# Two-way ANOVA for repeated measures using Python

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Previously I have shown how to analyze data collected using within-subjects designs using rpy2 (i.e., R from within Python) and Pyvttbl. In this post I will extend it into a factorial ANOVA using Python (i.e., Pyvttbl). In fact, we are going to carry out a Two-way ANOVA but the same method will enable you to analyze any factorial design. I start with importing the Python libraries that  are going to be use.

``````import numpy as np
import pyvttbl as pt
from collections import namedtuple```Code language: Python (python)```

Numpy is going to be used to simulate data. I create a data set in which we have one factor of two levels (P) and a second factor of 3 levels (Q). As in many of my examples the dependent variable is going to be response time (rt) and we create a list of lists for the different population means we are going to assume (i.e., the variable ‘values’). I was a bit lazy when coming up with the data so I named the independent variables ‘iv1’ and ‘iv2’. However, you could think of iv1 as two different memory tasks; verbal and spatial memory. Iv2 could be different levels of distractions (no distraction, synthetic sounds, and speech, for instance).

## Simulate data

``````N = 20
P = [1,2]
Q = [1,2,3]

values = [[998,511], [1119,620], [1300,790]]

sub_id = [i+1 for i in xrange(N)]*(len(P)*len(Q))
mus = np.concatenate([np.repeat(value, N) for value in values]).tolist()
rt = np.random.normal(mus, scale=112.0, size=N*len(P)*len(Q)).tolist()
iv1 = np.concatenate([np.array([p]*N) for p in P]*len(Q)).tolist()
iv2 = np.concatenate([np.array([q]*(N*len(P))) for q in Q]).tolist()

Sub = namedtuple('Sub', ['Sub_id', 'rt','iv1', 'iv2'])
df = pt.DataFrame()

for idx in xrange(len(sub_id)):
df.insert(Sub(sub_id[idx],rt[idx], iv1[idx],iv2[idx])._asdict())```Code language: Python (python)```

I start with a boxplot using the method boxplot from Pyvttbl. As far as I can see there is not much room for changing the plot around. We get this plot and it is really not that beautiful.

``df.box_plot('rt', factors=['iv1', 'iv2'])`Code language: Python (python)`

To run the Two-Way ANOVA is simple; the first argument is the dependent variable, the second the subject identifier, and then the within-subject factors. In two previous posts I showed how to carry out one-way and two-way ANOVA for independent measures. One could, of course, combine these techniques, to do a split-plot/mixed ANOVA by adding an argument ‘bfactors’ for the between-subject factor(s).

``````aov = df.anova('rt', sub='Sub_id', wfactors=['iv1', 'iv2'])
print(aov)```Code language: Python (python)```

The output one gets from this is an ANOVA table. In this table all metrics needed plus some more can be found; F-statistic, p-value, mean square errors, confidence intervals, effect size (i.e., eta-squared) for all factors and the interaction. Also, some corrected degrees of freedom and mean square error can be found (e.g., Grenhouse-Geisser corrected). The output is in the end of the post. It is a bit hard to read.  If you know any other way to do a repeated-measures ANOVA using Python please let me know. Also, if you happen to know that you can create nicer plots with Pyvttbl I would also like to know how! Please leave a comment.

Update (2017-07-03): If your installed version of Numpy is greater than 1.11.x, you will run into a Float and NoneType error. One quick solution for this is to downgrade Numpy to 1.11.x. I created a post, Step-by-step guide for solving the Pyvttbl Float and NoneType error, in which I show how to install Numpy 1.11.x in an virtual environment. This way, you can run your ANOVAs, without having to uninstall Numpy.

## Output ANOVA table

```rt ~ iv1 * iv2

TESTS OF WITHIN SUBJECTS EFFECTS

Measure: rt
Source                            Type III      eps      df         MS           F        Sig.      et2_G   Obs.     SE     95% CI    lambda    Obs.
SS                                                                                                         Power
=======================================================================================================================================================
iv1           Sphericity Assumed   4419957.211       -        1   4419957.211   324.248   2.128e-13   3.295     60   16.096   31.548   1023.941       1
Greenhouse-Geisser   4419957.211       1        1   4419957.211   324.248   2.128e-13   3.295     60   16.096   31.548   1023.941       1
Huynh-Feldt          4419957.211       1        1   4419957.211   324.248   2.128e-13   3.295     60   16.096   31.548   1023.941       1
Box                  4419957.211       1        1   4419957.211   324.248   2.128e-13   3.295     60   16.096   31.548   1023.941       1
-------------------------------------------------------------------------------------------------------------------------------------------------------
Error(iv1)    Sphericity Assumed    258996.722       -       19     13631.406
Greenhouse-Geisser    258996.722       1       19     13631.406
Huynh-Feldt           258996.722       1       19     13631.406
Box                   258996.722       1       19     13631.406
-------------------------------------------------------------------------------------------------------------------------------------------------------
iv2           Sphericity Assumed   5257766.564       -        2   2628883.282   206.008   4.023e-21   3.920     40   18.448   36.158    433.701       1
Greenhouse-Geisser   5257766.564   0.550    1.101   4777252.692   206.008   1.320e-12   3.920     40   18.448   36.158    433.701       1
Huynh-Feldt          5257766.564   0.550    1.101   4777252.692   206.008   1.320e-12   3.920     40   18.448   36.158    433.701       1
Box                  5257766.564   0.500        1   5257766.564   206.008   1.192e-11   3.920     40   18.448   36.158    433.701       1
-------------------------------------------------------------------------------------------------------------------------------------------------------
Error(iv2)    Sphericity Assumed    484921.251       -       38     12761.086
Greenhouse-Geisser    484921.251   0.550   20.911     23189.668
Huynh-Feldt           484921.251   0.550   20.911     23189.668
Box                   484921.251   0.500       19     25522.171
-------------------------------------------------------------------------------------------------------------------------------------------------------
iv1 *         Sphericity Assumed   1622027.598       -        2    811013.799    83.220   1.304e-14   1.209     20   22.799   44.687     87.600   1.000
iv2           Greenhouse-Geisser   1622027.598   0.545    1.091   1486817.582    83.220   6.085e-09   1.209     20   22.799   44.687     87.600   1.000
Huynh-Feldt          1622027.598   0.545    1.091   1486817.582    83.220   6.085e-09   1.209     20   22.799   44.687     87.600   1.000
Box                  1622027.598   0.500        1   1622027.598    83.220   2.262e-08   1.209     20   22.799   44.687     87.600   1.000
-------------------------------------------------------------------------------------------------------------------------------------------------------
Error(iv1 *   Sphericity Assumed    370327.311       -       38      9745.456
iv2)          Greenhouse-Geisser    370327.311   0.545   20.728     17866.175
Huynh-Feldt           370327.311   0.545   20.728     17866.175
Box                   370327.311   0.500       19     19490.911

TABLES OF ESTIMATED MARGINAL MEANS

Estimated Marginal Means for iv1
iv1    Mean     Std. Error   95% Lower Bound   95% Upper Bound
==============================================================
1     983.755       43.162           899.157          1068.354
2     599.917       21.432           557.909           641.925

Estimated Marginal Means for iv2
iv2     Mean     Std. Error   95% Lower Bound   95% Upper Bound
===============================================================
1      525.025       19.324           487.150           562.899
2      814.197       49.416           717.342           911.053
3     1036.286       43.789           950.459          1122.114

Estimated Marginal Means for iv1 * iv2
iv1   iv2     Mean     Std. Error   95% Lower Bound   95% Upper Bound
=====================================================================
1     1      553.522       24.212           506.066           600.978
1     2     1103.488       28.411          1047.804          1159.173
1     3     1294.256       19.773          1255.501          1333.011
2     1      496.528       29.346           439.009           554.047
2     2      524.906       20.207           485.301           564.512
2     3      778.317       21.815           735.560           821.073

```

## Alternative Data Analysis Techniques

In this section, you will find some blog posts that are covering other data analysis tecniques:

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### 6 thoughts on “Two-way ANOVA for repeated measures using Python”

1. Hi there. Thanks for your excellent blog. I’m trying to run a two-way repeated mesures ANOVA using pyvttbl as you explain. I use python 2.7 and installed pyvttbl via pip. I was able to import pyvttbl and create the dataframe just fine. However, when I run the test, I get this error: TypeError: unsupported operand type(s) for +: ‘float’ and ‘NoneType’. Can you help? Thanks in advance!

1. Hey Veronica,

Have you solved the problem? When I wrote this blog, this did not happen. However, I tried to run the script again and get the same problem. I am not sure what is going on here but I will try to find out given that you did not solve it.

Please let me know if and how you solved the problem.

Erik

1. Thanks for leaving your comment here, Damien!

I will update the posts later and reference to your solution.

Erik

2. Hi Erik,
I met the same problem when I ran my analysis. In my study, the design is a 2x3x3 repeated measure ANOVA. And my code is straightforward, import pyvttbl as pt
df = pt.DataFrame()

aov = df.anova(‘Z_score’, sub=’ID’, wfactors=[‘Task_types’, ‘conditions’, ‘Question’])
print(aov)
But it returned the error “unsupported operand type(s) for +: ‘float’ and ‘NoneType'”. I have checked my data file and found Z_score was stored as numpy.float64. Is it the reason I have this error message? Should I change my data from numpy.float64 to long or other data type?

1. Hey Shengjie,

Right now I don’t have a solution for this problem other than the one Damien gave here in the comments. It seems like you have to have Python 1.11.0 (maybe other versions work to but it worked for Damien). Hope it helps. I will update my post(s) with this solution. I don’t think Pyvttbl have been updated for 4 years, or so. Maybe someone should build a new package/update Pyvttbl. 🙂

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