Jul 24, 2019 Let’s enrich our previous discussion of linear regression by considering outcome data that is binary/dichotomous (e.g., success/failure, whether someone recovered or not) and data that is count-based (e.g., number of distinct species per survey site). In a regular linear regression, we model an outcome $$Y$$ as a linear function of explanatory variables $$X_i$$ and error $$e$$, like this $Y=\beta_0+\beta_1X_1+\dots+\beta_nX_n+e, \phantom{xxxxx} e\sim N(0,\sigma)$ Let’s focus on the case of just a single predictor, ...
Jul 14, 2019 One Within-Subjects Factor Partitioning the Total Sum of Squares (SST) Naive analysis (not accounting for repeated measures) Mixed-effects model of same data Checking Assumptions Effect size One between, one within (a two-way split plot design) Two within-subjects factors Real Example Hello again! In previous posts I have talked about one-way ANOVA, two-way ANOVA, and even MANOVA (for multiple response variables). But in practice, there is yet another way of partitioning the total variance in the outcome that allows you to account for repeated measures on the same subjects. ...
Jul 1, 2019 MANOVA MANOVA stands for Multivariate (or Multiple) Analysis of Variance, and it’s just what it sounds like. You have multiple response variables, and you want to test whether any of them differ across levels of your explanatory variable(s) (i.e., your groups). The null hypothesis here is that the means of each response variable are equal at every level of the explanatory variable(s); the alternative is that at least one response variable differs. ...
Jun 19, 2019 A couple months ago a colleague and I volunteered for some outreach at a local science museum. It was for an 21+ night (i.e., lots of nerdy adults drinking beers together) and the theme was “Get Your Game On”. As we are both faculty in the Department of Statistics and Data Sciences, we were spoiled for choice, but finally decided to set up a Monty Hall station where people would come by and play a version of the notoriously confusing Let’s Make a Deal game (the original car/goat scenario). ...
May 18, 2019 Two-Way ANOVA Assumptions Partitioning the Total Sum of Squares ($$SS_T$$) Two-Way ANOVA Table Mechanics of Two-Way ANOVA Interactions in Two-Way ANOVA: 8 Possibilities Full Example: Effect of Vitamin C on Guinea Pig Tooth Growth Sum of Squares Calculations Example: Two-Way ANOVA, no Interaction Calculation of Dose $$\times$$ Supplement Interaction Effect Example: Two-Way ANOVA with Interaction Full ANOVA table for Guinea Pig example Post Hoc Tests (only do comparisons of interest! ...
Jan 11, 2019 Why do we use sample means? The sample mean, $$\bar x = \frac 1 N\sum x_i$$, is a workhorse of modern statistics. For example, t tests compare two sample means to judge if groups are likely different at the population level, and ANOVAs compare sample means of more groups to achieve something similar. But does $$\bar x$$ actually deserve the great stature is has inherited? I recently heard a talk by Dr. ...
Jan 7, 2019 Enemy Tank Problem: Estimating Population Size In wartime, it is advantageous to know how well-equipped your adversary is. Ideally, you would like to have full knowledge of their armory: exactly how many tanks, planes, and ships they have that are currently battle-ready. Unfortunately, there is no good way to get this information. Imagine, however, that you have managed to capture $$k$$ tanks from the enemy. Furthermore, each enemy tank has a serial number stamped on it. ...
May 9, 2018 Correlation demonstration! Regression Why normally distributed errors? Normal-errors assumption and homoskedasticity Confidence (and prediction) intervals for a given X What’s the relationship between correlation and regression? Multiple regression: multiple predictors Multiple $$R$$ and $$R^2$$ $$R^2$$ in one bang using correlation matricies Semipartial and Partial Correlations Semipartial correlations: the residuals approach Increments to $$R^2$$ when adding predictors Multicollinearity Regression assumptions Partial correlations .table { margin-left:auto; margin-right:auto; width: 60%; text-align:center; } #Correlation People are often interested in how different variables are related to each other. ...
Sep 4, 2017 What is an Experiment? Research in a Perfect World What is Validity? Internal Validity Threats to Internal Validity Random Assignment External Validity Threats to External Validity Construct Validity Threats to Construct Validity Statistical Validity Threats to Statistical Validity Power! Ways to Increase Power In this post I reboot a page I had written for my old website back in 2013. It is based almost entirely on the excellent text by Shadish, Cook, and Campbell (2002). ...
Aug 27, 2017 One-Way ANOVA by way of Two-Sample T-Test Calculations Sums of Squares Plot Effect Size Assumptions Post Hoc Tests One-Way ANOVA by way of Two-Sample T-Test If the hero of our last post was William “Student” Gosset, then the hero of this and the next few posts will be Sir Ronald Fisher. After Student had derived the t distribution, he sent it to Fisher along with an historically ironic note: “I am sending you a copy of Student’s Tables as you are the only man that’s ever likely to use them! ...