Suppose we're given the results of a $2^4$ factorial design, which yields the following contrasts:
$$ A \leftarrow 0.4 \\ B \leftarrow -7.6 \\ C \leftarrow 14.1 \\ D \leftarrow 66.7 \\ AB \leftarrow 16.7 \\ AC \leftarrow 3.1\\ AD \leftarrow 5.2 \\ BC \leftarrow 8.3 \\ BD \leftarrow -3.6 \\ CD \leftarrow 14.3 \\ ABC \leftarrow -0.1 \\ ABD \leftarrow -4.7 \\ ACD \leftarrow 7.7 \\ BCD \leftarrow -2.3 \\ ABCD \leftarrow 3.9 \\ $$(This follows question 4.3 in Box and Draper.)
We'll use quantile plots to interpret the results - we can't go much deeper than that, since the problem does not give the vaues of input or response variables.
We'll start by importing some libraries, then populate a dictionary with the various main and interation effects (the only assumption is that the response has a mean of 0, x0 = 0.0).
In [1]:
%matplotlib inline
import pandas as pd
import numpy as np
from numpy.random import rand, seed
import seaborn as sns
import scipy.stats as stats
from matplotlib.pyplot import *
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effects = {}
effects[0] = {'x0': 0.0}
effects[1] = {'x1': 0.4,
'x2': -7.6,
'x3': 14.1,
'x4': 66.7}
effects[2] = {'x1-x2': 16.7,
'x1-x3': 3.1,
'x1-x4': 5.2,
'x2-x3': 8.3,
'x3-x4': 14.3}
effects[3] = {'x1-x2-x3': -0.1,
'x1-x2-x4': -4.7,
'x1-x3-x4': 7.7,
'x2-x3-x4': -2.3}
effects[4] = {'x1-x2-x3-x4': 3.9}
Now we can use that dictionary to create a dataframe (this is a bit more work than it needs to be, but this is showing you how might structure the data for more complicated interactions). The important thing is that we have a labeled list that's a Pandas DataFrame:
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master_dict = {}
for nvars in effects.keys():
effect = effects[nvars]
for k in effect.keys():
v = effect[k]
master_dict[k] = v
master_df = pd.DataFrame(master_dict,index=['dy']).T
master_df
Out[3]:
In [4]:
#print help(master_df.sort)
view = master_df.sort_values(by='dy',ascending=False)
view
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This table shows that one variable, $x_4$, has a huge effect that outshadows all other effects. In case it wasn't obvious already, the quantile-quantile plot makes it crystal clear:
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# Quantile-quantile plot of effects:
fig = figure(figsize=(4,4))
ax1 = fig.add_subplot(111)
stats.probplot(master_df['dy'], dist="norm", plot=ax1)
ax1.set_title('Effect Size: Quantile-Quantile Plot')
show()
The largest ordered value in the effects, which is the effect of $x_4$, is way off by itself. In this region of operational state-space, $x_4$ dominates the system's behavior.
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