Forget P-values? or just let it be what it is

P-value has always been controversial. It is required for certain publications, banned from some journals, hated by many, yet quoted widely. Not all p-values are loved equally. Because of a "rule" popularized some 90 years ago, small values below 0.05 have been the crowd's favorite.

When teaching hypothesis testing, we explain that the entire spectrum of p-value serves a single purpose: quantifying the "agreement" between an observed set of data and a statement (or claim) known as the null hypothesis.

Why are we "obsessed" with the small values then? Why can't we talk about any specific p-value the same way we talk about today's temperature? i.e., as a measure of something.

First of all, the scale of the p-value is hard to talk about. This is different from temperature. The difference between 0.21 and 0.20 is not the same as 0.02 and 0.01.

It almost feels like we should use the reciprocal of a p-value to discuss the likeliness of the corresponding observed "data" (represented by a summary/test statistic), assuming the null hypothesis is true.

If the null hypothesis is true, it takes, on average, 100 independent tests to observe a p-value below 0.01. The occurrence of a p-value under 0.02 is twice as likely, taking only about 50 tests to observe. Therefore 0.01 is twice as unlikely as 0.02. Using a similar calculation, 0.21 and 0.20 are almost identical in terms of likeliness under the null.

In Introductory Statistics, we teach that a test of significance has four steps:

1. stating the hypotheses and a desired level of significance;
2. computing the test statistics;
3. finding the p-value;
4. concluding given the p-value. 

It is step 4 that requires us to draw a line somewhere on the spectrum of p-value between 0 and 1. That line is known as the level of significance.

I never enjoyed explaining how one would choose a level of significance. Many of my students felt confused. Technically speaking, if a student derived a p-value of 0.28, she can claim it is significant at a significance level of 0.30. The reason why this is silly is that a chosen significance level should convey a certain sense of rare occurrence: so rare that it is deemed contradictory with the null hypothesis. No one of common sense would argue a chance that is close to 1 out of 3 represents rarity.

What common sense fails to deliver is how rare is contradictory enough. A recent HBR article showed that people have a wide variation in how they perceive a concept of likelihood such as "rare".

The solution?

"Use probabilities instead of words to avoid misinterpretation." 

P-value and significance level serve precisely this purpose.

Why 1/20 needs to be a universal choice? It doesn't. Statisticians are not quite bothered by "insignificant results" as we think 0.051 is just as interesting as 0.049. We, whenever possible, always just report the actual p-value instead of stating that we reject/accept the null hypothesis at a certain level. We use p-value to quantify the strength of evidence between variables and studies.

However, sometimes we don't have a choice so we got creative.

For any particular test between a null hypothesis and an alternative, a representative (i.e., not with selection bias) sample of p-values will offer a much better picture than the current published record of a handful of p-values under 0.05 out of who-knows-how-many trials. There have been suggestions on publishing insignificant results to avoid the so-called "cherry-picking" based on p-values. Despite the apparent appeal of such a reform, I cannot imagine it is practically possible. First of all, if we can assume that most people have been following the 0.05 "rule", publishing all the insignificant results will result in a 20-fold increase in the number of published studies. Yet it probably will create a very interesting data set for data mining. What would be useful is to have a public database of p-values on repeated studies of the precisely same test (not just the null hypothesis, as the test depends on what is the alternative as well). In such a database, p-value can finally be just what it is, a measure of agreement between a data set and a claim.

DSI Scholars Initiation Boot Camp - Project-based learning of the Data Science life cycle

"How do you give a Data Science bootcamp to a mixed cohort of undergraduate and graduate students from a wide range of disciplines?" This is a question I was facing earlier this year when I needed to design the initiation boot camp for our inaugural class of DSI Scholars.  Some of them are rising sophomores, and some of them are PhDs in English. Some of these students had years of coding experience, some of these students had taken a full curriculum in Statistics and Machine Learning, and some of them thought "we probably won't fit in a data science boot camp at all."

The DSI Scholars program is to promote data science research and create a learning community for student researchers, especially during the slow months of summer. The learning goals of the boot camp were designed to be less about specific ML tools or programming languages. Rather it was set out to be about the data science lifecycle, statistical thinking, data quality, reproducibility, interpretability of ML methods, data ethics ...

Therefore ...

We created a project-based bootcamp based on @ProPublica's recidivism story https://t.co/Pl88fscGb3. In 5 days, student teams carried out the tasks of map research question, apply DS tools, interpret results, discuss results in the light of the original research questions. 4/9

— Tian Zheng (@tz33cu) June 20, 2018

In 5 days, student teams carried out the tasks of 

• mapping research question - understand the domain question, identify data for answering the question and interpret the domain question to a data question;
• applying DS tools - for the data question, carry out exploratory data analysis, create data visualization and choose appropriate models and machine learning algorithms;
• interpreting and discussing results - what do the data results suggest in the light of the original research questions. 

Here is the approximate schedule of our bootcamp.

Day 1: studied the news coverage on @ProPublica story, discussed "what is fairness?" (taking notes from @mrtz's fairmlclass.github.io), played the Ultimatum Game, watched how monkey perceives fairness and failed to balance 2 definitions of fairness. 

Day 2a: teams studied @ProPublica's detailed account of their analysis (propublica.org/article/how-we…). Each team presented a part of the report. One team studied in detail how "recidivism" is defined in the study and commented "whether we should even be predicting recidivism." 

Day2b-Day3a: teams were then tasked to reproduce the analysis using data and codes shared (github.com/propublica/com…) and asked to explore the data more. 

Day 3b: we studied research on interpretable ML for recidivism prediction (arxiv.org/abs/1503.07810). We focused on how new ML tools were developed, evaluated and compared with conventional approaches; and how the evidence supporting new research presented and communicated.

Day 4-5: the teams worked on one idea they'd like to try with the data and presented their final projects. We didn't find a solution but we asked many good questions. Read more about our bootcamp at datascience.columbia.edu/scholars/bootc…