Removing the “I” and the “You” of it makes the inevitable (hopefully small) failures of both teacher and student less personal. It places the student and teacher together in a system, “The System”, if you like, attempting to do our jobs, fulfilling our roles, changing our brains – both of us.

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I found it on Andrew Gelman’s blog here. The red areas are based on an alpha (probability of rejecting a null hypothesis) of 0.05. A recent paper (preprint) endorsed/authored by a fat paragraph of statisticians recommends an alpha of .005 for “new discoveries.” I wanted to know how Dr. Gelman’s graph would change.

First here is my rendition of the original:

This is a normal curve with mean 2 and standard deviation (standard error) 8.1. It represents the true state of the world and is being compared to the null hypothesis with mean 0 and standard deviation 8.1. Assuming the null hypothesis with alpha of 0.05, the rejection (in the red) areas are set at plus or minus 1.96 times 8.1. Effect size here is the mean of the experiment divided by the standard error, in this case 2/8.1 around .25. The other information on the graph comes from a slight modification of the R code in this article. Power is the probability that an effect will be detected, that is, that we land in the red areas. This probability is .057 or 6 percent. Though the “real” mean is positive (2), 24% of the red area is to the left and negative. Thus the type S error. The exaggeration is got by simulating the process with mean 2 and standard deviation 8.1 ten thousand times and averaging the absolute values of those estimates that land in the red areas. Here the exaggeration ratio is nine.

Next I created this graph using the same methods with alpha = .005.

The rejection areas are barely visible. Finding a small effect using this method will occur much less often but if we do “find” it, there will still be a fair change of getting the sign wrong and effect will be much exaggerated.

All this is to say that it is hard to find a small effect in noisy data and if you do, the results will be deceptive. This is why scientific experimenters spend much of their time controlling for and eliminating extraneous variables. And yet sometimes discovering small effects can be important. Imagine discovering a drug that cures 0.1 percent of people with a common disease, say 0.1 percent of 10,000,000 – curing 10,000 people. The small effect exists but we can’t find it using reasonable sample sizes with statistical methods alone. This argues for putting more effort into understanding biological mechanisms and working to identify special subpopulations.

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A summer chore was to dust the book shelves in my home. As you can see, the shelves (There are two more bays of shelves to the right.) have an open design thus collecting their share of dust and incidentally also a thin film of cooking grease that wafts out of the kitchen. As I worked, I was struck by how much of what I touched is now obsolete.

The shelves hold other objects that just books. Some are artifacts from the past – pictures of my parents, a toy passenger rail car that my dad built, the odd gift or piece of art – not obsolete but providing mostly nostalgia value. There is a shelf of CD’s and DVD’s and video tapes all superseded by internet streaming services. There are board games. Do I really need three scrabble sets? The games are kept for possible interested visitors who never appear. The shelves also contain six or seven Go sets. I only need one. The others gather dust.

The books are arranged by type. There is a shelf of Go books, some in Japanese which I can’t read. I might dip into them in an idle moment but the totality is more than I could ever read and learn from for the rest of my life. There are reference books. We sent the Encyclopedia Britannica to the dump years ago but still there are dictionaries, atlases and the like – even the complete Shakespeare with print too small for old eyes. All these have been superseded by the internet. Speaking of atlases, part of a shelf has a pile of maps now made obsolete by their age and google maps. Nostalgic value only. There are how-to books – knitting, knotting, auto mechanics, hiking, carpentry, etc. Now if I want to know how-to, I search youtube.

We used to collect what I call idea books – The Black Swan or Consilience for example. These are good for lending without expectation of return but have not been and never will be reread. Valuable ideas sitting dormant. Other books – novels, popular science also lie fallow on the shelves.

Then there is the floor to ceiling collection of math books on the left. Many of them I have read but few have I mastered. My college notes are also there. These math books now exist as a reminder of knowledge I will never have – what I don’t have time to learn if I could. At this point if I need to know about a mathematical topic I go to the internet.

So what is the purpose of my book shelves. They provide wistful ambiance from my past life and a stab of regret for paths not taken. As my friend Barry the golfer remarked, his book shelves just provide a decorative background for his television set. Mine just adorn the north side of the living room.

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- Did you go to every single class?
- If not, what did you do during those hours?
- If not, how did learn the material you missed?
- Did you inform the professor prior to your absence?
- Did you make up, if possible, any points you missed by not being in class?
- When you are in class, where do you sit?
- How many questions do you ask each class session?
- Do you ever text during class?
- Do you ever cruise the internet during class?
- Do you ever scroll around your phone during class?
- Do you ever gossip with another student instead of engaging in the class?
- Can I see your notes? How many times have you read them over?
- Did you turn in every assignment on time? If not, why not?
- Can I see the best assignment you turned in?
- Can I see the worst assignment you turned in?
- Did you take advantage of every opportunity to get extra points?
- How many hours and how many days did you study for the midterm?
- Show me your graded midterm.
- Pick one question you missed and explain what happened.

As you can see, these questions are really about the student’s commitment and habits for success. To repeat,

**Five Habits of Highly Effective Students**

**They show up on time or early.****They show up ready to work and prepared for the day.****They have done their homework.****Their work shows they care.****They actively participate in the endeavors of the day.**

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Rubric data is commonly shown with a bar graph like this.

Note the gaps. I prefer a histogram-like bar graph like this.

I included the normal curve to make following point. If the scale on the horizontal axis has ordered meaning, then the bars on the histogram, if the same width, represent parts of a whole – think placing the bars end to end. This is what we are used to looking at with histograms – area = probability. The shape can be deceptive (see this post) but the idea gets across.

Yet overall I prefer a standard horizontal graph like so.

Here the redder the “badder” and the greener the “gooder” or for the color blind, “lefter” is “badder” and “righter” is “gooder.” One loses the ability to read frequencies directly but their relative size is easily seen. Comparisons work smoothly as can be seen in this graph from a forthcoming assessment committee report.

The problem with any of these choices is how to show prediction intervals (PI) This is a clunky graph from a previous post.

Now the rest of this saga.

The beauty of the Bayesian method as I gleaned from Statistical Rethinking by Richard McElreath was that one can get a distribution for each rubric category by repeated sampling using posterior probabilities. The ability to get means and PI’s and HPDI’s (prediction intervals with minimum width) and also densities comes naturally. I was proud of this graph for 15 minutes.

So proud I showed it to my wife. It is easy to read from afar – nice thick lines and color-coded. But it is deceptive (and mistitled). The overlap of the density plots have no meaning. The four graphs are just mushed together. The overlaps will be large or small depending on the sharpness of the density plots and also on their position. For instance two categories may have the same estimate for the mean.

I was also proud of this graph for a while.

This is similar in style to those in Statistical Rethinking except for larger plotted points and no shading extending between the bars for the PI’s. I particularly liked the contrast between posterior sampling mean and data using large open and filled disks. Yet the information seems to float in space and there is no sense that this is frequency data.

Maybe I could modify the horizontal stacked bar graph. I tried several ways of showing the “fuzz”, the uncertainty between the categories using transparent coloring and shades of gray. Not much success. Here is an example with the grey sections presenting uncertainty (95%).

The grey sections dominate the graph since the sample size was so small and are a bit deceptive since a smaller percentage for one category will necessarily cause a larger value in another category.

A cool feature of the Bayesian tools in Statistical Rethinking is the ability to sample from the posterior distribution. I decided to plot four thousand samples on one graph. I used forty narrow stacked graphs one above the other and iterated one hundred times with decreasing transparency. This gave a fair picture of the fuzz but no ability to visually quantify the sampling error. The last two graphs show how the uncertainty decreases with sample (original data) size.

Gradually changing the transparency one hundred times is overkill but I am out of ideas. So, at this point I will use the latter graphs to show fuzz and revert to a table to compare the various methods.

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Index to Rethinking – Formatted

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“Here’s what you don’t get to do. When you have children, you don’t get to say, “I was bad at math.” Because you are good at math. You could be better at math. You know how. Just work harder. Many people like to ‘brag’ that they are bad at math. Don’t do that to your kids. Don’t give them any excuses. You are good at math. I know. I grade your work and answer your good questions.”

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One group, looking for perfect square factors, had gotten this far,

One person asked the other, “Now what?” The other replied, “Just * deroot* the

What a great word! Dare I add “*deroot*” to the conversation next time we work such a problem? I think so..

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