.csv’,delimiter =’;’)

aov = df.anova(‘FLOW’, sub=’P_ID’, wfactors=[‘SETUP’])

print(aov)

The error I get is:

TypeError Traceback (most recent call last)

in ()

1 df = pt.DataFrame()

2 df.read_tbl(‘GEQ_DATA_ANALYSIS_CORE_CSV_SUMMARY_p.csv’,delimiter =’;’)

—-> 3 aov = df.anova(‘FLOW’, sub=’P_ID’, wfactors=[‘SETUP’])

4 print(aov)

C:\Program Files\Anaconda2\lib\site-packages\pyvttbl\base.pyc in anova(self, dv, sub, wfactors, bfactors, measure, transform, alpha)

1973 aov=stats.Anova()

1974 aov.run(self, dv, sub=sub, wfactors=wfactors, bfactors=bfactors,

-> 1975 measure=measure, transform=transform, alpha=alpha)

1976 return aov

1977

C:\Program Files\Anaconda2\lib\site-packages\pyvttbl\stats\_anova.pyc in run(self, dataframe, dv, wfactors, bfactors, sub, measure, transform, alpha)

708

709 if len(wfactors)!=0 and len(bfactors)==0:

–> 710 self._within()

711

712 if len(wfactors)==0 and len(bfactors)!=0:

C:\Program Files\Anaconda2\lib\site-packages\pyvttbl\stats\_anova.pyc in _within(self)

1131 for e in xrange(1,Ne+1):

1132 # code effects so we can build contrasts

-> 1133 cw = self._num2binvec(e,Nf)

1134 efs = asarray(factors)[Nf-1-where(asarray(cw)==2.)[0][::-1]]

1135 r=self[tuple(efs)] # unpack dictionary

C:\Program Files\Anaconda2\lib\site-packages\pyvttbl\stats\_anova.pyc in _num2binvec(self, d, p)

1238 d=floor(d/2.)

1239

-> 1240 return list(array(list(zeros((p-len(b))))+b)+1.)

1241

1242 ## def output2html(self, fname, script=”):

TypeError: ‘float’ object cannot be interpreted as an index

Which are the requirements for the CSV file ? How does it has to be formatted ?

Thank you in advance!

Best,

Luca

Erik MarsjaHey Luca,

your data should be stored in long format (See here, for instance.). I am not sure exactly what is going on with your ANOVA but if you could post your the first couple of rows of your data (‘df’, in your code) maybe we can find a solution.

Kind regards,

Erik

OlyHi – I am getting the same error. It don’t think it is about the data, but instead the packages installed in anaconda. My pyvttbl code will run perfectly until something is upgraded, then I get the identical error:

-> 1240 return list(array(list(zeros((p-len(b))))+b)+1.)

TypeError: ‘float’ object cannot be interpreted as an index

I don’t know how to tell which package is causing the crash. It was previously fixed again *magically* after upgrading some packages. Now I have the same problem again.

If anyone knows which package might be the culprit, or a work around for the error it would be very helpful. This is a great package for ANOVAs and I hope I can keep using it.

Erik MarsjaHey Oly,

You are right. After a comment by Damien we know what package is the culprit. It is Numpy. Pyvttbl is not maintained any more and seems only be compatible with Numpy version 1.1.x. I have created a step-by-step guide on how to run Pyvttbl within a virtual environment (https://www.marsja.se/solving-pyvttbl-error-float-nonetype-error/). I may see if I can come up with another solution if I have time. You also have the option to install the beta version of statsmodels. You can run repeated measures ANOVA with this version of statsmodels. However, you will only get an ANOVA table containing the degrees of freedeom, F-values, and p-values.

OlyGreat, thank you for your fast response! I will take a look at this information and post my solution in the future for others who may have the same problem.

Erik MarsjaNo worries. It may, of course, not be the same issue.

OlyHi Erik,

To follow up, I implemented the AnovaRM function in the beta version of statsmodels. It worked great! Results were identical to SPSS output (I’m running two-way, 2×2 levels, repeated measures ANOVAs).

Found your YouTube video very helpful as well: https://www.youtube.com/watch?v=xzET1rpvJ_A

Thank you so much for taking the time to do this.

Best,

Oly

Erik MarsjaHey Oly,

Thanks for your comment. That is great! I put together that video in a rush, glad it helped. I did not have time to figure out to get/calculate effect sizes. If, or when, I figure it out, I may put it together in a blog post and/or a YouTube video.

Best,

Erik

OlyThat would definitely be great! Should be relatively straight forward if you have the sum of squares, correct? I’m happy to test code on my own data, just let me know.

Looking forward to following your posts. Thanks again.

Best,

Oly

OlyHi Erik,

I just wanted to let you know that I did find a discrepancy in the AnovaRM function in the beta version of statsmodels when I ran a 1-way RM ANOVA with 2 levels (so akin to a paired t-test). I noticed the degrees of freedom of the error was off, and indeed the function gave different values than in SPSS. So in sum, the two-way RM ANOVAs I tested give identical output as SPSS, but the one-way RM ANOVAs do not. Let me know if you would like me to send the data I used?

Thanks in advance for all your help. The AnovaRM function implementation is great, so I hope this can be sorted out relatively easily.

Best,

Oly

Erik MarsjaHey Oly,

Thanks for letting me know. I am not involved in developing the AnovaRM method. Maybe you could open up an issue on their GitHub page (https://github.com/statsmodels/statsmodels). This way it may be fixed.

Erik

OlyHi Erik,

Ok thanks! Sorry, I thought you were one of the contributors.

Best,

Olympia

Leonardo VHi Erik! Thank you for your article!

I am a beginner in Python, I’m trying to use your guide for running a split plot anova (my goal is to determine the interaction between two within variables( AGE’, ‘ETHNICGROUP’) and one between variable( ‘SEXP’). and I obtained the output (thank you again).

Now I’m trying to use your function to extract the p-value obtained with the anova method, but I don’t understand how it works.

These are my aov.keys():

[(‘AGE’,),

(‘ETHNICGROUP’,),

(‘AGE’, ‘ETHNICGROUP’),

(‘SEXP’,),

(‘AGE’, ‘SEXP’),

(‘ETHNICGROUP’, ‘SEXP’),

(‘AGE’, ‘ETHNICGROUP’, ‘SEXP’),

(‘SUBJECT’,),

(‘TOTAL’,),

(‘WITHIN’,),

(‘AGE’, ‘SUBJECT’),

(‘ETHNICGROUP’, ‘SUBJECT’),

(‘AGE’, ‘ETHNICGROUP’, ‘SUBJECT’)]

How do I modify your script? ==> def extract_for_apa(factor, aov, values = [‘F’, ‘mse’, ‘eta’, ‘p’]):

results = {}

for key,result in aov[(factor,)].iteritems():

if key in values:

results[key] = result

return results

Erik MarsjaHey Leonardo,

I am glad you found it helpful. I realize that I don’t show an example of how to use that function. It can extract the values for each factor:

`print(extract_for_apa('condition', aov))`

Would print F, MSE, eta-squared, and the p-value (based on the example above). In your case you could loop through the 3 variables of interest or just runt the function 3 times with your variables as input. Let me know if it doesn’t work or if you need help.

/Erik

Leonardo VThank you, it works. These are the results:

print(extract_for_apa(‘AGE’, aov))

{‘p’: 0.38304275607253779, ‘mse’: 0.00071049005495488833, ‘eta’: 0.0047898426784019855, ‘F’: 0.77986179350784657}

print(extract_for_apa(‘ETHNICGROUP’, aov))

{‘p’: 0.019574151602473413, ‘mse’: 0.00069223374419013475, ‘eta’: 0.034659508263642044, ‘F’: 5.9711544610483269}

print(extract_for_apa(‘SEXP’, aov))

‘p’: 0.44921555871908303, ‘mse’: 0.00091254877064392932, ‘eta’: 0.0046182855882132079, ‘F’: 0.58533465964381803}

Due to my lack of knowledge in data analysis, I thought that if you want to explore the interaction between two wfactors and one bfactor you have to obtain just one p value (and not one for every variable.)…but as you said I was wrong!

Leonardo VSorry if I bother you again Erik, but I was wondering if there’s a way to print the p value of the interaction between these three factors. (Unfortunately you have to obtain just one p value to see if ‘AGE’ * ‘ETHNICGROUP’ * ‘SEXP’ is significant.

I tried to put in your function the “key of the interaction” (print(extract_for_apa((‘AGE’ , ‘ETHNICGROUP’ , ‘SEXP’), aov)) it doesn’t work.