Answer $11\mathrm{.}10$:

the exercises and what needs to be done step by step. We'll be working with the Palmer Penguins dataset to analyze the relationship between flipper length and body mass, without including interaction terms.

Here$$s how we$$ll proceed:

Exercise $11\mathrm{.}10($Penguins$!$ Main effects$)$

a$.$ Plot and summarize the observed relationships among these three variables.

We will start by plotting and summarizing the relationships between body mass, flipper length, and species. This can be done using scatter plots and summary statistics.

b$.$ Use R to simulate a posterior Normal regression model of body mass by flipper length and species, without an interaction term.

Here, we will fit a Bayesian regression model using the brms package in R$.$ The model will predict body mass using flipper length and species as predictors, but without any interaction terms.

c$.$ Create and interpret both visual and numerical diagnostics of your MCMC simulation.

We will generate diagnostic plots such as trace plots and posterior density plots to assess the convergence and performance of the MCMC simulation. Numerical diagnostics like the Gelman$-$Rubin statistic $($R$-$hat$)$ will also be used.

d$.$ Produce a summary of this model. Interpret the non$-$intercept coefficients' posterior median values in context.The model summary will provide the posterior distributions of the model parameters. We will interpret the coefficients for flipper length and species in terms of their impact on the body mass of the penguins.

e$.$ Simulate, plot, and describe the posterior predictive model for the body mass of an Adelie penguin that has a flipper length of $197$ mm$.$

Using the posterior samples, we will simulate the predicted body mass for an Adelie penguin with a flipper length of $197$ mm and visualize the distribution of these predictions.

Structure of the Submission:

$1)$Header:

Assignment Number: Assignment $11$

Week Number: Week $11$

Your Name: $[$Your Name$]$

$2)$ Sections and Subsections:

$11\mathrm{.}10$ Main Effects.

a$.$ Observed Relationships.

i$)$Plot$.$

ii$)$Summary Statistics.

b$.$ Bayesian Regression Model

i$)$Model Specification.

iModel Fitting.

c$.$ MCMC Diagnostics.

i$)$Trace Plots.

ii$)$Posterior Density Plots.

iii$)$Gelman$-$Rubin Statistic.

d$.$ Model Summary

i$)$Coefficients Interpretation.

e$.$ Posterior Predictive Model

i$)$Simulation$.$

ii$)$ Plot.

iii$)$Description$.$

Explanation:

$1)$References:

Reference sections from the textbook, lecture notes, and videos for priors and interpretation.

Code and Outputs:

We will use an R Markdown $(.$Rmd$)$ file to create the analysis, which will be rendered into an HTML document. The $.$Rmd file will contain all the code, explanations, outputs, and interpretations.

Let's start with the R code for each part of the exercise:

a$.$ Observed Relationships:

# Load necessary libraries

library$($ggplot$2)$

library$($dplyr$)$

library$($palmerpenguins$)$

# Plot observed relationships

ggplot$($penguins$,$ aes$($x $=$ flipper$\_$length$\_$mm$,$ y $=$ body$\_$mass$\_$g$,$ color $=$ species$))+$

geom$\_$point$()+$

labs$($title $=$ "Relationship between Flipper Length and Body Mass",

x $=$ "Flipper Length $($mm$)",$

y $=$ "Body Mass $($g$)")$

b$.$ Bayesian Regression Model:

# Load necessary libraries

library$($brms$)$

# Fit Bayesian regression model

model $<-$ brm$($body$\_$mass$\_$g ~ flipper$\_$length$\_$mm $+$ species, data $=$ penguins,

family $=$ gaussian$(),$ prior $=$ c$($set$\_$prior$("$normal$(0,10)",$ class $="$b$")))$

# Save the model

saveRDS$($model$,$ file $=$ "penguin$\_$model.rds$")$

c$.$ MCMC Diagnostics:

# Load model

model $<-$ readRDS$("$penguin$\_$model.rds$")$

# MCMC diagnostics

plot$($model$)$

d$.$ Model Summary

# Model summary

summary$($model$)$

e$.$ Posterior Predictive Model:

# Posterior predictive simulation for Adelie penguin with flipper length $197$ mm

newdata $<-$ data.frame$($flipper$\_$length$\_$mm $=197,$ species $=$ "Adelie"$)$

posterior$\_$predict$($model$,$ newdata $=$ newdata$)\%>\%$

as$.$data.frame$()\%>\%$

ggplot$($aes$($x $=$ value$))+$

geom$\_$histogram$($binwidth $=50)+$

labs$($title $=$ "Posterior Predictive Distribution of Body Mass for Adelie Penguin with Flipper Length $197$ mm$",$

x $=$ "Body Mass $($g$)",$

y $=$ "Frequency"$)$

You can then compile this into a structured R Markdown file and render it into HTML$.$ Let me know if you need further assistance or clarification on any part of this assignment.

AS PER ABOVE SITUATION, CREATE, CODE