Applied Univariate, Bivariate, and Multivariate Statistics. Daniel J. Denis
and predictions are much epistemologically “safer” words to use, less prone to critique ending in quicksand. For a brief, but enlightening discussion of causality in the social sciences, see Fox (1997, pp. 3–14). For a more thorough treatment of the subject as it relates to structural equation models, see Mulaik (2009, pp. 63–117). Even a brief study of the philosophy of science goes a long way to understanding the complexities involved in using “causal” statements in research. These issues are not nearly as simple as they may at first appear.
1.6 THE NATURE OF MATHEMATICS: MATHEMATICS AS A REPRESENTATION OF CONCEPTS
Ian Stewart (1995) said it best when he wrote that the mathematician is not a juggler of numbers, he is a juggler of concepts. The greatest ambivalence to learning statistical modeling experienced by students outside (and even inside, I suppose) the mathematical sciences is that of the presumed mathematical complexity involved in such pursuits. Who wants to learn a mathematically-based subject such as statistics when one has “never been good at math?”
The first step in this pursuit is to critically examine assumptions and prior learned beliefs that have become implicit. One way to help “demystify” mathematics and statistics is to challenge your perception of what mathematics and statistics actually are in the first place. It is of great curiosity that so many students claim to dislike mathematics and statistics, yet at the same time cannot verbalize just what mathematics and statistics actually are, and then even worse, proceed to engage in real‐life activities that utilize very much the same analytical cognitive capacities as would be demanded from doing mathematics and statistics!
More than likely, the “dislike” of these subjects has more to do with the perceptions one has learned to associate with these subjects than with an inherent ontological disdain for them. Human beings are creatures of psychological association. Any dislike of anything without knowing what that thing is in the first place is almost akin to disliking a restaurant dish you have never tried. You cannot dislike something until you at least know something about it and open your mind to new possibilities of what it might be that you are forming opinions about. Not to sound overly “Jamesian,” (the analogy isn't perfect, but it's close) but perhaps you are afraid of mathematics because of your fear of it rather than the mathematics itself. That is, you run, not because of the mathematics, but because of the fear. If you accept that you are yet unsure of what mathematics is, and will not judge it until you are knowledgeable of it, it may delay derogatory opinion about it. It is only when we assume we know something (to some extent, at least) that we usually feel free to judge and evaluate it. Keep your perceptions open to revision, and what you may find is that what was disliked yesterday curiously becomes likable today, simply because you have now learned more about what that something actually is. But to learn more about it, you need to first drop, or at minimum suspend, previously held beliefs about it. Have an open mind from the outset, and refresh that mindset each time you open a book or attend a lecture in a course.
The first point is that statistics is not mathematics. Statistics is a discipline unto itself that uses mathematics, the way physics uses mathematics, and the way that virtually all of the natural and social sciences use mathematics. Mathematics is the tool statisticians use to express their statistical ideas, and statistics is the tool that scientists use to help make sense of their research findings. The field of theoretical or mathematical statistics is heavily steeped in theorem‐building and proofs. Applied statistics, of the kind featured in this book, is definitely not. Thus, any fear of real mathematics can be laid to rest, because you will find no such mathematics in this book. Upon browsing this book, if you are of the opinion that it contains “lots of math,” then quite simply, you do not know what “lots of math” looks like. Rest assured, the mathematics in this book is simply used as a vehicle for understanding statistics.
Mathematics and statistics are not things “mysterious” that can only be grasped by those with higher mental faculties. A useful working definition might be that mathematics is a set of well‐defined and ever‐expanding rules and consequences about symbolic abstract objects based on fundamental assumptions called axioms. The axioms of mathematics are typically assumed to be true without needing to be proved. Theorems and other results built on such axioms usually require proof. What is a proof? It is an analytical argument for why a proposition should be considered true. Any given proof usually relies on other theorems that have already been proven to be true. Make no mistake, mathematics is a very deep field of intellectual endeavor and activity. However, expecting something to be deeper than it is can also lead you to just as well not understand it. Sometimes, if you are not understanding something, it may very well be that you are looking far beyond what there is to be understood. If you retreat in your expectations slightly of what there is to see, it sometimes begins to make more sense. Thinking “too deep” where such depth is not required or encouraged, is a peril. Many “bright” students have this “gift” of critical analysis, and to understand a concept, need to actually retreat somewhat in their depth of inquiry (at least for the moment).
For a general overview of the nature of mathematics, the reader is encouraged to consult Courant, Robbins, and Stewart (1996), and for an excellent introduction to basic mathematical analysis, Labarre (1961). Hamming (1985) is another good introduction to the field of mathematics, as well as Aleksandrov, Kolmogorov, and Lavrent’ev (1999). For more philosophical treatments, the reader should consult Dunham (1994) and Stewart (1995). For an in‐depth and very readable history of mathematics, consult Boyer and Merzbach (1991).
1.7 AS A SCIENTIST, HOW MUCH MATHEMATICS DO YOU NEED TO KNOW?
The answer to this question is, of course, as much as possible, for working through problems of any kind can only serve to hone your analytical and deductive abilities. Even working through problems completely unrelated to statistics can help your statistical abilities, because they give you practice in “figuring things out.” However, that answer is, of course, a naïve if not idealistic one, since there is only so much time available for study and the study of statistics must be balanced by your own study of your chosen field.
For example, if the biology student became immersed in mathematics and statistics full‐time, then that student would no longer be a student of biology. It can be exceedingly difficult to apply a statistical technique and interpret the results of such a technique in a field for which you are not familiar. If you are unaware of the substantive objects you are working with, that is, the “stuff” on which the statistics are being applied, then regardless of your quantitative expertise, you will often have difficulty interpreting the scientific result. Likewise, if spending too much time computing higher‐order derivatives, the student of animal learning, for instance, will have little time remaining to study the learning patterns of the rats he is conditioning, or to speculate on theoretical advancements in his field. Hence, a “happy medium” is required that will balance your study of your substantive area along with the technical quantitative demands of your field of study. Indeed, even for those who specialize exclusively in statistics, the American Statistical Association strongly advises aspiring statisticians to choose a field of application. As a researcher, you will be expected to apply modeling techniques that are quite advanced (entire courses are devoted to the statistical technique you may be applying), and so you will face the opposite problem, that of choosing to specialize in statistics (to some extent) so that you may better understand the phenomena of your own science. Hence, regardless of whether one is coming from a mathematics or science background, one should aspire for a healthy mix of scientific and statistical expertise. Computing experience (e.g., R, SPSS, SAS, etc.) should also be part of your “repertoire” of skills. As an applied scientist, you should probably find yourself in the data science or traditional research intersections in the following Venn diagram.