# Can a linear model reproduce a Welch t-test?

This doodle was motivated Jake Westfall’s answer to a Cross-Validated question. The short answer is yes but most R scripts that I’ve found on the web are unsatisfying because only the t-value reproduces, not the df and p-value. Jake notes the reason for this in his answer on Cross-Validated. To get the adjusted df, and the p-value associated with this, one can use the emmeans package by Russell Lenth, as he notes here.

# Normalization results in regression to the mean and inflated Type I error conditional on the reference values

Fig 1C of the Replication Study: Melanoma exosomes educate bone marrow progenitor cells toward a pro-metastatic phenotype through MET uses an odd (to me) three stage normalization procedure for the quantified western blots. The authors compared blot values between a treatment (shMet cells) and a control (shScr cells) using GAPDH to normalize the values. The three stages of the normalization are first, the value for the Antibody levels were normalized by the value of a reference (GAPDH) for each Set.

# A comment on the novel transformation of the response in " Senolytics decrease senescent cells in humans: Preliminary report from a clinical trial of Dasatinib plus Quercetin in individuals with diabetic kidney disease"

Motivation: https://pubpeer.com/publications/8DF6E66FEFAA2C3C7D5BD9C3FC45A2#2 and https://twitter.com/CGATist/status/1175015246282539009 tl;dr: Given the transformation done by the authors, for any response in day_0 that is unusually small, there is automatically a response in day_14 that is unusually big and vice-versa. Consequently, if the mean for day_0 is unusually small, the mean for day_14 is automatically unusually big, hence the elevated type I error with an unpaired t-test. The transformation is necessary and sufficient to produce the result (meaning even in conditions where a paired t-test isn’t needed, the transformation still produces elevated Type I error).

# What is the consequence of a Shapiro-Wilk test-of-normality filter on Type I error and Power?

Set up Normal distribution Type I error Power Right skewed continuous – lognormal What the parameterizations look like Type I error Power This 1990-wants-you-back doodle explores the effects of a Normality Filter – using a Shapiro-Wilk (SW) test as a decision rule for using either a t-test or some alternative such as a 1) non-parametric Mann-Whitney-Wilcoxon (MWW) test, or 2) a t-test on the log-transformed response.

# What is the bias in the estimation of an effect given an omitted interaction term?

Some background (due to Sewall Wright’s method of path analysis) Given a generating model: $$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3$$$ where $$x_3 = x_1 x_2$$; that is, it is an interaction variable. The total effect of $$x_1$$ on $$y$$ is $$\beta_1 + \frac{\mathrm{COV}(x_1, x_2)}{\mathrm{VAR}(x_1)} \beta_2 + \frac{\mathrm{COV}(x_1, x_3)}{\mathrm{VAR}(x_1)} \beta_3$$. If $$x_3$$ (the interaction) is missing, its component on the total efffect is added to the coefficient of $$x_1$$.

# Is the power to test an interaction effect less than that for a main effect?

I was googling around and somehow landed on a page that stated “When effect coding is used, statistical power is the same for all regression coefficients of the same size, whether they correspond to main effects or interactions, and irrespective of the order of the interaction”. Really? How could this be? The p-value for an interaction effect is the same regardless of dummy or effects coding, and, with dummy coding (R’s default), the power of the interaction effect is less than that of the coefficients for the main factors when they have the same magnitude, so my intuition said this statement must be wrong.

# Analyze the mean (or median) and not the max response

This is an update of Paired t-test as a special case of linear model and hierarchical model Figure 2A of the paper Meta-omics analysis of elite athletes identifies a performance-enhancing microbe that functions via lactate metabolism uses a paired t-test to compare endurance performance in mice treated with a control microbe (Lactobacillus bulgaricus) and a test microbe (Veillonella atypica) in a cross-over design (so each mouse was treated with both bacteria).
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#### R doodles. Some ecology. Some physiology. Much fake data.

Thoughts on R, statistical best practices, and teaching applied statistics to Biology majors.

Jeff Walker, Professor of Biological Sciences

University of Southern Maine, Portland, Maine, United States