Category Archives: ANOVA Analysis of Variance

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.

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Filed under ANOVA Analysis of Variance, correlation, Hypothesis Testing, Hypothesis Testing, Statistical Hypothesis Testing, t test

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!

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Filed under ANOVA Analysis of Variance, Core Concepts, Curriculum, effect size, Hypothesis Testing, Hypothesis Testing, non-parametric, Sampling Distribution, Statistical Hypothesis Testing, Statistical Tests, t test

Teaching hypothesis, design, analysis & inference as one thing

“So, the question I would like to pose for the sages and anyone else interested in commenting … for a first semester undergraduate applied statistics class … what are the most critical student learning outcomes that have to be mastered?”

First let me just comment on the blog Bonnie just posted today: I think her list of core concepts is excellent, and I agree that those concepts (all of which have to do with the ever-present error inherent in all our observations and measurements) should certainly be taught in the introductory statistics course.  Nevertheless, let me introduce a different perspective.

When I first saw the question posed by Bonnie  (reproduced above) I thought the answer would be an easy one to write. It turns out it is not quite so easy. My problem is that I see all the parts of the application of statistics as parts of an integrated whole. So my answer will appear to be a daunting one.

I hope that students can take away an appreciation (mastery would be too much to ask at this level) of how we use data to make inferences about the behavior under study. My typical homework problems were not, except for some initial ones, about calculations: finding means, t or F values, and p values. Rather an experimenter’s hypotheses would be stated along with how she collected the data to test those hypotheses, and (relatively simple) data would then be listed. The question posed was: what can you conclude from these data, and especially what can you conclude about the hypotheses? The appropriate way to answer such problems was to present the means and to interpret what the pattern tells us, with the statistical test of significance to guide us as to which differences we could assume due to the independent variable.

I understand that this is asking a lot of the students, but just getting statistics from data sets bores the heck out of me, and I don’t see why it would not be equally boring to the students. A few weeks into the semester we would be into the Analysis of Variance (Keppel’s book does a wonderful job facilitating early introduction of AOV), and the course especially emphasized factorial designs in which interpretation of patterns of means with the assistance of significance testing becomes, for me at least, most challenging and most interesting. The logic of the interplay of hypothesis, design, data, statistical analysis and inference is to me all one thing.

Such an integrated concept, satisfying to me, may or may not be an asset when applied to teaching the first undergraduate course in statistics.

 

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Filed under ANOVA Analysis of Variance, Core Concepts, Homework/ Assignments, Hypothesis Testing, Pedagogy, Significance (Statistical/ Practical), Statistical Hypothesis Testing