Graphing Sampling Error for Ordinal Data

In previous posts I have been discussing how to treat rubric data as sample data.  The next post along these lines is intended to show how to apply various Bayesian methods to such data.  My plan was to contrast these methods using some type of graph.  Thus the following digression.

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

Standard Bar Plot for Rubrics

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

Bar Graph with No Gaps

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.

Stacked Horizontal Bar Plot

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.

Multiple Stacked Horizontal Bars for Comparison

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

Vertical Bars with Errors on Each Bar

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.

Density Plots for Each Category

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.

Rubric Prediction Interval Plots

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%).

Horizontal Bars with Grey Errors

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.

Stacked Bar Errors by Sampling: Original Data n=39

Stacked Bar Errors by Sampling: Original Data n=390

Stacked Bar Errors by Sampling: Original Data n=3900

 

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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Command Index to Statistical Rethinking

I am having a great time working through Richard McElreath’s text, Statistical Rethinking: A Bayesian Course with Examples in R and Stan.  The focus is on building, testing, interpreting, and improving  statistical models.  It is absolutely empowering and I have begun to build simple models myself.  However, I have been frustrated as I have done so, by the book’s poor index particularly the sparsity of references to R commands.  A case of “I know there is a way to do it, but now I have to page through the book to find it.”  So I took an hour or so and built an index to all (most) of the R commands Dr. McElreath used and supplied with the Rethinking library.  I just finished.  It is very rough and untried and probably has some omissions.  Anyway it is pasted below in case anyone else might find it useful.

Index to Rethinking – Formatted

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FOIL is Verboten

My students and I have a habit of referring to the act of multiplying two binomial forms as FOILing, recalling the acronym FOIL (First, Outer, Inner, Last) for remembering the process.  My excuse for using the term  is that they use it all the time and their excuse is that I use it all the time.  And I really don’t mind using it in class.  They get the point  immediately. However I have officially banned the term from their written work.  “FOIL” doesn’t belong in formal English.  It is imprecise and sounds young.  I think what pushed me over the edge was when I found the term, “reverse FOIL,”  on a student’s paper.

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Here’s What You Don’t Get to Do

I started last Monday’s Precalculus II class with this little speech(edited).

“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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Derooting with Radicals

My students were working in pairs on this problem,

Radical Expression To Be Simplified

Radical Expression To Be Simplified

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

Radical Expression Simplified So Far

Radical Expression Simplified So Far

One person asked the other, “Now what?”  The other replied, “Just deroot the to the eight and the y to the eight.  The other said, “I get it.” and they got,

Simplified Expression

Simplified Expression

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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PEMDAS Strikes Again

Our pre-calculus class was going over how to combine sums and differences of logarithm expressions into a single logarithm when Steven Orton (He gave  permission to use his name.) asked, “What about PEMDAS?”  PEMDAS is an acronym used for remembering the order of operations of evaluating algebraic expressions.  I muttered a few sentences, said I probably didn’t understand the question, and offered to discuss it after class.

I woke up a 1:30 am the next morning knowing what Steven meant. At the beginning of the next class I warned my students to work left to right when combining logarithms and meet with Steven after class with the following explanation.

First let’s revisit the original issues remembering this codicil to PEMDAS :

Addition/subtraction are on the same hierarchical level of the order of operations and must be worked left to right.  The same holds for multiplication/division.

Here is the issue for addition/subtraction.

Addition-Subtraction Order of Operations

Addition-Subtraction Order of Operations

The final numbers differ.  Our agreement is that the first process, moving left to right,  gives the “correct” answer.  Correct in this context means what we have agreed such expressions should be evaluated left to right since order makes a difference.

Such expressions can be made unambiguous by remembering the definition of subtraction is just adding the opposite.  Thus,

How to "Fix" Addition - Subtraction

How to “Fix” Addition – Subtraction

Now any order of evaluation gives the same result since addition is associative.

The same argument works for multiplication/division.

Multiplication-Division Order of Operations

Multiplication-Division Order of Operations

The first evaluation is “correct” and we can “fix” the issue by remembering that division is just multiplying by the reciprocal.  Thus,

How to "Fix" Multiplication-Division

How to “Fix” Multiplication-Division

Now any order of evaluation gives the same result since multiplication is associative.

Now let’s use the properties of logarithms to simplify the following.

Logarithm Simplification Order of Operations

Logarithm Simplification Order of Operations

Again left to right gives the desired answer.  This can be “fixed” as follows.

How to "Fix" Logarithm Simplification

How to “Fix” Logarithm Simplification

This simplification uses a property of logarithms that we didn’t make explicit in class, namely,

A Property of Logarithms

A Property of Logarithms

We have an interesting application of PEMDAS for simplifying logarithm expressions addressing about both subtraction and division.

All of the above to fix (explain) a defect in standard algebraic expression notation.  Thanks to Steven for the excellent question.

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My Heart Aches

Yesterday I made my heart ache. I took a small action – made a small gesture at least in this day and age.  I can’t talk about it.  My students have an absolute right to privacy. I could have cried.

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