What is Logistic Regression?
Logistic Regression is a statistical technique capable of predicting a binary outcome. It's a well-known strategy, widely used in disciplines ranging from credit and finance to medicine to criminology and other social sciences. Logistic regression is fairly intuitive and very effective; you're likely to find it among the first few chapters of a machine learning or applied statistics book and it's usage is covered by many stats courses.
It's not hard to find quality logistic regression examples using R. This tutorial, for example, published by UCLA, is a great resource and one that I've consulted many times. Python is one of the most popular languages for machine learning, and while there are bountiful resources covering topics like Support Vector Machines and text classification using Python, there's far less material on logistic regression.
This is a post about using logistic regression in Python. I'll walk through the post using Yhat's Python IDE, Rodeo, but you could also run the code from your terminal, if you're so inclined. That said, if you do a lot of data analysis/visualization, Rodeo is a convenient way to code since it has a good text editor, a simple plot window and a terminal with autocomplete.
We'll use a few libraries in the code samples. Make sure you have these installed before you run through the code on your machine.
numpy: a language extension that defines the numerical array and matrix
pandas: primary package to handle and operate directly on data.
statsmodels: statistics & econometrics package with useful tools for parameter estimation & statistical testing
pylab: for generating plots
To install these packages, simply copy the code block below into the top left pane in Rodeo (this is called the text editor). Then, select the code block and either hit
on your keyboard or click
command + enter
You should see the result text in the bottom left pane in Rodeo (this is called the console.)
# hasthags indicate notes about code; the code below imports a few packages we will need for this analysis import pandas as pd import statsmodels.api as sm import pylab as pl import numpy as np
Example Use Case for Logistic Regression
We'll be using the same dataset as UCLA's Logit Regression in R tutorial to explore logistic regression in Python. Our goal will be to identify the various factors that may influence admission into graduate school.
The dataset contains several columns which we can use as predictor variables:
rankor prestige of an applicant's undergraduate alma mater
The fourth column,
admit, is our binary target variable. It indicates whether or not a candidate was admitted our not.
Load the data
Load the data using
pandas.read_csv. We now have a
DataFrame and can explore the data.
# read the data in df = pd.read_csv("http://www.ats.ucla.edu/stat/data/binary.csv")
In the top right pane of Rodeo, you should see that a dataframe is now avaiable in your environment.
Click on the blue icon to explore the dataframe.
Notice that one of the columns is called "
rank." This presents a problem since
rank is also the name of a method belonging to pandas
rank calculates the ordered rank (1 through n) of a
To make things easier, I renamed the rank column to "prestige". If you're working in Rodeo, go back to the Environment tab and reopen your DataFrame. You can see the change to the column name here. If you're not working in Rodeo, you can check the column names with
print df.columns or
# if you're not using <a href="https://www.yhat.com/products/rodeo" title="Python data science IDE, Rodeo" target="_blank">Rodeo</a>, you can check out the head of the dataset print df.head() # admit gre gpa rank # 0 0 380 3.61 3 # 1 1 660 3.67 3 # 2 1 800 4.00 1 # 3 1 640 3.19 4 # 4 0 520 2.93 4 # rename the 'rank' column because there is also a DataFrame method called 'rank' df.columns = ["admit", "gre", "gpa", "prestige"] print df.columns # array([admit, gre, gpa, prestige], dtype=object)
Summary Statistics & Looking at the data
Now that we've got everything loaded into Rodeo and named appropriately let's take a look at the data. We can use the
describe to give us a summarized view of everything--
describe is analogous to
summary in R. There's also function for calculating the standard deviation,
std. I've included it here to be consistent UCLA's tutorial, but the standard deviation is also included in
A feature I really like in
pandas is the
crosstab makes it really easy to do multidimensional frequency tables (sort of like
table in R). You might want to play around with this to look at different cuts of the data.
# summarize the data print df.describe() # admit gre gpa prestige # count 400.000000 400.000000 400.000000 400.00000 # mean 0.317500 587.700000 3.389900 2.48500 # std 0.466087 115.516536 0.380567 0.94446 # min 0.000000 220.000000 2.260000 1.00000 # 25% 0.000000 520.000000 3.130000 2.00000 # 50% 0.000000 580.000000 3.395000 2.00000 # 75% 1.000000 660.000000 3.670000 3.00000 # max 1.000000 800.000000 4.000000 4.00000 # take a look at the standard deviation of each column print df.std() # admit 0.466087 # gre 115.516536 # gpa 0.380567 # prestige 0.944460 # frequency table cutting presitge and whether or not someone was admitted print pd.crosstab(df['admit'], df['prestige'], rownames=['admit']) # prestige 1 2 3 4 # admit # 0 28 97 93 55 # 1 33 54 28 12 # plot all of the columns df.hist() pl.show()
Histograms are often one of the most helpful tools you can use during the exploratory phase of any data analysis project. They're normally pretty easy to plot, quick to interpret, and they give you a nice visual representation of your problem. Notice that the plots appear in the bottom right pane of Rodeo, in the plots tab (go figure!).
pandas gives you a great deal of control over how categorical variables are represented. We're going dummify the "prestige" column using
get_dummies creates a new
DataFrame with binary indicator variables for each category/option in the column specified. In this case,
prestige has four levels: 1, 2, 3 and 4 (1 being most prestigious). When we call
get_dummies, we get a dataframe with four columns, each of which describes one of those levels.
# dummify rank dummy_ranks = pd.get_dummies(df['prestige'], prefix='prestige') print dummy_ranks.head() # prestige_1 prestige_2 prestige_3 prestige_4 # 0 0 0 1 0 # 1 0 0 1 0 # 2 1 0 0 0 # 3 0 0 0 1 # 4 0 0 0 1 # create a clean data frame for the regression cols_to_keep = ['admit', 'gre', 'gpa'] data = df[cols_to_keep].join(dummy_ranks.ix[:, 'prestige_2':]) print data.head() # admit gre gpa prestige_2 prestige_3 prestige_4 # 0 0 380 3.61 0 1 0 # 1 1 660 3.67 0 1 0 # 2 1 800 4.00 0 0 0 # 3 1 640 3.19 0 0 1 # 4 0 520 2.93 0 0 1 # manually add the intercept data['intercept'] = 1.0
Once that's done, we merge the new dummy columns into the original dataset and get rid of the
prestige column which we no longer need.
Lastly we're going to add a constant term for our Logistic Regression. The
statsmodels function we're going to be using requires that intercepts/constants are specified explicitly.
Here's what the final dataframe looks like in Rodeo.
Performing the regression
Actually doing the Logistic Regression is quite simple. Specify the column containing the variable you're trying to predict followed by the columns that the model should use to make the prediction.
In our case we'll be predicting the
admit column using
gpa, and the prestige dummy variables
prestige_4. We're going to treat
prestige_1 as our baseline and exclude it from our fit. This is done to prevent multicollinearity, or the dummy variable trap caused by including a dummy variable for every single category.
train_cols = data.columns[1:] # Index([gre, gpa, prestige_2, prestige_3, prestige_4], dtype=object) logit = sm.Logit(data['admit'], data[train_cols]) # fit the model result = logit.fit()
Interpreting the results
One of my favorite parts about
statsmodels is the summary output it gives. If you're coming from R, I think you'll like the output and find it very familiar too.
# cool enough to deserve it's own comment print result.summary()
You get a great overview of the coefficients of the model, how well those coefficients fit, the overall fit quality, and several other statistical measures.
The result object also lets you to isolate and inspect parts of the model output. The confidence interval gives you an idea for how robust the coefficients of the model are.
# look at the confidence interval of each coeffecient print result.conf_int() # 0 1 # gre 0.000120 0.004409 # gpa 0.153684 1.454391 # prestige_2 -1.295751 -0.055135 # prestige_3 -2.016992 -0.663416 # prestige_4 -2.370399 -0.732529 # intercept -6.224242 -1.755716
In this example, we're very confident that there is an inverse relationship between the probability of being admitted and the prestige of a candidate's undergraduate school.
In other words, the probability of being accepted into a graduate program is higher for students who attended a top ranked undergraduate college (
prestige_1==True) as opposed to a lower ranked school with, say,
prestige_4==True (remember, a prestige of 1 is the most prestigious and a prestige of 4 is the least prestigious.
Take the exponential of each of the coefficients to generate the odds ratios. This tells you how a 1 unit increase or decrease in a variable affects the odds of being admitted. For example, we can expect the odds of being admitted to decrease by about 50% if the prestige of a school is 2. UCLA gives a more in depth explanation of the odds ratio here.
# odds ratios only print np.exp(result.params) # gre 1.002267 # gpa 2.234545 # prestige_2 0.508931 # prestige_3 0.261792 # prestige_4 0.211938 # intercept 0.018500
We can also do the same calculations using the coefficients estimated using the confidence interval to get a better picture for how uncertainty in variables can impact the admission rate.
# odds ratios and 95% CI params = result.params conf = result.conf_int() conf['OR'] = params conf.columns = ['2.5%', '97.5%', 'OR'] print np.exp(conf) # 2.5% 97.5% OR # gre 1.000120 1.004418 1.002267 # gpa 1.166122 4.281877 2.234545 # prestige_2 0.273692 0.946358 0.508931 # prestige_3 0.133055 0.515089 0.261792 # prestige_4 0.093443 0.480692 0.211938 # intercept 0.001981 0.172783 0.018500
Digging a little deeper
As a way of evaluating our classifier, we're going to recreate the dataset with every logical combination of input values. This will allow us to see how the predicted probability of admission increases/decreases across different variables. First we're going to generate the combinations using a helper function called
cartesian which I originally found here.
We're going to use
np.linspace to create a range of values for "gre" and "gpa". This creates a range of linearly spaced values from a specified min and maximum value--in our case just the min/max observed values.
# instead of generating all possible values of GRE and GPA, we're going # to use an evenly spaced range of 10 values from the min to the max gres = np.linspace(data['gre'].min(), data['gre'].max(), 10) print gres gpas = np.linspace(data['gpa'].min(), data['gpa'].max(), 10) print gpas #define the cartesian function def cartesian(arrays, out=None): arrays = [np.asarray(x) for x in arrays] dtype = arrays.dtype n = np.prod([x.size for x in arrays]) if out is None: out = np.zeros([n, len(arrays)], dtype=dtype) m = n / arrays.size out[:,0] = np.repeat(arrays, m) if arrays[1:]: cartesian(arrays[1:], out=out[0:m,1:]) for j in xrange(1, arrays.size): out[j*m:(j+1)*m,1:] = out[0:m,1:] return out # enumerate all possibilities combos = pd.DataFrame(cartesian([gres, gpas, [1, 2, 3, 4], [1.]])) # recreate the dummy variables combos.columns = ['gre', 'gpa', 'prestige', 'intercept'] dummy_ranks = pd.get_dummies(combos['prestige'], prefix='prestige') dummy_ranks.columns = ['prestige_1', 'prestige_2', 'prestige_3', 'prestige_4'] # keep only what we need for making predictions cols_to_keep = ['gre', 'gpa', 'prestige', 'intercept'] combos = combos[cols_to_keep].join(dummy_ranks.ix[:, 'prestige_2':]) # make predictions on the enumerated dataset combos['admit_pred'] = result.predict(combos[train_cols]) print combos.head()
Here's what you should see if you're following along within Rodeo.
Now that we've generated our predictions, let's make some plots to visualize the results. I created a small helper function called
isolate_and_plot which allows you to compare a given variable with the different prestige levels and the mean probability for that combination. To isolate prestige and the other variable I used a
pivot_table which allows you to easily aggregate the data.
def isolate_and_plot(variable): # isolate gre and class rank grouped = pd.pivot_table(combos, values=['admit_pred'], index=[variable, 'prestige'], aggfunc=np.mean) # make a plot colors = 'rbgyrbgy' for col in combos.prestige.unique(): plt_data = grouped.ix[grouped.index.get_level_values(1)==col] pl.plot(plt_data.index.get_level_values(0), plt_data['admit_pred'], color=colors[int(col)]) pl.xlabel(variable) pl.ylabel("P(admit=1)") pl.legend(['1', '2', '3', '4'], loc='upper left', title='Prestige') pl.title("Prob(admit=1) isolating " + variable + " and presitge") pl.show() isolate_and_plot('gre') isolate_and_plot('gpa')
Here's what that should look like in Rodeo.
The resulting plots in the bottom right pane of Rodeo show how gre, gpa, and prestige affect the admission levels. You can see how the probability of admission gradually increases as gre and gpa increase and that the different prestige levels yield drastic probabilities of admission (particularly the most/least prestigious schools).
Logistic Regression is an excellent algorithm for classification. Even though some of the sexier, black box classification algorithms like SVM and RandomForest can perform better in some cases, it's hard to deny the value in knowing exactly what your model is doing. Often times you can get by using RandomForest to select the features of your model and then rebuild the model with Logistic Regression using the best features.