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[The Monthly Mean] May/June 2009, Risk adjustment using Analysis of Covariance

You are viewing an early draft of the Monthly Mean newsletter for May/June 2009. I hope to send this newsletter out sometime between the first and the fifth of the month.

The monthly mean for May/June is 15.25.

Welcome to the Monthly Mean newsletter for May/June 2009. If you are having trouble reading this newsletter in your email system, please go to www.pmean.com/news/2009-05.html. If you are not yet subscribed to this newsletter, you can sign on at www.pmean.com/news. If you no longer wish to receive this newsletter, there is a link to unsubscribe at the bottom of this email. Here's a list of topics.

  1. Risk adjustment using Analysis of Covariance
  2. The first deadly sin of researchers: pride
  3. Is this a case control design?
  4. Monthly Mean Article: Design, analysis, and presentation of crossover trials
  5. Monthly Mean Blog: FiveThirtyEight: Politics Done Right
  6. Monthly Mean Book: Statistical Issues in Drug Development
  7. Monthly Mean Definition: What is a mosaic plot?
  8. Monthly Mean Quote: Two quotes this month
  9. Monthly Mean Website: Neural correlates of interspecies perspective taking in the post-mortem Atlantic Salmon: An argument for multiple comparisons correction
  10. Nick News: Newton says goodbye after 20 years
  11. Very bad joke: Two statistics are in a bar
  12. Tell me what you think.

1. Risk adjustment using Analysis of Covariance

In the January newsletter, I explained the difference between crude and adjusted estimates, and illustrated how to produce an adjusted estimate using Analysis of Covariance on a data set of housing prices in Albuquerque, New Mexico. I want to revisit this concept using a larger data set. In future newsletters, I want to discuss alternatives to Analysis of Covariance, such as reweighting, case mix index, and propensity scores. This work was part of a project I am helping with: the adjustment of outcome measures in the National Database of Nursing Quality Indicators. The work described in this article was supported in part through a grant of the American Nursing Association.

There's an interesting data set on the web, that shows average faculty salaries for all of the colleges and universities in each of the 50 states plus the District of Columbia. There is a wide disparity in salaries, with KS and WV having averages salaries of 35 thousand dollars and CA having an average salary of 54 thousand dollars. The colleges and universities are categorized as Division I, IIA, or IIB. Across the entire U.S., the percentage of I, IIA, and IIB colleges and universities are 16%, 32%, and 53%, respectively. Note that rounding causes the percentages to add up to more than 100%. Rounding will affect some of the calculations below slightly as well, but do not materially change any of the conclusions.

Note that each state has a different distribution of I, IIA, and IIB compared to the entire U.S., and this may account for some of the disparities seen in average salaries.  There are other factors that would account for some of these differences as well, but I wanted to show the process of getting an adjusted estimate in a very simple setting.

WV has a very high percentage of IIB (88%), far more than the national average. The salaries at IIB colleges and universities are lower than I and IIA. Could the very low average salary in WV be an artefact of the disproportionately high number of IIB colleges and universities?

Analysis of Covariance is one method to answer this question. I ran the ANCOVA model in SPSS, but it can be run in pretty much any decent statistical software program. In SPSS, I used the General Linear Model with SALARY as the dependent variable, STATE as a fixed factor, and indicator variables for IIA and IIB in the covariate box.

SPSS can produce a table of adjusted salaries, though it is not a default option. SPSS will also produce coefficients for each of the two indicators and these allow you to understand exactly how the adjustment is done.

The coefficients for the IIA / IIB indicators are -10,325 and -16,222. These values tell you that the salary for category IIA is about 10K dollars lower on average than for category I.

The salary for category IIB is about 16K dollars lower on average than for category I. Both of these estimates adjust for state-to-state differences.

The traditional interpretation of a regression coefficient is that it represents the estimated average change in the dependent variable when the independent variable increases by one unit. For an indicator, this means a shift from 0% in a particular category and 100% in the reference category to 100% in the particular category and 0% in the reference category. In most statistical adjustments we don�t want to shift by this extreme but rather by a smaller amount.

What would happen if we took the proportion of IIA schools in WV and increased it from 6% (the WV proportion) to 22% (the US proportion)? Such a change would lead to an estimated $2,600 decline in average salary.


What would happen if we decreased the proportion of IIB schools from 88% (the WV proportion) to 53% (the US proportion)? This would lead to an estimated $5,600 increase in average salary.

The total effect of both changes is to increase the average salary by $3,000. The adjusted salary for WV, $38,000 is still below the national average, but not by as much as the raw data ($35,000) would have you believe.

2. The first deadly sin of researchers, pride

Two years ago, I wrote on my webpages about the seven deadly sins of researchers.

Here I want to elaborate on each of these sins in detail. The first deadly sin of researchers is pride. Researchers often think they know better than anyone else how to do research, so they ignore existing precedents in designing their studies.

One of the best examples of this appears in a large scale review by Ben Thornley and Clive Adams of research on schizophrenia. You can find the full text of this article on the web at bmj.com/cgi/content/full/317/7167/1181 and it is well worth reading. Thornley and Adams looked at the quality of clinical trials for treating schizophrenia and summarized 2,000 studies published between 1948 through 1997. The research covered a variety of therapies: drug therapies, psychotherapy, policy or care packages, or physical interventions like electroconvulsive therapy. Thornley and Adams found that researchers did not measure these patients consistently. In the 2,000 studies, the researchers used 640 ways to measure the impact of the interventions. There were 369 measures that were used in one study and never used again. Granted, there are a lot of dimensions to the schizophrenia and there were measures of symptoms, behavior, cognitive functioning, side effects, social functioning, and so forth. Still, there is no justification for using so many different measurements.

When someone comes to me with a survey, I ask if there is an existing survey that can be adapted to this particular research setting. It may not be a perfect survey, but it is "battle tested" and it simplifies the task of summarizing the entire research record down the road.

3. Is this a case-control design?

Here's a question I got by email.

I have a stats study design question. If I were to look at the association of curly hair for instance with a rash on the forehead, I pick a case control study design. When I analyze this I find that 45% of kids in the clinic (surprise) had curly hair. But I look at two groups curly vs non curly and the outcome of interest is the rash on the forehead, instead of cases vs controls so now, has this become an observational study instead of case control? Hope I am making sense, this is only a theoretical question.

You're confusing observational and cohort study, I think. Both case control and cohort studies are observational studies, as are cross-sectional and historical control studies.

Let's review the terminology. There are two types of variables in an observational study. Exposure variables describe some of the potential causes and outcome variables describe some of the potential effects. When you select a group of patients who have a rash, you are selecting according to an outcome, not an exposure. So you might think that this is a case control design.

But wait! Where's your control group. Did you select a control group? In a case control study, you would have selected a group of patients who do NOT have a rash. You didn't do this (naughty, naughty you!). You just noted that in the case group, the proportion of curly hair was extremely high (45%). Much too high to be due to chance, or so you think, because the incidence of curly hair is actually much lower in the general population. When you compare a group of cases (or a cohort group for that matter) to numbers in the general population, you are using a historical controls design.

Now all of the sudden the experiment morphs. You are now comparing curly hair kids to straight hair kids. Except, you're not thinking about the outcome here. You're still looking at kids who show up at the clinic with a rash, so 100% of the curly hair kids in your data set have a rash and 100% of the straight hair kids in your data set have a rash. That doesn't lead to a very interesting comparison.

Now, perhaps what you were thinking of doing was selecting all patients in your clinic, finding which ones have rashes, which ones don't, which ones have curly hair, and which ones have straight hair. Since you are selecting a single group and assessing both exposure and outcome at the same time, it's a cross-sectional study.

No, that wasn't it either? What you were really thinking of doing was selecting a group of kids who have rashes, finding a comparable number of matched controls at your clinic, and then looking at their hair? Okay, now that's a classic case-control study.

The difference is subtle. The terminology is frequently used incorrectly, even by seasoned professionals. I offer a few more hints about this at www.pmean.com/09/CaseControl.html.

4. Monthly Mean Article: Design, analysis, and presentation of crossover trials.

Mills E, Chan A, Wu P, et al. Trials. 2009;10(1):27. Available at: www.trialsjournal.com/content/10/1/27 [Accessed May 20, 2009]. Excerpt: Reports of crossover trials frequently omit important methodological issues in design, analysis, and presentation. Guidelines for the conduct and reporting of crossover trials might improve the conduct and reporting of studies using this important trial design.

5. Monthly Mean Blog: FiveThirtyEight: Politics Done Right, Nate Silver, Sean Quinn.

This looks like a blog about U.S. politics, but it's really a blog about political polling. During the 2008 U.S. presidential campaign, this site aggregated all the state polls to come up with a simulated electoral college result. The methodology is interesting and represents a meta-analysis of sorts for political polls. The site is currently tracking the 2010 U.S. Senate elections to try to forecast which state elections are likely to lead to a change in party. www.fivethirtyeight.com

6. Monthly Mean Book: Statistical Issues in Drug Development, 2nd ed., by S. Senn

I wrote a review of this book for the Journal of Biopharmaceutical Statistics and then paid a hefty fee to get the review published under an open source license. You can find the review at

Here's a few excerpts from the review:

The first five chapters of Statistical Issues in Drug Development offer a very general perspective, including a nice historical overview. The discussion of the proper role of a statistician in a pharmaceutical company will help others to appreciate the depth and breadth of our contributions, but Dr. Senn also holds us to a very high standard. I was humbled, for example, by a discussion on page 58 of how a statistician familiar with pulmonary function testing might be well-positioned to discuss the implications on design and sample size when peak expiratory flow is substituted for forced expiratory volume. I've worked with such measures for more than two decades, but I doubt that I have sufficient medical appreciation of these tests to discuss this topic at the level suggested here.

The remaining 20 chapters of the book cover specific topics such as baseline adjustments, subgroup analysis, multiplicity, intention-to-treat analysis, multicenter trials, equivalence studies, meta-analysis, cross-over trials, n-of-1 trials, sequential trials, dose-finding, pharmacokinetics/dynamics, pharmacoepidemiology, and pharmacoeconomics. A new chapter in the second edition covers pharmacogenetics.

One needs to be careful in describing the audience for a book like this. A practicing statistician will find that this book does not cover any particular topic in the level of detail that such a person needs. You won't be able to properly design and analyze a group sequential trial, for example, after reading the chapter on this topic. What this book provides is a gentle introduction to an area, an outline of the major controversies in that area, and references for anyone who wants to dig further. If you are moving into an area of drug development that is new to you, (say, dose-finding) then this book can jump start your transition, but this won't be the book that you constantly reach for as you hone your skills.

Statisticians with limited experience in drug development will greatly benefit from seeing the careful layout of controversies that are unique to this arena. I especially loved the description of the controversies associated with intention-to-treat analysis and random effects in a multicenter trial.

Another possible audience is researchers who want to develop a greater degree of sophistication in their work by better understanding the statistical issues associated with the design and analysis of drug development studies. You may want this book, just to help answer questions from some of your more sophisticated clients. To help some of the math-phobic clients, Dr. Senn segregates most formulas to an appendix at the end of the chapter.

It won't help, though, for your unsophisticated clients. This is not a Statistics for Idiots book. Even with the mathematics removed, the intellectual caliber required to appreciate this book is still substantial.

7. Monthly Mean Definition: What is a mosaic plot?

A mosaic plot is a graphical display that allows you to examine the relationship among two or more categorical variables. This type of plot does not appear commonly in the research literature, but it should be used more often.

The mosaic plot starts as a square with length one. The square is divided first into vertical bars whose widths are proportional to the probabilities associated with the first categorical variable. Then each bar is split horizontally into bars that are proportional to the conditional probabilities of the second categorical variable. Additional splits can be made if wanted using a third, fourth variable, etc.

Here is an example of a simple mosaic plot. There is a publicly available data set on the mortality rates aboard the Titanic, which are influenced strongly by age, sex, and passenger class. If you wanted to compare the mortality rates between men and women using a mosaic plot, you would first divide the unit square according to the overall proportion of males and females.

Roughly 35% of the passengers were female, so the first split of the mosaic plot is 35/65. Next, split each bar vertically according to the proportion who lived and died.

If the two horizontal lines were touching, then that shows that the proportion surviving is the same in each gender. Here there is a large displacement which reflects the fact that 2/3 of the women survived and only 1/6 of the men survived.

Most implementations of the mosaic plot offer as a default a small margin around each cell to make the graph easier to read.

You should consider carefully the choice of which variable to split the unit square first. Here is the same mosaic plot where the unit square is split first by survival status and then by gender.

About two thirds of the Titanic passengers died. The fatalities were mostly men (82%) and the survivors were mostly women (68%). The choice here is not too much different than the choice of using row percentages or column percentages in a cross-tabulation.

Here's a different mosaic plot that looks at mortality by passenger class.

The probability of surviving went down as passenger class went from 1st to 2nd to 3rd. The third class passengers were in the lower and less easily evacuated areas of the ship and some of the gates in the third class area were locked while 1st and 2nd class passengers were boarding the lifeboats.

You can get a three dimensional view of sex, passenger class, and survival by splitting vertically by sex, horizontally by passenger class, and vertically again by survival.

This plot is worth staring at for a while. It shows that while women fared better than men, this was far more true in first and second class than in third class. You will get a different perception of the patterns in the data if you exchange the order of categorical variables, so it worth trying the three dimensional mosaic plot several different ways.

Mosaic plots are not available in most statistical software packages, which is a shame. Like boxplots, they can be a bit confusing when you first see them, but they can be very helpful in assessing the trends and patterns among multliple categorical variables.

This definition is taken almost verbatim from a page at my old website:

It's not self-plagiarism, it's trusting the only reliable source.

8. Monthly Mean Quote: Two quotes this month.

When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind. Lord Kelvin as cited at physicsworld.com/cws/article/indepth/32214.

Sackettisation ... the artificial linkage of a publication to the evidence-based medicine movement in order to improve sales. David Sackett as cited at bmj.bmjjournals.com/cgi/content/full/320/7244/1283

9. Monthly Mean Website: Neural correlates of interspecies perspective taking in the post-mortem Atlantic Salmon: An argument for multiple comparisons correction. Craig M. Bennett, Abigail A. Baird, Michael B. Miller, and George L. Wolford.

This is a poster presentation intended to make you laugh and make you think at the same time. It discusses a technique known as functional magnetic resonance imaging (fMRI). This technique provides a non-invasive way to measure brain activity. The data from an fMRI scan is four dimenional, the three spatial dimensions of the brain plus the time dimension. Those regions of the brain that show a large change in activity between an intervention (the display of an image perhaps) and a resting state are highlighted in bright colors.

Source: Robinson R. fMRI Beyond the Clinic: Will It Ever Be Ready for Prime Time? PLoS Biol. 2004;2(6):e150. Available at: dx.doi.org/10.1371/journal.pbio.0020150 [Accessed July 1, 2009].

The problem is that there is so much data in an fMRI scan that statistical tests may tend to yield too many false positives. Bennett et al illustrate this by putting a dead fish (post-mortem Atlantic Salmon) into an fMRI experiment. The dead salmon was forced to watch a series of images and the fMRI highlighted certain regions of the fish brain where differential activity was found.

A re-analysis of the data using two popular corrections for multiple comparisons yielded fully negative results. The lesson is to tread cautiously when analyzing high dimensional data.

10. Nick News: Newton says goodbye after 20 years

While I have spent most of my time talking about Nicholas, he is not the only member of the "family." When we got married in 2002, Cathy brought a dog, Shauna, into the relationship, and Steve brought a cat, Newton. At the time, I joked that we had a blended family. Both cat and dog were "only children" so it took a while for them to get used to one another. After a year, though, they became best friends. Newton had always had trouble adjusting to other cats, but Shauna seemed to bring out the best in her. Shauna liked it because the food we got for Newton was a whole lot more interesting than her stuff.

Nicholas joined the mix in 2004, so he's really the third child. He learned pretty quickly how to pet a dog and how to pet a cat, and even took Shauna on some walks.

Both pets were quite old, and Shauna died in 2006, leaving Newton as the only animal member of the family. Newton did seem to miss Shauna but at her age, it would have been cruel to introduce a young energetic puppy or kitten into the mix. It was better to let Newton nap through her retirement years.

Newton, who was small for a cat at 6 pounds started to decline in health, and lost much of her weight. She dropped down to 4 pounds, and was just skin and bones. Still, she seemed to hold up well with the help of the Johnson County Cat Clinic.

She celebrated her 20th birthday in February 2009. In April, she had a bad stroke. Her hind legs were very weak and she couldn't stand without a pronounced lean to the left. We took Newton in to be put to sleep.

Here's a picture of Newton soaking up the sun.

Here's a picture of Shauna resting on Mom and Dad's bed.

We probably won't be getting any new pets until Nicholas is older and better able to assume most of the responsibilities of pet ownership.

11. Very bad joke: Two statistics are in a bar

Two statistics are in a bar, talking and drinking. One statistic turns to the other and says "So how are you finding married life?" The other statistic responds, "It's okay, but you lose a degree of freedom."

This joke has been around for a while. I cited it on my old website back in 1999

but you can also find variants at

12. Tell me what you think.

How did you like this newsletter? I have three short open ended questions that I'd like to ask. It's totally optional on your part. Your responses will be kept anonymous, and will only be used to help improve future versions of this newsletter.

I got several nice comments about the March/April newsletter. There was appreciation for the explanation of GEE models, the rule of 15 for logistic regression, and the loss of power caused by dichotomization. Someone pointed out that the Java application on dichotomization is also available at www.bolderstats.com/jmsl/doc/medianSplit.html.

I probably need to explain GEE models in more detail, though, and further clarify the Bayesian approach. There was a suggestion to explain what heterogeneity is. I presume this is heterogeneity in meta-analysis.

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Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 United States License. This page was written by Steve Simon and was last modified on 2017-06-15.