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Often, we have to compare data from three or more groups to see if one or more of them is different from the others. To do this, scientists use a statistic called the *F*- statistic, defined as $F=\frac{Var_{between}}{Var_{within}}$. The variability between groups is the sum of the squared differences between the group means and the grand mean. The variability within groups is the sum of the group variances. In LS 40, we will use a variation of the F-statistic that does not require squaring, which is called the *F*-like statistic.

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- Import pandas, Numpy and Seaborn.

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In [42]:

#TODO import pandas as pd import numpy as np import seaborn as sns

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In this lab, we will examine the results of a pharmaceutical company's study comparing the effectiveness of different pain relief medications on migraine headaches. For the experiment, 27 volunteers were selected and 9 were randomly assigned to one of three drug formulations. The subjects were instructed to take the drug during their next migraine headache episode and to report their pain on a scale of 1 = no pain to 10 =extreme pain 30 minutes after taking the drug.

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- Using the pandas
`read_csv`

function, import the file`migraines.csv`

and show the data.

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In [43]:

#TODO migraines=pd.read_csv("migraines.csv") migraines

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Drug A | Drug B | Drug C | |
---|---|---|---|

0 | 4 | 7 | 6 |

1 | 5 | 8 | 7 |

2 | 4 | 4 | 6 |

3 | 2 | 5 | 6 |

4 | 2 | 4 | 7 |

5 | 4 | 6 | 5 |

6 | 3 | 5 | 6 |

7 | 3 | 8 | 5 |

8 | 3 | 7 | 5 |

- Visualize the data. Use as many different plots as you need to get a sense of the distributions. What are your initial impressions?

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In [44]:

#TODO p=sns.stripplot(data=migraines, palette="PuBu") p.set(ylabel="Count") p.set_title("Migraine Treatment Study", fontsize=20)

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Text(0.5,1,'Migraine Treatment Study')

In [45]:

p2=sns.violinplot(data=migraines, palette="PuBu", orient="horizontal") p2.set(ylabel="Count") p2.set_title("Migraine Treatment Study", fontsize=20)

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Text(0.5,1,'Migraine Treatment Study')

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In the next several exercises, you will compute the *F*-like statistic for your data.

*F*-like=$\frac{n_{a}|\widetilde{a}-\widetilde{G}|+n_{b}|\widetilde{b}-\widetilde{G}|+n_{c}|\widetilde{c}-\widetilde{G}|}{\Sigma|a_{i}-\widetilde{a}|+\Sigma|b_{i}-\widetilde{b}|+\Sigma|c_{i}-\widetilde{c}|}$

Each quantity you compute in this section should be assigned to a variable.

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- Find the median of each column. HINT: pandas data frames have a built-in function for columnwise medians, so you can just use
`df.median()`

, where`df`

is the name of your data frame.

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In [46]:

#TODO druga=migraines["Drug A"] drugb=migraines["Drug B"] drugc=migraines["Drug C"]

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In [47]:

medians=migraines.median() medians

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Drug A 3.0
Drug B 6.0
Drug C 6.0
dtype: float64

In [48]:

medians[2]

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6.0

- Find the grand median (the median of the whole sample). HINT: Use
`np.median`

.

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In [49]:

#TODO gmedian=np.median(medians) gmedian

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6.0

- Find the numerator of the
*F*-like statistic (variation among groups). HINT: When working with data frames and NumPy arrays, you can do computations like addition and multiplication directly, without for loops (unlike in regular Python). Also, Numpy has`abs`

and`sum`

functions.

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In [50]:

d=medians-gmedian d

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Drug A -3.0
Drug B 0.0
Drug C 0.0
dtype: float64

In [51]:

#TODO a=len(druga) b=len(drugb) c=len(drugc)

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In [52]:

numerator=((a*abs(medians[0]-gmedian))+(b*abs(medians[1]-gmedian))+(c*abs(medians[2]-gmedian)))

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In [53]:

numerator=np.sum(len(migraines)*abs(medians-gmedian))

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In [54]:

numerator

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27.0

- Compute the denominator of the
*F*-like statistic. This represents variation within groups. HINT: Numpy and pandas will handle columns automatically.

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In [55]:

#TODO denominator=np.sum(np.sum(abs(migraines-migraines.median()))) denominator

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24.0

- Compute
*F*-like, which is the ratio $\frac{\text{variation among groups}}{\text{variation within groups}}$.

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In [82]:

#TODO flikeog=numerator/denominator flikeog

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1.125

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We now want to find a p-value for our data by simulating the null hypothesis. This, of course, means computing the *F*-like statistic each time, which takes a lot of code and would make a mess in the bootstrap loop. Instead, we will package our code into a function and call this function whereever necessary.

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- Write a function that will compute the
*F*-like statistic for this dataset or one of the same size.

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In [69]:

#TODO def fstat(df): medians=df.median() gmedian=np.median(df) numerator=np.sum(len(df)*abs(medians-gmedian)) denominator=np.sum(np.sum(abs(df-df.median()))) Flike=numerator/denominator return Flike

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We now want to simulate the null hypothesis that there is no difference between the groups. To do this, we have to make all the data into one dataset, sample pseudo-groups from it, and compute the *F*-like statistic for the resampled data.

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- Use the code
`alldata = np.concatenate([migraine["Drug A"], migraine["Drug B"], migraine["Drug C"]])`

(this assumes your data frame is called "migraine") to put all the data into one 1-D array.

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In [70]:

#TODO alldata=np.concatenate([migraines["Drug A"], migraines["Drug B"], migraines["Drug C"]])

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- Sample three groups of the appropriate size from
`alldata`

. Assign each to a variable.

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In [71]:

#TODO a=np.random.choice(alldata,len(migraines)) b=np.random.choice(alldata,len(migraines)) c=np.random.choice(alldata,len(migraines))

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- Make the three samples into a data frame. To do this, use the NumPy function
`column_stack`

to put the 1-D arrays side by side and then use the pandas function`DataFrame`

to convert the result into a data frame.

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In [72]:

#TODO data=pd.DataFrame(np.column_stack([a,b,c]))

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- Compute the
*F*-like statistic for your resampled data.

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In [73]:

#TODO fstat(df=data)

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0.9090909090909091

- Do the above steps 10,000 times to simulate the null hypothesis, storing the results.

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In [84]:

#TODO flikelist=np.zeros(10000) alldata=np.concatenate([migraines["Drug A"], migraines["Drug B"], migraines["Drug C"]]) for i in range(10000): a=np.random.choice(alldata,len(migraines)) b=np.random.choice(alldata,len(migraines)) c=np.random.choice(alldata,len(migraines)) data=pd.DataFrame(np.column_stack([a,b,c])) flikelist[i]=fstat(data) flikelist

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array([0.62068966, 0.5 , 0.64285714, ..., 0.54545455, 0.39130435,
0.36 ])

In [89]:

plot=sns.distplot(flikelist, kde=False, color="pink") plot.set(xlabel="Flike", ylabel="Count") plot.axvline(flikeog,color="red")

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<matplotlib.lines.Line2D at 0x7f4a5840fb00>

- Find the p-value for your data. What do you conclude about the migraine treatments? Use $\alpha$=0.05 for NHST.

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In [88]:

# TODO d1=np.sum(flikelist>=flikeog) pvalue=(d1)/10000 pvalue

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0.0095

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Having obtained a significant result from the omnibus test, we can now try to track down the source of the significance. Which groups are actually different from which?

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Perform pairwise two-group comparisons and record the p-value for each.

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In [19]:

# TODO medianlistdiff=[] medarray=np.zeros(10000) for i in range(10000): e=np.random.choice(migraines,len(migraines)) g=np.random.choice(migraines,len(migraines)) mediandiff=(np.median(e)-np.median(g)) medarray[i]=mediandiff medarray

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If more than one comparison comes out significant at the \alpha=0.05α=0.05 level, divide \alphaα by the number of significant tests. Which ones are still significant?

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In [ ]:

# TODO

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Write a sentence or two describing your findings.

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In [ ]:

#TODO

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