This project will assure you have mastered the subjects covered in the statistics lessons. The hope is to have this project be as comprehensive of these topics as possible. Good luck!

• Introduction
• Part I - Probability
• Part II - A/B Test
• Part III - Regression

A/B tests are very commonly performed by data analysts and data scientists. It is important that you get some practice working with the difficulties of these

For this project, you will be working to understand the results of an A/B test run by an e-commerce website. Your goal is to work through this notebook to help the company understand if they should implement the new page, keep the old page, or perhaps run the experiment longer to make their decision.

As you work through this notebook, follow along in the classroom and answer the corresponding quiz questions associated with each question. The labels for each classroom concept are provided for each question. This will assure you are on the right track as you work through the project, and you can feel more confident in your final submission meeting the criteria. As a final check, assure you meet all the criteria on the RUBRIC.

To get started, let's import our libraries.

import pandas as pd
import numpy as np
import random
import matplotlib.pyplot as plt
%matplotlib inline
#We are setting the seed to assure you get the same answers on quizzes as we set up
random.seed(42)

1. Now, read in the ab_data.csv data. Store it in df. Use your dataframe to answer the questions in Quiz 1 of the classroom.

a. Read in the dataset and take a look at the top few rows here:

df=pd.read_csv('ab_data.csv')

b. Use the below cell to find the number of rows in the dataset.

print('Number of rows in dataset :: ',df.shape[0])
#find no of columns and non-empty fields in dataset
print('dataset information :: ')
df.info()

Number of rows in dataset ::  294478
dataset information :: 
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 294478 entries, 0 to 294477
Data columns (total 5 columns):
user_id         294478 non-null int64
timestamp       294478 non-null object
group           294478 non-null object
landing_page    294478 non-null object
converted       294478 non-null int64
dtypes: int64(2), object(3)
memory usage: 11.2+ MB

print few rows in dataset

df.head(10)

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c. The number of unique users in the dataset.

print('Number of unique users in the dataset :: ',df['user_id'].nunique())

Number of unique users in the dataset ::  290584

d. The proportion of users converted.

print('Proportion of users converted :: ',df['converted'].mean())

Proportion of users converted ::  0.11965919355605512

e. The number of times the new_page and treatment don't line up.

row_len=df.query("(group == 'control' and landing_page == 'new_page') or (group == 'treatment' and landing_page == 'old_page')").shape[0] 
print('Number of times new_page and treatment dont line up :: ',row_len)

Number of times new_page and treatment dont line up ::  3893

Another way to find out number of times the new_page and treatment don't line up

#find correct landing for group treatment on new_page
grp1=df.query("group == 'treatment' and landing_page == 'new_page'")
print('Total rows for correct landing of treatment group',len(grp1))
#find correct landing for group control on old_page
grp2=df.query("group == 'control' and landing_page == 'old_page'")
print('Total rows for correct landing of control group',len(grp2))
#total rows minus correct landings for both groups gives rows where treatment group doesn't line up with new_page
print('Number of times new_page and treatment dont line up :: ',df.shape[0]-(len(grp1)+len(grp2)))

Total rows for correct landing of treatment group 145311
Total rows for correct landing of control group 145274
Number of times new_page and treatment dont line up ::  3893

f. Do any of the rows have missing values?

#counts null values in all columns 
df.isnull().sum()

user_id         0
timestamp       0
group           0
landing_page    0
converted       0
dtype: int64

From above output we can see that there are no columns with missing values

2. For the rows where treatment is not aligned with new_page or control is not aligned with old_page, we cannot be sure if this row truly received the new or old page. Use Quiz 2 in the classroom to provide how we should handle these rows.

a. Now use the answer to the quiz to create a new dataset that meets the specifications from the quiz. Store your new dataframe in df2.

df2=df
df2.drop(df.query("(group == 'treatment' and landing_page == 'old_page') or (group == 'control' and landing_page == 'new_page')").index, inplace=True)

# Double Check all of the correct rows were removed - this should be 0
df2[((df2['group'] == 'treatment') == (df2['landing_page'] == 'new_page')) == False].shape[0]

0

3. Use df2 and the cells below to answer questions for Quiz3 in the classroom.

a. How many unique user_ids are in df2?

print('Number of unique users in the dataset :: ',df2['user_id'].nunique())

Number of unique users in the dataset ::  290584

b. There is one user_id repeated in df2. What is it?

df2[df2.duplicated(['user_id'], keep=False)]['user_id'].unique()

array([773192])

c. What is the row information for the repeat user_id?

df2[df2.duplicated(['user_id'], keep=False)]

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d. Remove one of the rows with a duplicate user_id, but keep your dataframe as df2.

df2=df2.drop_duplicates(['user_id'])

4. Use df2 in the below cells to answer the quiz questions related to Quiz 4 in the classroom.

a. What is the probability of an individual converting regardless of the page they receive?

df2['converted'].mean()

0.11959708724499628

b. Given that an individual was in the control group, what is the probability they converted?

df2[df2['group'] == 'control']['converted'].mean()

0.1203863045004612

c. Given that an individual was in the treatment group, what is the probability they converted?

df2[df2['group'] == 'treatment']['converted'].mean()

0.11880806551510564

d. What is the probability that an individual received the new page?

#find rows with landing page as new page
new_page_rows = len(df2.query("landing_page == 'new_page'"))
#find total no of rows
total_rows = df2.shape[0]
print('New page probability :: ',new_page_rows/total_rows)

New page probability ::  0.5000619442226688

e. Consider your results from a. through d. above, and explain below whether you think there is sufficient evidence to say that the new treatment page leads to more conversions.

From above results we can find below information :

a) probability of conversion : 0.11959708724499628

b) probability of conversion when individual was in the control group : 0.1203863045004612

c) probability of conversion when individual was in the treatment group : 0.11880806551510564

d) probability of individual receiving a new page : 0.5000619442226688

From results we can see that treatment group(landing_page=new_page) has less probability than control group(landing_page=old_page). However above information is not from original dataset and difference in probability is less which means we can't say for sure that new page leads to conversions.Also probability of new page is roughly 50% which doesn't prove that all new treatment page will be more converted.

Notice that because of the time stamp associated with each event, you could technically run a hypothesis test continuously as each observation was observed.

However, then the hard question is do you stop as soon as one page is considered significantly better than another or does it need to happen consistently for a certain amount of time? How long do you run to render a decision that neither page is better than another?

These questions are the difficult parts associated with A/B tests in general.

1. For now, consider you need to make the decision just based on all the data provided. If you want to assume that the old page is better unless the new page proves to be definitely better at a Type I error rate of 5%, what should your null and alternative hypotheses be? You can state your hypothesis in terms of words or in terms of $p_{old}$ and $p_{new}$, which are the converted rates for the old and new pages.

$H_{0}$ : $p_{new}$ <= $p_{old}$

$H_{1}$ : $p_{new}$ > $p_{old}$

2. Assume under the null hypothesis, $p_{new}$ and $p_{old}$ both have "true" success rates equal to the converted success rate regardless of page - that is $p_{new}$ and $p_{old}$ are equal. Furthermore, assume they are equal to the converted rate in ab_data.csv regardless of the page.

Use a sample size for each page equal to the ones in ab_data.csv.

Perform the sampling distribution for the difference in converted between the two pages over 10,000 iterations of calculating an estimate from the null.

Use the cells below to provide the necessary parts of this simulation. If this doesn't make complete sense right now, don't worry - you are going to work through the problems below to complete this problem. You can use Quiz 5 in the classroom to make sure you are on the right track.

a. What is the convert rate for $p_{new}$ under the null?

p_new = df2['converted'].mean()
print('Convert rate for p_new under the null :: ',p_new)

Convert rate for p_new under the null ::  0.11959708724499628

b. What is the convert rate for $p_{old}$ under the null?

p_old = df2['converted'].mean()
print('Convert rate for p_old under the null :: ',p_old)

Convert rate for p_old under the null ::  0.11959708724499628

c. What is $n_{new}$?

n_new = len(df2.query("landing_page == 'new_page'"))           
print('n_new :: ', n_new)

n_new ::  145310

d. What is $n_{old}$?

n_old = len(df2.query("landing_page == 'old_page'"))           
print('n_old :: ', n_old)

n_old ::  145274

e. Simulate $n_{new}$ transactions with a convert rate of $p_{new}$ under the null. Store these $n_{new}$ 1's and 0's in new_page_converted.

new_page_converted = np.random.binomial(n_new,p_new)
print('new_page_converted :: ',new_page_converted)

new_page_converted ::  17373

f. Simulate $n_{old}$ transactions with a convert rate of $p_{old}$ under the null. Store these $n_{old}$ 1's and 0's in old_page_converted.

old_page_converted = np.random.binomial(n_old,p_old)
print('new_page_converted :: ',old_page_converted)

new_page_converted ::  17284

g. Find $p_{new}$ - $p_{old}$ for your simulated values from part (e) and (f).

#re-calculate p_new,p_old and do substraction
(new_page_converted/n_new) - (old_page_converted/n_old)

0.000583008007667557

h. Simulate 10,000 $p_{new}$ - $p_{old}$ values using this same process similarly to the one you calculated in parts a. through g. above. Store all 10,000 values in a numpy array called p_diffs.

p_diffs = []
for _ in range(10000):
    new_page_converted = np.random.binomial(n_new,p_new)
    old_page_converted = np.random.binomial(n_old, p_old)
    p_diff = new_page_converted/n_new - old_page_converted/n_old
    p_diffs.append(p_diff)

i. Plot a histogram of the p_diffs. Does this plot look like what you expected? Use the matching problem in the classroom to assure you fully understand what was computed here.

plt.xlabel('p_diff value')
plt.ylabel('Frequency')
plt.title('Plot of Simulated p_diffs');
plt.hist(p_diffs)

(array([3.000e+00, 4.600e+01, 2.360e+02, 1.103e+03, 2.491e+03, 3.040e+03,
        2.112e+03, 8.050e+02, 1.520e+02, 1.200e+01]),
 array([-0.00512973, -0.00417508, -0.00322042, -0.00226576, -0.0013111 ,
        -0.00035645,  0.00059821,  0.00155287,  0.00250753,  0.00346218,
         0.00441684]),
 <a list of 10 Patch objects>)

j. What proportion of the p_diffs are greater than the actual difference observed in ab_data.csv?

compute p_diff from ab_data.csv

p_diff_orig = df[df['landing_page'] == 'new_page']['converted'].mean() -  df[df['landing_page'] == 'old_page']['converted'].mean()
print('p_diff from ab_data.csv :: ',p_diff_orig)

p_diff from ab_data.csv ::  -0.0015790565976871451

p_diffs = np.array(p_diffs)
p_diff_proportion = (p_diff_orig < p_diffs).mean()
print('proportion of p_diffs greater than p_diffs from ab_data.csv :: ',p_diff_proportion)

proportion of p_diffs greater than p_diffs from ab_data.csv ::  0.9048

k. In words, explain what you just computed in part j. What is this value called in scientific studies? What does this value mean in terms of whether or not there is a difference between the new and old pages?

**We are calculating pvalue. If null hypothesis($H_{0}$) is true pvalue gives the probability of statistics tested.In this case, the new page doesn't have better conversion rates than the old page **

l. We could also use a built-in to achieve similar results. Though using the built-in might be easier to code, the above portions are a walkthrough of the ideas that are critical to correctly thinking about statistical significance. Fill in the below to calculate the number of conversions for each page, as well as the number of individuals who received each page. Let n_old and n_new refer the the number of rows associated with the old page and new pages, respectively.

import statsmodels.api as sm
import warnings
warnings.filterwarnings("ignore")

convert_old = sum(df2.query("landing_page == 'old_page'")['converted'])
convert_new = sum(df2.query("landing_page == 'new_page'")['converted'])
n_old = len(df2.query("landing_page == 'old_page'"))
n_new = len(df2.query("landing_page == 'new_page'"))

m. Now use stats.proportions_ztest to compute your test statistic and p-value. Here is a helpful link on using the built in.

z_score, p_value = sm.stats.proportions_ztest([convert_old, convert_new], [n_old, n_new], alternative='smaller')
print('z_score :: ',z_score)
print('p_value :: ',p_value)

z_score ::  1.3109241984234394
p_value ::  0.9050583127590245

n. What do the z-score and p-value you computed in the previous question mean for the conversion rates of the old and new pages? Do they agree with the findings in parts j. and k.?

from scipy.stats import norm
# significant of z-score
print(norm.cdf(z_score))

# for our single-sides test, assumed at 95% confidence level, we calculate: 
print(norm.ppf(1-(0.05)))

0.9050583127590245
1.6448536269514722

**z_score is a measure of how many standard deviations below or above the population mean a raw score is. We find that the z-score of 1.3109241984234394 is less than the critical value of 1.6448536269514722 which means we can't reject the null hypothesis($H_{0}$).we find that old page conversions are slightly better than new page conversions. Eventhough the values are different from findings in parts j. and k but it suggests there is no significant difference between old page and new page conversions **

1. In this final part, you will see that the result you acheived in the previous A/B test can also be acheived by performing regression.

a. Since each row is either a conversion or no conversion, what type of regression should you be performing in this case?

Logistic Regression

b. The goal is to use statsmodels to fit the regression model you specified in part a. to see if there is a significant difference in conversion based on which page a customer receives. However, you first need to create a column for the intercept, and create a dummy variable column for which page each user received. Add an intercept column, as well as an ab_page column, which is 1 when an individual receives the treatment and 0 if control.

df2['intercept'] = 1
df2[['control', 'ab_page']]=pd.get_dummies(df2['group'])
df2.drop(labels=['control'], axis=1, inplace=True)
df2.head()

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c. Use statsmodels to import your regression model. Instantiate the model, and fit the model using the two columns you created in part b. to predict whether or not an individual converts.

import statsmodels.api as sm
import scipy.stats as stats
logit = sm.Logit(df2['converted'],df2[['intercept' ,'ab_page']])
results = logit.fit()

Optimization terminated successfully.
         Current function value: 0.366118
         Iterations 6

d. Provide the summary of your model below, and use it as necessary to answer the following questions.

stats.chisqprob = lambda chisq, df: stats.chi2.sf(chisq, df)
results.summary()

e. What is the p-value associated with ab_page? Why does it differ from the value you found in Part II?

Hint: What are the null and alternative hypotheses associated with your regression model, and how do they compare to the null and alternative hypotheses in the Part II?

**p-value is 0.190 .The p-value here suggests that that new page is not statistically significant as 0.19 > 0.05 . In this section it was a two sided test and in Part II it was a one sided test.Here we test for not equal in our hypotheses whereas in Part II it was for different. **

Hypothesis in Part III ::

$H_{0}$ : $p_{new}$ - $p_{old}$ = 0

$H_{1}$ : $p_{new}$ - $p_{old}$ != 0

Hypothesis in Part II ::

$H_{0}$ : $p_{new}$ <= $p_{old}$

$H_{1}$ : $p_{new}$ > $p_{old}$

f. Now, you are considering other things that might influence whether or not an individual converts. Discuss why it is a good idea to consider other factors to add into your regression model. Are there any disadvantages to adding additional terms into your regression model?

**There might be many factors that can effect individual converts . age group and gender can play a significant change. Also usage and access may effect the rate of conversion. We can find new trends using other factors but there may be some disadvantages like even with new factors we may miss some other influencing factors which lead to unreliable and contradictory results compared to previous results **

g. Now along with testing if the conversion rate changes for different pages, also add an effect based on which country a user lives. You will need to read in the countries.csv dataset and merge together your datasets on the approporiate rows. Here are the docs for joining tables.

Does it appear that country had an impact on conversion? Don't forget to create dummy variables for these country columns - Hint: You will need two columns for the three dummy variables. Provide the statistical output as well as a written response to answer this question.

countries_df = pd.read_csv('./countries.csv')
df_new = countries_df.set_index('user_id').join(df2.set_index('user_id'), how='inner')
df_new.head()

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### Create the necessary dummy variables
df_new[['CA','UK','US']]=pd.get_dummies(df_new['country'])
df_new.head()

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mod = sm.Logit(df_new['converted'], df_new[['intercept', 'CA', 'UK']])
results = mod.fit()
results.summary()

Optimization terminated successfully.
         Current function value: 0.366116
         Iterations 6

h. Though you have now looked at the individual factors of country and page on conversion, we would now like to look at an interaction between page and country to see if there significant effects on conversion. Create the necessary additional columns, and fit the new model.

Provide the summary results, and your conclusions based on the results.

### Fit Your Linear Model And Obtain the Results
mod = sm.Logit(df_new['converted'], df_new[['intercept', 'CA', 'UK','ab_page']])
results = mod.fit()
results.summary()

Optimization terminated successfully.
         Current function value: 0.366113
         Iterations 6

We can accept Null Hypothesis as there is no significant difference in conversion rates. We can reject alternate hypothesis.These results are based on given dataset. There may be limitations due to incorrect data or missing columns etc.