# Final project

Lesia Nestorowich final project for MATH 1590 3.0 - April 9, 2006

## Math is Everywhere: Interviewing a Neuroscientist

In our daily lives, the importance of mathematics is immeasurable. Whether we are calculating our change at a grocery store or negotiating our mortgage rate at the bank, we are continually exposed to some form of mathematics. This requirement extends beyond daily life into the workplace. Many, if not all, occupations deal with some type of mathematics. Teachers use mathematics to determine student’s grades, physicists use mathematics to determine various theories, and even little Susie uses mathematics to make the right concentration of lemonade to sell on the street corner. In many situations, the most basic principles of mathematics are even utilized unconsciously. As a result, it is safe to say with confidence that the discipline of mathematics is extremely important in our everyday lives.

In order to further investigate the role of mathematics in society, I have chosen to interview a neuroscientist at the University of Western Ontario. A neuroscientist is a researcher who studies the mechanisms of the brain at the molecular, physiological, or behavioural levels. The neuroscientist that I chose to interview specifically studies the hormonal basis of cognitive differences between the sexes. Male and female hormones differ considerably not only in terms of the types and levels present in each sex, but also in the specific effects that they exert throughout the body and especially in the brain. These differential hormonal effects have been shown to result in a sex difference in a variety of cognitive tasks, such as a male advantage in spatial ability and a female advantage in linguistic and mathematical ability. These sex differences have been replicated in numerous studies and remain robust across species as well. The study that this neuroscientist recently completed involved studying the effects of the female estrous cycle (equivalent to the menstrual cycle in humans) on spatial learning and memory in rats. Different levels of female hormones (estrogen and progesterone) were administered to female rats in order to assess the degree to which they contributed to a spatial learning task.

Academic, or scientific, research relies heavily on mathematical statistics and analyses. Though the most important and obvious use of mathematic in research is involved in the analysis of the data obtained from the experiment, I found out that mathematics is also extensively used throughout the experimental procedure. In this particular experiment, simple mathematical calculations were used to prepare hormonal injections for the female rats, which consisted of a mixture of one of the female hormones (either estrogen or progesterone) and a vehicle (a solution in which the hormones were dissolved; in this case, it was sesame oil). In addition, a statistical equation was used in order to estimate the number of rats required per treatment group in order to obtain results that could be reliably analyzed for their significance. The use of this mathematical equation allowed the researcher to pre-determine the number of rats needed to conduct the experiment, thereby allowing him to plan his experiment and efficiently conduct it. This estimation was also important in terms of finances, as the research conducted required the purchase of rats, hormones/solutions, and other necessary materials. There was also a budget with which to conduct the research, yet another example of how mathematics played a role in the experimental procedure.

The most important and complicated mathematics, as I soon found out, is used to analyze the data that is obtained throughout the experiment. In order to analyze the data to determine if there are any significant effects due to the different treatment conditions, a statistical analysis software package called SPSS (Statistical Package for the Social Sciences) is used. In SPSS, the researcher inputs the data into the program, selects the appropriate analysis to be used, and determines and sets a variety of parameters that are integral to the selected analytical test. After this is completed, the SPSS program provides an output with the results. My interest in this statistical program, which appears to magically produce results out of thin air, necessitated a further explanation of the underlying principles of some of the tests used to statistically analyze the experimental data. And that is exactly what I got!

One of the statistical techniques used to analyze the data is called ANOVA (Analysis of Variance). The ANOVA is used to compare two or more group means (averages) to see if there are any reliable differences among them. This statistical technique is the first one used to draw comparisons and conclusions regarding all of the groups (the groups receiving the different hormonal treatments and the control groups receiving no treatment). It is used to determine if any of the groups differ from each other and if so, if these differences are statistically significant. Though slightly more complicated in its calculation, the general ANOVA equation looks like this:

$F = \frac{ n E \alpha_a^2 + \sigma_e^2}{\sigma_e^2}$

The first term ($n E \alpha_a^2$) represents the variance estimate that comes from the differences between group means and is considered a reflection of the group differences or treatment effects. The second term ($\sigma_e^2$) represents the variance estimate that comes from the differences in scores within each group and is considered a reflection of random error. The random error term in the above equation will remain the same in the numerator and the denominator for a given test. If there are no differences between any of the group means ($n E \alpha_a^2$), the numerator and the denominator are relatively equal to each other and it is concluded that any slight differences are attributed to random error as opposed to differences in the treatment effects. However, if there are significant differences between the group means ($n E \alpha_a^2$), then the numerator is larger than the denominator and a significant level for the ANOVA may be obtained. This means that there are differences present between the groups that can be reliably attributed to the treatment effects.

In order to obtain a significant result, I found that you needed to set a limit that would distinguish between a significant and non-significant value. In many cases, this level is set at 0.05 (5%) and is referred to as an α level. The α level, or Type I error, refers to the probability of obtaining a significant result that is not significant. Thus, the researcher allots a 5% error for obtaining the result. The lower this value is (1% for example), the stronger the criterion for determining a significant result is and thus, the stronger the conclusions that can be drawn based on the treatment effects.

The logical extension of the ANOVA is to test each of the groups individually (in pairs) in order to determine which groups differ from each other. This type of test is called a t-test. I found out that you can perform a t-test if you have a significant result in the ANOVA and also if you do not have a significant result. You can perform an a priori t-test, where you decide beforehand which groups you would like to compare (in this situation, you do not need a significant ANOVA to perform the t-test, but you can only compare the groups that you previously decided to compare) or a post-hoc test, which is performed with a significant ANOVA and include all of the group comparisons. A t-test assumes the same level of significance as previously described (for an ANOVA), and determines if one treatment effect (group) differs significantly from another treatment effect (group). If there is a significant difference, you can practically say the treatment that you administered either contributed to an improvement or impairment in spatial learning and memory (in this particular study). Thus, these independent t-tests are extremely important in allowing the researcher to draw accurate and interpretable conclusions regarding the behaviour of the rats in the different treatment groups in the numerous learning and memory tasks.

My interview with a neuroscientist was able to show the importance of mathematics in the field of scientific research. Different analyses require different statistical techniques, and all of these mathematical forms help to conduct an experiment and analyze the results for interpretation and application to real-life situations. This is just one example of a profession that relies heavily on mathematics. As previously stated, there are many occupations that similarly involve different elements of mathematics in order to complete their objectives. Whether the applied mathematics is simple addition or completed procedures as outlined by my interviewee, it is nevertheless extremely important to study and learn mathematics.