## Introduction

On this project I will analyse an experiment conducted by Udacity. When the experiment was made, the Udacity courses featured two options to experience the courses.

• “start free trial” - the student was asked their credit card information and then will be enrolled in a 14 day free trial of the paid course version. After the 14 days, the credit card would be automatically charged if the student the not cancel the course subscription before.

• “access course materials” - the student is able to view the course videos and solve the quizzes for free, but do not get support from coaches, do not get a verified certificate and cannot submit their final project for feedback.

For the experiment, a change was tested, where after a student clicked “start free trial”, they would be asked how much time they could commit to the course. If they had more than 5 hours per week, they would be taken to the checkout as usual. If they had less than 5 hours per week a message would inform that the Udacity courses require a greater time commitment for successful completion and suggest the free version instead. The student would have the option to continue the enrollment anyway.

The screenshot below shows how the experiment looked like:

The hypothesis behind the experiment was that this might set clearer expectations to the students, reducing the amount of frustrated students who left the free trial because they did not have enough time and at the same time not reducing significantly the number of students who stay enrolled and complete the course. If the hypothesis turns out to be true, then the overall student experience could improve because coaches would be able to better support the students which will complete the course.

The unit of diversion on this experiment is a cookie. However, after enrollment in the free trial, the students are tracked by the user-id from that point forward. A user-id cannot be enrolled twice in the free trial.

## Experiment Design

### Metric Choice

The metrics below were considered for this experiment. $$d_{min}$$ is the practical significance boundary for each metric.

• Number of cookies: That is, number of unique cookies to view the course overview page. ($$d_{min}=3000$$)

• Number of user-ids: That is, number of users who enroll in the free trial. ($$d_{min}=50$$)

• Number of clicks: That is, number of unique cookies to click the “Start free trial” button (which happens before the free trial screener is trigger). ($$d_{min}=240$$)

• Click-through-probability: That is, number of unique cookies to click the “Start free trial” button divided by number of unique cookies to view the course overview page. ($$d_{min}=0.01$$)

• Gross conversion: That is, number of user-ids to complete checkout and enroll in the free trial divided by number of unique cookies to click the “Start free trial” button. ($$d_{min}= 0.01$$)

• Retention: That is, number of user-ids to remain enrolled past the 14-day boundary (and thus make at least one payment) divided by number of user-ids to complete checkout. ($$d_{min}=0.01$$)

• Net conversion: That is, number of user-ids to remain enrolled past the 14-day boundary (and thus make at least one payment) divided by the number of unique cookies to click the “Start free trial” button. ($$d_{min}= 0.0075$$)

#### Invariant metrics

First we need to decide which metrics to use as invariant metrics. That is, which metrics should stay constant in both control and experimental groups.

• Number of cookies: since we will randomly assign a cookie to either the experimental and control group, we expect that both groups have the same number of cookies.

• Number of clicks: a visitor needs to click the “start free trial” button before he gets to the point of the experiment. So we also expect that the both groups have the same number of clicks.

• Click-through probability: the click through probability is the number of cookies which actually click the button divided by the number of cookies which view the course page. Since the click happens before the experiment, we should also expect it to be constant across groups.

#### Evaluation metrics

• Gross conversion: the gross conversion tells us the percentage of students that enroll on the course after clicking the “Start free trial button”. This is a good metric for how experiment because it allows us to measure if there is an effect of the experiment screener on the enrolling. If our hypothesis turns out to be true, we expect the gross conversion to decrease, since students which to not have so much time to commit are less likely to enroll.

• Net conversion: Net conversion tells us which students stay enrolled and end up making at least a payment. We want our experiment to not reduce the net conversion. We want to keep the same or increase the proportion of students that stay enrolled and pay.

#### Unused metrics

• Number of user-ids: the number of user-ids is not a good metric because it can change due to differences in traffic which are not related to the effect of the experiment.

• Retention: This would be a good metric, because it tells us exactly, from the students who enroll in the course, the proportion which made a payment. However, because the unit of analysis is the number of students which enrolled, we need a higher amount of pageviews to be able to compute this metric. This would make the experiment exposure time too long for Udacity goals.

### Measuring Variability

To calculate the standard deviation of a proportion we can use the following formula:

$\sigma = \sqrt{\frac{p * (1-p)}{n}}$

where $$p$$ is the expected proportion and $$n$$ is the sample number.

Based on the baseline values, we know that $$p$$ for the gross conversion, is the probability of enrolling given click, $$p=0.20625$$, and the net conversion, is the probability of payment given click, $$p=0.1093125$$. We have a sample size of 5000 cookies, which correspond to the following amount of unique cookies which click the “start free trial”:

#unique cookies per day
u_clicks <- 3200

n
## [1] 400

Then the standard deviations are:

#stdev net conversion
sd_proportion(0.20625, n)
## [1] 0.0202306
#stdev gross conversion
sd_proportion(0.1093125, n)
## [1] 0.01560154

The unit of analysis of both our evaluation metrics is the number of unique cookies. Since this matches our unit of diversion, it is safe to assume that the empirically variability is similar to the analytical variability.

### Sizing

#### Number of Samples vs. Power

To calculate the number of necessary pageviews for our experiment, we can use this online calculator.

The calculator gives the following results:

Metric Baseline conversion rate Minimum Detectable Effect Sample Size
Gross Conversion 20.63% 1% 25835
Net Conversion 10.931% 0.75% 27413

The numbers given by the calculator are the sample size per variation. Since we will have two groups (control and experiment) we need to double those amounts (51670 and 54826 respectively).

To have these amounts of cookies clicking the “Start free trial button” (our unit of analysis), we need more users visiting the homepage. We know that 8% of the unique cookies which view the page end up clicking the button. So, the needed sample sizes become 645875 for gross conversion and 685325 for net conversion. We need then 685325 total pageviews to achieve our desired power.

#### Duration vs. Exposure

In order to have 685325 samples for our experiment, we need more total pageviews on our webpage depending on how much traffic we decide to divert.

The change we are testing is not so critical, since the enrolling process for the free trial already existing before which involves credit card information already existed before.

There is not chance of anyone getting hurt because of the duration of our experiment. The experiment is just an extra question asking how much time they have to dedicate to the course. We also do not collect any extra sensitive data than before.

We may experience a few loss of enrollments, but it seems reasonable to use around all of our traffic to the experiment. Note that only 8% of the unique cookies that visit the site are actually clicking the “Start free trial” button, so we’re not exposing so many users to the change.

Given the baseline value of 40000 pageviews per day, we can expect a duration of 18 days for our experiment, which corresponds to roughly 2.5 weeks.

sample_size <- 685325
percent_diverted <- 1
daily_pageviews <- 40000

#duration is:
sample_size/percent_diverted/daily_pageviews
## [1] 17.13312

## Experiment Analysis

control <- read.csv("Control.csv", stringsAsFactors = FALSE)
experiment <- read.csv("Experiment.csv", stringsAsFactors = FALSE)

### Sanity checks

Lets create a function to calculate the confidence interval for counts:

#Calculate confidence interval and observed value for counts
calculate_ci_count <- function(n_control, n_experiment, alpha=0.05, p=0.5){
n <- n_control + n_experiment

z <- -qnorm(alpha/2)

#standard error for the proportion
SE <- sqrt((p * (1 - p))/n)
#margin of error
margin <- SE * z

#observed p
p_obs <- n_control/n

#Confidence interval
CI <- c(p - margin, p+margin)
return(list("p_obs"=p_obs, "Confidence Interval"=CI))

}

We see that the observed proportion of cookies in the control group is 0.5006, and the confidence interval is [0.4988, 0.5012]. Therefore, the cookies are splitted as expected between both groups.

#expect fraction to be 0.5
ncookies_cont <- sum(control$Pageviews) ncookies_exp <- sum(experiment$Pageviews)

calculate_ci_count(ncookies_cont, ncookies_exp)
## $p_obs ## [1] 0.5006397 ## ##$Confidence Interval
## [1] 0.4988204 0.5011796

#### Number of Clicks

We have a observed proportion of 0.5005 clicks in the control group and the confidence interval is [0.4959, 0.5041]. This metric also passed the sanity check.

nclicks_cont <- sum(control$Clicks) nclicks_exp <- sum(experiment$Clicks)

calculate_ci_count(nclicks_cont, nclicks_exp)
## $p_obs ## [1] 0.5004673 ## ##$Confidence Interval
## [1] 0.4958846 0.5041154

#### Click-through-probability

For the click through probability we need a different formula:

calculate_ci_proportion <- function(Xcont, Xexp, Ncont, Nexp, alpha=0.05, metric="invariant"){
#XCont - sucess on control
#Xexp - sucess on experiment
#Ncont - number of observations in control
#Nexp - number of observations in experiment
#
p_cont <- Xcont/Ncont
p_exp <- Xexp/Nexp
diff <- p_exp - p_cont

if(metric == "invariant"){
diff_obs <- 0
}else if(metric == "evaluation"){
diff_obs <- diff
}else{
stop(paste(metric, "is not a valid value for metric. Try 'invariant' or 'evaluation'" ))
}

p_pool <- (Xcont+Xexp)/(Ncont+Nexp)

SEpool <- sqrt(p_pool * (1- p_pool) * (1/Nexp+1/Ncont))

#alpha and z
alpha <- 0.05
z <- -qnorm(alpha/2)

#margin of error
margin <- z * SEpool

CI <- c(diff_obs-margin, diff_obs+margin)

return(list("diff"=diff, "Confidence Interval"=CI))

}

calculate_ci_proportion(nclicks_cont, nclicks_exp, ncookies_cont, ncookies_exp)
## $diff ## [1] 5.662709e-05 ## ##$Confidence Interval
## [1] -0.001295655  0.001295655

The observed difference in click-through-probability was 0.0001 and the confidence interval is [-0.0013, 0.0013]. The observed difference is inside the confidence interval, therefore this metric also passes the sanity check.

### Result Analysis

#### Effect Size Tests

Analysing the effect of the experiment in the gross conversion we see significant decrease in gross conversion. The confidence interval puts the estimated difference lower than zero and lower than the practical significance value. The difference in gross conversion is statistically and practically significant.

#Gross conversion
control_subset <- control[complete.cases(control), ]
experiment_subset <- experiment[complete.cases(experiment), ]

Xcont <- sum(control_subset$Enrollments) Ncont <- sum(control_subset$Clicks)
Xexp <- sum(experiment_subset$Enrollments) Nexp <- sum(experiment_subset$Clicks)

calculate_ci_proportion(Xcont, Xexp, Ncont, Nexp, metric="evaluation")
## $diff ## [1] -0.02055487 ## ##$Confidence Interval
## [1] -0.02912320 -0.01198655

For the net conversion rate there is no significant change. The confidence interval includes zero. The change in conversion rate is not statistically significant and not practically significant. However, because the lower margin of confidence interval is lower than the negative practical significance, there’s a chance that the net conversion may be reduced to a level that matters to the business.

#Net Conversion
Xcont <- sum(control_subset$Payments) Ncont <- sum(control_subset$Clicks)
Xexp <- sum(experiment_subset$Payments) Nexp <- sum(experiment_subset$Clicks)

calculate_ci_proportion(Xcont, Xexp, Ncont, Nexp, metric="evaluation")
## $diff ## [1] -0.004873723 ## ##$Confidence Interval
## [1] -0.011604501  0.001857055

#### Sign Tests

The sign tests corresponds to make a binomial test. For the gross conversion we have a p value of 0.0026, which indicates statistical significance. There is a significant difference between the gross conversion of the control and experimental group.

For the net conversion, the p value is 0.6776. There is no significant difference of net conversion between the control and experimental group.

control_subset$gross_conversion <- control_subset$Enrollments / control_subset$Clicks control_subset$net_conversion <- control_subset$Payments / control_subset$Clicks

experiment_subset$gross_conversion <- experiment_subset$Enrollments / experiment_subset$Clicks experiment_subset$net_conversion <- experiment_subset$Payments / experiment_subset$Clicks

sign_test_gross <- control_subset$gross_conversion - experiment_subset$gross_conversion

ammount_plus_gross <- sum(sign_test_gross > 0)
n <- length(sign_test_gross)
binom.test(ammount_plus_gross, n)
##
##  Exact binomial test
##
## data:  ammount_plus_gross and n
## number of successes = 19, number of trials = 23, p-value =
## 0.002599
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
##  0.6121881 0.9504924
## sample estimates:
## probability of success
##               0.826087
sign_test_net <- control_subset$net_conversion - experiment_subset$net_conversion

ammount_plus_net <- sum(sign_test_net > 0)
n <- length(sign_test_net)
binom.test(ammount_plus_net, n)
##
##  Exact binomial test
##
## data:  ammount_plus_net and n
## number of successes = 13, number of trials = 23, p-value = 0.6776
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
##  0.3449466 0.7680858
## sample estimates:
## probability of success
##              0.5652174

#### Summary

It would make sense to use the Bonferroni correction if the decision for launching our experiment was based on a significant change of any of the evaluation metrics. The Bonferroni correction goal is to reduce the chance of false positives (type I error) in the individual metrics, but it does so at the expense of increasing the chance of false negatives (type II error). In this experiment we are only interested in implementing a change if all our metrics meat the criteria (significant decrease of gross conversion, no significant decrease in net conversion). Requiring all metrics makes our experiment already vulnerable to type II errors, which would not be helped by using the Bonferroni correction. (Wikipedia, 2016, Davies, 2010, Discussions.udacity.com, 2016)

From the effect size tests we observe a significant difference in the gross conversion and no significant difference in the net conversion. The sign tests agree with this conclusions.

### Recommendation

Let us go back to our initial hypothesis:

The hypothesis was that this might set clearer expectations for students upfront, thus reducing the number of frustrated students who left the free trial because they didn’t have enough time - without significantly reducing the number of students to continue past the free trial and eventually complete the course. If this hypothesis held true, Udacity could improve the overall student experience and improve coaches’ capacity to support students who are likely to complete the course.

So, the gross conversion was reduced but the net conversion stayed the same. It seems we achieved our goal of reducing the number of students which enroll and my become frustrated and we did not reduce the number of students to continue past the free trial.

However, having a look at the net conversion results, we see that the lower margin of the confidence interval is lower than the negative practical significance boundary. This means there is a chance that there is a decrease of net conversion that may be relevant to the Udacity business. Given this fact, we should be careful and make further experiments to understand if this risk is real. So, my recommendation at this stage is to not launch the experiment!

## Follow-up Experiment

To reduce the number of frustrated students who cancel early in the course we need to think a bit about the reasons that are behind the early cancellations. Maybe even make a survey! However, if the students were not so engaged in the course, maybe they would also not be willing to take part in a survey.

From my point of view, students my feel frustrated if they feel they are not making progress or if they struggle in the new part of the material.

One idea for an experiment would be to send an email to the students after the first week of enrollment reminding them of the support options they have. Something along the lines of:

“Hey, if you’re having trouble with the course, don’t hesitate to ask for help in the forums! Our coaches are there to help!”

The hypothesis is that if the students are reminded of support options they will become more engaged after exposing their questions and getting good support. This change could increase the retention of the course. The metric to measure is therefore the Retention (number of students who pay / number of students who enrolled). A invariant metric for this experiment would be the unique user id, since we want to have the same amount of users in the control and experiment group. The unit of diversion would be the unique user id, because we only want to send emails to students which have already enrolled.