NAME: Basketball free throws (sequence of successes and failures) TYPE: Time series of discrete dichotomous data SIZE: 200 observations, one variable DESCRIPTIVE ABSTRACT: This is a simple sequence of dichotomous observations from a series of 200 consecutive basketball free throw attempts. SOURCES: I collected the data myself. VARIABLE DESCRIPTIONS: S = success (free throw made), F = failure (free throw missed) SPECIAL NOTES: (none) STORY BEHIND THE DATA: I teach a Biostatistics class and I like to occasionally give the students a data set for statistical analysis before they have learned the appropriate test. I wanted to obtain a simple data set involving a time sequence of dichotomous data, because there are a variety of questions that can be asked of such a data set. In our campus gymnasium I attempted 200 consecutive basketball free throw shots under standard conditions (basket rim 18 inches in diameter and 10 feet above the ground; horizontal distance from free throw line to basket 15 feet; leather basketball 9 inches in diameter). I recorded whether each shot was a success or failure. PEDAGOGICAL NOTES: I use these data for an in-class exercise in my undergraduate Biostatistics class, about halfway through a 14-week semester.* The students work in groups of 3 or 4 for a full 50-minute class period. I give them a 1-page handout that shows the data along with the following 4 questions to guide them: 1. How might you determine whether there was a pattern to this sequence? 2. What is your null hypothesis? 3. Can you think of a statistic whose value would vary depending on whether there was a sequential pattern? 4. Is there more than one type of non-random, sequential pattern one could observe with data like these? (i.e., is there more than one way the data could deviate from a random sequence?) I don't need to provide any motivation for the students; they dive right in. While students work I circulate around the room to field questions, discuss ideas, and gently steer them down more productive avenues. I strongly encourage students not to consult their texts (we use Zar 1998) unless they need to look up a formula or table of critical values that they already are familiar with. (This is so they do not immediately discover and apply the runs test before they get an opportunity to thoroughly think about the data.) Towards the end of the class session, each group of students briefly describes the approach they used. Most students do not complete their analysis during the class period. Typically, students have tried diverse analyses. Some of the common approaches that students come up with are: 1. Analyzing whether the data are consistent with a success rate of 50% (using a binomial test, usually with a normal approximation). Although common, this approach does not address the main questions. 2. Calculating the probability of streaks of a given length (either makes or misses), again using the binomial distribution. Somehow they have to relate these individual calculations to the overall expected distribution under the null hypothesis that each free throw is independent. This is difficult for them. 3. Analyzing whether the success rate varies in time. Students do this by breaking up the data into blocks of 10 or 20 observations, then testing for heterogeneity of successes vs. failures among blocks (as it turns out, there is significant heterogeneity by a G-test or chi-square test). This is a peripheral issue but is straightforward to analyze, and is also interesting to connect to the issue of streaks. 4. I've never had students come up with the simple concept of runs (streaks of 1 or more consecutive misses or makes, without distinguishing between long and short streaks) on their own. However, students sometimes develop more complicated versions of this idea, so I can steer them towards the simpler idea of runs, and how the number of runs would make a convenient test statistic. 5. On rare occasions a student has realized that each pair of successive observations could be considered as a transition (make-make, make-miss, miss-make, and miss-miss), and that the numbers of each type of transition are informative about whether the data are sequentially independent. Using this method, one constructs a table of the observed frequencies of the four transitions and compares it to the expected frequencies under the null hypothesis that each observation is independent. Thus, this data set can be used to teach students a little bit about the simplest type of Markov process. During the following class period, we revisit this data set as I illustrate the various ways one could analyze it. I present the runs test, a contingency-table analysis of transition frequencies, and a contingency-table test for heterogeneity in time. I try to connect each of these analyses to the approaches students had come up with on their own during the previous class. This leads to a broader discussion of the different ways one can analyze certain data sets. When we discuss the runs test, it also provides an opportunity to discuss where tables of critical values came from. I use a simple MATLAB simulation to generate a null distribution of runs using the same data set, and we find that the observed number of runs is very close to the median of this null distribution. This provides an example of the value of randomization approaches, which I like to emphasize often in my course. * Note on student background: at this point the students are familiar with standard introductory concepts in frequentist statistics, including estimation, sampling distributions, and hypothesis testing, and have done both parametric and non-parametric analyses of continuous and discrete data. They have worked with one-, two-, and paired-sample tests. They are also familiar with the concept of randomization tests, which I introduce within the first week of the course because the students have found that it gives them a much better understanding of what a P value means. Immediately prior to this exercise, the students have been analyzing data involving discrete categorical variables, including work with binomial and Poisson distributions, R X C contingency tables, and goodness-of-fit tests. We do this exercise before the students have learned any methods for analyzing temporal sequences of data (e.g., runs tests) because I want them to experience this as a novel problem that they are unfamiliar with. REFERENCES: Zar, Jerrold H. 1998. Biostatistical Analysis. 4th ed. Prentice Hall SUBMITTED BY: Stephen C. Adolph Department of Biology Harvey Mudd College 301 Platt Blvd. Claremont, CA 91711 USA adolph@hmc.edu