## Algebra and Statistics

Daniel Kaplan’s interesting and provocative article, “Calculus and Statistics,” which appeared in the July issue of *Amstat News*, raised several questions in my mind, one of which I’d like to mention and discuss: What are university statistics teachers to do with “methods” courses in social science fields like psychology and education, where most students do not have any mathematics background beyond high school? Of course this is a much-discussed problem, to which I found an unusual solution.

At an early point in my career (1969–1976), I held a joint appointment in psychology (in the School of Behavioral Sciences) and statistics (in the School of Economic and Financial Studies) at the then-new Macquarie University in Sydney. I taught statistics in both schools—part of the mathematical statistics sequence in economics and part of the sophomore “methods” year in psychology.

The university had from its beginning a small common freshman introductory statistics course for all students, which caused a great deal of dissatisfaction. At the sophomore level and above, each school (equivalent to a faculty) developed its own statistics courses.

I was responsible for the development of statistics courses in psychology at a time when the university was expanding (along with other new universities in Australia and elsewhere). The earliest students were beginning their junior year when I began, and there was an obvious need for a strong regression and ANOVA course in this year to prepare students for their final senior year course.

The departmental emphasis at the time was on experimental psychology, but the social psychology strand was expanding, and both experimental and survey data structures needed to be dealt with. This was the period when card-programmable electronic calculators were coming onto the market in Australia, and we considered developing courses based on these cards. This would have required a major department investment in machines.

A major advance occurred in 1971 with the university’s computer center acquiring and mounting the U.S. National Bureau of Standards OMNITAB statistical package for the UNIVAC 1108 machine. This was a revolution in statistical software for the time, providing a simple English-language spreadsheet structure with powerful statistical functions, including a general (normal) regression routine, which provided residual plots and a hierarchical partition of the total sum of squares into orthogonal components determined by the order of variables specified by the user. (The design of OMNITAB was closely followed by the commercial MINITAB package, though without the hierarchical partitioning of the ANOVA table.) The university system provided remote card-deck input, another major advance over paper tape.

The arrival of this package on our computer system coincided with my research leave year at the Educational Testing Service as a visitor to Frederic Lord’s Psychometric group, supported by a Fulbright Senior Fellowship. I spent much of the year writing a textbook exposition of regression, ANOVA and ANCOVA through the general linear model, with an integrated OMNITAB handbook for the examples. I began to teach this approach in a new junior linear models course when I returned in 1972 to a combined group of psychology and statistics students.

A difficulty I had to face was the same one Kaplan refers to, but one which was and still is endemic in the social sciences: the lack of mathematical background. I took a full-frontal approach to this. The course *must* be mathematically based, and therefore I would give an intensive short course in the foundational mathematics necessary in the first four weeks, with intensive tutorial back-up from other members of the psychology staff.

This may seem daring, or [like] madness, as some of my statistics colleagues thought at the time. My view was that a student coming into the third year who intended to be a professional psychologist had the cognitive ability to cope with the necessary high-school algebra, linear and quadratic functions, and the coordinate geometry of points in the plane. (As Dr. Kaplan notes, these are what we require of mathematics for regression courses.) The psychology students’ fear of their inadequacy could be dealt with by supportive tutorials, and (in later years) by the fact that no one had failed this course. In addition, the course book and its computer manual support were provided from the beginning to students, and the lectures were essentially tutorial expositions of the examples in the text.

I should say that I had pressured the psychology department to make passing this course a prerequisite for the final year in psychology. All final-year psychology students had to do an empirical project, with appropriate analysis, which counted for 25% of the final grade. There was quite strong opposition to my proposal (which did not apply to sociology or anthropology students), especially from the social psychologists. In the department meeting [during] which [we were] to vote on this, I was asked, “Murray, can you put your hand on your heart and say that *every* student in psychology, in whatever sub-field, *must* have this course to successfully complete the professional year?” I replied, “No, I cannot say that. What I can say though is that if a senior-year student asks me for help on the senior project, and the student has not taken my course, then I *will not* assist them. That will be the responsibility of the student’s advisor.” There was silence. The vote was taken, and passed by a large majority.

The course was very successful and expanded year by year, from 25 students in 1972 to 65 in 1976, the last year I taught it. It quickly acquired the reputation of the most difficult course in psychology. At student advising before each year began, I was regularly approached by three or four psychology students with the same question: “Dr. Aitkin, I’m considering taking the honors (professional year) in psychology. But I know I have to take your course and I’ve heard that it’s very difficult. Do you think I’ll be able to cope with it?” I would reply:

This course is very difficult. It’s the most difficult course you’ll meet in your university experience. In the first four weeks of the course, you’ll sweat blood—you’ll weep! But if you have gotten this far in your course and you want to be a professional psychologist, you can manage this course. Why did I make it so difficult? Because to understand statistical analysis you have to think like a mathematician, and you’re not a mathematician. But we can train you to think like one, as far as is necessary. That’s the hard part. The payoff for you is that you will understand statistical analysis in a way no student from any other psychology course in Australia does. You’ll be able to work out analyses of very complex structures. You’ll be an analysis leader in any group of your contemporaries.

This was perhaps boastful, but it was no idle boast. The practical exam (50% of the assessment—there was no theory exam) required students to analyze a real observational study of hostility and affection in the husbands of wives who had attempted suicide (unsuccessfully) by overdoses of sedatives and compare their affect levels with those of husbands of wives who had been admitted to hospital with critical organic abdominal conditions from which they had recovered.

Husbands were cross-classified by nationality and whether there had been a previous occurrence of the suicide attempt or acute abdominal condition. There were multiple measures of affect for hostility and affection.

The data structure was a severely unbalanced three-way cross-classification, with multiple covariates. Each student had a unique data set with a single affect response measure. They had four weeks for the take-home analysis and could discuss their work with any other student.

This analysis was, at the time, beyond the range of most statistics undergraduate programs. The hierarchical partitioning of the ANOVA table, leading to several orthogonal decompositions, was a logical but major innovation, and was not accepted later by many applied statisticians, who sought instead a single “ANOVA” table constructed by various adjustment arguments. I published years later an argument for the hierarchical approach as a discussion paper in *Journal of the Royal Statistical Society*, “The analysis of unbalanced cross-classifications.”

Designing and teaching this course gave me great professional satisfaction. It also improved my teaching abilities: Macquarie had a heavy concentration of part-time mature-age students, many of them teachers or public servants aiming for promotion in their professions. These students were intolerant of time-wasting and lack of clarity in exposition and were not afraid to be critical.

I submitted the course-book to several publishers under the title *Linear Models with Applications in the Social Sciences*. Their referees’ reports were similarly negative: the book approach was not sufficiently mathematical for statistics students, was not particularly social science oriented, and was far too difficult for social science students. This approach eventually appeared in a different computational framework, with three co-authors (Aitkin et al 1989, *Statistical Modelling in GLIM*. Oxford University Press). The GLM approach was by then completely standard.

*Murray Aitkin, Honorary Professorial Fellow
Department of Mathematics and Statistics
University of Melbourne *

Josiesaid:You say the course was successful – how did you measure ‘success’? What about the success rate for the final exam?

Elliot M. Cramersaid:I recall my first course in statistics as a math major. It included an extensive development of Analysis of Variance using mimeographed notes for a forthcoming book. I recall being completely clueless. Many years later I came across the book in a thrift shop and thought it was the most wonderful thing I had ever read – very clear and well motivated; I had learned a bit in the meantime. I think one should have very modest goals for an undergraduate course in statistics, particularly in the social sciences; those few who will ever have the occasion to use statistics will probably go on to graduate school where they will hopefully develop the kind of facility that Murray wishes them to have. I think that if students can gain some appreciation for what can be done with statistics and some critical thinking with regard to what they read in magazines and newspapers, the course will have been a success.

Murray Aitkinsaid:The pass rate for the course was 100% in every year. That isn’t my criterion (though it was important for students to know); to me the course was successful because

a) the students were highly motivated by their professional futures;

b) the staff were well motivated to support and assist students (many of whom they already knew) wlth mathematical issues;

c) the philosophy of basing it mathematically paid off for students, giving them the technical ability to understand, formulate and analyse complex cross-classified models with covariates.

Keith O'Rourkesaid:The option which until very recently was not available would be to avoid math almost all together.

This may seem strange to statisticians but some very bright statisticians recently overcame serious barriers of not being able to derive likelihoods by using Approximate Bayesian Calculation (ABC).

Here one simply draws parameters from priors, draws possible data given those parameters and subsets the parameters drawn that generated the same actual data or summaries of actual data (conditions). http://en.wikipedia.org/wiki/Approximate_Bayesian_computation#The_ABC_rejection_algorithm

Now if one uses this for simple introductory examples most of statistical inference can be demonstrated (explained) without (much) math at all. The sample sizes need to be very small unless one takes advantage of knowing the likelihood. I have had some success in doing this (ASA webinar last February and one for non-statisticians last month http://cseb.ca/conferences/webinar.php ).

As for how complicated the examples can be, I did do an observational study with a risk factor and gender. Here I drew heavily on Pearl’s causal inference formalisms in underscoring haw the data generating process needed to mimic Nature’s.

This is an animation of the idea here http://phaneron0.files.wordpress.com/2012/11/animation3.pdf where conceptually one posits how Nature generated the unknown and then provided data from it and an analyst has to represent this process and carry out the analysis using that representation.