Category Archives: Communication

Simulations toolbox: The beauty of permutation tests

A “Choose-your-own-adventure” hypothesis test

The last three posts focused on revealing our human mistakes in interpreting statistics and providing solutions to overcome those pitfalls. Now, we will focus on tools that you might consider using in every statistical project.

First up is the permutation test, an alternative to the t-test for comparing two populations. “An alternative,” you say, “why would we ever need that?” Unfortunately, real-world experiments do not always yield perfect data. Sometimes you only have 15 samples. Sometimes the populations were unevenly sampled. Often there is no guarantee that the underlying distributions are normal with equal variances in the two populations. While the t-test is robust, you may be uncomfortable with violating assumptions of the t-test. For example, you might have samples from two populations that look like the samples below:

SampleHIsts

Permutation tests shine here because they make fewer assumptions about your data. Rather than assume any underlying distribution, the first step in a permutation test is to construct a null distribution from the data by shuffling (or “permuting”) the data so that the population labels are scrambled. After all, if the two populations are the same, then the values should be exchangeable and shuffling the labels should be meaningless. Repeating this shuffle a 1000 times or so and calculating the difference in means each time provides a null distribution against which to assess the unusualness of the observed (unpermuted) difference. The proportion of values in the null distribution that are more extreme than the actual difference is the p-value of the permutation test. Below, the red lines indicate the observed difference in means for the comparison shown above. A comparison of the two observed means yields a p-value of 0.004.

permuteMean

Basic steps of a permutation test (a.k.a. Randomization Test):

  1. Calculate the observed test statistic, for example the difference between the means of Sample 1 versus Sample 2.
  2. Permute (i.e. shuffle) the sample labels of the observations to simulate a new Sample 1 and Sample 2 from the same data.
  3. Calculate the same test statistic for the permuted data. This gives one example of what the test statistic would look like if the null hypothesis of no difference between populations were true.
  4. Repeat steps 2 and 3 a thousand times. With each repetition, you get an additional example of what the test statistic looks like, by chance, when the null hypothesis is true.
  5. Compare the observed test statistic (step 1) to the distribution of values that describe how things would be if the null hypothesis were true. The proportion of the permuted statistics that is more extreme than the observed value is the p-value for the permutation test.

Comparing populations in other ways can also be useful. Because you define your test statistic as part of the permutation test, you can design practically any statistic you want to compare. Instead of looking at the difference in means, for example, you can calculate the difference in variances for each permutation. As seen below, there is a significant difference in the variances of the two samples we used earlier. You can even look at the difference in ex, where x is the measured data, if you wanted. The choices are endless and the same procedure applies. Simply substitute in your custom metric wherever you would calculate a test statistic.

PermuteVar

Of course, permutation tests are not without limitations. The nature of shuffling labels in the data set to form a null distribution assumes that there is no structure in your data – for example, no correlations or grouping among samples that would be lost when permuting. Additionally, permutation tests only provide p-values. As we have discussed in a previous installment, confidence intervals and effect sizes are also important metrics for judging the statistical and practical significance of observed patterns. These metrics can also be captured using simulation approaches.

Nevertheless, when conditions are right, permutation tests are a powerful tool for hypothesis testing that circumvents some of the assumptions of parametric tests and allows increased flexibility in making insightful comparisons between populations. Here is the R code that generated all the graphics and tests in this article. You can see how a permutation test compares to a t-test and try creating test statistics other than the mean. Feel free to experiment!

Thanks again for reading, and hope you will join us in a few weeks for the final post of this series! It will cover power analyses, a simulation-based tool that can aid in study design and provide insight into the strength of your statistical tests. Stay tuned!

 

Sources

Ong, D. C. (2014) “A primer to bootstrapping; and an overview of doBootstrap.” URL: https://web.stanford.edu/class/psych252/tutorials/doBootstrapPrimer.pdf

Past Articles in the Series

  1. Your Brain on Statistics
  2. Patterns from Noise
  3. P-hacking and the garden of forking paths

 

I am working with E. Ashley Steel and Rhonda Mazza at the PNW Research Station to write short articles on how we can improve the way we think about statistics. Consequently, I am posting a series of five blogs that explores statistical thinking, provides methods to train intuition, and instills a healthy dose of skepticism. Subscribe to this blog or follow me @ChenWillMath to know when the next one comes out!

Ideas in this series are based on material from the course, “So You Think You Can Do Statistics?” taught by Dr. Peter Guttorp, Statistics, University of Washington with support from Dr. Ashley Steel, PNW Station Statistician and Quantitative Ecologist, and Dr. Martin Liermann, statistician and quantitative ecologist at NOAA’s Northwest Science Center.

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P-hacking and the garden of forking paths

Multiple comparisons can lead to spurious conclusions

This time, we continue our discussion around p-values by discussing a common trap that people fall into when analyzing data.

Imagine the following: you are interested in the influence of hydrology on the abundance of a particular fish species. Because there are many aspects of the hydrological cycle that might play a role in regulating this species, you perform multiple tests to see if there is a statistically significant relationship between fish abundance and hydrological flow metrics such as summer flow magnitude, timing of flooding, duration of low flows, and more. And lo and behold, you find that summer flow magnitude is positively correlated with fish abundance, with a p­-value less than 0.05! Great, time to write up the paper describing how summer flows are important in regulating the abundance of this species, right?

Well… hold on a minute. Unfortunately, this would be a textbook example of p-hacking – sorting through numerous statistical tests to determine which aspects of a particular hypothesis (e.g. that hydrology influences fish abundance) are significant, and reporting on only the significant findings. The issue when looking at multiple comparisons like this is that the more tests you perform, the more likely you are to see a “weird” response.

Let us consider a hypothetical scenario where we know the null hypothesis, e.g. there is no effect of a particular flow metric on fish abundance, to be true. Setting our significance level to α = 0.05, how often do we falsely reject the null hypothesis and incorrectly conclude that there really is an effect when making different numbers of comparisons?

Probability of falsely rejecting the null hypothesis (red) when it is true

MultipleComparisonsRisk

Unfortunately, as we do more and more comparisons, we are more and more likely to get a significant p-value simply by chance alone. For a single statistical test using a significance threshold of 0.05, there is a 5% chance to reject the null hypothesis even when it is true (known as a Type I error). With five comparisons using the same significance level, there is only a 0.955 = 77.4% chance of correctly concluding that there is no relationship or almost a 25% chance of finding a significant relationship even though none exists. The chance of finding at least one statistically significant result simply by chance climbs to 40.1% with ten tests. This is a hazard because, as you might recall from the last installment, it is impossible to differentiate between a truly significant relationship versus simply a weird occurrence.  By the time you’ve conducted over 20 tests, including exploratory graphs used as unofficial tests, on data where there is no true relationship, you’ve got over a 60% chance of at least one “weird” result that leads you to wrongly conclude that there is a relationship.

For a more hands-on experience, try p-hacking in this interactive from FiveThirtyEight, which puts you in the statistician’s seat of a study on the health of the United States economy under the two major political parties. After playing around with their interactive visualization, you will find that you can “obtain evidence” for literally any hypothesis you want about how the United States economy is affected by whether Democrats or Republicans are in office. It just takes some tweaking of predictors and responses.

The garden of forking paths

Now, you might be thinking, “Jeez, with all these challenges around multiple tests, why not just focus on the hypotheses and statistical tests that are most likely to be interesting?” The problem comes when you realize that there are a multitude of ways to approach a hypothesis, and there are numerous informal decisions we make when looking at our data about which tests or data we use. For example, which covariates to focus on, how data is transformed, how many categorical bins to divide data into, and so on. These endless choices lead to what statistician Andrew Gelman calls the “garden of forking paths”, where researchers may explore their data extensively, but only report a subset of the statistical methods they ended up utilizing (e.g. the models that fit the data better).

Avoiding this pitfall

There are several possibilities for mitigating the hazard of falsely rejecting null hypotheses when making multiple comparisons. First, each time you conduct multiple statistical tests related to a single overarching hypothesis, you can adjust your significance level from α = 0.05 to α* = 0.05/n, where n is the number of tests you are performing. This is known as a Bonferroni correction, and it brings the probability of a false rejection over the whole suite of tests in line with the rate you would see if you were only conducting one test.

Distinguish between hypothesis testing and exploratory data analysis

We often learn that exploratory data analysis should be precursor to statistical analysis. Indeed, checking for erroneous observations and distributional assumptions needs to come before statistical testing. Other forms of exploratory analysis, however, can be conducted after formal hypothesis testing.  In the above example, what if the researcher had identified, from previous research, the two strongest hypotheses about how hydrology might impact fish abundance and only formally conducted those two tests. The scientist would then be free to continue graphing and exploring the data to identify other facets of the hydrological cycle that appear to have a relationship with fish abundance as ideas for future research or to spark new mental models of how the ecological system might be structured. These ideas and models could then be tested with new, independent data.

Establish hypotheses and statistical methodology early

This leads to the crucial importance of establishing hypotheses and methods, including statistical models from the outset, ideally before data collection even begins. You can formally “preregister” your hypotheses or you can at least be mindful of not altering your methodology based on data exploration. This practice will dissuade you from unintentional p-hacking and from digging too hard for expected results. Consequently, you will fortify your statistical assertions and lend more credence to your science when you follow up with the full results of your analyses.

Hopefully, this has been helpful for being cognizant of p-hacking and will prevent you from being an unintentional p-hacker. The article that accompanies the FiveThirtyEight interactive above is also a great read for how p-hacking is not the end of credible science.

Thanks again for reading, and hope you’ll be back for the next post in this series! We will be diving into simulations to aid in your statistical analyses.

 

Sources

Aschwanden, C. (2015) “Science Isn’t Broken.” URL: https://fivethirtyeight.com/features/science-isnt-broken/

Center for Open Science (2017) “Preregistration Challenge” URL: https://cos.io/prereg/

Gelman, A. and Loken, E. (2013) “The garden of forking paths: Why multiple comparisons can be a problem, even when there is no ‘fishing expedition’ or ‘p-hacking’ and the research hypothesis was posited ahead of time.” URL: http://www.stat.columbia.edu/~gelman/research/unpublished/p_hacking.pdf

Past Articles in the Series

  1. Your Brain on Statistics
  2. Patterns from Noise

 

 

I am working with E. Ashley Steel and Rhonda Mazza at the PNW Research Station to write short articles on how we can improve the way we think about statistics. Consequently, I am posting a series of five blogs that explores statistical thinking, provides methods to train intuition, and instills a healthy dose of skepticism. Subscribe to this blog or follow me @ChenWillMath to know when the next one comes out!

Ideas in this series are based on material from the course, “So You Think You Can Do Statistics?” taught by Dr. Peter Guttorp, Statistics, University of Washington with support from Dr. Ashley Steel, PNW Station Statistician and Quantitative Ecologist, and Dr. Martin Liermann, statistician and quantitative ecologist at NOAA’s Northwest Science Center.

Patterns from Noise

What does the p-value really tell us?

Welcome back! If you missed the previous installment, you can find it here.

Continuing the series, we’ll be talking about the p-word. That’s right, “p-values”. A concept so central to statistics, yet one of the most often misunderstood.

Not too long ago, the Journal of Basic and Applied Psychology straight up banned p-values from appearing in their articles. This and other controversies about the use and interpretation of p-values led the American Statistical Association (ASA) to voice their thoughts on p-values; writing such recommendations for the fundamental use of statistics was unprecedented for the organization.

Part of the confusion stems from the complacency with which we teach p-values, leading to blind applications of p-values as the litmus test for significant findings.

Q: Why do so many colleges and grad schools teach p = 0.05?
A: Because that’s still what the scientific community and journal editors use.

Q: Why do so many people still use p = 0.05?
A: Because that’s what they were taught in college or grad school.

– George Cobb

Snide comments aside, let us unpack what a p-value does and does not tell us. First, take a look at the following twenty sets of randomly generated data:PatternFromNoise.png

Each one of the boxes contains 50 points whose x-y coordinates were randomly generated from a normal distribution with mean 0 and variance 1. Yet, we see that there is occasionally a set of points that appears to have a trend, such as the one highlighted in red, which turns out to exhibit a correlation of 0.45. If even random noise can display patterns, how do we discern when we have a real mechanism influencing some response versus simply random data? P-values provide this support by giving us a measure of how “weird” an observed pattern is, given a proposal of how the world works.

More formally, the definition of a p-value is “the probability under a specified statistical model that a statistical summary of the data would be equal to or more extreme than its observed value” (taken from the ASA). Note that this says nothing about the real world. Rather, it measures how much doubt we have about one particular statistical view of the world. If our null hypothesis were true and our model of the world pretty accurate, a “statistically significant p-value”, means that something unlikely has happened (where unlikely could be defined as a 1 in 20 chance). So unlikely that it throws significant doubt into whether that null hypothesis is a very good model of the world after all. It is important to note, however, that this does not mean that your alternative hypothesis is true.

Conversely, an insignificant p-value is not an indication that your null hypothesis is true. Rather, it suggests a lack of evidence as to whether your null hypothesis is an inaccurate model of the world. The null hypothesis may well be accurate or you may simply not have collected enough evidence to throw significant doubt on an inaccurate null hypothesis. A common trap is to argue for a practical effect because of some perceived pattern even though the p-value is insignificant. Resist this temptation, as the insignificant p-value indicates that the pattern is not particularly unusual even under the null hypothesis.  Also resist the temptation to state or even imply that the insignificant p-value indicates (a) there is no effect; (b) there is no difference; or (c) the two populations are the same. Absence of evidence is not evidence of absence.

Ultimately, the p-value is only one aspect of statistical analyses, which is, in turn, only one step in the life-cycle of science. P-values only describe how likely it might be to get data like yours if the null hypothesis were really how the world worked.

There are, however, some practices that can supplement p-values:

  1. Graph the data. For example, how different do two groups look when you make box plots of their responses? How much data do you really have? Large sample sizes can help elucidate significant differences (a topic we will dive into more in a later installment about statistical power). Are there unusual observations?
  2. More formally, estimate the size of the effect that you are seeing (e.g. via a confidence interval). Is it a potentially large effect that is not significant or a very small effect that is statistically significant? Is the effect size you see relevant to potential real-world decisions? A 95% confidence interval of [0.01, 0.05] may be significantly different from zero, but if that interval represents say the increase in °C of river temperature after a wildfire, is it a relevant difference to whatever decision is at hand?
  3. Conduct multiple studies testing the same hypothesis. Real world data is noisy. Each additional study allows you to update prior information and possibly provide more conclusive support for or against a hypothesis. This is, in fact, the basic idea behind Bayesian statistics, which we do not have the space to cover here, but go here for an introduction on the topic.
  4. Use alternative metrics to corroborate your p-values, such as likelihood ratios or Bayes factors

Hopefully, we have provided significant enlightenment on p-values. Next time, we will continue thinking about p-values, specifically the risks involved with testing multiple hypotheses in the same analysis.

Thanks for reading and hope you will join us for the next installment in a few weeks!

Sources

Etz, A. (2015) “Understanding Bayes: A Look at the Likelihood.” URL: https://alexanderetz.com/2015/04/15/understanding-bayes-a-look-at-the-likelihood/

Kurt, W. (2016) “A Guide to Bayesian Statistics.” URL: https://www.countbayesie.com/blog/2016/5/1/a-guide-to-bayesian-statistics

Trafimow, D. and Marks, M. (2015) “Editorial.” URL: http://www.tandfonline.com/doi/abs/10.1080/01973533.2015.1012991

Wasserstein, R.L., and Lazar, N.A. (2016) “The ASA’s statement on p-values: context, process, and purpose.” URL: http://www.tandfonline.com/doi/full/10.1080/00031305.2016.1154108

 

Past Articles in the Series

  1. Your Brain on Statistics

 

Bonus Article: A different type of p-value…

 

I am working with E. Ashley Steel and Rhonda Mazza at the PNW Research Station to write short articles on how we can improve the way we think about statistics. Consequently, I am posting a series of five blogs that explores statistical thinking, provides methods to train intuition, and instills a healthy dose of skepticism. Subscribe to this blog or follow me @ChenWillMath to know when the next one comes out!

Ideas in this series are based on material from the course, “So You Think You Can Do Statistics?” taught by Dr. Peter Guttorp, Statistics, University of Washington with support from Dr. Ashley Steel, PNW Station Statistician and Quantitative Ecologist, and Dr. Martin Liermann, statistician and quantitative ecologist at NOAA’s Northwest Science Center.

 

 

Your Brain on Statistics

Are apparent patterns indicative of population differences or simply caused by different sample sizes?

I am working with E. Ashley Steel and Rhonda Mazza at the PNW Research Station to write short articles on how we can improve the way we think about statistics. Consequently, I am posting a series of five blogs that explores statistical thinking, provides methods to train intuition, and instills a healthy dose of skepticism. Subscribe to this blog or follow me @ChenWillMath to know when the next one comes out!

We begin by looking at how the wiring of the brain interferes with our ability to process statistics. The way we internalize information and make decisions can be broken down into two categories:

  • System 1 thinking that is automatic and intuition-based
  • System 2 thinking that is more deliberate and analytic

Unfortunately, the impulsive nature of System 1 thinking tends to get us into trouble when we interpret statistics. For example, look at the following map of the lower 48 United States.

HighCancerRate.png

It illustrates the counties that exhibit the highest 10% of kidney cancer rates (i.e. number of per capita kidney cancer cases), colored by whether they are predominantly rural or urban. Note that there are more rural counties represented on the map than urban counties and that many of the cancer-prevalent counties are in the South or Midwest.

Why might that be? Perhaps rural areas tend to have less access to clean water, which could adversely affect kidney function? Perhaps there are more factories in these areas leading to more health issues?

Before you get too far, let me show you another map, this time of the counties in the bottom 10% of kidney rate incidence.

LowCancerRate

Interestingly, rural areas appear over-represented among the counties with the lowest kidney cancer rates as well! What is going on?

This was the conundrum that Howard Wainer delved into in an article titled “The most dangerous equation”, published in the American Scientist in 2007. Wainer explained how trends can appear even when the underlying probability of an event occurring is constant. Using data from the United States Census Bureau, we have simulated that scenario in the maps above.

The effect you are seeing has nothing to do with rural versus urban, though it would make a believable headline. The real culprit is population size. It turns out that smaller samples, such as less populous counties, are more prone to exhibiting extreme results. Let us explore this further.

Imagine you flipped 3 (fair) coins. The chance of getting either all heads or all tails is 25%. Now what is the chance of getting all heads or all tails when flipping 30 coins? Less than 1 in 10,000. Despite the identical chance for any one coin to turn up heads (or tails), larger collections of coin flips are less likely to all be heads.

The take home point: our brains are predisposed to look for and interpret patterns. However, strong patterns, regardless of tempting explanations, can be caused by random chance. Here, sample-size differences across counties are responsible for observed kidney cancer rate differences, despite the constant individual risk of kidney cancer (which is likely not the case, but that is a different discussion).

So, what should scientists and science readers do? The first step is to remain vigilant. When confronted with apparent patterns, consider whether they might be due to chance alone.  For data like these, ask if the more extreme responses are exhibited by the samples that contain fewer individuals or cover smaller areas. You might also consider using simulations to assess how much random chance contributes to apparent patterns. Simulations will be discussed in future installments of this summer statistical thinking series.

If you would like to know more about how the brain tricks you into false statistical conclusions, Amos Tversky and Daniel Kahneman discusses this and many other pitfalls.

Thanks for reading and stay tuned for the next installment! We’ll be talking about the p-word!

 

Sources

Bhalla, J. “Kahneman’s Mind-Clarifying Strangers: System 1 & System 2”. URL: http://bigthink.com/errors-we-live-by/kahnemans-mind-clarifying-biases. Accessed 27 May 2017.

Tversky, A. & Kahneman, D. (1974) Judgment under Uncertainty: Heuristics and Biases. Science 185 (4157). URL: http://science.sciencemag.org/content/185/4157/1124. Accessed 27 May 2017.

United States Census Bureau. “Geography: Urban and Rural”. URL: https://www.census.gov/geo/reference/urban-rural.html. Accessed 27 May 2017.

Wainer, H. (2007). The Most Dangerous Equation. American Scientist. 95 (3). URL: http://www.americanscientist.org/issues/pub/the-most-dangerous-equation. Accessed 27 May 2017.

 

Ideas in this series are based on material from the course, “So You Think You Can Do Statistics?” taught by Dr. Peter Guttorp, Statistics, University of Washington with support from Dr. Ashley Steel, PNW Station Statistician and Quantitative Ecologist, and Dr. Martin Liermann, statistician and quantitative ecologist at NOAA’s Northwest Science Center.

Geek Heresy and EarthGames

I’ve recently started reading a fantastic book on a friend’s recommendation called Geek Heresy: Rescuing Social Change from the Cult of Technology. The book takes a look at the culture of technology in human society, with the premise of delving into how technology came to be so highly-regarded as a tool for social change and why this view can be problematic. I’m only one chapter in, but Geek Heresy has already got me thinking about what is likely a central theme: technology does little for social change without the right people to support the change.

Over the weekend, I helped represent EarthGames UW at the second annual Seattle Youth Climate Action Network (Seattle Youth CAN) Summit. During the lunch hour, we let the eager high-schoolers explore some of the games that EarthGames designed over the past year. We followed this up with an activity-packed hour where we guided a dozen students in developing a concept for their very own environmental game!

The event ended up being the highlight of my weekend. I met a young woman who had already designed her own game about pollution using HTML/Javascript, and within the hour-long game jam, we already had a game concept down (tower defense style game about overfishing)! I got to meet a bunch of really smart kids that were excited to bring about environmental change.

Now, you might be wondering why these two pieces are in the same blog post. Throughout the event, I kept thinking back to Geek Heresy and how these games are like the teaching tools presented at the beginning of the book. While EarthGames UW was founded on the motivation to teach people about climate change and the environment, the games that we make are just as likely to see the same downfalls as the laptops-in-the-wall presented in Geek Heresy’s first chapter — a lack of mentoring or guidance means less effective or a complete lack of social change.

I’m glad that EarthGames is taking on more opportunities to engage with youth with games and game design. There’s a lot of potential in using games to engage with the public, and even more in using game design to let the public engage with us and each other. I hope EarthGames will continue to foster collaborations with engagement groups to enable change in our society. If I get the chance (and time!), I hope to be able to foster these collaborations myself.

What do you think is essential for social change? How do you go about engaging your community? Let me know in the comments section!

 

 

ENGAGEing graduate student research talks coming to Town Hall Seattle!

I’ve had the excellent opportunity to be participating in University of Washington’s ENGAGE seminar this year. I encourage you to look around their website, but in short, it is a science communication seminar aimed at giving science graduate students the skills to translate their research into a form that is digestible by a general audience.

To show that we can “walk the talk”, so to speak, we will be giving twenty-minute presentations at Town Hall Seattle. Topics this year range from the ethics of social media data to bio-engineered crops to alien life within Arctic glaciers!

I’ll be presenting my research on dam management, and how math and statistics help both human society and rivers have the water they need even as fresh water becomes more scarce. Be sure to be at Town Hall Seattle on May 12 if you want to hear my talk, but talks will be happening throughout March, April, and May. More details to come soon!

ENGAGE Seminar Blog

I’m currently enrolled in a seminar on science communication called ENGAGE, and it’s been incredibly informative! The goal of the seminar is to teach graduate students how to effectively communicate their scientific research to the general public. It culminates in a presentation at Town Hall Seattle! Dates are to be determined, but be on the lookout for exciting and accessible talks in a few months!

In the meantime, I did a guest blog for the seminar, which you can find here. While I focused on relating the class assignments to my recent board game exhibition, the same lessons applies to scientific presentations as well. There is only so much time in a presentation, so a real challenge is how to pack everything you want to say into an engaging package without skimping on details? Often in scientific presentations, the temptation is to cram every detail in; after all, we don’t want to misconstrue any aspect of our work, right? Unfortunately, when we do that, our audience only gets the sense that there’s a lot of details, and really loses sight of the story.

I recently gave an hour-long presentation to an audience of quantitatively-focused professors and students. Generally, this crowd appreciates seeing the details behind mathematical models, so at first I thought, “Hey, there’s this really cool method that I’m incorporating, and I should talk in-depth about it so others can appreciate how cool it is too!” Upon further reflection though, I realized that my story wasn’t really about this cool method. It was how I used this cool method to show an even cooler framework for solving a central conflict in dam management – namely, how do we allocate fresh water so that human society can benefit from rivers, without drastically harming the river itself? In the end, I cut most my discussion of the “cool method” and focused on the “cool story” where the “cool method” was a supporting actor.

The result? I got to talk about the “cool method” without it interfering with the overall “cool story”. For the people that were interested in the math, I offered up a novel tool. For the people that were interested in dams and applications, I offered up a success story for an incredibly challenging problem. By dialing back on the accuracy just a bit, I was able to engage my audience a little more, and everyone ended up winning.