# Multiple regression

Multiple regression designs apply to at least two scenarios.

• Analyzing a single subject’s data when trials are associated with one or multiple continuous variables.

• Analyzing group data when subjects are associated with one or multiple individual difference variables.

Here the analysis is illustrated for a simulated dataset from a single subject. Group analysis works analogously, except that each case in the datatset would represent a different subject rather than a different trial.

For such group analysis, it is necessary to reduce each subject’s data to a single case first because multiple regression assumes a fixed effects model. Such designs are described under Two-stage test.

## Simulated data

```# sphinx_gallery_thumbnail_number = 2
from eelbrain import *

data = datasets.simulate_erp(snr=0.5)
```

The data represents 80 trials from a simulated word reading paradigm, where each word is associated with a word length (n_chars) and predictability (cloze).

```p = plot.Scatter('cloze', 'n_chars', data=data, h=3)
```

Plot the average of each level on n_chars to illustrate the linear increase of the response around 130 ms with n_char.

```p = plot.TopoButterfly('eeg', 'n_chars', data=data, t=.130, axh=2, w=6)
```

## Multiple regression

Fit a multiple regression model. Estimate \$p\$-values using a cluster-based permuatation test, using a cluster-forming threshold of uncorrected \$p\$=.05 and 10,000 permutations (the default).

```lm = testnd.LM('eeg', 'n_chars + cloze', data=data, tstart=0.050, tstop=0.500, pmin=0.05)
```
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```

A quick plot suggests an early effect related to n_chars around 130 ms, and a later effect of cloze around 400 ms:

```p = plot.TopoButterfly(lm, t=0.130, axh=2, w=6)
```

Detailed plots of the two effects show that the cluster is quite variable due to noise in the data.

```p = plot.TopoArray(lm.masked_parameter_map('n_chars'), t=[0.110, 0.130, 0.150], title='N Chars')
p = plot.TopoArray(lm.masked_parameter_map('cloze'), t=[0.380, 0.400, 0.420], title='Cloze')
```

To access some of the test results we need to know the index of different effects:

```lm.effects
```
```('intercept', 'n_chars', 'cloze')
```

Create a plot that shows the spatial cluster extent across time.

```EFFECTS = [
('n_chars', (0.110, 0.150)),
('cloze', (0.380, 0.420)),
]

t_maps = []
for effect, time in EFFECTS:
# t-maps are retrieved by effect name
t = lm.t(effect).mean(time=time)
# p-maps are stored in a list, so we need to know the index of te effect
index = lm.effects.index(effect)
# We are interested in the maximal spatial cluster extent, i.e., any sensor that is part of the cluster at any time
p = lm.p[index].min(time=time)
# Create a masked average t map