Introduction

• We have discussed how to do regression analyses
  • Presenting analyses to communicate results just as important
  • Maximizes the impact of your work
• Introduced ways to communicate results, continue this discussion
  • Focus on metrics to use to quantify results from analyses

Correlation analyses

• Recall our previous visualizations

Correlation analyses

• Correlations are an example of an effect size metric
  • Metric has standardized units of measure of the relationship
  • Strength is same regardless of scale for \(X\), \(Y\)

\[ \text{Cor}(X,Y)=\frac{\text{Cov}(X,Y)}{\text{Var}(X)\text{Var}(Y)} \]

Group differences

• Can use T-test and F-test to evaluate pairwise differences or multi-group differences:
  • \(H_0: \mu_1=\mu_2\)
  • \(H_1: \mu_1\neq\mu_2\)

\[ T=\frac{\bar{X_1}-\bar{X_2}}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \sim \text{T}_{(n_1+n_2-2)} \]

• \(H_0: \mu_1=\mu_2=\ldots=\mu_K\)
• \(H_1: \text{At least one = is } \neq\)

\[ F=\frac{\text{between-group variance}}{\text{within-group variance}} \sim \text{F}_{(K-1, N-K)} \]

Group differences

• \(\bar{X_1}-\bar{X_2}\) depends on units of variables
  • How big of a difference is ``big’’?
  • Effect size: Cohen’s D

\[ d=\frac{\bar{X_1}-\bar{X_2}}{s_p} \]

• I.e., just standardizing difference by spread

Group differences

• How about for F-Test?
  • Cohen’s \(f^2\):

\[ \begin{align} &f^2=\frac{\eta^2}{1-\eta^2} \\ &\text{ where } \eta^2=\frac{SS_{group}}{SS_{total}}=\frac{SS_{group}}{SS_{group}+SS_{error}} \end{align} \]

• I.e., how much of the variability in variable is related to groups?
• Easy to calculate from lm function in R

Summary statistics

• Can create easily formatted summary stats tables in code
• Add in effect sizes using add_stat

Summary statistics in regression

• ANOVA

  • Recall: ANOVA = F-test for multi-group differences
  • Model:

\[ \begin{align} &MSEL = \beta_0+\beta_1*I(Group=\text{HR-ASD})+\beta_2*I(Group=\text{HR-Neg})+\epsilon \\ & I(Group=\text{x}) \text{ is dummy variable for group x} \\ & \rightarrow \text{LR is reference group} \end{align} \]

• Now can express group difference test in terms of \(\beta\)

\[ \begin{align} &H_0: \mu_{LR}=\mu_{HR-ASD}=\mu_{HR-Neg} \leftrightarrow\\ &H_0: \beta_0=\beta_0+\beta_1=\beta_0+\beta_2 \leftrightarrow\\ &H_0: \beta_1=\beta_2=0 \end{align} \]

• \(\rightarrow\) can use Cohen’s \(f^2\) for effect size

Summary statistics in regression

• General regression
  • Consider model

\[ MSEL = \beta_0+\beta_1*I(Group=\text{HR-ASD})+\beta_2*I(Group=\text{HR-Neg})+\beta_3*TCV +\epsilon \]

• How to define effect sizes for \(\beta\) estimates?
• Metrics:
  • Semi-partial correlation $ = _{y,x|z}$ using ppcor in R
  • Adjusted Cohen’s D $ = = $
  • \(R^2\) = \(\frac{SS_{total}-SS_{error}}{SS_{total}}\)
  • Recall sum of squares (SS) is

\[ \begin{align} &SS_{total}=\sum_{i=1}^{n}(y_i-\bar{y})^2 \\ &SS_{error}=\sum_{i=1}^{n}(\epsilon_i)^2 \end{align} \]

Summary statistics in regression

• Mixed models
  • Consider model

\[ MSEL = \beta_0+\beta_1*I(Group=\text{HR-ASD})+\beta_2*I(Group=\text{HR-Neg})+\beta_3*TCV+\beta_4*Age+\delta_{0,i}+\delta_{1,i}*Age +\epsilon \]

• Looking at changing MSEL over time by group and TCV
• Random effects: intercept (\(\delta_0\)), slope for age (\(\delta_1\)), residual (\(\epsilon\))
• How to compute effect sizes
  • Group differences $ = _{mixed} = $
  • Semi-partial marginal \(R^2\) from r2glmm in R

Regression diagnostics

• With modeling need to assess model fit
  • Residual distribution and variance
  • Outliers and their effects on results
• Can easily create visuals using ggfortify and olsrr

Regression diagnostics

Customized visualizations

• Can use flextable package with results stored as data frame
  • Create any table you want
• Can use ggplot with ggpubr to combine figures