**Prerequisites**

This course assumes that you already know Python. The scipy-lectures give an introduction to Python.

Tip

This page gives an introduction to statistics with Python. It is useful to get acquainted with data representations in Python. In terms of experimental psychology, the patterns demonstrated here can be applied to simple dataset that arise from psychophysics, or reaction time experiments. We will see how to process efficiently more complex data as produced by brain imaging in the next course.

Tip

We will be working in the interactive Python prompt, IPython. You can either start it by itself, or in the spyder environment.

In this document, the Python prompts are represented with the sign “>>>”. If you need to copy-paste code, you can click on the top right of the code blocks, to hide the prompts and the outputs.

We have the reaction times in a psychophysical experiment:

474.688 506.445 524.081 530.672 530.869 566.984 582.311 582.940 603.574 792.358

In Python, we will represent them using a numerical “array”, with the
`numpy` module:

```
>>> import numpy as np
>>> x = np.array([474.688, 506.445, 524.081, 530.672, 530.869, 566.984, 582.311, 582.940, 603.574, 792.358])
```

We can access the elements of x with **indexing, starting at 0**:

```
>>> x[0] # First element
474.68799999999999
>>> x[:3] # Up to the third element
array([ 474.688, 506.445, 524.081])
>>> x[-1] # Last element
792.35799999999995
```

Mathematical operations on arrays are done using the numpy module:

```
>>> np.max(x)
792.35799999999995
>>> np.sum(x[:3])
1505.2139999999999
>>> np.mean(x)
569.49219999999991
>>> np.log(x)
array([ 6.16265775, 6.22741573, 6.26164625, 6.27414413, 6.27451529,
6.34033108, 6.36700467, 6.36808427, 6.40286865, 6.67501331])
```

**Finding help**

- The reference documentation can be found on http://docs.scipy.org
- http://scipy-lectures.github.io contains tutorials on Python, numpy, and the scientific Python ecosystem.
- Typing np.mean? in IPython will display help on
`numpy.mean()`. `numpy.lookfor()`can be used to search for keywords in functions help.

**Exercise**

- Display the last two entries of x.
- Compute the standard deviation of x (how will you find the function to do so?).

Basic plotting is done with matplotlib:

```
>>> from matplotlib import pyplot as plt
>>> plt.boxplot(x)
```

Hint

If a window doesn’t display, you need to call plt.show().

Under IPython, type %matplotlib to have plots display automatically.

See also

Matplotlib is very rich and can be controlled in detail. See the scipy lectures for more details.

We have a CSV file giving observations of brain size and weight and IQ (Willerman et al. 1991):

"";"Gender";"FSIQ";"VIQ";"PIQ";"Weight";"Height";"MRI_Count" "1";"Female";133;132;124;"118";"64.5";816932 "2";"Male";140;150;124;".";"72.5";1001121 "3";"Male";139;123;150;"143";"73.3";1038437 "4";"Male";133;129;128;"172";"68.8";965353

The data are a mixture of numerical and categorical values. We will use pandas to manipulate them:

```
>>> import pandas
>>> data = pandas.read_csv('examples/brain_size.csv', sep=';', na_values=".")
>>> print data
Unnamed: 0 Gender FSIQ VIQ PIQ Weight Height MRI_Count
0 1 Female 133 132 124 118 64.5 816932
1 2 Male 140 150 124 NaN 72.5 1001121
2 3 Male 139 123 150 143 73.3 1038437
3 4 Male 133 129 128 172 68.8 965353
4 5 Female 137 132 134 147 65.0 951545
...
```

Warning

**Missing values**

The weight of the second individual is missing in the CSV file. If we don’t specify the missing value (NA = not available) marker, we will not be able to do statistics on the weight.

data is a pandas dataframe, that resembles R’s dataframe:

```
>>> print data['Gender']
0 Female
1 Male
2 Male
3 Male
4 Female
...
>>> gender_data = data.groupby('Gender')
>>> print gender_data.mean()
Unnamed: 0 FSIQ VIQ PIQ Weight Height MRI_Count
Gender
Female 19.65 111.9 109.45 110.45 137.200000 65.765000 862654.6
Male 21.35 115.0 115.25 111.60 166.444444 71.431579 954855.4
>>> # More manual, but more versatile
>>> for name, value in gender_data['VIQ']:
... print name, np.mean(value)
Female 109.45
Male 115.25
>>> # Simpler selector
>>> data[data['Gender'] == 'Female']['VIQ'].mean()
109.45
```

**Exercise**

What is the mean value for VIQ for the full population?

How many males/females were included in this study?

**Hint**use ‘tab completion’ to find out the methods that can be called, instead of ‘mean’ in the above example.What is the average value of MRI counts expressed in log units, for males and females?

Pandas comes with some plotting tools (that use matplotlib behind the scene) to display statistics on dataframes:

```
>>> from pandas.tools import plotting
>>> plotting.scatter_matrix(data[['Weight', 'Height', 'MRI_Count']])
```

```
>>> plotting.scatter_matrix(data[['PIQ', 'VIQ', 'FSIQ']])
```

**Exercise**

Plot the scatter matrix for males only, and for females only. Do you think that the 2 sub-populations correspond to gender?

For simple statistical tests, we will use the stats sub-module of scipy:

```
>>> from scipy import stats
```

See also

Scipy is a vast library. For a tutorial covering the whole scope of scipy, see http://scipy-lectures.github.io/

`scipy.stats.ttest_1samp()` tests if observations are drawn from a
Gaussian distributions of given population mean. It returns the T
statistic, and the p-value (see the function’s help):

```
>>> stats.ttest_1samp(data['VIQ'], 0)
(30.088099970849338, 1.3289196468727784e-28)
```

Tip

With a p-value of 10^-28 we can claim that the population mean for the IQ (VIQ measure) is not 0.

**Exercise**

Is the test performed above one-sided or two-sided? Which one should we use, and what is the corresponding p-value?

We have seen above that the mean VIQ in the male and female populations
were different. To test if this is significant, we do a 2-sample t-test
with `scipy.stats.ttest_ind()`:

```
>>> female_viq = data[data['Gender'] == 'Female']['VIQ']
>>> male_viq = data[data['Gender'] == 'Male']['VIQ']
>>> stats.ttest_ind(female_viq, male_viq)
(-0.77261617232750124, 0.44452876778583217)
```

PIQ, VIQ, and FSIQ give 3 measures of IQ. Let us test if FISQ and PIQ are significantly different. We need to use a 2 sample test:

```
>>> stats.ttest_ind(data['FSIQ'], data['PIQ'])
(0.46563759638096403, 0.64277250094148408)
```

The problem with this approach is that is forgets that there are links between observations: FSIQ and PIQ are measure on the same individuals. Thus the variance due to inter-subject variability is confounding, and can be removed, using a “paired test”, or “repeated measure test”:

```
>>> stats.ttest_rel(data['FSIQ'], data['PIQ'])
(1.7842019405859857, 0.082172638183642358)
```

This is equivalent to a 1-sample test on the difference:

```
>>> stats.ttest_1samp(data['FSIQ'] - data['PIQ'], 0)
(1.7842019405859857, 0.082172638183642358)
```

T-tests assume Gaussian errors. We can use a Wilcoxon signed-rank test, that relaxes this assumption:

```
>>> stats.wilcoxon(data['FSIQ'], data['PIQ'])
(274.5, 0.034714577290489719)
```

Note

The corresponding test in the non paired case is the Mann–Whitney U
test,
`scipy.stats.mannwhitneyu()`.

**Exercice**

- Test the difference between weights in males and females.
- Use non parametric statistics to test the difference between VIQ in males and females.

Given two set of observations, x and y, we want to test the hypothesis that y is a linear function of x. In other terms:

where e is observation noise. We will use the statmodels module to:

- Fit a linear model. We will use the simplest strategy, ordinary least squares (OLS).
- Test that coef is non zero.

First, we generate simulated data according to the model:

```
>>> x = np.linspace(-5, 5, 20)
>>> np.random.seed(1)
>>> # normal distributed noise
>>> y = -5 + 3*x + 4 * np.random.normal(size=x.shape)
>>> # Create a data frame containing all the relevant variables
>>> data = pandas.DataFrame({'x': x, 'y': y})
```

Specify an OLS model and fit it:

```
>>> # The following will work only with a recent version of statsmodels
>>> from statsmodels.formula.api import ols
>>> model = ols("y ~ x", data).fit()
>>> print(model.summary())
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.804
Model: OLS Adj. R-squared: 0.794
Method: Least Squares F-statistic: 74.03
Date: ... Prob (F-statistic): 8.56e-08
Time: ... Log-Likelihood: -57.988
No. Observations: 20 AIC: 120.0
Df Residuals: 18 BIC: 122.0
Df Model: 1
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
Intercept -5.5335 1.036 -5.342 0.000 -7.710 -3.357
x 2.9369 0.341 8.604 0.000 2.220 3.654
==============================================================================
Omnibus: 0.100 Durbin-Watson: 2.956
Prob(Omnibus): 0.951 Jarque-Bera (JB): 0.322
Skew: -0.058 Prob(JB): 0.851
Kurtosis: 2.390 Cond. No. 3.03
==============================================================================
```

**Exercise**

Retrieve the estimated parameters from the model above. **Hint**:
use tab-completion to find the relevent attribute.

Consider a linear model explaining a variable z (the dependent variable) with 2 variables x and y:

Such a model can be seen in 3D as fitting a plane to a cloud of (x, y, z) points.

**Example: the iris data**

```
>>> data = pandas.read_csv('examples/iris.csv')
>>> model = ols('sepal_width ~ name + petal_length', data).fit()
>>> print(model.summary())
OLS Regression Results
==============================================================================
Dep. Variable: sepal_width R-squared: 0.478
Model: OLS Adj. R-squared: 0.468
Method: Least Squares F-statistic: 44.63
Date: ... Prob (F-statistic): 1.58e-20
Time: ... Log-Likelihood: -38.185
No. Observations: 150 AIC: 84.37
Df Residuals: 146 BIC: 96.41
Df Model: 3
===============================================================================...
coef std err t P>|t| [95.0% Conf. Int.]
-------------------------------------------------------------------------------...
Intercept 2.9813 0.099 29.989 0.000 2.785 3.178
name[T.versicolor] -1.4821 0.181 -8.190 0.000 -1.840 -1.124
name[T.virginica] -1.6635 0.256 -6.502 0.000 -2.169 -1.158
petal_length 0.2983 0.061 4.920 0.000 0.178 0.418
==============================================================================
Omnibus: 2.868 Durbin-Watson: 1.753
Prob(Omnibus): 0.238 Jarque-Bera (JB): 2.885
Skew: -0.082 Prob(JB): 0.236
Kurtosis: 3.659 Cond. No. 54.0
==============================================================================
```

In the above iris example, we wish to test if the petal length is different between versicolor and virginica, after removing the effect of sepal width. This can be formulated as testing the difference between the coefficient associated to versicolor and virginica in the linear model estimated above (it is an Analysis of Variance, ANOVA). For this, we write a vector of ‘contrast’ on the parameters estimated: we want to test “name[T.versicolor] - name[T.virginica]”, with an ‘F-test’:

```
>>> print(model.f_test([0, 1, -1, 0]))
<F test: F=array([[ 3.24533535]]), p=[[ 0.07369059]], df_denom=146, df_num=1>
```

Is this difference significant?

**Exercice**

Going back to the brain size + IQ data, test if the VIQ of male and female are different after removing the effect of brain size, height and weight.