Femi Oyedokun

7/1/2020 Analyze_ab_test_results_notebook Analyze A/B Test Results 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! Table of Contents Introduction Part I - Probability Part II - A/B Test Part III - Regression Introduction 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 (https://review.udacity.com/#!/projects/37e27304-ad47-4eb0-a1ab-8c12f60e43d0/rubric). Part I - Probability 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 s et 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: file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 1/16 7/1/2020 Analyze_ab_test_results_notebook In [2]: df=pd.read_csv('ab_data.csv') df.head() Out[2]: user_id timestamp group landing_page converted -:11:- control old_page 0 -:01:- control old_page 0 -:55:- treatment new_page 0 -:28:- treatment new_page 0 -:52:- control 1 old_page b. Use the below cell to find the number of rows in the dataset. In [3]: df.info() 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 c. The number of unique users in the dataset. In [4]: n=df['user_id'].nunique() n Out[4]: 290584 d. The proportion of users converted. In [5]: n_con = df[df['converted']==1]['user_id'].count() p=n_con/n p Out[5]:- e. The number of times the new_page and treatment don't line up. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 2/16 7/1/2020 Analyze_ab_test_results_notebook In [6]: df[((df['group'] == 'treatment') == (df['landing_page'] != 'new_page'))==True] ['user_id'].count() Out[6]: 3893 f. Do any of the rows have missing values? In [7]: # check if any columns have null values; it should print false df.isnull().sum().any() Out[7]: False 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 [8]: remove=df[((df['group'] == 'treatment') == (df['landing_page'] != 'new_page')) ==True] remove.head() Out[8]: user_id 22 timestamp group landing_page converted -:58:- control new_page 0 -:11:- control new_page 0 -:34:- treatment old_page 0 -:26:- treatment old_page 0 -:29:- treatment old_page 0 In [9]: df2=df.drop(remove.index,axis=0) df2.head() Out[9]: user_id timestamp group landing_page converted -:11:- control old_page 0 -:01:- control old_page 0 -:55:- treatment new_page 0 -:28:- treatment new_page 0 -:52:- control 1 old_page file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 3/16 7/1/2020 Analyze_ab_test_results_notebook In [10]: df2['user_id'].count() Out[10]: 290585 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]: 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? In [12]: df2['user_id'].nunique() Out[12]: 290584 b. There is one user_id repeated in df2. What is it? In [13]: df2[df2.duplicated(['user_id'], keep=False)] Out[13]: user_id timestamp group landing_page converted -:37:- treatment new_page 0 -:55:- treatment new_page 0 c. What is the row information for the repeat user_id? In [14]: # The landing_page for the non-unique id is "new_page". The group for the nonunique id is "treatment". # The value of converted column for the non-unique id is 0 d. Remove one of the rows with a duplicate user_id, but keep your dataframe as df2. In [15]: df2.drop_duplicates(inplace=True) 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? file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 4/16 7/1/2020 Analyze_ab_test_results_notebook In [16]: npc=df2[df2['converted']==1]['user_id'].count() prc = npc/n prc Out[16]:- b. Given that an individual was in the control group, what is the probability they converted? In [17]: cong=df2[df2['group']=='control'] ng=cong.user_id.count() cong_conv=cong[cong['converted']==1]['user_id'].count() pc=cong_conv/ng pc Out[17]:- c. Given that an individual was in the treatment group, what is the probability they converted? In [18]: cont=df2[df2['group']=='treatment'] nt=cont.user_id.count() cont_conv=cont[cont['converted']==1]['user_id'].count() pt=cont_conv/nt pt Out[18]:- d. What is the probability that an individual received the new page? In [19]: new_page=df2[df2['landing_page']=='new_page']['user_id'].count() new_page/n Out[19]:- 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 the metrics above, although half of the population received the new page, there is no significant evidence to conclude that the new treatment page led to more conversions. There were as many conversions with those who landed on the old page as were with the new page. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 5/16 7/1/2020 Analyze_ab_test_results_notebook Part II - A/B Test 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 pold and pnew , which are the converted rates for the old and new pages. Null Hypothesis: pnew - pold ≤ 0 Alternative Hypothesis: pnew - pold ≥ 0 2. Assume under the null hypothesis, pnew and pold both have "true" success rates equal to the converted success rate regardless of page - that is pnew and pold 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 pnew under the null? file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 6/16 7/1/2020 Analyze_ab_test_results_notebook In [20]: npc=df2[df2['converted']==1]['user_id'].count() p_new = npc/n p_new Out[20]:- b. What is the convert rate for pold under the null? In [21]: npc=df2[df2['converted']==1]['user_id'].count() p_old = npc/n p_old Out[21]:- c. What is nnew ? In [22]: new_df=df2.query('group=="treatment"') nnew=new_df.query('landing_page=="new_page"').user_id.nunique() nnew Out[22]: 145310 d. What is nold ? In [23]: old_df=df2.query('group=="control"') nold=old_df.query('landing_page=="old_page"').user_id.nunique() nold Out[23]: 145274 e. Simulate nnew transactions with a convert rate of pnew under the null. Store these nnew 1's and 0's in new_page_converted. In [32]: new_page_converted = np.random.binomial(1, p_new,nnew) In [25]: n_new=len(new_page_converted) n_new Out[25]: 145310 f. Simulate nold transactions with a convert rate of pold under the null. Store these nold 1's and 0's in old_page_converted. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 7/16 7/1/2020 Analyze_ab_test_results_notebook In [31]: old_page_converted = np.random.binomial(1, p_old,nold) old_page_converted Out[31]: array([1, 0, 0, ..., 0, 0, 0]) In [27]: n_old=len(old_page_converted) n_old Out[27]: 145274 g. Find pnew - pold for your simulated values from part (e) and (f). In [33]: n_old_converted=(old_page_converted==1).sum() n_new_converted=(new_page_converted==1).sum() obs_diff=n_new_converted/n_new-n_old_converted/n_old obs_diff Out[33]:-e-05 h. Simulate 10,000 pnew - pold 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 [40]: p_diffs = [] for _ in range(10000): new_pages_converted = np.random.binomial(1, p_new,nnew) n_news=len(new_pages_converted) old_pages_converted = np.random.binomial(1, p_old,nold) n_olds=len(old_pages_converted) n_olds_converted=(old_pages_converted==1).sum() n_news_converted=(new_pages_converted==1).sum() obs_diffs=n_news_converted/n_news-n_olds_converted/n_olds p_diffs.append(obs_diffs) 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. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 8/16 7/1/2020 Analyze_ab_test_results_notebook In [41]: plt.hist(p_diffs); j. What proportion of the p_diffs are greater than the actual difference observed in ab_data.csv? In [42]: ndiff=len(p_diffs) In [43]: obsdiff=pt-pc grt = [i for i in p_diffs if i>float(obsdiff)] n_grt=len(grt) p_grt = n_grt/ndiff p_grt Out[43]: 0.9031 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 index calculated in "j" above is the p-value, which is the probability of observing the statistic in favor of the alternative hypothesis if the null hypothesis is true.The null hypothesis suggests that there is no difference in conversion based on the landing page. With a p-value of 0.9, suggests the null hypothesis should not be rejected. 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. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 9/16 7/1/2020 Analyze_ab_test_results_notebook In [ ]: import statsmodels.api as sm convert_old = df2.query("landing_page == 'old_page' and converted == 1").shape [0] convert_new = df2.query("landing_page == 'new_page' and converted == 1").shape [0] n_old = n_old n_new = n_new m. Now use stats.proportions_ztest to compute your test statistic and p-value. Here (http://knowledgetack.com/python/statsmodels/proportions_ztest/) is a helpful link on using the built in. In [ ]: z_score, p_value = sm.stats.proportions_ztest([convert_new, convert_old], [nne w, nold],alternative='larger') In [ ]: p_value In [ ]: z_score 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.? The p-value, as in indicated in the response to j above, is the probability of observing our statistic provided the null hypothesis is true. Similar to parts j and k, the p-value obtained in part l suggests that the landing page has no impact on the conversion rates.The z-score is a measure of how many standard deviations are below or above the population mean. Part III - A regression approach 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 file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 10/16 7/1/2020 Analyze_ab_test_results_notebook 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 [45]: import statsmodels.api as sm ##df2['group'].replace('treatment','ab_page',inplace=True) C:\Users\User\Anaconda3\lib\site-packages\statsmodels\compat\pandas.py:56: Fu tureWarning: The pandas.core.datetools module is deprecated and will be remov ed in a future version. Please use the pandas.tseries module instead. from pandas.core import datetools In [46]: ## Getting duration colum import datetime as dt from datetime import datetime df2['timestamp'] = df2['timestamp'].apply(lambda x : pd.to_datetime(str(x))) In [47]: df2['hours'] = df2['timestamp'].dt.hour + (df2['timestamp'].dt.minute)/60 + (d f2['timestamp'].dt.second)/3600 In [48]: df2[['control','treatment']]=pd.get_dummies(df2['group']) df2['ab_column']=df2['treatment'] df2=df2.drop(['control','treatment'],axis=1) df2.head() Out[48]: user_id timestamp group landing_page converted hours ab_column - - control 22:11:- old_page 0 - - - control 08:01:- old_page 0 - - - treatment new_page 16:55:- 0 - - - treatment new_page 18:28:- 0 - - - control 01:52:- 1 - old_page 0 0 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. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 11/16 7/1/2020 Analyze_ab_test_results_notebook In [49]: df2['intercept']=1 logit_mod=sm.Logit(df2['converted'],df2[['intercept','ab_column']]) results=logit_mod.fit() Optimization terminated successfully. Current function value:- Iterations 6 d. Provide the summary of your model below, and use it as necessary to answer the following questions. In [50]: results.summary() Out[50]: Logit Regression Results Dep. Variable: converted No. Observations: 290585 Model: Logit Df Residuals: 290583 Method: MLE Df Model: 1 Date: Wed, 24 Jan 2018 Pseudo R-squ.: 8.085e-06 Time: 11:05:35 Log-Likelihood: -1.0639e+05 converged: True LL-Null: -1.0639e+05 LLR p-value: 0.1897 coef intercept std err z - ab_column - P>|z| [-] - -2.005 -1.973 -1.312 0.190 - 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_column is 0.19 > new threshold rate. Thus suggesting that the null hypothesis should not be rejected. The null hypothesis suggests that there is no significant relationship between conversion and page, while the alternative hypothesis indicates that there is significant relationship between the new page and conversion rates; the two hypotheses are similar to Part II. And the results of this analysisis similar to what was observed in Part II above. The simulation and the z-test performed above were implemented as one-tailed tests while the regression is a two-tailed test; this resulted in the difference in the value of p-value obtained in this regression part. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 12/16 7/1/2020 Analyze_ab_test_results_notebook 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? Other possible factors what considering include the country of residents of the individual converts, profession, age, and gender. With more factors to consider, there could be possible problems such as multicolinearity, nonlinear relationship existence among explanatory variables, outliers, and many others. 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 (https://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.join.html) 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 [51]: 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() Out[51]: country timestamp group landing_page converted hours ab_col user_id 630000 US - treatment new_page 06:26:- 0 - 1 630001 US - treatment new_page 03:16:- 1 - 1 630002 US - control 19:20:- 0 - 630003 US - treatment new_page 10:09:- 0 - 630004 US - treatment new_page 20:23:- 0 - old_page file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 13/16 7/1/2020 Analyze_ab_test_results_notebook In [52]: ### Create the necessary dummy variables df_new[['US','UK','CA']]=pd.get_dummies(df_new['country']) df_new=df_new.drop('UK',axis=1) df_new.head() Out[52]: country timestamp group landing_page converted hours ab_col user_id 630000 US - treatment new_page 06:26:- 0 - 1 630001 US - treatment new_page 03:16:- 1 - 1 630002 US - control 19:20:- 0 - 630003 US - treatment new_page 10:09:- 0 - 630004 US - treatment new_page 20:23:- 0 - old_page 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 [53]: ### Fit Your Linear Model And Obtain the Results df_new['intercept']=1 logit_mod2=sm.Logit(df_new['converted'],df_new[['intercept','US','CA']]) results2=logit_mod2.fit() Optimization terminated successfully. Current function value:- Iterations 6 file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 14/16 7/1/2020 Analyze_ab_test_results_notebook In [54]: results2.summary() Out[54]: Logit Regression Results Dep. Variable: converted No. Observations: 290585 Model: Logit Df Residuals: 290582 Method: MLE Df Model: 2 Date: Wed, 24 Jan 2018 Pseudo R-squ.: 1.521e-05 Time: 11:05:48 Log-Likelihood: -1.0639e+05 converged: True LL-Null: -1.0639e+05 LLR p-value: 0.1983 coef std err z P>|z| [-] intercept - - -2.009 -1.964 US - -1.786 0.074 - CA - -0.746 0.456 - In [55]: np.exp(-0.0507) Out[55]:- In [56]: np.exp(-0.0099) Out[56]:- From the results above, the conversion rates from each country are equally possible. Thus suggesting that conversion rate is not tilted in favor of a particular country. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 15/16 7/1/2020 Analyze_ab_test_results_notebook Conclusions Congratulations on completing the project! Gather Submission Materials 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 (don't include the "" 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. Submit the Project 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- In the meantime, you should feel free to continue on with your learning journey by beginning the next module in the program. file:///C:/Users/User/Documents/Documents/FST-TAMU/ANTAEUS/Personal Documents/Upwork/Data Analysis with Python Project 2.html 16/16