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!

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.

In [1]:

```
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:

In [2]:

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

Out[2]:

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

In [3]:

```
row_count = df.shape[0]
print('row count = ' + str(row_count))
```

c. The number of unique users in the dataset.

In [4]:

```
unique_users = df.user_id.nunique()
print('unique user count = ' + str(unique_users))
```

d. The proportion of users converted.

In [5]:

```
unique_converted_users = df[df.converted == 1].user_id.nunique()
print('unique converted user count = ' \
+ str(unique_converted_users))
print('proportion of unique users that are converted = ' \
+ str(unique_converted_users/unique_users))
```

e. The number of times the `new_page`

and `treatment`

don't line up.

In [6]:

```
df.groupby(['group', 'landing_page']).count()
```

Out[6]:

In [7]:

```
treatment_old_page = \
df[(df.group == 'treatment') & \
(df.landing_page == 'old_page')].shape[0]
control_new_page = \
df[(df.group == 'control') & \
(df.landing_page == 'new_page')].shape[0]
print('# times treatment group receives old_page: ' \
+ str(treatment_old_page))
print(' # times control group receives new_page: ' \
+ str(control_new_page))
print(' Sum of both: ' \
+ str(treatment_old_page + control_new_page))
```

f. Do any of the rows have missing values?

In [8]:

```
df.info()
```

In [9]:

```
print(df.isnull().sum())
print('\nTherefore, no.')
```

`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**.

In [10]:

```
drop_index = df[( (df.group == 'treatment') & \
(df.landing_page == 'old_page') ) | \
( (df.group == 'control') & \
(df.landing_page == 'new_page') ) ].index
df2 = df.drop(drop_index)
```

In [11]:

```
# 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]
```

Out[11]:

`3.`

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

a. How many unique **user_id**s are in **df2**?

In [12]:

```
df2_unique_users = df2.user_id.nunique()
print('df2 unique user count = ' + str(df2_unique_users))
```

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

In [13]:

```
print(df2[df2.duplicated('user_id')].user_id)
repeat_user_id = str(df2[df2.duplicated('user_id')].user_id)[8:14]
```

In [14]:

```
print('repeat user id = ' + repeat_user_id)
```

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

In [15]:

```
df2[df2.user_id == int(repeat_user_id)]
```

Out[15]:

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

In [16]:

```
print('shape before dropping = ' + str(df2.shape))
df2.drop([1899], inplace=True)
print(' shape after dropping = ' + str(df2.shape))
```

`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?

In [17]:

```
df2.converted.mean()
```

Out[17]:

b. Given that an individual was in the `control`

group, what is the probability they converted?

In [18]:

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

Out[18]:

c. Given that an individual was in the `treatment`

group, what is the probability they converted?

In [19]:

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

Out[19]:

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

In [20]:

```
new_page_count = df2[df2.landing_page == 'new_page'].user_id.count()
new_page_count / df2.shape[0]
```

Out[20]:

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.

**There does not appear to be sufficient evidence to conclude that the new
treatment page produces more conversions than the current control page. The probability that a user who receives the treatment page will convert is actually slightly less than the probability that a user who receives the control page will convert.
**

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.

$$H_0: p_{new} - p_{old} \leq 0$$ $$H_1: p_{new} - p_{old} > 0$$

Or, verbally:

$H_{0}$: The likelihood of conversion for a user receiving the new page is less than or equal to the likelihood of conversion for a user receiving the old page.

$H_{1}$: The likelihood of conversion for a user receiving the new page is greater than the likelihood of conversion for a user receiving the old page.

`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?

In [21]:

```
p_new = df2.converted.mean()
print(p_new)
```

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

In [22]:

```
p_old = df2.converted.mean()
print(p_old)
```

c. What is $n_{new}$?

In [23]:

```
n_new = df2[df2.landing_page == 'new_page'].user_id.count()
print(n_new)
```

d. What is $n_{old}$?

In [24]:

```
n_old = df2[df2.landing_page == 'old_page'].user_id.count()
print(n_old)
```

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**.

In [25]:

```
new_page_converted = \
np.random.choice([0, 1], size=n_new, p=[(1-p_new), p_new])
```

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**.

In [26]:

```
old_page_converted = \
np.random.choice([0, 1], size=n_old, p=[(1-p_old), p_old])
```

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

In [27]:

```
new_page_converted.mean() - old_page_converted.mean()
```

Out[27]:

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**.

In [28]:

```
new_converted_simulation = \
np.random.binomial(n_new, p_new, 10000)/n_new
old_converted_simulation = \
np.random.binomial(n_old, p_old, 10000)/n_old
p_diffs = new_converted_simulation - old_converted_simulation
```

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.

**This boils down to a computation of the "spread" of the data, assuming that the probability of converting a given user is the same whether they see the treatment page or the control page.**

In [29]:

```
obs_diff = treatment_conv - control_conv
plt.hist(p_diffs);
plt.axvline(x=obs_diff, color='red');
```

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

In [30]:

```
p_diffs = np.array(p_diffs)
(p_diffs > obs_diff).mean()
```

Out[30]:

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?

**The histogram plotted in part i contains the sampling distribution under the null hypothesis, namely, that the conversion rate of the control group is equal to the conversion rate of the treatment group. Part j involves calculating what proportion of the conversion rate differences were greater than the actual observed difference, which was calculated from the conversion rate data. The special name given to the proportion of values in the null distribution that were greater than our observed difference is the "p-value."**

**A low p-value (specifically, less than our alpha of 0.05) indicates that the null hypothesis is not likely to be true. Since the p-value is very large at 90%, it is likely that our statistic is from the null, and therefore we fail to reject the null hypothesis. Ultimately, this indicates that it would be best for Audacity to keep the current 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.

In [31]:

```
import statsmodels.api as sm
convert_old = df2[df2['group'] == 'control'].converted.sum()
convert_new = df2[df2['group'] == 'treatment'].converted.sum()
n_old = df2[df2['group'] == 'control'].converted.size
n_new = df2[df2['group'] == 'treatment'].converted.size
```

m. Now use `stats.proportions_ztest`

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

In [32]:

```
from scipy.stats import norm
z_score, p_value = \
sm.stats.proportions_ztest( [ convert_new, convert_old ], \
[ n_new, n_old ], \
alternative='larger' )
print('Z-score critical value (95% confidence) to \n' \
+ ' reject the null: ' \
+ str(norm.ppf(1-(0.05/2))))
print('z_score = ' + str(z_score))
print('p_value = ' + str(p_value))
```

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.**?

**Since the magnitude of the z-score of 1.31 falls within the range implied by the critical value of 1.96, we fail to reject the null hypothesis. The null hypothesis is that there is no statistical difference between the conversion rates for the control and treatment groups.**

**Additionally, since the p_value of 0.90 (note, approximately the same value as was calculated manually) is larger than the alpha value of 0.05, we fail to reject the null hypothesis.**

**Thus, for both the foregoing reasons, the built-in method leads to the same conclusion as the manual method, the results of which are summarized in parts j and k, above.**

`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**.

In [33]:

```
df2.head()
```

Out[33]:

In [34]:

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

Out[34]:

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.

In [35]:

```
import statsmodels.api as sm
logit_mod = sm.Logit(df2['converted'], df2[['intercept', 'ab_page']])
```

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

In [36]:

```
results = logit_mod.fit()
results.summary()
```

Out[36]:

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**?

**The p-value associated with ab_page in this regression model is 0.19. The p-value that was returned from the built-in ztest method was ~0.90. The p-value that I calculated manually was also ~0.90.**

**The null hypothesis associated with a logistic regression is that there is no relationship between the dependent and independent variables. In this case, this means there is no relationship between which page a user is shown and the conversion rate. The alternative hypothesis would therefore be that there is a relationship of some sort.**

**The null hypothesis from part 2 is that the likelihood of conversion for a user receiving the new page is less than or equal to the likelihood of conversion for a user receiving the old page.
The alternative hypothesis from part 2 is that the likelihood of conversion for a user receiving the new page is greater than the likelihood of conversion for a user receiving the old page.**

**The factor that accounts for the large difference in the p-values may be that part 2 hypothesized one of the pages (specifically, the new_page the treatment group received) would lead to more conversions than the other. This is different from the hypotheses of part 3, which merely predicted a difference of some sort.**

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?

**Additional factors may make the model more predictive, yielding greater understanding. It may also result in business insights that would not have been evident in this simpler anlysis. For example, it would be possible to have different versions of the website for different locations. It is likely that people from different countries might have different tastes in website layout.**

**Possible disadvantages include increased risk of human error, especially misinterpretation, as well as possibly obscuring the message the data is really trying to tell (decreasing the so-called signal-to-noise ratio).**

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.

In [37]:

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

In [38]:

```
df_new.head()
```

Out[38]:

In [39]:

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

In [40]:

```
logit_mod_new = sm.Logit(df_new['converted'],\
df_new[['intercept', 'ab_page', 'US', 'UK']])
results_new = logit_mod_new.fit()
results_new.summary()
```

Out[40]:

In [41]:

```
np.exp(0.0408), np.exp(0.0506)
```

Out[41]:

**The interpretation of the foregoing variables is counterintuitive. In this case, Canada is the baseline since it was the one out of three variables that wasn't included in the regression. We would say that US users are 1.04 times as likely (or 4% more likely) to convert as Canadians users. Similarly, we would say that UK users are 1.05 times as likely (or 5% more likely) to convert as Canadian users.**

**The effect is not statistically significant, given the fairly large P-values. Even if it were, it is not clear that such a small difference between the different countries would be practically significant.**

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.

In [42]:

```
# These columns indicate that a given user received both the new page
# and lived in the country shown.
df_new['new_CA'] = df_new['ab_page']*df_new['CA']
df_new['new_UK'] = df_new['ab_page']*df_new['UK']
df_new['new_US'] = df_new['ab_page']*df_new['US']
```

In [43]:

```
df_new.head()
```

Out[43]:

In [44]:

```
### Fit Your Linear Model And Obtain the Results
lin_mod = sm.OLS(df_new['converted'], \
df_new[['intercept', 'ab_page', 'US', 'new_US', 'UK', 'new_UK']])
results = lin_mod.fit()
results.summary()
```

Out[44]:

In [45]:

```
log_mod2 = sm.Logit(df_new['converted'], \
df_new[['intercept', 'ab_page', 'US', 'new_US', 'UK', 'new_UK']])
results_log2 = log_mod2.fit()
results_log2.summary()
```

Out[45]:

**The foregoing present both a linear (included to comply with the comment "#Fit your linear model") and logistic regression for the case with the interaction terms. Neither effect is statistically significant, given the high P-values shown in the results. Additionally, in the case of the linear plot, the R-squared value is zero, implying a terrible fit.**

Congratulations on completing the project!

Once you are satisfied with the status of your Notebook, you should save it in a format that will make it easy for others to read. You can use the **File -> Download as -> HTML (.html)** menu to save your notebook as an .html file. If you are working locally and get an error about "No module name", then open a terminal and try installing the missing module using `pip install <module_name>`

(don't include the "<" or ">" or any words following a period in the module name).

You will submit both your original Notebook and an HTML or PDF copy of the Notebook for review. There is no need for you to include any data files with your submission. If you made reference to other websites, books, and other resources to help you in solving tasks in the project, make sure that you document them. It is recommended that you either add a "Resources" section in a Markdown cell at the end of the Notebook report, or you can include a `readme.txt`

file documenting your sources.

When you're ready, click on the "Submit Project" button to go to the project submission page. You can submit your files as a .zip archive or you can link to a GitHub repository containing your project files. If you go with GitHub, note that your submission will be a snapshot of the linked repository at time of submission. It is recommended that you keep each project in a separate repository to avoid any potential confusion: if a reviewer gets multiple folders representing multiple projects, there might be confusion regarding what project is to be evaluated.

It can take us up to a week to grade the project, but in most cases it is much faster. You will get an email once your submission has been reviewed. If you are having any problems submitting your project or wish to check on the status of your submission, please email us at [email protected] In the meantime, you should feel free to continue on with your learning journey by beginning the next module in the program.