1. HW1 due Wednesday 9/5/18; SPSS formatted data available here
   • Comments: For those unfamiliar with some of the concepts described in this first homework, here are some explanatory comments that might help. They are completely optional; only look at them if any confusion or curiosity arises.
   • HW1 Google Form
2. HW2 due Thursday 9/20/18; SPSS formatted data available here
   • Comments: Instead of clicking on OKAY when ready to run an analysis, remember you can click on PASTE to get the commands in a syntax window and run them from there. The advantage of that is you'll be able to simply copy and repaste the commands you just compiled to run the next analysis, and simply change the occurrences of, say, "1939" to "1970" as appropriate. Saves a lot of redundant clicking around if you're inclined to do that.
   • HW2 Google Form
3. HW3 due Friday 9/28/18; SPSS formatted data available here
   • Comments: This homework is about t-tests -- both independent samples and paired samples -- and though we haven't covered the topic in lecture yet it's probably trivial to follow the instructions and complete it with just a little info coming this week in class. Friday 9/28 is a convenient due date since homeworks are being submitted electronically instead of in class; if that date changes due to lecture being too out-of-sync, I'll let you know.
   • HW3 Google Form
4. HW4 due Friday 10/5/18
   • Comments: There is no SPSS component to this homework. Note that after the six homework questions, I've appended some extra questions listed as "THINGS THAT ARE NOT PART OF THIS ASSIGNMENT THAT YOU SHOULD THINK ABOUT ANYWAY". These are NOT part of the homework and should NOT be turned in, but may be helpful for you to work through in fully understanding the material.
   • HW4 Google Form
5. HW5 due Friday 10/19/18 (after Wednesday's exam); SPSS formatted data available here
   • Comments:
     Here is some SPSS translation that you either understand already or don't need for this homework, but which may be helpful to know about in the long run:
   • Note that SPSS gives different names to your Sources of Variance in the output: A = "group" (your independent variable name), S/A = "error". As we'll soon see, the sum of those two gives a Total for both SS and df, and the Total is listed in the output not as just plain "Total", but as "Corrected Total"!
     • The way those labels work is something like this. The "corrected model" row refers to the total of all the factors present in your experiment. For now we have only one factor (A) so that IS the whole model, thus the rows for "corrected model" and "group" have the same information. Soon enough we will also have a second factor (B) and its interaction with the first (A*B), and then the "corrected total" will refer to the three of those effects summed together, and each will be listed separately in its own row in place of the sole factor we now have called "group".
     • In SPSS, the so-called "total" SS (which is NOT the Total we're interested in!) computes the SS around an origin of zero, rather than around the grand mean of all the scores, and its degrees of freedom is the total number of observations. The "corrected total" (the one we ARE interested in!) finds the SS around the grand mean, which is after all an estimate of the population mean, and you may remember that in estimating a parameter from the data we lose a degree of freedom. And indeed, the df for the "corrected total" is the number of observations minus 1. You may think only the "corrected total" makes any sense - who bothers finding the "sum of squared deviations from zero" instead of "... from the mean"? And I agree with you completely, but read on...
     • The "intercept" represents the grand mean of all the observations, i.e., your estimate of the population grand mean, and it will almost always be highly significant, and will always have df = 1: that's the 1 df that you lost above by estimating the population grand mean from your data. What significance test are you doing on it? You're testing whether it's different from 0! Who knows why. Read p. 37 of Keppel and Wickens, you'll see that it's not especially useful or interesting, it's just there for some reason. This seems nonsensical to me, but... It has SS = 135.809 because if your grand mean were, say, 2.1277 (which it is on this homework), its squared deviation from a hypothesized mean of 0 would be 2.1277 squared or 4.5270, and if you summed that number over all 30 of your observations, well it'd be the same for each of them - there's only one grand mean so the 2.1277 and the 0 are the same for everyone. And 4.5270 x 30 = 135.81. Voilà! - and no one cares. But there it is (which is pretty much what "voilà" means in the first place; those of you who think the word is "viola" are beyond help). Notice that if you add the "intercept" SS and df to the "corrected total" SS and df, you get what SPSS labels the "total" SS and df.
     • Bottom line: it's the "corrected total" you'll care about all semester, so ignore the "intercept" and the (uncorrected) "total".
   • Why do we call the within-groups variance (S/A) effect the "error"? That's because it's the denominator of the F ratio, representing the experimental error (individual differences, measurement error, etc) that is the variability present among subjects who have all received the same treatment but still differ from each other. In more complicated designs the "error" term will not always be S/A; in fact, we will use different error terms to test different effects within the same experiment. Fun stuff.
   • The vertical axis of your means plot is labeled "Estimated Marginal Means", which you should just read as saying "the means of the groups"!
   • The output column labeled "Type III Sums Of Squares" is indeed your SS for each effect; why it's called "Type III" is best saved for the spring semester course on regression, though I'll be happy to share before then if you like. Let it be said that Type I SS may be of interest, but you will rarely if ever encounter Type II and Type IV. Don't worry, they're still calculated the same, it's just which data they're calculated from that might differ. For now, don't even think about it at all, just recognize that Type III is what we do here.
   • The R-squared value is printed underneath the "tests of between subjects effects" output box, and there's also something called "adjusted R-squared". The latter is an estimate of the population value that you may safely ignore until you look at R-squared in multiple regression next semester, at which point all will become clear.
   • HW5 Google Form
6. HW6 due Friday 10/26/18 SPSS formatted data available here AND here (you need BOTH HW6Af18.sav and HW6Bf18.sav!); the power analysis program GPower 3 is available here.
   • Comments:
   • HW6 Google Form
7. HW7 due Friday 11/9/18; SPSS formatted data available here
   • Comments: see the Analytic Contrasts summary for a review.
   • HW7 Google Form
8. HW8 due Friday 11/16/18; SPSS formatted data available here
   • Comments: see the Two Factor Design: Interactions and Main Effects summary.
   • HW8 Google Form
9. HW9 due Friday 11/30/18; SPSS formatted data available here
   • Comments:
   • HW9 Google Form
10. HW10 due Friday 12/7/18
    • Comments: see Recognizing Higher Order Interactions From Graphs And Means Tables.
      There is no data file because for once I'm letting you enter the data yourselves, because 1) there's very little of it and 2) welcome to real life. Except, in real life, you'd get an undergrad to do it. (And of course you'd make it a worthwhile overall educational experience for them.) Note three points about repeated measures designs on this homework: the additional new assumption for repeated measures designs called "sphericity"; Mauchly's test for sphericity which according to SPSS "tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix" (now you know!); and what to do in case of violations of sphericity (the Greenhouse-Geisser, Huynh-Feldt, and lower-bound corrections). Furthermore there are some post hoc tests to be done after the omnibus, but they're the easiest you've seen so far (equivalent to paired samples t-tests with a Bonferroni correction for alpha). OPTIONAL EXTRA CREDIT: The handout is worth looking at but is not required, because HW10's second question is only for extra credit, as described on the assignment. Don't stress over trying to figure out the three-way graphs if it's too annoying, but honestly, if you can assign some numbers for the cell means as the handout advises, you would find it a worthwhile exercise and possibly get 3 more points giving you a possible 13 points out of 10. Lucky!
    • HW10 Google Form
11. OPTIONAL 5 POINT EXTRA CREDIT HW11 due FRIDAY 12/14/18; SPSS formatted data available here
    • Comments: see Finding Sources of Variance (PDF).
      This is worth 5 points extra credit if you do it. You won't do it because you care about the points, though -- you'll do it because it'll take you through how to analyze a mixed design (containing both between and within subjects factors), which will be helpful in your data analysis future. Because it's optional, the due date is not until the Friday after the final exam so concentrate on studying first. If you don't have time after the exam, no worries, but if you can fit it in it won't be incredibly hard, just useful.
    • Actual comment on this homework from a tenured professor of psychology (who, okay, is also a good friend since grad school), after she emailed me for advice on analyzing a mixed design and I sent her this homework link: "You perfect person, you! This is the best homework handout ever. I need to come take your stat classes for fun and refresher-ness." How do you not want to do it now?
    • HW11 Google Form

Ten simple rules for structuring papers, which you might find to be sort of a useful guide to how to write a good research report. Not about APA style, but about what makes a good paper. Handy for your own reference and maybe for when you're trying to teach students how to do it when you're a TA, or later on, a graduate advisor.

Some miscellaneous topics explained: I wrote this for my undergrad research methods class (so when it refers to exams, it means THEIRS, not OURS!) but it's consistent with things I say in the grad course and some of it is useful to see addressed explicitly.

The Worst Example Of A Three Way Interaction Ever: This is a ludicrous example inspired by a ludicrous blog post and includes a ludicrous graph, and is still an accurate description of a three-way interaction in ANOVA.

How do you pronounce "Likert scale"?: The definitive answer.

Why the sample variance has a denominator of N-1 instead of N: a proof that dividing the sample sum of squares by N-1 instead of N gives an unbiased estimate (i.e. accurate in the long-run average) of the population variance. This is purely for the mathematically inclined -- others should steer clear. (Believe it or not, I've seen other proofs that are more complicated and thus probably more thorough.) The "expectation" operator notated as E(X) means roughly the long-run average of X or the mean of all X's in the population, but note that doesn't necessarily indicate a mean of some score -- X could be a variance for instance, and then E(X) would be the population value of that variance, as it is in this proof. If that helps clear anything up.
Here is an alternative proof from a book on mathematical statistics. Other pages from the same book follow but are unrelated to this topic.

Confidence Intervals in Howell ch. 7 pp. 181-183
Notes on the meaning and interpretation of Confidence Intervals: Howell's discussion is very good, so the somewhat lengthy little essay that I've included here is more than I intended to write; still, it may be helpful to hear it expressed in more than one way.

ANOVA vs Regression: this post describes some general situations in which you might prefer one to the other, even though they're equivalent expressions of the General Linear Model. Then there's a bit about why dichotomizing continuous variable scores (or grouping scores into more than two categories, as well) is not a great idea, and why it's better to treat continuous variables with regression-based techniques instead of trying to cram them into possibly more familiar ANOVA-type techniques.

An illustration of the three types of kurtosis which I've also incorporated into an informative web page about everyone's favorite monotreme.
Sadly it must be noted that the diagram is based on a traditional misunderstanding of what kurtosis looks like -- see the next link...

What kurtosis actually means: To correct the diagram presented above, this paper gets the message across in its first lines: "The incorrect notion that kurtosis somehow measures "peakedness" (flatness, pointiness or modality) of a distribution is remarkably persistent, despite attempts by statisticians to set the record straight. This article puts the notion to rest once and for all. Kurtosis tells you virtually nothing about the shape of the peak - its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution)." That is, leptokurtic really only means more observations in the tails relative to the normal distribution, and platykurtic means less in the tails. It concludes decisively, "[K]urtosis should never be defined in terms of peakedness. To do so is counterproductive to the aim of fostering statistical literacy. The relationship of peakedness with kurtosis is now officially over." It's a short article but between those sentences there's probably way more than you want to know: Westfall, P. H. (2014). Kurtosis as Peakedness, 1905-2014. R.I.P. The American Statistician, 68(3), 191-195.

Mean, Median, and Skew: Correcting a Textbook Rule: interesting to note that the cliché picture of how skewness draws the mean in the direction of the skew, and the median less so, is not always true, especially for discrete distributions. The point is probably made adequately if you just read the brief "Abstract" and "What To Teach" sections.

Reliability is described adequately here in Wikipedia, as are several types of validity -- among them Internal, External, Construct, and Statistical Conclusion validity. See especially the respective threats to each, for aspects of research designs to pay special attention to.

Distributions and distribution free tests: a brief listing of the most common sampling distributions, used to evaluate the probabilities of observing various values of a statistic calculated from a sample. When the assumptions of a statistical test aren't met, those distributions don't describe the probabilities of getting those values. Instead you should use a test that is free of any distribution assumptions. Hence the term "distribution-free" tests, or "non-parametric" in that they don't specify the parameters of the distribution ahead of time -- though those terms are not strictly synonymous. Some of the more popular distribution free tests are listed at the end as suggestions to try out, without further explanation.

Odds and Probabilities: a primer on definitions, interpretations, and calculations

How to talk about odds and odds ratios in English, duplicating the post from the chi-square section in the syllabus above

Exponents and logarithms a primer on some basic mathematics that comes up in statistical contexts such as: logarithmic data transformations; loglinear models of categorical data with multiple IV's; the log(odds) transformation in logistic regression; the log likelihood (or "deviance" or "-2LL") in model comparison analyses like Structural Equation Modeling.

The Secretary Problem, or how to choose a spouse. In case you're interested in the underlying math or something, apart from the illustration of how mathematical assumptions determine the applicability of models.

A diagram of a "quincunx", sometimes called a "Galton Board" after its inventor Francis Galton, which models the way multiple causation results in a normal distribution. It's a wooden board with pins inserted into it, and when a ball is dropped into the top it will bounce randomly either right or left at each pin it encounters. Most of the balls will bounce about an equal number of times in both directions, canceling out the left and right directions and landing in the middle. By chance, some of them will bounce to the left or the right more times, landing further from the middle. The end result is the accumulation of balls forming a normal distribution, which shows the decreasing likelihood of extreme patterns of bouncing (or of multiple causes all pushing the outcome in the same direction). Here's a video that shows a quincunx in action, where something more sand-like than ball-like is poured through the opening.

The opening scene of Rosencrantz And Guildenstern Are Dead by Tom Stoppard, in which an unlikely extended run of coin flips gives rise to some existential angst. Note that even though each coin flip is perfectly in line with the "laws" of probability, we still don't quite believe this run of events should occur. (The play is a modern comedic take on two minor characters from Shakespeare's Hamlet who are unwittingly involved in a plot to kill Hamlet; this 1966 update focuses on their misadventures before their own eventual deaths.)

Deriving the estimate of the standard error of the mean: something you don't need to be able to do at all but may be curious about, and if you are, it's explained clearly in section 10.17 of this text by Glass and Hopkins.

Bayes's Theorem article in Wikipedia: I'm pretty sure it's legitimate to phrase the theorem this way: The probability of A being true given that B is true is equal to the probability that B actually does occur due to A, divided by the probability that B actually does occur due to any possible reason it might occur -- that is, that B occurs at all under any circumstances. This denominator is sometimes expressed as the sum of two other probabilities: that B occurs due to A, and that B occurs due to every reason other than A, which do in fact account for all occurrences of B since "A and not-A" pretty much covers every possible reason for B. You can substitute the observations of interest into this formula: A = a hypothesis being true, and B = data bearing on that hypothesis. Examples listed on this link are pretty illuminating, if you follow them closely. The trick with Bayesian statistics is coming up with those probabilities that are the ingredients in the formula, e.g., of B occurring due to any possible reason -- it's educated guesswork at best (which can be pretty good after all).
Bayes's Theorem excerpt from Howell ch. 5: a very good basic treatment.

Always Use Welch's t-test instead of the traditional Student's t-test: the rationale for this recommendation is explained in this blog post, and a published reference is cited too. "The idea that a two-step procedure (first performing Levene's test, then deciding which test statistic to report) should be replaced by unconditionally reporting Welch's t-test is generally accepted by statisticians, but the fool-hardy stick-to-what-you-know 'I'm not going to change if others are not changing' researcher community seems a little slow to catch on. But not you." Also included: the formula for the modified df in the Welch test, since you're interested.

Logic Of ANOVA summary

Proof that the expected value of F isn't actually 1 but rather is df_error/(df_error-2), with bonus proof that t²=F. This is good for a laugh, or else for some extended highly motivated effort.

Understanding ANOVA Visually: a fun bit of Flash animation; related teaching tools are listed at http://old.psych.utah.edu/learn/statsampler.html

G*Power Home Page: free software for power calculations.

Some interesting demonstration apps can be found at this site, but many of them may no longer work due to Java being (appropriately) shunned by browsers these days. Among them is this Statistical Power Applet, a visual demonstration of the relations among the various quantities related to power (α, β, N, and effect size), which does require Java to be enabled in the browser.

Correlation article in Wikipedia: whether or not the math explained here is of interest (correlations as cosines, etc.), the two images depicting sets of scatterplots are very important to understand. One of the diagrams on this page shows some scatterplots and the correlation coefficients calculated from them, just to give you an idea of what typical correlations might look like, but also of how unpredictable they might be if you don't look at your data in a scatterplot. This point is made even more obvious by this diagram further down the same page known as Anscombe's quartet, which shows some very different sets of data that all give the exact same value for the correlation coefficient r (as well as for some other descriptive statistics). Along those lines, this game is also quite illuminating.

All the XKCD comics I always try to look up for a stats class (and then some)

Analytic Contrasts summary

Glossary of some statistical and experimental design terms: Many terms and concepts encountered in the latter part of the course are easily confusable, so I've tried to lay out their definitions and relationships here.

Keppel's ANOVA notation system (PDF)
Keppel's ANOVA notation system (Microsoft Word)
This is a handy summary of how to compute Degrees of Freedom for any Source of Variance. Keppel and Wickens (2004) use an ANOVA notation system that provides a simple way to compute Sums Of Squares: by converting Sources of Variance into Degrees of Freedom, and then into a combination of "bracketed" quantities, where the brackets indicate some further adding and dividing. But since no one in their right mind computes Sums Of Squares by hand, the only remaining useful part of this page is the part describing how to get Degrees of Freedom. That is quite useful though.

Recognizing Higher Order Interactions From Graphs And Means Tables

Finding Sources of Variance (PDF)
Finding Sources of Variance (Microsoft Word)

Expected Mean Squares (PDF)
Expected Mean Squares (Microsoft Word)

Excel spreadsheet for calculating values of the z, t, F, and chi-square distributions and their probabilities

Table of Selected Values of the t Distribution:

Table of Selected Values of the F Distribution:

• Cohen (1994).pdf: The Earth Is Round (p<.05).
• Cohen (1990).pdf: Things I Have Learned (So Far).
• Cowles & Davis (1982).pdf: On The Origins Of The .05 Level Of Statistical Significance.
• Wilkinson and APA Task Force (1999).pdf: Statistical Methods In Psychology Journals: Guidelines And Explanations.
• Kehle Bray Chafouleas Kawano (2007): Lack Of Statistical Significance [further criticism of significance testing]
• Killeen (2005) [pp. 15-18]: tea-Tests [an overview of the p-rep statistic or "probability of replication"]

• Howell Ch.4 [excellent and up-to-date treatment of the logic and controversies of hypothesis testing, possibly more accessible than Cohen's (1994) paper]
• Howell Ch.6 [excellent reference for chi-square and related topics]
• Howell Ch.7 [excellent treatment of the logic of the t-test, applied to the cases of a single sample mean, two related sample means, and two independent sample means; makes explicit the transition from hypothesis tests using the z distribution to those using the t distribution when the population standard error is unknown, as well as the basic similarity of all the t-test equations to the basic z-score formula; confidence intervals are described accurately on pp. 181-183]
• Howell Ch.9 Excerpt on some significance tests for correlation coefficients [describes how to test for significant differences between correlation coefficients using a t-test for two independent r's, for comparing r to a hypothesized population value, and for two non-independent r's]
• Howell Ch. 5 Excerpt on Bayes's Theorem [provides a brief accurate description of Bayes's Theorem, which is the only way to really go from the probability of the data given the null hypothesis (i.e., p-values), to the probability of the null hypothesis given the data]
• Howell Ch.15 Excerpts on Mediation & Moderation, and Logistic Regression [the earlier part is an alternative and/or supplement to the Keith pages and other listed resources on mediation and moderation; the latter part is an alternative and/or supplement to the Grimm and Yarnold chapter on Logistic Regression]