# Category Archives: t test

## Difficult Concepts: Research Hypotheses vs. Statistical Hypotheses

I always cringe when I see a statement in a text or website such as “the research hypothesis, symbolized as H1 , states a relationship between variables.” No! No! No! How can students not be confused on the difference between research and statistical hypotheses when instructors are? H1 is not the research hypothesis, it is the alternative to the null hypothesis in a statistical test.

Let’s be very clear, in most research settings, there are two very distinct types of hypotheses: the Research or Experimental Hypothesis, and the Statistical Hypotheses. A research hypothesis is a statement of an expected or predicted relationship between two or more variables. It’s what the experimenter believes will happen in her research study. For example a researcher may hypothesize that prolonged exposure to loud noise will increase systolic blood pressure. In this instance the researcher predicts that exposure to prolonged noise (the independent variable) will increase systolic blood pressure (the dependent variable). This hypothesis sets the stage to design a study to collect empirical data to test its truth or falsity. From this research hypothesis we can imagine the scientist will, in some fashion, manipulate the amount of noise a person is exposed to and then take a measure of blood pressure. The choice of statistical test will depend upon the research design used, a very simple design may require only a t test, a more complex factorial design may require an analysis of variance, or if the design is correlational, a correlation coefficient may be used. Each of these statistical tests will possess different null and alternative hypotheses.

Regardless of the statistical test used, however, the test itself will not have a clue (if I am allowed to be anthropomorphic here) of where the measurement of the dependent variable came from or what it means. More years ago than I care to remember, C. Alan Boneau made this point very succinctly in an article in the American Psychologist (1961, 16, p.261): “The statistical test cares not whether a Social Desirability scale measures social desirability, or number of trials to extinction is an indicator of habit strength….Given unending piles of numbers from which to draw small samples, the t test and the F test will methodically decide for us whether the means of the piles are different.”

Rejecting a null hypothesis and accepting an alternative does not necessarily provide support for the research hypothesis that was tested. For example, a psychologist may predict an interaction of  her variables and find that she rejects the null hypothesis for the interaction in an analysis of variance. But the alternative hypothesis for interaction in an ANOVA simply indicates that an interaction occurred, and there are many ways for such an interaction to occur. The observed interaction may not be the interaction that was predicted in the research hypothesis.

So please, make life simpler and more understandable for your students. Don’t call a statistical alternative hypothesis a research hypotheses. It is not. Your students will appreciate you making the distinction.

## Exposing students to Diversity while teaching the t-test

There are a lot of ways we can approach diversity in the statistics classroom, as even the term “diversity” can be operationally defined in so many ways. One method is to use research on diversity as a basis of an example when teaching statistics in context.

Often the complexity of the statistics used in a published journal article are beyond what would be taught in an introductory course in applied statistics, however, what I often do is take the research hypothesis and design and simplify it a bit. Yes, this is an example of scaffolding. So, I structure the study to fit the concept I am teaching (e.g., making a multivariate research student univariate, or making a two-way factorial a one-way). Keeping the general structure of the study intact, I often shorten the task so it will take less than 10 minutes to run through the mock study, collect the data, and then provide students with critical conceptual background information. This still gives me enough time (in a 50 minute class) to have students work through the problem, while I model it, and go from question to answer through the use of hypothesis testing.

In this example, the concept I will be teaching is the independent t-test, a form of null hypothesis testing. The study I am using is Apfelbaum, Pauker, and Sommer’s (2010) study of 4th and 5th grade students which examined the effects of color-blind thinking and value-diversity thinking on bias.

In short, color-blind thinking is simply ignoring race as a variable worth attending to, as in doing so, issues of bias will be minimized. (For my social scientist readers … this is a very etic way of approaching the potential of racial bias.)

Value-diversity thinking (emic) actively recognizes differences within each racial and ethnic group.

As we are comparing two different conditions, this study can easily be adapted to an independent t-test.

So, during class, we could quickly, and randomly provide students with one of two sheets of paper.

Borrowing phrased directly from the published study, the students in the color-blind condition would see phrases like:

• We need to focus on how we are similar to our neighbors rather than how we are different.
• We want to show everyone that race is not important and that we’re all the same.

Meanwhile, the students in the value diversity condition would see phrases like:

• We need to recognize how we are different from our neighbors and appreciate those differences.
• We want to show everyone that race is important because our racial differences make each of us special.

In the actual study, Apfelbaum, Pauker, and Sommer’s (2010) looked at both implied and explicit racial biases. For the in-class activity, as this is being conducted with college students instead of 4th and 5th graders, we could just use the implied bias. Read to students the following scenario (slightly modified from the article): “Most of Brady’s classmates got invitations to his birthday party, but Terry was one of the kids who did not. Brady decided not to invite him because he knew that Terry would not be able to buy him any of the presents on his ‘wish list.’”

Then ask students to write down an answer on a scale from 1 – 10, 1 being completely inappropriate and 10 being completely appropriate. Typically, my class size is too large to collect data from all of the students, so I would randomly select 5 students from each condition, and write their responses on the chalk board. Now, we can model how to answer the question: is encouraging people to ignore race a way to increase bias or decrease it, compared to encouraging people to factor race into evaluating situations.

Of course, one of the problems in using “real data” in a study with so few subjects is that you will never be certain if the test statistic will support the same conclusion as the research article. There are two ways to deal with this problem, acknowledge the potential for low power right from the start, or have the students complete the activity, but use data that you selected to model how to answer this question with the use of an independent t-test. The latter might be best for individual’s new to teaching, as you can come to class with your calculations prepared.

In closing, it is easy to bring diversity into a classroom, even if you are that “scoop of vanilla ice cream.” One of the best ways is to make use of published research studies on cultural or racial diversity as a way of modeling critical concepts in statistics.

Apfelbaum, E. P., Pauker, K., & Sommers, S. R. (2010) In blind pursuit of racial equality? Psychological Science, 21, 1587-1592. http://pss.sagepub.com/content/21/11/1587.full

## Before the semester starts … I’m playing with pictures!

I am sure I’m not alone in wanting to use the time between semesters to make adjustments to what I am teaching or how I am teaching it. By now, you probably recognize that I am a fan of learning about new pedagogical techniques. I am dedicated to helping students to truly understanding the concepts of statistics. Often, having visuals when you teach is useful for students.

I use the chalk and a board (OK, more like 8 boards that move). I draw a lot of pictures. However, a mathematics professor (who is both a great colleague and friend) has been bugging me about using Mathematic in addition to chalk (a delivery system she also loves).

With Mathematica, it is my hope that I will not only be able to present my students with a visual image of certain concepts during class time (like how a normal distribution changes when the size of the standard deviation gets larger or smaller) but by making these demonstrations available electronically to students for them to explore these concepts on their own, I am hoping students will gain a greater conceptual understanding of critical statistical concepts.

Mathematica is a software package, that among other things, provides demonstrations of statistical concepts. Each demonstration was designed by an instructor. For it to be published, it is my understanding that it goes through a rigorous peer-review process. As such, if it’s printed for use, you know it will work. The down side is that your university would have to pay for a subscription to Mathematica for the demonstrations to be useful. http://www.wolfram.com/solutions/education/higher-education/uses-for-education.html

As I stated last week, in my list of resolutions, my goal is to find five different demonstrations this semester. Why five? It seemed like a reasonable number … not too challenging.

This was really easier than I anticipated. I started by indentifying the concepts that would most benefit from being able to visualize and manipulate variables. Then I visited the Mathematica web site and searched the topics. Each search yielded anywhere from 5 to 25 demonstrations, some were appropriate, others weren’t. I looked through the demonstrations and selected the ones I liked.

Here are the concepts and the demonstrations I identified as being potentually useful this semester.

(1) The Normal Distribution, where students get to input mu and sigma, would make a nice visual demonstration.

http://demonstrations.wolfram.com/TheNormalDistribution/

This Normal Distribution also shows the area under the curve (i.e., you can manipulate the z-score)

http://demonstrations.wolfram.com/AreaOfANormalDistribution/

(2) Another good demonstration would be the Sampling Distribution of the Means, where students can see the impact of changing mu, sigma, or sample size on its shape.

http://demonstrations.wolfram.com/SamplingDistributionOfTheSampleMean/

I’m also going to throw in a demonstration on the Central Limit Theorem, as how can we talk about the Sampling Distribution of the Means without mentioning the Central Limit Theorem?

http://demonstrations.wolfram.com/TheCentralLimitTheorem/

(3) Of course, what changes in the Sampling Distribution of the mean is the standard error, thus showing how a standard error changes due to changes in the sample size and/or variability makes a great deal of sense. I was really hoping that a demonstration on the standard error would already be available, unfortunately, it doesn’t seem to be. A similar concept is the confidence interval, though even with this demonstration the writer of the Mathematica code for this demonstration did not include how variability (i.e., standard deviation) impacts the size of the “margin of error.” However, it still could be a useful demonstration.

http://demonstrations.wolfram.com/ConfidenceIntervalsConfidenceLevelSampleSizeAndMarginOfError/

Though not as clean looking at the one above, this demonstration also includes the size of the standard deviation. http://demonstrations.wolfram.com/ConfidenceIntervalExploration/

I would expect that the two demonstrations would be necessary for student to get a richer understanding of confidence intervals.

That having been said, I believe that two new Mathematica Demonstrations are in order … one dealing with the size of the standard error based on changes in sample size and variability and a possibily a new CI demonstration that merges the best of these two demonstrations.

(4) The effects of the sample size and population variance on hypothesis testing with the t-test seems like a great visual demonstration.

(5) How changes in the variables impact correlation’s (depending on how they are calculated) should be useful for my students.

http://demonstrations.wolfram.com/CorrelationAndRegressionExplorer/

(6) Those of you who know me, are probably not surprised that I can’t just stop at 5 examples for this first semester … so here is a great demonstration on Power. Though I can get students to define power, and identify threats to power, I am never fully certain that they truly get the beauty (and hassle) of power. This demonstration may help.

http://demonstrations.wolfram.com/ThePowerOfATestConcerningTheMeanOfANormalPopulation/

Of course, without proper instruction during class time and an accompanying explanation following class instruction, these demonstrations may end up being little more than pretty pictures to students.

In a few weeks, especially after I actually try these demonstrations with my students, I will provide for you the information I attached with the demonstrations as well as feedback as to what worked and what didn’t. After all … anyone who has taught long enough knows, even the best planned lessons and demonstrations some times flop.

Though not specifically having to do with teaching statistics … I found a nice article at Chronicle of Higher Education on Iphones, Blackberries, etc … and apps that could help professors. The attendance and learning students’ names apps look promising. http://chronicle.com/article/College-20-6-Top-Smartphone/125764/

I look forward to hearing from any of you who have used Mathematica Demonstrations (or others) during class and for homework.

## How wonderful and I wish, I wish …

As I type this, I have one fifty minute class left to teach, and my time with my statistics class will be over. As with anything, each semester is varied. Some semesters I cover more information than other semesters. I liken this semester to driving through the city and hitting all green lights! As such, I believe my students were able to master additional information based on what is probably mostly good fortune.

So, here is my list of things I’m so thrilled I covered:

(1) Effect size statistics, like eta squared: Sure effect size statistics are not used that much, and lets face it, they are super easy to calculate, but my biggest reason for wanting to teach effect size statistics is it helps students to understand what a t-test or F-test can tell us (is there a difference) and what it can’t tell (how big is the effect). In fact, by spending about 20 minutes on the teaching of effect size statistics, students were better able to understand why the “p-value” for an observed t or F score provides us with no information. All we need to know is, did we pass the threshold.

(2) We find the critical value BEFORE calculating the observed value: This discussion helps focus student on the logic of statistical hypothesis testing. Specifically, statistical hypothesis testing works because we assume that the null hypothesis is true, that there is no effect of the independent variable on the dependent variable. With this assumption, we are able to generate the sampling distribution that provides us with information on the standard error. Now, if our sample mean is too extreme, we reject our initial hypothesis, the null, and accept the alternative hypothesis, that is the means are different. By finding the critical value prior to calculating the statistic, it helps focus students on that “line in the sand” to say … my observations are too extreme for me to stay with my current hypothesis. Students are far less likely to fall victim to equating p-value with the strength of the effect of the independent variable, or to conclude … the data is trending because I have a p-value of .07 or some other funky thing far too many people do with null hypothesis testing. By spending a bit more time on the steps involved in hypothesis testing, I think students are less likely to fall victim to the common misconceptions surrounding Statistical Null Hypothesis Testing.

(3) Though not a specific concept, I am pleased that for almost every concept I taught this semester I used new examples. Sure, I’m still a sage in training, no grey hair and all, but I was beginning to find myself using the same examples. As this is the third semester my supplement instructor, Amy, is taking notes in class, I felt I owed it to her, at least, to “keep it fresh.” I also found thinking about this blog helped spur my mind toward different examples. In doing so, I found some worked even better than my “old stand by” examples, but the great things was, when the new example flopped, I just quickly switched to the example I knew helped students.

Now for my Wish List of things I always wished I could have covered, but didn’t.

(1) Though I do get to cover the concepts of the F-test. I teach a three credit class, and only have time to cover the one-factor between subject ANOVA. If only I could cover a two-factor between subject ANOVA and a one-factor within subject ANOVA, I would feel my students would really understand the F-test (and as such, be less incline to misuse or over use it).

(2) Yet, I feel if I could cover non-parametrics, students would better understand the role of the assumptions in parametric tests, and issues like Power and random error could be even better understood. Plus they would get the benefit of learning about a really important class of statistics. Sadly, another semester has passed without me being able to cover this topic with the depth I think it deserves.

(3) I fear I don’t emphasize the weakness of statistics, and that they are only as good as the quality of the theories being tested in the design. They are also only as good as the quality of the sample and the quality of the measure. At least the latter two concepts get covered in classes that will follow the statistics class. But so few people speak of the topic of equifinity, that the same outcome can have multiple explanations. Again, though I touch on this, the idea of developing the alternative rival hypotheses that could explain the same empirical evidence is one I simply don’t have time to cover to the extent I would like. If you have a weak theory or haven’t taken into account the alternative rival hypotheses when designing your study, cool statistics will not improve the quality of your findings.

(4) Though I tell students the hypothesis drive everything, from the selection of the measure and research design, to the specific statistic one would select, and though there are example problems in the textbook (Integrating Your Knowledge) that students have to complete, I really wish we could spend more time on this.

Maybe next semester, I can find a way to reach my wish list … maybe!