In the past, I have questioned the extent to which we really suck at math in the U.S. While I’m still a bit skeptical that the average child in developing countries is doing dramatically better than children in the U.S., one thing is pretty clear from our results to date, and that is that the average child living in poverty** in the U.S. is doing pretty darn bad when it comes to math.

About a week ago, I discussed the results from a test on fractions given as part of our Fish Lake game evaluation. The pretest score was around 22% correct. Not terribly good.

There were also two questions where children had to explain their answers:

   Zoongey Gniw ate 1/3 of a fish. Willow ate 1/3 of a different fish. Zoongey Gniw said that he ate more fish. Willow says that he ate the same amount as she did, because they both ate 1/3 of a fish. Explain to Willow how Zoongey Gniw could be right.

3 different ways of showing 1/4

 

Explain why each of the above figures represents ONE-FOURTH.

Answers were scored 2 points if correct, 1 if partially correct and 0 if incorrect.

Out of 4 points possible, the mean for 260 students in grades 3 through 7 was .42. In other words, they received about 10% of the possible points.

These two questions test knowledge that is supposed to be taught in 3rd grade and 96% of the students we tested were in fourth grade or higher.
PUH-LEASE don’t say,

 

“Well, those are hard questions. I’m not sure I could explain that.”

 

If that is the case, feel sad! These are easy questions if you understand basic facts about fractions. “Understand” is the key word in that sentence.

SO many people, including me, when I was young, simply memorize facts and repeat them when prompted, like some kind of trained parrot, and with no more understanding.

When understanding of mathematics is required, they fail. Yes, some of the items tested under the new Common Core standards are harder. That doesn’t show a failure of the standards or tests, but rather of the students’ knowledge.

This is one of those cases where “teaching to the test” is not a bad idea.

** The reason I limited my statement to children living in poverty is that the schools in our study had from 72% -98% of their students receiving free lunch. Being a good little statistician I don’t want to extrapolate beyond the population from which our sample was drawn.

 

 

I tell clients on our statistical consulting side all of the time that if your conclusion is only valid if you look at this specific subset of your sample, with this particular statistical technique. You need to look for a convergence or results. Does the mean score increase? Does the proportion of people passing a test increase? Do the test scores still increase when you co-vary for the pretest score?

(This is for my friend, Dr. Longie, who tells me I always put too many numbers in things and should get to the point – no matter how we sliced it, the scores of students who played Fish Lake improved over 30% from their pretest. Analysis is continuing on Spirit Lake and other data from Fish Lake. There! Are you happy now?) 

We are just at the very beginning stages of analyzing data from the second phase of our research grant funded by the U.S. Department of Agriculture. Coincidentally, we are in Maryland at the National SBIR Conference this week and got the chance to meet in person all of the folks whose email we have been receiving for years.)

Me, our CMO and USDA staff

When we were in the middle of developing and testing Fish Lake, one of the interns in our office asked me,

“Are you sure this is going to work?”

I told her,

“No, I’m not sure. That’s why they call it research.”

School has now ended at all of our test sites and I have just completed cleaning the data for analysis from the first data set, which is the pre- and post-test data for Fish Lake, our game that teaches fractions as your avatar retraces the Ojibwe migration – canoeing, hunting and fishing your way across the continent.

So … what happened?

The first thing I did was compute the mean and standard deviation for the students who completed the pretest and the posttest. Then, I merged the datasets together and did a paired t-test for the 61 students who took the post-test and pre-test both. I didn’t show you any of those results because I assumed (correctly) that the merge would have to be reviewed because some people would have misspelled their username on the pretest or posttest. Surprisingly, I only found two of those, as well as one record that was just testing the software by one of our interns. The programs that I developed to clean the data (programs presented at a couple of regional SAS software conferences) worked pretty well.

Then, I re-ran the analysis.

Result 1

Pre-test mean = 22.4%, SD= 16.5% N=260

Post-test mean = 30.8% SD =17.4% N=63

So far, so good. We were not surprised by the low scores on the pretest. We knew that the majority of students in several of our test schools were achieving a year or two below grade level. The improvement from pre-test to post-test of 8.4% represented an improvement in test scores of 37.5%

BUT …. what if the students who did not take the post-test were the lower performing students? Shouldn’t we do a pretest and post-test comparison only including matched pairs?

This brings us to ….

Result 2 – With Matched Pairs

Pre-test mean = 23.6%, SD= 17.4% N=63

Post-test mean = 30.8% SD =17.4% N=63

As hypothesized, the students who completed the post-test scored higher on the pretest than the average, but not dramatically so. The difference was still statistically significant (p < .01)

What about outliers? That standard deviation seems awfully high to me and when I look at the raw data I find five players who have a 0 on the pretest or post-test and one who had one of the highest scores on the pretest whose test is blank after the first few questions.

Now, it is possible that those students just knew none of the questions – but it appears they just entered their username and (almost) nothing else. I deleted those 6 records and got this

Result 3 – Matched Pairs with Outliers Deleted

Pre-test mean =24.4 SD =16.2 N=57

Post-test mean = 32.7 SD = 16.7 N=57

With a difference of 8.3 percentage points, this presents an improvement of 34% (p<.001 )

Conclusion ? Well, we are not even close to a conclusion because we have a LOT of more data still to analyze, but what I can say is that the results are looking promising.

I’m preparing a data set for analysis and since the data are scored by SAS I am double-checking to make sure that I coded it correctly. One check is to select out an item and compare the percentage who answered correctly with the mean score for that item. These should be equal since items are scored 0=wrong, 1=correct.

When I look at the output for my PROC MEANS it says that 31% of the respondents answered this item correctly, that is, mean = .310.

However, the correct answer is D and when I look at the results from my PROC FREQ it shows that 35% of the respondents gave ‘D’ as the correct answer.

What is going on here? Is my program to score the tests off somewhere? Will I need to score all of these tests by hand?

Real hand soaps

I am sure those of you who are SAS gurus thought of the answer already (and if you didn’t, you’re going to be slapping your head when you read the simple solution).

By default, PROC FREQ gives you the percentage of non-missing records. Since many students who did not know the answer to the question left it blank, they were (rightfully) given a zero when the test was automatically scored. To get your FREQ and MEANS results to match, use the MISSING option, as so

PROC FREQ DATA =in.score ;
TABLES  item1 / MISSING ;

You will find that 31% of the total (including those who skipped the question) got the answer right.

Sometimes it’s the simplest things that give you pause.

Do you have a bunch of sites bookmarked with articles you are going to go back and read later? It’s not just me, is it?

One of my (many) favorite things at SAS Global Forum this year was the app. It included a function for emailing links to papers you found interesting. Perhaps the theory is that you would email these links to your friends to rub it in that their employer did not like you well enough. I emailed links to myself to read when I had time. Finally catching up on coding, email and meetings, today, I had a bit of time.

I was reading a paper by Lisa Henley

A Genetic Algorithm for Data Reduction.

It’s a really cool and relatively new concept – from the 1970s – compared to 1900 for the Pearson chi-square, for example.

In brief, here is the idea. You have a large number of independent variables.  How do you select the best subset? One way to do it is to let the variables fight it out in a form of natural selection.

Let’s say you have 40 variables. Each “chromosome” will have 40 “alleles” that will randomly be coded as 0 or 1, either included in the equation or not.

You compute the equation with these variables included or not and assess each equation based on a criterion, say, Akaike Information Criterion or the Root Mean Square Error.

You can select the “winning” chromosome/ equation either head to head, whichever has the higher AIC/ RMSE , although there are other methods of determination, like giving those with the higher criterion a higher probability of staying.

You do this repeatedly until you have your winning equation. Okay, this is a bit of a simplification but you should get the general idea. I included the link above so you could check out the paper for yourself.

Then, while I was standing there reading the paper, the ever-brilliant David Pasta walked by and mentioned the name of another paper on use of Genetic Algorithm for Model Selection that was presented  at the Western Users of SAS Software conference a couple of years back.

I don’t have any immediate use for GA in the projects I’m working on at this moment. However, I can’t even begin to count the number of techniques I’ve learned over the years that I had no immediate use for and then two weeks later turned out to be exactly what I needed.

Even though I knew the Genetic Algorithm existed,  I wasn’t as familiar with its use in model selection.

You’ll never use what you don’t know – which is a really strong argument for learning as much as you can in your field, whatever it might be.

We’ve looked at data on Body Mass Index (BMI) by race. Now let’s take a look at our sample another way. Instead of using BMI as a variable, let’s use obesity as a dichotomous variable, defined as a BMI greater than 30. It just so happened (really) that this variable was already in the data set so I didn’t even need to create it.

The code is super-simple and shown below. The reserved SAS keywords are capitalized just to make it easier to spot what must remain the same.  Let’s look at this line by line

LIBNAME  mydata “/courses/some123/c_1234/” ACCESS=READONLY;
PROC FREQ DATA = mydata.coh602 ;
TABLES race*obese / CHISQ ;
WHERE race NE “” ;
RUN ;

LIBNAME  mydata “/courses/some123/c_1234/” ACCESS=READONLY;

Identifies the directory where the data for your course are stored. As a student, you only have read access.
PROC FREQ DATA = mydata.coh602 ;

Begins the frequency procedure, using the data set in the directory linked with mydata in the previous statement.

TABLES race*obese / CHISQ ;

Creates a cross-tabulation of race by obesity and the CHISQ following the option statistic produces the second table you see below of chi-square and other statistics that test the hypothesis of a relationship between two categorical variables.
WHERE race NE “” ;

Only selects those observations where we have a value for race (where race is not equal to missing)
RUN ;

Pretty obvious? Runs the program.

Cross-tabulation of race by obesity

 

Similar to our ANOVA results previously, we see that the obesity rates for black and Hispanic samples are similar at 35% and 38% while the proportion of the white population that is obese is 25%. These numbers are the percentage for each row. As is standard practice, a 0 for obesity means no, the respondent is not obese and a 1 means yes, the person is obese.

The CHISQ option produces the table below. The first three statistics are all tests of statistical significance of the relationship between the two variables. Table with chi-square statistics

You can see from this that there is a statistically significant relationship between race and obesity. Another way to phrase this might be that the distribution of obesity is not the same across races.

The next three statistics give you the size of the relationship. A value of 1.0 denotes perfect agreement (be suspicious if you find that, it’s more often you coded something wrong than that everyone of one race is different from everyone of another race). A value of 0 indicates no relationship whatsoever between the two variables. Phi and Cramer’s V range from -1 to +1 , while the contingency coefficient ranges from 0 to 1. The latter seems more reasonable to me since what does a “negative” relationship between two categorical variables really mean? Nothing.

From this you can conclude that the relationship between obesity and race is not zero and that it is a fairly small relationship.

Next, I’d like to look at the odds ratios and also include some multivariate analyses. However, I’m still sick and some idiot hit my brand new car on the freeway yesterday and sped off, so I am both sick and annoyed.  So … I’m going back to bed and discussion of the next analyses will have to wait until tomorrow.

So far, we have looked at

  1. How to get the sample demographics and descriptive statistics for your dependent and independent variable.
  2. Computing descriptive statistics by category 

Now it’s time to dive into step 3, computing inferential statistics.

The code is quite simple. We need a LIBNAME statement. It will look something like this. The exact path to the data, which is between the quotation marks, will be different for every course. You get that path from your professor.

LIBNAME mydata “/courses/ab1234/c_0001/” access=readonly;

DATA example ;
SET mydata.coh602;
WHERE race ne “” ;
run ;

I’m creating a data set named example. The DATA statement does that.

It is being created as a subset from the coh602 dataset stored in the library referenced by mydata. The SET statement does that.

I’m only including those records where they have a non-missing value for race. The WHERE statement does that.

If you already did that earlier in your program, you don’t need to do it again. However, remember, example is a temporary data set (you can tell because it doesn’t have a two level name like mydata.example ) . It resides in working memory. Think of it as if you were working on a document and didn’t save it. If you closed that application, your document would be gone.  Okay, so much for the data set. Now we are on to ….. ta da da

Inferential Statistics Using SAS

Let’s start with Analysis of Variance.  We’re going to do PROC GLM. GLM stands for General Linear Model. There is a PROC ANOVA also and it works pretty much the same.

PROC GLM DATA = example ;

CLASS race ;

MODEL bmi_p = race ;

MEANS race / TUKEY ;

The CLASS statement is used to identify any categorical variables. Since with Analysis of Variance you are comparing the means of multiple groups, you need at least one CLASS statement with at least one variable that has multiple groups – in this case, race.

MODEL dependent = independent ;

Our model is of bmi_p  – that is body mass index, being dependent on race. Your dependent variable MUST be a numeric variable.

The model statement above will result in a test of significance of difference among means and produce an F-statistic.

What does an F-test test?

It tests the null hypothesis that there is NO difference among the means of the groups, in this case, among the three groups – White, Black and Hispanic . If the null hypothesis is accepted, then all the group means are the same and you can stop.

However, if the null hypothesis is rejected, you certainly also want to know which groups are different from which other groups. After that significant F-test, you need a post hoc test (Latin for “after that”. Never say all those years of Catholic school were wasted).

There are a lot to choose from but for this I used TUKEY. The last statement requests the post hoc test.

Let’s take a look at our results.

I have an F-value of 300.10 with a probability < .0001 .

Assuming my alpha level was .o5 (or .01, or .001, or .ooo1) , this is statistically significant and I would reject my null hypothesis. The differences between means are probably not zero, based on my F-test, but are they anything substantial?

If I look at the R-square, and I should, it tells me that this model explains 1.55% of the variance in BMI – which is not a lot. The mean BMI for the whole sample is 27.56.

You can see complete results here. Also, that link will probably work better with screen readers, if you’re visually impaired (Yeah, Tina, I put this here for you!).

ANOVA table

 

Next, I want to look at the results of the TUKEY test.

table of post hoc comparisons

 

We can see that there was about a 2-point difference between Blacks and Whites, with the mean for Blacks 2 points higher. There was also about a 2-point difference between Whites and Hispanics. The difference in mean BMI between White and Black samples and White and Hispanic samples was statistically significant. The difference between Hispanic and Black sample means was near zero with the mean BMI for Blacks 0.06 points higher than for Hispanics.

This difference is not significant.

So …. we have looked at the difference in Body Mass Index, but is that the best indicator of obesity? According to the World Health Organization, who you’d think would know, obesity is defined as a BMI of greater than 30.

The next step we might want to take is examine our null hypothesis using categorical variable, obese or not obese. That, is our next analysis and next post.

In the last post, I posed the following null hypothesis as an example:

There is no difference in obesity among Caucasians, African-Americans and Latinos.

You can see the results from the statistical analyses here.

Since my question only pertains to those three groups, let’s begin by creating a data set with just those subjects.

libname mydata “/courses/ab1234/c_0001/” access=readonly;

Data example ;
set mydata.coh602;
where race ne “” ;

Don’t forget to run the program!

Now, let’s do something new and use something relatively new, the tasks in SAS Studio. On the left screen, click on TASKS, then on STATISTICS, then click DATA EXPLORATION.

 

tasks in left pane

Once you click on DATA EXPLORATION, in the right window pane you’ll see several boxes, but the first thing you need to do is select the correct data set. To do that, click on the thing that looks like a sort of spreadsheet.

tasks2When you do that, you’ll see the list of libraries available to you. You need to scroll all the way down to the WORK library. This is where temporary data sets that you create are stored. Click on the WORK library to see the list of data sets in it.

Dataset example in work librarySelect the EXAMPLE data set and click on OK. Now that you have your data set, it is time to select your variables.

Window with task rolesClick on the +  next to the variables and you’ll get a list of variables from which you can select.  Scroll down and select the variable you want. First, as shown above, I selected RACE for the classification variable.

Selecting variables

This gives me a chart, and it appears that whites have a lower body mass index than black or Latino respondents in this survey.

bar chart of BMIMy next analysis is to do the summary statistics. I simply click on SUMMARY STATISTICS under the statistics tab (it’s right under data exploration) and select the same two variables. You can click here to see the results. Mean BMI for both the black and Hispanic samples was 29, while for whites it was 27. Standard deviations for the three groups ranged from 5.7 to 6.9 which was actually less than I expected.

So, there are differences in body mass index by race/ ethnicity, but that leaves a few questions left:

  1. Do those differences persist when you control for age and gender?
  2. While there are differences in body mass index, that doesn’t necessarily mean more people are obese. Maybe there are more underweight white people. Hey, it’s possible.

Well, now you have a chart and a table to add to the table you created in the first analyses. In the next post, we can move on to those other questions.

 

I get asked this question fairly often so I thought I would do a few posts on it. The most common problem is that a student who is new to statistics has no idea where to even start.

These examples use SAS but you could use any package you like.

My recommendation to students beginning to learn statistics is to start with some type of publicly available data set, getting some experience with real data.

1. IDENTIFY THE VARIABLES YOU HAVE AVAILABLE

The first thing to do is examine the contents of the dataset. Look at the variables you have available. With SAS, you would do this with PROC CONTENTS.

Your program at this point is super simple

LIBNAME mydata “path to where your data are” ;

PROC CONTENTS DATA = mydata.datasetname ;

Normally, you would come up with a hypothesis first and then collect the data. The advantage of working with public use data sets is you don’t have to go to the time and expense of interviewing 40,000 people. The disadvantage is that you are limited to the variables collected.

2. GENERATE A HYPOTHESIS

Looking at the California Health Interview Survey data, I came up with the following null hypothesis:

There is no difference in obesity among Caucasians, African-Americans and Latinos.

3. RUN DESCRIPTIVE STATISTICS

You need descriptive statistics for three reasons. First, if you don’t have enough variance on the variables of interest, you can’t test your null hypothesis. If everyone is white or no one is obese, you don’t have the right dataset for your study. Second, you are going to need to include a table of sample statistics in your paper. This should include standard demographic variables – age, sex, education, income and race are the main ones. Last, and not necessarily least, descriptive statistics will give you some insight into how your data are coded and distributed.

proc freq data = mydata.coh602 ;
tables race obese srsex aheduc ;
where race ne “” ;

proc means data= mydata.coh602 ;
var ak22_p srage_p ;

where race ne “” ;
run ;

You can see the results from the code above here.

Notice something about the code above – the WHERE statement. My hypothesis only mentioned three groups – Caucasians, African-Americans and Latinos. Those were the only three groups that had a value for the race variable. (This example uses a modified subset of the CHIS , if you are really into that sort of thing and want to know.) Since that is the population I will be analyzing, I do not want to include people who don’t fall into one of those three groups in my computation of the frequency distributions and means.

4. PUT TOGETHER YOUR FIRST TABLE

Using the results from your first analysis, you are all set to write up your sample section, like this

Subjects

The sample consisted of 38,081 adults who were part of the 2009 California Health Interview Survey. Sample demographics are shown in Table 1.

<Then you have a Table 1>

Variable …………N….     %

Race

  • Black 2,181 5.7
  • Hispanic ,4926 13.0
  • White 30,974 81.3

Gender

  • Male 15,751 41.4
  • Female 22,330 58.6

Variable ……N ….. Mean… SD

Age…………38,081 55.4 18.0

Income  37,686 $69,888  $63,586

 

I’ll try to write more soon, but for now The Invisible Developer is pointing out that it is past 1 a.m. and I should get off my computer.

 

UPDATE: Click here for step 2

For many students just learning statistics, the relationship of z-scores and probability is confusing.

Let’s try this concrete example. Here is a chart of the distribution of height in a sample of over 2,800 women.

distribution of height of women

 

Notice that the peak, the mode is around 62-63 inches.

You can see the frequency table here, as well as a larger picture of the histogram. You’ll notice the median is between 62 and 63 inches

The mean is 62.7 – between 62 and 63 inches.

Looks like a normal distribution in that mean = median = mode.

Let’s go back to that mean of 62.7 inches. The standard deviation in this population is 2.46.   What would 2 standard deviations above the mean be? Let’s round our 2 x 2.46 = 4.92 up to 5.

The mean + 2 standard deviations = 62.7 + 5 = 67.7

height distribution of women with 2 sd marked

If you look at the frequency distribution you’ll see that 97.3% of the people measured had a height of 67 inches or less.

So, this is a perfect demonstration of what we mean when we say that 97.5% of the people fall below 2 standard deviations above the mean.

First off, the good news. You can find all of the papers from SAS Global Forum 2015 online.  This is good news if you are anything like me (and you should be, because, let’s face it, I’m awesome) because even if you went to Dallas there were no doubt several papers you wanted to attend scheduled at the same time.

I liked everything I attended but there were two that stood out as really interesting. The first one was …

Taking the Path More Travelled – SAS Visual Analytics and Path Analysis
Falko Schulz
You can download it here
http://support.sas.com/resources/papers/proceedings15/SAS1444-2015.pdf

My idea of path analysis is a series of regression coefficients where you calculate direct and indirect effects. That is not the path analysis discussed in this paper.

He literally means what path did the customer (critter, whatever) take ?

For example, your path in using this website could be you went to the home page then blog page home then the previous entry on the blog.

While websites are an obvious use for this type of path analysis, there could be many others – customer experience in a call center, where people go in a huge department store, migration of humans or animals, path to achieving a job at a start-up.

Drop-off is often of interest in a path analysis – did they fall out of the path before the endpoint you wanted, e.g., sale, employment, customer support problem solved?

You can also look at weight in a path, not only whether they buy a widget but how much money did they spend?

Visual analytics allows for path segmentation. You can combine items or exclude items.
In the example, Schulz discussed using path analysis to see how effective your different online marketing methods are. Since many people will come from typing your name into a search engine, you may want to delete those paths and only include ads from Google adwords, blogher, your corporate website and other paid marketing efforts.

You can click on events and select Exclude to filter out all paths beginning with those events that are not of interest to you.

Sankey diagrams are available in visual analytics. Although these have their origin in uses like energy flow, they are now being applied all over the place.

Here is a sample from Schulz’s paper

sample sankey diagram

Sample Sankey Diagram

A Sankey diagram, FYI, shows the direction and quantity of the flow along a path. There is a blog devoted to Sankey diagrams here.
http://www.sankey-diagrams.com/sankey-definitions/

(This wasn’t mentioned in the paper, I just found that interesting. I’m sure there’s a blog out there devoted to Gantt charts that I could find if I looked, which I didn’t.)

Once the path analysis roles of interest are defined:

  • Event
  • Sequence
  • ID

… one of the first things to do would be drop the number of paths displayed. Just imagine the mess you would be looking at if you tried to visually display all of the paths someone took in navigating a website with even a few hundred pages.

You can edit the minimum path frequency, e.g., only show a path if at least 250 people took it.

This is just a brief, brief taste of what you can do with path analysis using SAS Visual Analytics and the coolness of SAS Global Forum. There was a lot, lot more and I’ll try to post about the second paper I really liked this week,

tractor in dirt field

but for now perhaps I should quit looking out the window and pay attention in this training session I’m sitting in at Fort Berthold (don’t tell Bruce I wrote my blog during it).

If you missed out on SAS Global Forum, you don’t need to wait until next year for your fix of networking, instruction (and possibly drinking). You can go to the Western Users of SAS Software conference in San Diego in September.

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