Here goes the first step in a long journey — an attempt to cover an introductory statistics course with a visuals-based (and eventually programming-based) approach. I hope you like it.

To change things up, I’ll use these short posts when uploading a video instead of having a dedicated piece of writing. Maybe I’ll continue with longer posts uncoupled from videos, but no plans there.

See you in Chapter 2.

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated

In this issue…

I wanted to share my goals for the Very Normal YouTube channel for 2026. I did a similar post for 2025, and I wanted to do the same thing this year.

A lot has changed since then. I’ve graduated and I made a huge move from San Diego. I started my first full-time job, and I’m grappling with how to balance that with my side ventures. It’s tough, and it’s certainly shown in my reduced output in the latter half of 2025.

A major goal in 2025 was to grow the channel to 100K subscribers. I experienced what it was like to focus on growth and the negative aspects that came along with it. Thanks to you all, I was able to achieve that goal — but now I’m ready to set my sights on a greater goal

2026 is the year of “heavy” things.

To borrow a phrase from a fantastic post, “to make something heavy” is to make something with weight. Something that reflects “quality, durability, presence and permanence”.

Now that the channel has reached a nice round number of subscribers, I feel more secure in shifting how I approach my videos. Before, I had consciously focused on videos to try to encourage channel growth.

Now, I’d like to try to make something heavy.

The driving force behind this channel has always been to “make you better at statistics.” For a long time, my efforts have catered to a rather small subset of people — people with some statistics education. And

As a first step towards that lofty gold button, I think I need to cast a wider net and start making resources for those who have yet to start their statistics journey.

I’ve thought a lot about how I started learning statistics and wondered how I might make things better. I’m not quite sure what form(s?) my heavy thing will take, but I know that it’ll necessitate some changes to the channel and the videos I put out. Of course, I’ll still make videos on cool, technical stuff, but you can expect more beginner friendly content too.

Thanks for sticking around for however long you have. See you in the next one.

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated

In this issue…

I wanted to share a little piece of advice that I heard recently. I’d been struggling to write consistently since I started my full-time job, and I’d been trying out different strategies to get more consistent effort.

When I was in my Ph.D, I had a luxury of long stretches of time to dedicate to writing — stretches that I don’t have anymore.

Dr. K (of HealthyGamerGG fame) has been in my YouTube autoplay for a while now, and he said something that really stuck with me:

“Do what you can do. Not what you want to do.”

I don’t quite remember the video that it comes from. But I just happen to be in the perfect place and the perfect mindset to take in this piece of advice.

For the past few months, I’ve been stressing out because I just cannot meet my own standards for writing.

I wanted an hour of time to write YouTube scripts. I wanted an hour of time for course writing.

But time and time again, I failed to do this.

After hearing Dr. K say this, I genuinely tried to ask myself, “How much can I write for these tasks?” I decided to allocate 15 minutes everyday for both of these writing tasks and commit to that. I could take 30 minutes away from my phone for writing.

And honestly, I’m so happy with the results. I’m actually ahead with both efforts, and I hope to be able to release another video before the end of 2025.

Just another reminder to not let the perfect be the enemy of the good.

Until the next one.

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated

In this issue…

I wanted to chat a little bit about a major difference between academia and industry that I’ve observed in my 3 months at my new job.

During my Ph.D, I had the chance to just read and think about statistics almost purely in a theoretical form. My research was in clinical trials, so I would hypothesize about the benefits of new design features.

I could simulate it, show some empirical benefits, and publish a paper. It wasn’t easy per se, but it was comfortable.

In contrast, industry statisticians are not just thinking about statistics. They’re thinking about business interests. They’re juggling demands from other parts of their team. Statistics becomes steeped in company strategy.

It was hard to adjust to at first, but it puts everything I’ve learned in a new light. Designs and analyses have to be backed up by not just by sound theory, but also: historical data, competitors, regulator demands, priorities against other drugs within the company, and so many other factors.

But above all else, I’ve learned to think about statistics through one main lens:

Speed.

Everything I do must get my projects to completion as fast as possible. Designs must get results as fast as possible. Designs might even react to other events in the company.

I had no idea that I would be dealing with speed in industry, but it’s been a crucial forcing function for more learning than I’ve ever done before, even during my Ph.D.

Being an industry statistician is great. And I hope to share more of my learnings with you all in the future.

Until the next one.

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated

In this issue…

I wanted to talk about some recent struggles I’ve been having with making videos. Since starting my job, making videos had turned into a slog. Between having to plan, write and edit before and after work, I’d been starting to feel like I’d stopped really leveling up as a statistics educator.

After the mixed effects video, I swore to change my approach. I had been trying to do things the same way as when I was in school:

1. Write through the entire script

2. Work through the edit and think of visuals along the way

This process worked when I had large swaths of time during my Ph.D. Admittedly, it was moreso a process I fell into rather than really thought out. Looking back, it wasted a lot of time because I would often forget what I’d planned when writing the script, so I’d get stuck thinking while editing.

This is the new process:

With work, I often only have half an hour or an hour to work on my personal projects. Gone are the days of long stretches of time. I try to write and edit in shorter scenes nowadays. This helps keep up momentum with a project and saves time on making visuals. A nice advantage to this new approach is that it’s more conducive to letting me experiment and polish my manim skills. I was able to try so many new animations and techniques in the FDR video, and it made making content fun again.

I had lost the chance to learn new things with each video, which led me to dread the process. I was just making what I made before and neglected my own creativity. The last two videos took a month to make, but I was able to make this most recent one in half the time, even with increasingly more responsibility at work.

I can say I’m excited to continue making statistics content for you all.

See you in the next one!

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

📺 What’s coming up?

A cool (and money winning) approach to handling multiple testing

In this issue…

I wanted to write a more personal post and talk about some thoughts I had about the channel going forward. I’ve slowly been figuring out how to balance a full-time job and YouTube, and I think I’m getting better.

I’ve been feeling a bit lost after I achieved my initial goal for the channel — to reach 100K subscribers before I graduated from my Ph.D. I did what I set out to do, but I neglected to think about next steps.

I like having a goal, but I’m having a hard time thinking about a new target that feels achievable, yet just out of reach with my current ability. I would love to get that golden plaque, but the timeline to 1 million subscribers too long to be achievable in the near future.

I really believe that this number is doable for YouTube, even with an esoteric subject like statistics. No matter what, I think that the key is just to keep providing value to people with each video. To just keep being helpful. This is a lesson I learned as a new employee and it applies just as well to getting attention on YouTube.

At first, I catered my material to people like me — statistics students. The next horizon is to figure out how to expand the appeal of my videos beyond the journeymen and make it accessible to beginners as well. I heard this on a recent Colin and Samir podcast, and I couldn’t stop thinking about it.

Maybe it’ll take a few years, but it’s not like I’m in a rush.

Before I reach 1 million subscribers, a good intermediate goal could be to reach 1 million views in a short amount of time. That’s what I’m currently working towards right now, and I hope you’ll enjoy that video when it comes out.

See you in the next one

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

Question

You are in charge of analyzing the data from a randomized controlled clinical trial. Some people in the intervention group stopped treatment due to side effects. Furthermore, you found out that some people in the placebo were given treatment due to worsening conditions. How should you handle these switchers in the analysis? How would you justify your approach?

Discussion

TLDR: Randomized controlled trials get much of their power from randomization. You need to analyze people how they are randomized, not how they end up in the trial.

Randomized controlled trials are considered to be the gold standard in terms of providing evidence that a treatment has a relationship to some disease. That is, they provide evidence for a causal relationship between the treatment and the disease. For a more detailed explanation behind this, you can check out this old video of mine.

Randomization is one of those things that is nice to have in theory, but it’s also an area where reality often has other things to say. Sometimes, you’ll have treatments whose side effects are too much for the participants to bear and they will stop taking the treatment mid-trial. Other times, people in the placebo groups may realize they’re in the placebo group and will seek out alternative treatments. With human participants, we have to make sure they are treated ethically.

When all is said and done, you’ll have data to analyze. Assuming that your data is rigorously maintained, you’ll know what group each person was randomized to as well as any other treatment-related events that happen to them.

On one hand, it’s tempting to analyze people based on the treatment they ended up in. The outcome you’re measuring is related to them being on what they’re taking, so that should be good right?

Wrong.

This is called a “as-treated” analysis, and it is very likely to bias your results. In our example, the treatment group is likely to drop because of side effects, which may or may not be related to the disease itself. We should strive to put the treatment in the most honest light, not the one that we personally want. This way of analyzing the data breaks the randomization, so it is not preferred.

Another way you might approach the analysis is to remove anyone who stopped their assigned treatment. They did not perfectly adhere to the trial protocol, so they bias the results. Hence, only including the completers should be reasonable… right?

Nope!

This is called a “per-protocol“ analysis, and it also has the potential to bias your results. If people are doing worse because of the treatment, then this should rightly be accounted for in the analysis. Similarly, if people drop out of placebo due to how severe the disease is, ignoring them may inadvertently make the treatment look more good than it actually is. When people drop out, it is important that we strive to keep collecting follow up data from them or have a solid way to impute their missing data.

The best way to analyze the data would be to preserve the randomization and keep each person in the group that they were randomized to, even if they switch or discontinue. Randomization plays a central role in breaking any and all confounding that may be present, enabling us to get to a causal effect. Even if people switch treatment, we are in effect analyzing the intent to treat. As such, this is called an “intent-to-treat analysis”, and it is the way that we should analyze data (and in fact it’s what the FDA demands).

To end this issue, I’ll end with a quote by Sir Austin Bradford Hill, who helped to pioneer the modern randomized clinical trial as we know it:

In many trials the original careful randomization of patients to treatment and control can be later disturbed by selective withdrawals of patients who cease to take a treatment or are proved sensitive to it so that they have to be withdrawn. The experiment is necessarily weakened – indeed we may on occasions have to assess the value of an intent to treat rather than a treatment.

Were you able to answer it correctly? Was there anything that surprised you? Let me know in the comments! See you in the next question.

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

Question

Power is defined as the probability of rejecting the null hypothesis, given that it is actually false. In some simple cases, we can calculate it directly, but this may not always be the case. Alternatively we can turn to simulation studies. Explain the process of calculating power this way.

Discussion

TLDR: Generate many independent datasets and leverage the law of large numbers to estimate power via the proportion of correctly rejected null hypotheses.

Power is an important tool for statisticians to more efficiently plan experimental studies. Although we’d like all of our studies to have large sample sizes, it would be helpful to figure out exactly what is “large” enough.

Despite its importance, I’ve found that it students don’t get a lot of practice with power or sample size calculations it in university coursework. So, we’re gonna cover it today.

I’ve covered simulation studies extensively on the Very Normal channel. If you’d like a more detailed explanation, go watch this video. Here, you’ll get a more condensed version of it here.

A simulation study involves the following:

1. Generate a dataset according to a specific alternative hypothesis

2. Perform the appropriate hypothesis test on this data

3. Record whether or not the null hypothesis was rejected

4. Repeat steps 1-3 a large number of times

5. Calculate the proportion of simulations where the null was rejected. This is your simulated (Monte Carlo) power.

Here’s what this would look like in code form. We’re doing a one-sample proportion test where the null hypothesis is 20%, and the alternative hypothesis is that the difference is 40%. This might correspond to a simple Phase 2 study where we’re trying to figure out if a new treatment is actually effective enough (40%) or not (20%).

set.seed(1)				   # for reproducibility
n_sims = 100000            # number of simulated datasets/tests
rejected = logical(n_sims) # for storing results

# Step 4: Repeat steps 1-3 a large number of times (here, 100,000)
for (i in 1:n_sims) {

  # Step 1: Gather the "data"
  data = rbinom(1, size = 30, prob = 0.4)
  
  # Step 2: Perform the test
  test = prop.test(data, n = 30, p = 0.2, alternative = "greater")
  
  # Step 3: Record the result (reject if p-value less than 0.05)
  rejected[i] = test$p.value < 0.05
}

# Step 5: Calculate power via proportion of rejected results
mean(rejected) # = 0.70698

There are more efficient ways to program this, but I’ve found that a for-loop better drives home the point of generating independent datasets. There’s just no way that a dataset from one iteration is going to affect one from another.

How does that work?

In the end, power is a probability. According to the frequentist paradigm, a probability is just the “long-run frequency” of a given event, which in this case is just rejecting the null hypothesis, given that it’s false.

We can directly control this in programming because we can choose to generate the dataset in specific way. Our specified null hypothesis is that the success probability is 20%, but I generated it with a probability of 40%. By definition, the null hypothesis is false for this data.

To establish a long-run frequency, we need a large number of simulations, so I simulated 100,000 datasets. Thanks to the power of modern computing, this is relatively fast and easy. There’s no “right” number to establish a long-run frequency, but in my experience, the number I see the most is actually 10,000.

What connects the proportion of rejected nulls to the probability of rejected nulls is the Law of Large Numbers. This tells us that the proportion will converge to the true underlying probability if the number of trials is large. It’s 100,000, which is plenty here.

One advantage of the simulation aka Monte Carlo approach is that we can employ hypothesis tests that don’t traditionally have nice calculations. Think nonparametric tests. This lets you be more flexible in the planning phase or if you expect your data to violate classic statistical assumptions.

One disadvantage with this approach is that your calculated power is always an approximation. That’s usually okay since power is moreso a tool for planning, but it’s just something to keep in mind.

Were you able to answer it correctly? Was there anything that surprised you? Let me know in the comments! See you in the next question.

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

Question

You are working on a team with a doctor. You’ve recently analyzed the primary endpoint and found that it wasn’t statistically significant. The doctor suggests that you redo the analysis with a different subgroup, which you politely reject. In response, the doctor suggests that you “do your job” and “just get a number” that the team can publish. How should you respond?

Discussion

TLDR: Some people may not understand the need for principled statistical analysis, but they might better understand what will happen if you don’t have it.

In school, we often focus on the technical aspects of statistics: knowing assumptions, understanding concepts, and implementing models in code. As a statistician, it’s your job to understand all of this on a deep level. But something that’s often missed in school is that statisticians also need strong communication and collaboration skills to work well on a team. Statistics is infamously opaque and is easily misunderstood even by highly educated people — including doctors.

In light of this, I wanted to mix in some behavioral questions into the Statistical 52 to give you something to prepare against. I received many types of questions during my job interviews, but collaboration questions were always asked.

In the particular scenario I described, a physician collaborator has disrespected you and suggested that you “p-hack”(fishing for favorable p-values with slightly different results).

It is not your fault that the experiment did not yield a significant result, but it is common that statisticians will be asked to redo work so that something can be published. You know that p-hacking is wrong, but others with less statistics exposure may not appreciate why this is the case. You may be viewed as just “the numbers person” who does not really understand the underlying science or medical knowledge that motivated an experiment.

So what do you do?

What not to do

Just to get the obvious out of the way, the wrong thing to do is to get defensive and attack or criticize the physician. You and the physician are on the same team, and it’s in no one’s best interest that there’s beef.

What to do

There’s a lot of different ways that this can be approached diplomatically, so I’ll just lay out some key points that you should hit in your answer if you’re asked this.

1. Focus on a shared goal

As a team, you and the physician are trying to publish the results of your experiment. This is a shared goal. The physician is suggesting something that might push the team closer to this goal, but you know that this actually works against this.

Statistical analyses for primary endpoints are typically pre-planned in advance. This is exactly to prevent people from “moving the goalposts”. If it gets out to a reviewer that you redo the analysis simply because it didn’t produce a specific p-value, then your paper is as good as dead. Even more so if you’re asked to provide your data so that your results are being replicated.

You should explain that the team should still report the null result since it represents the result of the pre-planned analysis. It might still be interesting to result the directionality (i.e. was it positive? negative?) of the estimate. By doing what the physician has asked, you are endangering the project and potential manuscript overall.

2. Be polite, respectful and understanding

Maybe this is obvious, but you should stay level-headed in your response. People may have all sorts of feelings about statistics, but you are the expert on it. You know what needs to be done since it’s your responsibility. You may need to take a moment to collect your thoughts and answer. This can help you avoid reflex defensiveness as well. The situation I described is more relevant in an in-person scenario, but I’ve seen it unfold in emails as well.

Given the pressure to publish, it’s natural that you might be viewed with some frustration if you seem like the person standing in the way of that. But you know that even deeper troubles wait even if you oblige this physician collaborator, so it’s good to be firm and polite. If you don’t do it, then a future statistical reviewer will — and your team will have lost a bunch of time waiting for this to happen.

3. Lean towards non-technical explanations

Even if you refuse, a collaborator may still want to know the statistical reason why you refused. This is where good communication skills really come into play. It’s very easy for statisticians to just jump back to technical definitions about p-values and p-hacking, but you might just lose someone’s attention if you do this.

P-values have a very specific definition, and they will lose this meaning if you repeat the analysis. It would be good to explain that even if you get a “good p-value” with a repeated analysis, it will not be the same as the original p-value you got. Even if it’s good, the process that led to it totally invalidates any planning that came before it. This physician understands it as a requirement to publish, but you know otherwise.

As a last note, I acknowledge that there are so many “what abouts” that can accompany hypothetical situations. These may change the answer, but I hope I’ve laid out some basic ideas that you can fall back on when developing your own answer. Rather than focus on a hypothetical, it would be good to try to think about an actual situation that you ran into.

Were you able to answer it correctly? Was there anything that surprised you? Let me know in the comments! See you in the next question.

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

📺 What’s coming up?

Wrapping up the longitudinal series

In this issue…

I wanted to talk about work-life balance again. I’ve been at my new job for about a month now, and it’s been a big adjustment from life as a Ph.D student. It’s very interesting, but it’s abundantly clear to me now that it’s very hard to balance a full-time job with YouTube.

In the last issue, I speculated that it would take me about a month to write, edit and upload a video. This turned out to be the case, but only after putting in some careful controls on my own life. I wanted to share the concrete things I did to be able to balance work and YouTube. None of this advice is groundbreaking, but it comes from my own lived experience

1. Establish clear defined periods for business

In my first week of work, I gained a greater appreciation for my parents. After 8 hours of work, the last thing I wanted to do was think hard about statistics and polish scripts. I essentially got nothing done for YouTube in that week, and it got me scrambling for a solution.

If the evening wasn’t the answer, then the only thing left was the mornings. This was tough because I had already dedicated my mornings to going to the gym, so I needed to balance the two. It forced me to push both my waking and sleeping times earlier so that I could fit both. Every morning, I get up, work on the YouTube channel for exactly an hour, go to the gym, and then get ready for work. I do my best to give 100% of my attention to my business because it’s often the only time I can give to it each day.

I don’t change my schedule for the weekends either. I can work a bit more, but it’s also important for me to give time to my wife, so I can’t be a total weekend warrior. It’s hard, and I’m definitely not perfect, but at least it gives YouTube a minimum of 7 dedicated hours of attention each week.

2. Accept reduced output

Establishing defined periods of time for YouTube also came with major attitude shifts. When I started making an hour each day for YouTube, I felt deeply unsatisfied with only having an hour to work on it. Back during my Ph.D, I could essentially dedicate as much time as I could to it outside of my research. I was used to having large swaths of time to dedicate to YouTube. If I felt like I couldn’t get that, I often procrastinated because I felt like it wasn’t even worth it to try for “just an hour.”

But with work, this attitude wasn’t going to cut it. I needed to convince myself that even an hour of deeply productive work was still worthwhile. If I were to be really honest, a lot of my time during my “large swaths” was distracted and unproductive. But I was so used to just having a comfortable period like that, that I just got used to it.

Even though I can’t work as much as I could back then, I can still make good progress each day. But I needed to change my personal definition of a “productive session” to really get the most out of my mornings. I’m still working on it, but this perspective change has already been helpful.

3. Do things even though you’re tired

This time around I had a deadline to deal with: submitting before the Summer of Math Exposition deadline. This meant that I would need to do some writing and editing in the evenings after work. This was tough at first, but it was necessary to get the video over the finish line.

After a few days of this, I realized that my “tiredness” was the strongest at the beginning and would fade away if I just stuck to my task for more than a few minutes. I was still able to do a little bit of my business after work, but it was just more difficult to get started. There’s certainly a point where I can’t do any productive thinking anymore, but I can squeeze out some more effort out of myself despite this tiredness.

The experience made me realize that this in itself is a skill, and it has broad application outside of just personal projects. If I can do one or two extra chores after work, then this totally frees up a weekend where I might have originally done them. It’s possible to work some tiredness, and it can be worth it to resist the urge to shut off your brain immediately after work. I know there are so many “what abouts” that can be said about this, but after freeing up consecutive weekends for myself and my wife, it’s well worth it to consider pushing through “tiredness”.

Until the next one.

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

Question

You’re working on a randomized, controlled trial with a binary outcome (i.e. symptoms resolved). Describe two methods you can use to estimate the response rates and confidence intervals for both of the placebo and treatment groups.

Discussion

TLDR: You can do a simple proportion for each of the two groups, or you can use your estimates from logistic regression and perform the necessary back calculations.

We are often interested in investigating how different interventions influence the probability that some events will happen. Will a new treatment increase the chance that my symptoms go away? Will adding more information to the checkout page increase the chance that a customer finishes their purchase? This is why methods for binary endpoints are important.

Binary endpoints are often modeled with Binomial/Bernoulli distributions, so our parameter of interest is the probability of a “success”, usually defined as some event happening.

The two ways that we can estimate this probability is by:

1. Calculating simple proportions (i.e. what percent of a group had the event?)

2. Using model-based estimates (i.e. start with logistic regression, go to desired unit)

For the purposes of this article, I’ll generate some toy data to simulate an RCT:

set.seed(4) # replicability

data = tibble(
  X = rep(c(0, 1), each = 20),
  Y = rbinom(40, size = 1, prob = 0.3 + 0.2*X)
)

People on placebo have a response rate of 30%, and people on treatment have a response rate of 50%. Sample size is 40 people.

Method 1: Calculating simple proportions

This is probably the most direct way to get what we want. Just take each group, calculate their proportion, and do what you want with the results.

# Quickly tabulate group-wise proportions and 95% CIs
data |> 
  group_by(X) |> 
  summarize(p = mean(Y),
            L_CI95 = p - qnorm(0.975) * sqrt((p)*(1-p)/20),
            U_CI95 = p + qnorm(0.975) * sqrt((p)*(1-p)/20))

# A tibble: 2 × 4
      X     p L_CI95 U_CI95
  <dbl> <dbl>  <dbl>  <dbl>
1     0  0.45  0.232  0.668
2     1  0.65  0.441  0.859

In this particular dataset, the estimated response rates for the placebo and treatment groups were 45% and 60%. Not the best, but we only have 20 people per group.

The 95% confidence intervals for the placebo and treatment groups were (23.2% - 66.8%) and (44.1% - 85.9%), respectively. These confidence intervals are sometimes called “Wald confidence intervals” because they rely on the Central Limit Theorem to be (approximately) correct.

Method 2: Logistic Regression

Another way would be to start with logistic regression and then do the appropriate calculations to get back to the scale of the response rate. For a standard logistic regression using only treatment as the covariate, we get the following:

The intercept is interpreted as the “log-odds of the event” for someone in the reference (read: placebo) group. The slope is the change in the log-odds or “log-odds ratio” for someone in the treatment group. The parameters are in terms of the log-odds, but we can do some calculations to get an expression for the response rate.

Then, you can get the group-specific response rates by conditioning on different covariate values.

# Logistic regression
fit = glm(Y ~ X, family = "binomial", data = data)

betas = coef(fit) # store estimated parameters
p_X0 = exp(betas[1]) / (1 + exp(betas[1]))     # 0.45
p_X1 = exp(sum(betas)) / (1 + exp(sum(betas))) # 0.65

So both methods come to the same response rates for each group. But what about the confidence intervals?

The estimated parameters for logistic regression are calculated via maximum likelihood, so — as discussed in Question 3 — they have an approximately Normal distribution whose variance is the inverse of the Fisher Information:

V = vcov(fit) # inverse Fisher information

            (Intercept)          X
(Intercept)   0.2020202 -0.2020202
X            -0.2020202  0.4218004

We’ll need these variances and covariances for the intervals.

To get the confidence interval for the placebo group, you have to do the calculation on the scale of the log-odds first, and then make the transformation:

# Calculate 95% CI for log-odds of placebo group
X0_logodds_CI = betas[1] + qnorm(c(.025, 0.975)) * sqrt(V[1,1])

exp(X0_logodds_CI) / (1 + exp(X0_logodds_CI)) 
[1] 0.2532017 0.6637984 

But the log-odds for the treatment group is actually a linear combination of the estimated parameters:

The calculation for the confidence interval needs to take this into account. Using the properties of Normal distributions, we get:

# Note the linear combination c(1, 1) on point estimate and variance
X1_logodds_CI = c(1, 1) %*% betas + qnorm(c(.025, 0.975)) * sqrt(t(c(1, 1)) %*% V %*% c(1,1))

exp(X1_logodds_CI) / (1 + exp(X1_logodds_CI))
[1] 0.4256049 0.8231570 

From the logistic regression approach, the rate and interval for the placebo group was 45% (25.3% - 66.3%). For the treatment group, it was 65% (42.5% - 82.3%). Both intervals hinge on an asymptotic normality, which is questionable with our sample size.

Were you able to answer it correctly? Was there anything that surprised you? Let me know in the comments! See you in the next question.

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

Question

What is maximum likelihood estimation, and what benefits do we get from using it?

Discussion

TLDR: Maximum likelihood estimators are consistent, have a convenient asymptotic distribution and are asymptotically efficient. These translate into efficient confidence intervals and p-values.

Data is random. When we collect data, we often assume it comes from a mathematical approximation called a statistical model. More specifically, we often make assumptions on the probability distribution that is responsible for the randomness in the data.

Continuous data are often modeled with the Normal distribution. Binary data are often modeled with the Bernoulli or Binomial distribution.

The randomness in distributions like these is decided by just a few values called parameters. Different choices of parameters will change what data we are likely and unlikely to see. They also often have intuitive interpretations.

When we collect data, we try to infer the most likely value of the parameter that could have generated the data we saw. We convert the data into an estimator, which is our guess for the parameter value.

Maximum likelihood estimation is a procedure for producing estimates for statistical models. And it produces pretty good estimates too.

We take advantage of the benefits of these estimators all the time, but it’s easy to forget what these benefits are. This was a literal interview question I had one time, and I’m lucky that I had brushed up on the topic beforehand.

The benefits of maximum likelihood estimation can be summarized in three points.

1. Maximum likelihood estimators (MLE) are consistent. This means that — with large amounts of data — the value of the MLE will be very close to the theoretical value of the parameter that generated the data. Assuming your model is right (huge assumption) and with enough, your estimate will give you a good guess for the parameter and for a value of interest.

2. Maximum likelihood estimators have an asymptotic Normal distribution. Distributions for statistics like the MLE are important for hypothesis testing. Generally, they’re hard to figure out, given how diverse data is. In this case, the process of maximum likelihood actually gives us a really convenient distribution for the MLE: a Normal one. This makes it easy to calculate p-values and confidence intervals with standard statistical programming.

3. Maximum likelihood estimators are asymptotically efficient. With large enough samples, MLEs will achieve the smallest variance possible by an (unbiased) estimator. Smallest possible variance means that the resulting confidence intervals we create will be as small as possible, which means that we’re more likely to get statistically significant results.

Commonly used models like GLMS (i.e. logistic regression) and mixed-effects models all incorporate use maximum likelihood, which gives us the benefits above.

Important caveat: maximum likelihood estimators are the best among unbiased estimators. There are estimators that are biased, but can achieve lower variance than the MLE.

Were you able to answer it correctly? Was there anything that surprised you? Let me know in the comments! See you in the next question.

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

Question

You have finished recruiting for a randomized controlled trial (RCT). Even though you planned for a 50-50 split, you have found that 60% of the sample are in the treatment group due to drop out. How much less efficient is your trial compared to a trial with a 50-50 split?

Discussion

TLDR: Deviating from a 50-50 split increases the variance in our estimator. You can measure loss of efficiency by taking a ratio of the variance you have and the ideal variance.

Sometimes experiments don’t go as planned.

Even though a 50-50 split is optimal for a two-sample experiment like a randomized clinical trial, we might end up with slightly imbalanced group sizes. With this question, I wanted to make sure my readers understood the downstream effects of this imbalance.

Here, I’m assuming that we’re using the difference in sample means to estimate the difference in population means. If it’s discernibly different from zero, it means the treatment has a causal effect on creating this difference. A huge factor in this discernibility is the variance of this statistic:

Assuming that you have a fixed sample size you can work with, this variance is minimized when the sample size for both groups is equal. We can equivalently express the above variance in terms of the proportion of the sample size that’s in the treatment group.

When the split is 50-50, we get the following value for the variance of the estimator:

But when the split is 60-40, we get this value instead:

You can see that this slight imbalance in the treatment groups leads to a slight increase in the variance of the estimator. It may look small, but it could mean the difference between a published paper and having nothing to show for your data.

When we talk about efficiency in statistics, we are usually referring to relative variances. We want the lowest variance that we can get, and in this case, we can derive an easy expression for what that looks like. To compare variances, we take their ratio:

The variance under a 60-40 split is about 104% of the variance under a 50-50 split. Thus, there is about a 4% loss in efficiency due to this imbalance. We can either eat this efficiency loss or try to recruit more people to get parity with the ideal case.

Were you able to answer this question correctly? Were there concepts you didn’t know about before? Let me know in the comments!

See you in the next question.

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

Question

In a randomized clinical trial (RCT) or AB test, we strive to achieve a 50-50 sample split between the treatment and placebo/control group. Why is this the case?

Discussion

TLDR: We strive for 50-50 splits in RCTs because it is optimal in a statistical sense. It minimizes the variance of the difference in sample means, which gives us the highest efficiency possible.

RCTs (or AB tests) are an experiment for comparative efficacy. We compare a treatment group against a control or placebo group to assess if there is a meaningful difference between these two groups. Thanks to the benefits of randomization, a non-zero difference suggests that the treatment is causing this difference.

More specifically, we are interested in looking for a difference in the population means of these two groups. So, we use the difference in the sample means as an educated guess for the difference in population means:

Where A and B indicate treatment and control group, respectively.

Both of these means will vary slightly in value depending on the data we collect. So, there is a degree of variability in the difference we will actually observe. It’s in our best interest to minimize this variance.

Below is an expression of this variance:

The first line hinges on the Central Limit Theorem. The result in the second line comes from assuming that the variances of the two groups are the same (i.e. homoskedasticity).

Since we are running an experiment, we have total sample size to recruit for:

Taking a step back, let’s say that we don’t necessarily want to go for a 50-50 split yet. Instead, we want to figure out directly what proportion of the sample size should be dedicated to the treatment group. We’ll designate some notation for this:

We can substitute these expressions back into the variance equation:

What’s relevant here is that the variance of the difference in sample means is a function of the proportion of people assigned to the treatment group.

Our goal is to minimize the variance, so an equivalent goal here is to maximize the expression in the denominator:

Hence, the value of pi that maximizes the denominator and minimizes the variance is 0.5. This is why a 50-50 split in an RCT is in a sense optimal.

Were you able to answer it correctly? Were you able to learn anything new? Let me know in the comments!

See you in the next question.

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are the best and easiest ways to support me, but if you feel like going the extra mile, this would be the place. Always appreciated!

📺 What’s coming up?

SoME4 entry, new interview question series on newsletter

In this issue…

I just wanted to give a quick life update for those that keep up with the channel and newsletter. I’ve successfully moved to my new city, and I start work in exactly a week!

Before, I was balancing my Ph.D with YouTube, but now I have a full-time job to answer to. This will be a new experience for me, and there will be growing pains, but I am more convinced than ever that I need to keep this platform growing. My supervisor told me I’d be learning new things from day one, and I want to share these lessons with you all.

One tactic I plan to deploy first is having a dedicated period of time that is purely towards progressing this channel. I don’t know exactly when it’ll be yet, but it’s probably going to be in the morning before my wife wakes up and before the gym. I’ve tried doing this before, and it’s helpful when I’m able to commit to it. I know from experience that I tend to allow my attention to slip during these dedicated moments. I know that the likes of Cal Newport and Ali Abdaal have employed this tactic before, so I know there’s some value to it.

In other news, I’m also taking on a new challenge for myself. For one year, I plan to publish a weekly statistics-based question and answer on this newsletter.

The goal of this challenge is to develop a repository of questions that student statisticians can use to prepare for future interviews and work. My wife, also a statistician, has been preparing for interviews herself, and I realized that there’s not a lot of resources out there to help people polish up on necessary topics. I originally made a video to serve this purpose, but my goal at the time was to rush towards 100K subscribers, so I tabled it. It seemed like a great time to bring it back.

This means you’ll get much more emails from me, so I totally understand if you unsubscribe to maintain the sanctity of your inbox.

That’s it for this one, see you in the next one!

Christian

📦 Check out my other stuff!

• Read through my Statistical Garden on the Very Normal website! This digital garden houses all the knowledge I gained as a biostatistics graduate student. It’ll grow as I learn more, and it’s free for you to look through.

• You can support me on Ko-fi! YouTube and Substack are by far the best (and easiest) ways to support me, but if you feel like going the extra mile, this would be the place. It is always appreciated!

📺 What’s coming up?

Part 2 of the longitudinal series, SoME4 entry

In this issue…

I wanted to briefly take a break from packing up my stuff and talk a bit about what I’d like to do with this newsletter.

With June ending, I have about a month left until I start my full-time job. I plan to close out my longitudinal series and get my Summer of Math Exposition entry finished before then.

But those are videos. I still have plans for this newsletter.

A while back, I tried to do a video on questions that I thought statistics students should know to keep track of their mastery of the material. It didn’t perform that well, so I didn’t pursue it further, but the concept has always stuck in my mind.

It might not be great material for YouTube, but it could possibly be great content for this newsletter. I was thinking of turning this newsletter into a question-and-answer bank for statistics content. I don’t have much to say on this newsletter nowadays, but I believe this will be a great new phase to bring it into.

I don’t know about frequency, format, or content yet, but I’m pretty excited to do this. It also gives me a chance to touch on topics that I don’t have the time to cover in video format. It fits perfectly with the channel’s mission to make you better at statistics, but just in a different form.

I’ll still be using it to notify you about new videos, but expect more posts from the newsletters in the future. If that’s not what you want, then please feel free to ignore them. Either way, thanks for letting me into your inbox.

See you in the next one.

Christian

📦 Other stuff of mine

• Read through my Statistical Garden on the Very Normal website! This digital garden contains the biostatistics knowledge I gained in graduate school, and it’s free for you to peruse.

• You can support me on Ko-fi! YouTube and Substack are by far the best (and easiest) ways to support me, but if you feel like going the extra mile, this would be the place. It is always appreciated!

What software do you use to make your videos?

• Editing, Final Cut Pro X: I don’t think there’s anything special about it; it just happens to be the software that I was taught and stuck to. I use Mac, so this is also their dedicated software.

• Design, Figma: I use Figma to design all of my thumbnails and some of the visuals that appear in my videos. I used to use Midjourney to help me generate simple icons, but now I try to make my own via the pen tool. Not as professional looking, but there’s a sense of satisfaction with making it myself. I’m also teaching myself Affinity Designer, but am still learning.

• Animation, Manim (Community Edition): For notation, formulas and graphs, I use the community edition of manim, the animation engine created by Grant Sanderson of 3Brown1Blue fame.

What kind of equipment do you use?

• Machine: I edit on a M2 Pro Mac Mini with 32GB of unified memory. I think it handles my type of videos okay, but I have come into problems exporting 15+ minute videos. Not that big of an annoyance since I can chunk things into smaller parts, but just a heads up from my own experience

• Mic: My videos are faceless, so I try make sure my audio is as high quality as possible. When I can make videos at home, I use a Elgato Wave 3 on a low profile mic arm to make it as convenient as possible to do voiceover and redo lines. I also have a pop filter on it to help guard against bad voices. If I am traveling, I bring along my DJI Lavalier Mics for portability. After I do voiceover, I add some audio filters on it to further improve the quality.

Full disclosure: the links here are my affiliate links, so I will get some compensation if you choose to buy them from. Much appreciated if you do!

Can you recommend textbooks for self-study?

I find this question difficult to answer because different people may have different goals. For this question, I’ll list out a few books that I think are useful for picking up biostatistics as a skillset. I have a preference for books that are easily found/accessible and have at least partial solutions to them.

One of the hardest aspects of self-study vs enrolling in a program is feedback; it is often hard for self-studiers to know if they actually have a handle of a concept if they do not have the chance to be wrong and be corrected.

If you need to learn the basics of a model for an applied problem, then I recommend Bernard Rosner’s Fundamentals of Biostatistics.

If you’re looking to bulk up on your foundational knowledge on statistics and probability, then I recommend George Casella and Roger Berger’s Statistical Inference.

Finally, I also recommend Richard McElreath’s Statistical Rethinking. Most statistics is taught from the frequentist perspective, but current technology has made it such that Bayesian statistics are the most approachable they’ve ever been. He gives a fresh and new perspective on Bayesian statistics, and he even offers videos on it!

Please list out a set of textbooks that would cover your MS/Ph.D studies

While I think that self-study should be dedicated to gaining useable skills for your needs, I do respect the desire for more knowledge for the sake of it. People have asked me about resources for gaining the knowledge base that you’d get for a graduate program, so this is the answer for that.

For more context, watch my video on the The Big Picture of Statistics to see a breakdown of what content is expected of biostatistics graduate students. Here, I’m listing out most of the textbooks that my courses used, as well as the topic they focused on. Most courses only use a small part of a textbook.

• Probability Fundamentals: Casella and Berger’s Statistical Inference

• Mathematical Statistics: Shao’s Mathematical Statistics, van der Vaart’s Asymptotic Statistics, Wainwright’s High Dimensional Statistics

• Basic hypothesis tests: Rosner’s Fundamentals of Biostatistics

• Linear Regression: Kutner’s Applied Linear Statistical Models

• GLMs: Agresti’s Categorical Data Analysis

• Longitudinal Data Analysis: Diggle’s Analysis of Longitudinal Data

• Survival Data: Cox’s Analysis of Survival Data

• Machine Learning: James’ Introduction to Statistical Learning

• Clinical Trials: Friedman’s Fundamentals of Clinical Trials, Piantadosi’s Clinical Trials (A Methodologic Perspective)

What are the key components of your knowledge management system?

I used Obsidian to manage all the information that I gathered in my Ph.D. I also use it to power the website behind The Statistical Garden. I like Obsidian, and I support them, but any place that you can write notes and (cruically) make links between these notes is good.

My system is simple. I have one folder to hold all my notes, which each hold digestible bits of information. I also have another folder for reference notes, notes that are specifically dedicated to representing sources of information like textbooks or papers. I use templates to help make sure that notes have a consistent structure.

To make sure that related notes are easily grouped together, I use hierarchical tags. For example:

• #project will be used to denote a note that pertains to a project in general, #personal denotes that it contains information relevant to my personal life

• #project/paper2 will denote a note that is related to my second manuscript; #personal/finance means the note deals with my financial planning

Hierarchical tags are nice since they enable more nuanced searches. Even if I search for just #project tags, the results will also include any notes that have further “subtags”. Likewise, searching #project/paper2 will only look at notes with this specific subtag.

Sometimes I’ll dedicate pages to act as hubs for projects as well, just to make it easier to navigate.

📺 What’s coming up?

100K Q&A; more videos for the longitudinal series; working on me SoME4 entry

In this issue…

I wanted to celebrate the fact that my channel has finally reached 100K subscribers! I uploaded my first video on April 9th, 2023, and I reached the milestone on June 4th, 2025, meaning that it took 787 days to achieve this goal.

At the start of 2025, I made it a goal to reach 100K before I graduate, and I barely got there. 3 months ago, I realized that things needed to change fast if I was going to make it, so I put myself through grueling weekly uploads. After it was all said and done, I don’t think the weekly pace helped, but rather a more mindful approach to the thumbnail and title. As much as I hate to admit it, they matter almost as much as the actual effort I put into the video.

Some Very Normal Lore

Getting that YouTube plaque has been a dream of mine since high school. YouTube started in 2008, and I started taking film classes just as it started to blow up. During lunch, I would watch Corridor Digital and Epic Meal Time. This was where I learned how to edit, and I had always wanted to make a name for myself on this new medium from a company that was strangely near my house.

The YouTube Creator Awards didn’t come out until 2012-2013, when I was in college, but I instantly knew that I wanted one for myself. I kept up my editing skills in college thanks to student clubs, always keeping the award in the back of my head.

I even got to do some editing during my Master’s degree! I had a class where we needed to make a video demo, and I had the perfect skillset to make our project stand out.

If you had asked me what I would be getting the plaque for, I would have NEVER guessed it would be for teaching statistics on YouTube. My 2008 self would have never believed it because I’d never even been in a statistics class before.

One million feels so far away right now, so I’m not going to think about my subscriber count for a while. I’ll just keep focusing on making helpful statistics content while I adjust to a new life in industry.

I’m proud of what I’ve been able to make, and I’m glad that you’ve been a part of the journey. Thank you for watching and even going the extra mile to read this newsletter.

What’s next for the channel?

I’ll have a dedicated section to this in the upcoming 100K Q&A, so keep an eye out for that!

Until the next one.

Christian.

📦 Other stuff of mine

• You can support me on Ko-fi! YouTube and Substack are by far the best (and easiest) ways to support me, but if you feel like going the extra mile, this would be the place. It is always appreciated!

A while back, I made a video on the idea that helped win the “Nobel Prize in Statistics”, check it out will ya?

📺 What’s coming up?

A video on everything I’ve learned in my two biostatistics degrees

In this issue…

I just wanted to reflect a little bit on the past three weeks and share some thoughts I had in the interim. Since the last issue, I:

• got my Ph.D

• finalized a lease for a cross-country move

• kind of went wild with shopping for myself

In all that time, I was thinking about what I had actually accomplished in my Ph.D.

Would I do it again? Should I have done anything differently? Was it worth it?

Firstly, I would not go get a second Ph.D. I am so ready to work a more consistent schedule and better delineate my work and personal lives.

The only thing I would have done differently for my Ph.D would be to develop a knowledge management system (i.e. Zettelkasten, Second Brain) much sooner. I was still throwing away notes even as a first year, but it would have been much better to keep and refine them further as I grew as a researcher.

The question I want to focus on the most in this issue is that last question: Was it worth it?

This is a hugely personal question, and it would be disingenuous for me to pretend that I can answer it for everyone.

But, what I can do here is to offer my insight on a specific situation: getting a Ph.D after a Masters.

What many people outside of biostatistics may not know is that many Ph.D programs will prefer to admit students who already have Masters degrees.

Given that an MS in biostatistics is already considered a terminal degree, what is the value in getting a Ph.D afterwards?

1. Higher earning power

I’ll get the easy one out of the way first: having a Ph.D generally comes with earning more money over the course of one’s career. In the context of biostatistics, this stems from higher responsibilities expected from Ph.Ds.

In pharma, Ph.Ds are expected to help with the design of possibly multi-million dollar experiments. They are also expected to make sure that the statistical methodology is appropriate. Given the sheer diversity of data from various therapeutic areas, this is a lot of responsibility to bear. With more responsibility comes more money.

2. More job opportunities

This isn’t to say that for MS graduate can’t do this, but the gap in experience and expertise is often too large. After interning and talking to various companies, I hate to say that many of the higher job opportunities for biostatistics are locked behind a degree.

From personal experience, I can speak to why this might be the case. A typical biostatistics Ph.D lasts 4-5 years, while an MS lasts 2. Ph.D students and MS students often take the same classes, but Ph.D students may take more difficult versions.

So the literal time difference is 3 years. That doesn’t seem like a lot, but this is also 3 years of full-time learning, studying and research. The end result of these extra 3 years is more robust statistical expertise. From an employers perspective, this is 3 years of extra training that they don’t have to pay for, and they get a working employee right away.

This is to say, one form of value for a Ph.D is simply time. Time where a student can fully immerse themselves in getting better at statistics, without having to worry about money.

3. The skill of independence

If you apply for Ph.D programs, it’s inevitable that you will be asked why you want to pursue a Ph.D, especially if you already have an MS.

My answer back then was and still is: independence. As an MS, I felt like I could do the statistics if someone laid out what I needed to do. But I knew that this meant that I would inherently need a boss to always tell me what to do. I didn’t like that feeling, and I wanted the ability to do my own thing and create value on my own.

In the last point, I mentioned the 3 year difference between a Ph.D and an MS. This 3 year period is a trial by fire on learning how to be independent via research.

Research is really hard! You have to — independently — find a problem to solve, figure out existing solutions, imagine a better solution better than the current ones, demonstrate it’s better, and convince others of its use too. The first year after coursework is often a painful one for Ph.D students: it’s the first time that they have all the time in the world, and they need to be the ones to decide how to use that to further their degree. It’s a painful growing period.

But the payoff is huge. Being able to move and work independently is useful not just for research, but for pushing personal projects. When starting something new, there is a lot of power in having a framework for going from start to finish.

Feel free to riff of my answer in your own interviews. I think it plays well to admission committees.

That’s it for this one, see you in the next one.

📦 Other stuff of mine

• You can support me on Ko-fi! YouTube and Substack are by far the best (and easiest) ways to support me, but if you feel like going the extra mile, this would be the place. It is always appreciated!