---
title: "ipsRdbs: Introduction to Probability, Statistics and R for Data-Based Sciences"
output: 
   BiocStyle::html_document
bibliography: REFERENCES.bib
number_sections: true
date: "`r format(Sys.Date(), format='%B %d,  %Y')`"
author: 
- name: Sujit K. Sahu
  affiliation:  University of Southampton
  email: S.K.Sahu@soton.ac.uk 
package: ipsRdbs
abstract: >  
 This is a vignette for the `R` package `r  BiocStyle::CRANpkg("ipsRdbs")`. This package contains data  sets, programmes and illustrations discussed in the book,  "Introduction to Probability, Statistics and R:
 Foundations for Data-Based Sciences" by Sahu (2023). 
 <br>
keywords: Anova, butterfly drawing, t-test, Factorial effects, Multiple Regression modelling, Simple linear regression. 
vignette: >
  %\VignetteEncoding{UTF-8}
  %\VignetteIndexEntry{ipsRdbs: Introduction to Probability, Statistics and R for Data-Based Sciences}
  %\VignetteEngine{knitr::rmarkdown}
---
```{r style, echo = FALSE, results = 'asis'}
  BiocStyle::markdown()
```
```{css, echo=FALSE}
.watch-out {
  background-color: lightpurple;
  border: 3px solid purple;
  font-weight: bold;
}
```

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  class.source="watch-out",
  comment = "#>")
```

```{r setup,  eval=TRUE, echo=FALSE, include=FALSE}
library(ggplot2)
require(ipsRdbs)
knitr::opts_chunk$set(eval = FALSE)

figpath <- system.file("figures", package = "ipsRdbs") 
print(figpath)
```

# Introduction 
This package complements the book, `Introduction to Probability, Statistics and R for Data-Based Sciences' by Sahu (2023). The package distributes the data sets used in the book and provides code illustrating the statistical modeling of the data sets. In addition, the package provides code for illustrating various results in probability and statistics. For example, it provides code to simulate the Monty python game illustrating conditional probability, and gives simulation based examples to illustrate the central limit theorem and the weak law of large numbers. Thus the package helps a beginner reader in enhancing understanding of a few elementary concepts in probability and statistics, and introduces them to perform linear statistical modelling, i.e., regression and ANOVA which are among the key foundational concepts of data science and machine learning, more generally data-based sciences.        

```{r coverplot, eval=TRUE, echo=FALSE, fig.cap="ipsRdbs book cover", fig.width =1.5, fig.height=2.5}
library(ipsRdbs)
figpath <- system.file("figures", package = "ipsRdbs") 
knitr::include_graphics(paste0(figpath, "/cover.jpg"))
```

<!-- A fuller version of this document with additional graphical illustrations 
<a href="https://www.sujitsahu.com/bmbook/bmstdr-full_vignette.html" > is also available.  </a> -->

## Installing the required software packages 

The reader is first instructed to install the `R` software package by searching for `CRAN` in the internet. The reader shoud then go onto the web-page <a href="https://cran.r-project.org/"> https://cran.r-project.org/ </a> and install the correct and latest version of the package on their own computer. Please note that `R` cannot be installed on a mobile phone. 
Once `R` has been installed, the next task is to install the frontend software package <a href="https://posit.co/download/rstudio-desktop/"> Rstudio, </a>
which provides an easier interface to work with `R`.

After installing `R` and `Rstudio`, the reader should launch the `Rstudio` programme in their computer. This will open up a four pane window with one named 'Console' or 'Terminal'. This window accepts commands (or our instructions) and prints results. For example, the reader may type 2+2 and then hit the Enter button to examine the result. The reader is asked to search the internet for gentler introductions and videos. 

In order to getting started here, thereader is aked to install the add-on `R` package `ipsRdbs` simply by issuing the `R` command
```{r echo=TRUE, eval=FALSE}
install.packages("ipsRdbs", dependencies=TRUE)
```
without committing any typing mistakes.  If this installation is successful, the reader can issue the following two commands to list all `R` objects (data sets and programmes) included in the package. 
```{r}
library(ipsRdbs)
ls("package:ipsRdbs")
```
Note that this command will only produce the intended results if only the package has been successfully installed in the first place. 

## How to learn more about the objects included in the package 

All the listed objects, as the output of the `ls` command in the previous section, have associated help files. The reader can gain information for each of those object by asking for help by typing the question mark immediately followed by the object name, e.g. `?butterfly` or by issuing the command `help(butterfly)`.  

The help files provide details about the objects and the user is able to run all the code included as illustrations at the end of the help file. This cam be done either by clicking the `Run Examples` link or simply by copy-pasting all the commands onto the command console in Rstudio. This is a great advantage of `R` as it allows the users to reproduce the results without having to learn all the commands and syntax correctly. After gaining this confidence, a beginner user can examine and experiment with the commands further. More details regarding the objects are provided in the book Sahu (2023).          


The remainder of this vignette simply elaborates the help files for all the main  objects and programmes included in the package. The main intention here is to enable the reader to reproduce all the results by actually running the commands and the code included already included in the help files.     

Section \@ref(data-sets) discusses all the data sets.
All the `R` functions are discussed in \@ref(illustrated-r-functions).  
Some summary remarks are provided in Section \@ref(discussion). 



# Data sets

## `beanie`: Age and value of  beanie baby toys.
  
This data set contains the age and the value of 50 beanie baby toys. Source: Beanie world magazine. This data set has been  used as an example of simple linear regression modellinhg where the exercise is to predict the value of a beanie baby toy by knowing it's age. 

```{r, eval=TRUE}
head(beanie)
summary(beanie)
plot(beanie$age, beanie$value, xlab="Age", ylab="Value", pch="*", col="red")
```

## `bill`:  Wealth and age of world billionaires
This data set contains wealth, age and region of 225 billionaires in 1992 as reported in the Fortune magazine. This data set can be  used to illustrate exploratory data analysis by producing side-by-side box plots of wealth for billionaires from different continents of the world.  It can also be used for multiple linear regression models, although  such tasks have 
not been undertaken here.   


```{r bill, eval=TRUE}
head(bill)
summary(bill)
library(ggplot2)
gg <- ggplot2::ggplot(data=bill, aes(x=age, y=wealth)) +
geom_point(aes(col=region, size=wealth)) +
geom_smooth(method="loess", se=FALSE) +
xlim(c(7, 102)) +
ylim(c(1, 37)) +
labs(subtitle="Wealth vs Age of Billionaires",
y="Wealth (Billion US $)", x="Age",
title="Scatterplot", caption = "Source: Fortune Magazine, 1992.")
plot(gg)
```

## `bodyfat` : Body fat percentage and skinfold thickness of athletes 

This data set contains body fat percentage data for 102 elite male athletes
training at the Australian Institute of Sport. This data set has been used to illustrate simple linear regression in Chapter 17 of the book by Sahu (2023). 

```{r bodyfat, eval=TRUE}
summary(bodyfat)
plot(bodyfat$Skinfold,  bodyfat$Bodyfat, xlab="Skin", ylab="Fat")
plot(bodyfat$Skinfold,  log(bodyfat$Bodyfat), xlab="Skin", ylab="log Fat")
plot(log(bodyfat$Skinfold),  log(bodyfat$Bodyfat), xlab="log Skin", ylab="log Fat")
# Keep the transformed variables in the data set 
bodyfat$logskin <- log(bodyfat$Skinfold)
bodyfat$logbfat <- log(bodyfat$Bodyfat)
bodyfat$logskin <- log(bodyfat$Skinfold)
 # Create a grouped variable 
bodyfat$cutskin <- cut(log(bodyfat$Skinfold), breaks=6) 
boxplot(data=bodyfat, Bodyfat~cutskin, col=2:7)
require(ggplot2)
p2 <- ggplot(data=bodyfat, aes(x=cutskin, y=logbfat)) + 
geom_boxplot(col=2:7) + 
stat_summary(fun=mean, geom="line", aes(group=1), col="blue", linewidth=1) +
labs(x="Skinfold", y="Percentage of log bodyfat", 
title="Boxplot of log-bodyfat percentage vs grouped log-skinfold")  
plot(p2)

n <- nrow(bodyfat)
x <- bodyfat$logskin
y <- bodyfat$logbfat
xbar <- mean(x)
ybar <- mean(y)
sx2 <- var(x)
sy2 <- var(y)
sxy <- cov(x, y)
r <- cor(x, y)
print(list(n=n, xbar=xbar, ybar=ybar, sx2=sx2, sy2=sy2, sxy=sxy, r=r))
hatbeta1 <- r * sqrt(sy2/sx2) # calculates estimate of the slope
hatbeta0 <- ybar - hatbeta1 * xbar # calculates estimate of the intercept
rs <-  y - hatbeta0 - hatbeta1 * x  # calculates residuals
s2 <- sum(rs^2)/(n-2)  # calculates estimate of sigma2
s2
bfat.lm <- lm(logbfat ~ logskin, data=bodyfat)
### Check the diagnostics 
plot(bfat.lm$fit, bfat.lm$res, xlab="Fitted values", ylab = "Residuals", pch="*")
abline(h=0)
### Should be a random scatter
qqnorm(bfat.lm$res, col=2)
qqline(bfat.lm$res, col="blue")

# All Points should be on the straight line 
summary(bfat.lm)
anova(bfat.lm)
plot(bodyfat$logskin,  bodyfat$logbfat, xlab="log Skin", ylab="log Fat")
abline(bfat.lm, col=7)
title("Scatter plot with the fitted Linear Regression line")
# 95% CI for beta(1)
# 0.88225 + c(-1, 1) * qt(0.975, df=100) *  0.02479 
# round(0.88225 + c(-1, 1) * qt(0.975, df=100) *  0.02479, 2)
# To test H0: beta1 = 1. 
tstat <- (0.88225 -1)/0.02479 
pval <- 2 * (1- pt(abs(tstat), df=100))
x <- seq(from=-5, to=5, length=500)
y <- dt(x, df=100)
plot(x, y,  xlab="", ylab="", type="l")
title("T-density with df=100")
abline(v=abs(tstat))
abline(h=0)
x1 <- seq(from=abs(tstat), to=10, length=100)
y1 <- rep(0, length=100)
x2 <- x1
y2 <- dt(x1, df=100)
segments(x1, y1, x2, y2)
abline(h=0)
# Predict at a new value of Skinfold=70
# Create a new data set called new
newx <- data.frame(logskin=log(70))
a <- predict(bfat.lm, newdata=newx, se.fit=TRUE) 
# Confidence interval for the mean of log bodyfat  at skinfold=70
a <- predict(bfat.lm, newdata=newx, interval="confidence") 
# a
#          fit      lwr     upr
# [1,] 2.498339 2.474198 2.52248
# Prediction interval for a future log bodyfat  at skinfold=70
a <- predict(bfat.lm, newdata=newx, interval="prediction") 
a
#          fit      lwr      upr
# [1,] 2.498339 2.333783 2.662895
#prediction intervals for the mean 
pred.bfat.clim <- predict(bfat.lm, data=bodyfat, interval="confidence")
#prediction intervals for future observation
pred.bfat.plim <- suppressWarnings(predict(bfat.lm, data=bodyfat, interval="prediction"))
plot(bodyfat$logskin,  bodyfat$logbfat, xlab="log Skin", ylab="log Fat")
abline(bfat.lm, col=5)
lines(log(bodyfat$Skinfold), pred.bfat.clim[,2], lty=2, col=2) 
lines(log(bodyfat$Skinfold), pred.bfat.clim[,3], lty=2, col=2) 
lines(log(bodyfat$Skinfold), pred.bfat.plim[,2], lty=4, col=3) 
lines(log(bodyfat$Skinfold), pred.bfat.plim[,3], lty=4, col=3) 
title("Scatter plot with the fitted line and prediction intervals")
symb <- c("Fitted line", "95% CI for mean", "95% CI for observation")
## legend(locator(1), legend = symb, lty = c(1, 2, 4), col = c(5, 2, 3))
# Shows where we predicted earlier 
abline(v=log(70))
summary(bfat.lm)
anova(bfat.lm)
```

## `bombhits`: Number and frequency of bombhits in London 

This data set contains the number of bomb hits in 576 areas in London during World War II. Data sourced from Shaw and Shaw (2019), see also Hand {\it et al.} (1993). [@Handetal1993] [@bombhits] This data set has been used to illustrate elementary concepts of statistical inference in Chapter 9 of the book Sahu (2023). 

```{r bombhits, eval=TRUE}
 summary(bombhits)
 # Create a vector of data 
 x <- c(rep(0, 229), rep(1, 211), rep(2, 93), rep(3, 35), rep(4, 7), 5)
 y <- c(229, 211, 93, 35, 7, 1) # Frequencies 
 rel_freq <- y/576
 xbar <- mean(x)
 pois_prob <- dpois(x=0:5, lambda=xbar)
 fit_freq <- pois_prob * 576
  #Check 
  cbind(x=0:5, obs_freq=y, rel_freq =round(rel_freq, 4),  
  Poisson_prob=round(pois_prob, 4), fit_freq=round(fit_freq, 1))
  obs_freq <- y
  xuniques <- 0:5
  a <- data.frame(xuniques=0:5, obs_freq =y, fit_freq=fit_freq)
  barplot(rbind(obs_freq, fit_freq), 
  args.legend = list(x = "topright"), 
  xlab="No of bomb hits",  
  names.arg = xuniques,  beside=TRUE, 
  col=c("darkblue","red"), 
  legend =c("Observed", "Fitted"), 
  main="Observed and Poisson distribution fitted frequencies 
  for the number of bomb hits in London")
```

## `cement`: Breaking strength of cement  

Contains data regarding breaking strength of cement. This data set has been used to illustrate analysis of variance in Chapter 19 of the book Sahu (2023). 

```{r, eval=TRUE}
summary(cement)
boxplot(data=cement, strength~gauger, col=1:3)
boxplot(data=cement, strength~breaker, col=1:3)
```

## `cfail`: Number of weekly computer failures

This data set contains weekly number of failures of a university computer 
system over a period of two years. Source: Hand {\it et al.} (1993). 
[@Handetal1993] Like the `bombhits` example this data set has been used to illustrate elementary concepts of statistical inference in Chapter 9 of the book Sahu (2023). 

## `cheese` : Taste of cheese 

This data set is from a testing of cheese experiment. 
This data set has been used to illustrate multiple linear regression modeling 
in Chapter 18 of the book Sahu (2023). 

```{r, eval=TRUE, message=FALSE, warning=FALSE}
summary(cheese)
pairs(cheese)
GGally::ggpairs(data=cheese)
cheese.lm <- lm(Taste ~ AceticAcid +  LacticAcid + logH2S, data=cheese, subset=2:30)
 # Check the diagnostics 
 plot(cheese.lm$fit, cheese.lm$res, xlab="Fitted values", ylab = "Residuals")
 abline(h=0)
 # Should be a random scatter
 qqnorm(cheese.lm$res, col=2)
 qqline(cheese.lm$res, col="blue")
 summary(cheese.lm)
 cheese.lm2 <- lm(Taste ~ LacticAcid + logH2S, data=cheese)
 # Check the diagnostics 
 plot(cheese.lm2$fit, cheese.lm2$res, xlab="Fitted values", ylab = "Residuals")
 abline(h=0)
 qqnorm(cheese.lm2$res, col=2)
 qqline(cheese.lm2$res, col="blue")
 summary(cheese.lm2)
 # How can we predict? 
 newcheese <- data.frame(AceticAcid = 300, LacticAcid = 1.5, logH2S=4)
 cheese.pred <- predict(cheese.lm2, newdata=newcheese, se.fit=TRUE)
 cheese.pred
 # Obtain confidence interval 
 cheese.pred$fit + c(-1, 1) * qt(0.975, df=27) * cheese.pred$se.fit
 # Using R to predict  
 cheese.pred.conf.limits <- predict(cheese.lm2, newdata=newcheese, interval="confidence")
 cheese.pred.conf.limits
 # How to find prediction interval 
 cheese.pred.pred.limits <- predict(cheese.lm2, newdata=newcheese, interval="prediction")
 cheese.pred.pred.limits
```

## `emissions` : Exhaust emissions of cars  

Data on the nitrous oxide content of exhaust emissions from a set of 
cars was collected by the Australian Traffic Accident Research Bureau 
to explore the relationship between several measures of nitrous 
oxide emissions. Like the `cheese` data set, this has been used to illustrate multiple linear regression modeling  in Chapter 18 of the book Sahu (2023). 

```{r emissions, eval=TRUE}
summary(emissions)
 
 rawdata <- emissions[, c(8, 4:7)]
 pairs(rawdata)
# Fit the model on the raw scale 
raw.lm <- lm(ADR37 ~ ADR27 + CS505  + T867 + H505, data=rawdata) 
old.par <- par(no.readonly = TRUE)
par(mfrow=c(2,1))
plot(raw.lm$fit, raw.lm$res,xlab="Fitted values",ylab="Residuals", main="Anscombe plot") 
abline(h=0)
qqnorm(raw.lm$res,main="Normal probability plot", col=2)
qqline(raw.lm$res, col="blue")
# summary(raw.lm)
logdata <- log(rawdata)
# This only logs the values but not the column names!
# We can use the following command to change the column names or you can use
# fix(logdata) to do it. 
dimnames(logdata)[[2]] <- c("logADR37", "logCS505", "logT867", "logH505", "logADR27")
pairs(logdata)
log.lm <- lm(logADR37 ~ logADR27 + logCS505  + logT867 + logH505, data=logdata) 
plot(log.lm$fit, log.lm$res,xlab="Fitted values",ylab="Residuals", main="Anscombe plot") 
abline(h=0)
qqnorm(log.lm$res,main="Normal probability plot", col=2)
qqline(log.lm$res, col="blue")
summary(log.lm)
log.lm2 <- lm(logADR37 ~ logADR27 + logT867 + logH505, data=logdata) 
summary(log.lm2)
plot(log.lm2$fit, log.lm2$res,xlab="Fitted values",ylab="Residuals", main="Anscombe plot") 
abline(h=0)
qqnorm(log.lm2$res,main="Normal probability plot", col=2)
qqline(log.lm2$res, col="blue")
par(old.par)
#####################################
# Multicollinearity Analysis 
######################################
mod.adr27 <-  lm(logADR27 ~ logT867 + logCS505 + logH505, data=logdata) 
summary(mod.adr27) # Multiple R^2 = 0.9936,
mod.t867 <-  lm(logT867 ~ logADR27 + logH505 + logCS505, data=logdata)  
summary(mod.t867) # Multiple R^2 = 0.977,
mod.cs505 <-  lm(logCS505 ~ logADR27 + logH505 + logT867, data=logdata)  
summary(mod.cs505) # Multiple R^2 = 0.9837,
mod.h505 <-  lm(logH505 ~ logADR27 + logCS505 + logT867, data=logdata)  
summary(mod.h505) # Multiple R^2 = 0.5784,
# Variance inflation factors 
vifs <- c(0.9936, 0.977, 0.9837, 0.5784)
vifs <- 1/(1-vifs) 
#Condition numbers 
X <- logdata 
# X is a copy of logdata 
X[,1] <- 1
# the first column of X is 1
# this is for the intercept 
X <- as.matrix(X) 
# Coerces X to be a matrix
xtx <- t(X) %*% X # Gives X^T X
eigenvalues <- eigen(xtx)$values
kappa <- max(eigenvalues)/min(eigenvalues)
kappa <- sqrt(kappa)
# kappa = 244 is much LARGER than 30!

### Validation statistic
# Fit the log.lm2 model with the first 45 observations  
# use the fitted model to predict the remaining 9 observations 
# Calculate the mean square error validation statistic 
log.lmsub <- lm(logADR37 ~ logADR27 + logT867 + logH505, data=logdata, subset=1:45) 
# Now predict all 54 observations using the fitted model
mod.pred <- predict(log.lmsub, logdata, se.fit=TRUE) 
mod.pred$fit # provides all the 54 predicted values 
logdata$pred <- mod.pred$fit
# Get only last 9 
a <- logdata[46:54, ]
validation.residuals <- a$logADR37 - a$pred  
validation.stat <- mean(validation.residuals^2)
validation.stat
```

## `err_age` : Error in guessing ages from photographs 

This data set contains the errors in guessing ages of 10 Southampton 
mathematicians. This data set has been used as an exercise in obtaining summary statistics and performing exploratory data analysis in Chapter 2 of the book 
Sahu (2023). 

```{r, eval=TRUE}
summary(err_age)
```

## `ffood` : Service times in a fast food restaurant  

Service (waiting) times (in seconds) of customers at a fast-food restaurant. Source: Unknown. 
```{r ffood, eval=TRUE}
summary(ffood)
 # 95% Confidence interval for the mean waiting time usig t-distribution
 a <- c(ffood$AM, ffood$PM)
 mean(a) + c(-1, 1) * qt(0.975, df=19) * sqrt(var(a))/sqrt(20) 
 # Two sample t-test for the difference between morning and afternoon times
 t.test(ffood$AM, ffood$PM)
```

## `gasmileage` : Gas mileage of cars 

This data set contains gas mileage obtained from four models of a car. 
This data set has been used to illustrate the concepts of analysis of variance  
in Chapter 19 of the book Sahu (2023). 

```{r gasmileage , eval=TRUE}
summary(gasmileage)
y <- c(22, 26,  28, 24, 29,   29, 32, 28,  23, 24)
xx <- c(1,1,2,2,2,3,3,3,4,4)
# Plot the observations 
plot(xx, y, col="red", pch="*", xlab="Model", ylab="Mileage")
# Method1: Hand calculation 
ni <- c(2, 3, 3, 2)
means <- tapply(y, xx, mean)
vars <- tapply(y, xx, var)
round(rbind(means, vars), 2)
sum(y^2) # gives 7115
totalSS <- sum(y^2) - 10 * (mean(y))^2 # gives 92.5 
RSSf <- sum(vars*(ni-1)) # gives 31.17 
groupSS <- totalSS - RSSf # gives 61.3331.17/6
meangroupSS <- groupSS/3 # gives 20.44
meanErrorSS <- RSSf/6 # gives 5.194
Fvalue <- meangroupSS/meanErrorSS # gives 3.936 
pvalue <- 1-pf(Fvalue, df1=3, df2=6)

#### Method 2: Illustrate using dummy variables
#################################
#Create the design matrix X for the full regression model
g <- 4
n1 <- 2 
n2 <- 3
n3 <- 3
n4 <- 2
n <- n1+n2+n3+n4
X <- matrix(0, ncol=g, nrow=n)       #Set X as a zero matrix initially
X[1:n1,1] <- 1    #Determine the first column of X
X[(n1+1):(n1+n2),2] <- 1   #the 2nd column
X[(n1+n2+1):(n1+n2+n3),3] <- 1    #the 3rd
X[(n1+n2+n3+1):(n1+n2+n3+n4),4] <- 1    #the 4th 
#################################
####Fitting the  full model####
#################################
#Estimation
XtXinv <- solve(t(X)%*%X)
betahat <- XtXinv %*%t(X)%*%y   #Estimation of the coefficients
Yhat <- X%*%betahat   #Fitted Y values
Resids <- y - Yhat   #Residuals
SSE <- sum(Resids^2)   #Error sum of squares
S2hat <- SSE/(n-g)   #Estimation of sigma-square; mean square for error
Sigmahat <- sqrt(S2hat)

##############################################################
####Fitting the reduced model -- the 4 means are equal #####
##############################################################
Xr <- matrix(1, ncol=1, nrow=n)
kr <- dim(Xr)[2]
#Estimation
Varr <- solve(t(Xr)%*%Xr)
hbetar <- solve(t(Xr)%*%Xr)%*%t(Xr)%*% y   #Estimation of the coefficients
hYr = Xr%*%hbetar   #Fitted Y values
Resir <- y - hYr   #Residuals
SSEr <- sum(Resir^2)   #Total sum of squares
###################################################################
####F-test for comparing the reduced model with the full model ####
###################################################################
FStat <- ((SSEr-SSE)/(g-kr))/(SSE/(n-g))  #The test statistic of the F-test
alpha <- 0.05
Critical_value_F <- qf(1-alpha, g-kr,n-g)  #The critical constant of F-test
pvalue_F <- 1-pf(FStat,g-kr, n-g)   #p-value of F-test

modelA <- c(22, 26)
modelB <- c(28, 24, 29)
modelC <- c(29, 32, 28)
modelD <- c(23, 24)

SSerror = sum( (modelA-mean(modelA))^2 ) + sum( (modelB-mean(modelB))^2 ) 
+ sum( (modelC-mean(modelC))^2 ) + sum( (modelD-mean(modelD))^2 )
SStotal <-  sum( (y-mean(y))^2 ) 
SSgroup <- SStotal-SSerror

####
#### Method 3: Use the  built-in function lm directly

#####################################
aa <- "modelA"
bb <- "modelB"
cc <- "modelC"
dd <- "modelD"
Expl <- c(aa,aa,bb,bb,bb,cc,cc,cc,dd,dd)
is.factor(Expl)
Expl <- factor(Expl)
model1 <- lm(y~Expl)
summary(model1)      
anova(model1)
###Alternatively ###

xxf <- factor(xx)
is.factor(xxf)
model2 <- lm(y~xxf)
summary(model2)
anova(model2)
```

## `possum` : Body weight and length of possums in Australian regions  
This data set has been used to illustrate multiple linear regression modeling and analysis of variance  in Chapter 19 of the book Sahu (2023). 

The data set contains body weight and length of possums (tree living furry animals who are mostly nocturnal (marsupial) caught in 7 different regions of Australia.
Source: Lindenmayer and Donnelly (1995). [@Lindenmayer1995]

```{r possum, eval=TRUE}
 head(possum)
 dim(possum)
 summary(possum)
 ## Code lines used in the book
 ## Create a new data set   
 poss <- possum 
 poss$region <- factor(poss$Location)
 levels(poss$region) <- c("W.A", "S.A", "N.T", "QuL", "NSW", "Vic", "Tas")
 colnames(poss)<-c("y","z","Location", "x")
 head(poss)
 # Draw side by side boxplots 
 boxplot(y~x, data=poss, col=2:8, xlab="region", ylab="Weight")
 # Obtain scatter plot 
 # Start with a skeleton plot 
 plot(poss$z, poss$y, type="n", xlab="Length", ylab="Weight")
 # Add points for the seven regions
 for (i in 1:7) {
    points(poss$z[poss$Location==i],poss$y[poss$Location==i],type="p", pch=as.character(i), col=i)
    }
## Add legends 
 legend(x=76, y=4.2, legend=paste(as.character(1:7), levels(poss$x)),  lty=1:7, col=1:7)
 # Start  modelling 
 #Fit the model with interaction. 
 poss.lm1<-lm(y~z+x+z:x,data=poss)
 summary(poss.lm1)
 plot(poss$z, poss$y,type="n", xlab="Length", ylab="Weight")
 for (i in 1:7) {
 lines(poss$z[poss$Location==i],poss.lm1$fit[poss$Location==i],type="l",
 lty=i, col=i, lwd=1.8)
 points(poss$z[poss$Location==i],poss$y[poss$Location==i],type="p",
 pch=as.character(i), col=i)
 }
 poss.lm0 <- lm(y~z,data=poss)
 abline(poss.lm0, lwd=3, col=9)
 # Has drawn the seven parallel regression lines
 legend(x=76, y=4.2, legend=paste(as.character(1:7), levels(poss$x)), 
 lty=1:7, col=1:7)
 
 n <- length(possum$Body_Weight)
 # Wrong model since Location is not a numeric covariate 
 wrong.lm <- lm(Body_Weight~Location, data=possum)
 summary(wrong.lm)
 
 nis <- table(possum$Location)
 meanwts <- tapply(possum$Body_Weight, possum$Location, mean)
 varwts <- tapply(possum$Body_Weight, possum$Location, var)
 datasums <- data.frame(nis=nis, mean=meanwts, var=varwts)
 datasums <- data.frame(nis=nis, mean=meanwts, var=varwts)
 modelss <- sum(datasums[,2] * (meanwts - mean(meanwts))^2)
 residss <- sum( (datasums[,2] - 1) * varwts)
 
 fvalue <- (modelss/6) / (residss/94)
 fcritical <- qf(0.95, df1= 6, df2=94)
 x <- seq(from=0, to=12, length=200)
 y <- df(x, df1=6, df2=94)
 plot(x, y, type="l", xlab="", ylab="Density of F(6, 94)", col=4)
 abline(v=fcritical, lty=3, col=3)
 abline(v=fvalue, lty=2, col=2)
 pvalue <- 1-pf(fvalue, df1=6, df2=94)
 
 ### Doing the above in R
 # Convert  the Location column to a factor
 
 localpossum <- possum
 localpossum$Location <- as.factor(localpossum$Location)
 summary(localpossum)  # Now Location is a factor 
  
 # Put the identifiability constraint:
 options(contrasts=c("contr.treatment", "contr.poly"))
 # Change to have easier column names 
 colnames(localpossum) <- c("y", "z", "x")
 # Fit model M1
 possum.lm1 <- lm(y~x, data=localpossum)
 summary(possum.lm1)
 anova(possum.lm1)
 possum.lm2 <- lm(y~z, data=localpossum)
 summary(possum.lm2)
 anova(possum.lm2)
 # Include both location and length but no interaction 
 possum.lm3 <-  lm(y~x+z, data=localpossum)
 summary(possum.lm3)
 anova(possum.lm3)
 # Include interaction effect 
 possum.lm4 <-  lm(y~x+z+x:z, data=localpossum)
 summary(possum.lm4)
 anova(possum.lm4)
 anova(possum.lm2, possum.lm3)
 #Check the diagnostics for M3
 plot(possum.lm3$fit, possum.lm3$res,xlab="Fitted values",ylab="Residuals", 
 main="Anscombe plot")
 abline(h=0)
 qqnorm(possum.lm3$res,main="Normal probability plot", col=2)
 qqline(possum.lm3$res, col="blue")
 rm(localpossum)
 rm(poss)
```


## `puffin` : Nesting habits of puffins in Newfoundland 

This object contains data regarding nesting habits of common puffin. Source: Nettleship (1972).  [@puffindata]
This data set has been used to illustrate multiple linear regression modeling 
in Chapter 18 of the book Sahu (2023). 

```{r puffin, eval=TRUE}
head(puffin)
dim(puffin)
summary(puffin)
pairs(puffin)
puffin$sqrtfreq <- sqrt(puffin$Nesting_Frequency)
puff.sqlm <- lm(sqrtfreq~ Grass_Cover + Mean_Soil_Depth + Slope_Angle 
+Distance_from_Edge, data=puffin) 
old.par <- par(no.readonly = TRUE)
par(mfrow=c(2,1))
qqnorm(puff.sqlm$res,main="Normal probability plot", col=2)
qqline(puff.sqlm$res, col="blue")
plot(puff.sqlm$fit, puff.sqlm$res,xlab="Fitted values",ylab="Residuals", 
main="Anscombe plot", col="red")
abline(h=0)
summary(puff.sqlm)
par(old.par)
#####################################
# F test for two betas at the  same time: 
######################################
puff.sqlm2 <- lm(sqrtfreq~ Mean_Soil_Depth + Distance_from_Edge, data=puffin) 
anova(puff.sqlm)
anova(puff.sqlm2)
fval <-  1/2*(14.245-12.756)/0.387 # 1.924 
qf(0.95, 2, 33) # 3.28
1-pf(fval, 2, 33) # 0.162
anova(puff.sqlm2, puff.sqlm)
```

## `rice` : data set on rice yield

This data set has been used to illustrate multiple linear regression modeling 
in Chapter 18 of the book Sahu (2023).  This data set has been obtained from the journal research article Bal and Ojha (1975). [@BAL1975353]

```{r rice, eval=TRUE}
 summary(rice)
 plot(rice$Days, rice$Yield, pch="*", xlab="Days", ylab="Yield")
 rice$daymin31 <- rice$Days-31
 rice.lm <- lm(Yield ~ daymin31, data=rice)
 summary(rice.lm)
 # Check the diagnostics 
 plot(rice.lm$fit, rice.lm$res, xlab="Fitted values", ylab = "Residuals")
 abline(h=0)
 # Should be a random scatter
 # Needs a quadratic term
 
 qqnorm(rice.lm$res, col=2)
 qqline(rice.lm$res, col="blue")
 rice.lm2 <- lm(Yield ~ daymin31 + I(daymin31^2) , data=rice)
 old.par <- par(no.readonly = TRUE)
 par(mfrow=c(1, 2))
 plot(rice.lm2$fit, rice.lm2$res, xlab="Fitted values", ylab = "Residuals")
 abline(h=0)
 # Should be a random scatter 
 # Much better plot!
 qqnorm(rice.lm2$res, col=2)
 qqline(rice.lm2$res, col="blue")
 summary(rice.lm2)
 par(old.par) # par(mfrow=c(1,1))
 plot(rice$Days,  rice$Yield, xlab="Days", ylab="Yield")
 lines(rice$Days, rice.lm2$fit, lty=1, col=3)
 rice.lm3 <- lm(Yield ~ daymin31 + I(daymin31^2)+I(daymin31^3) , data=rice)
 #check the diagnostics 
 summary(rice.lm3) # Will print the summary of the fitted model 
 #### Predict at a new value of Days=31.1465
 
 # Create a new data set called new
 new <- data.frame(daymin31=32.1465-31)
 
 a <- predict(rice.lm2, newdata=new, se.fit=TRUE) 
 # Confidence interval for the mean of rice yield  at day=31.1465
 a <- predict(rice.lm2, newdata=new, interval="confidence") 
 a
 #          fit      lwr      upr
 # [1,] 3676.766 3511.904 3841.628
 # Prediction interval for a future yield at day=31.1465
 b <- predict(rice.lm2, newdata=new, interval="prediction") 
 b
 # fit      lwr      upr
 #[1,] 3676.766 3206.461 4147.071
```

## `wgain`: Weight gain of students starting college 

This contains weight gain data from 68 first year students during their first 12 
weeks in college. Source: Levitsky {\it et al} (2004). [@Levitsky2004]
This data set has been used to illustrate confidence intervals and $t$-tests  
in part I of the book Sahu (2023). 

```{r wgain, eval=TRUE}
summary(wgain)
# 95% Confidence interval for mean weight gain 
x <- wgain$final - wgain$initial
mean(x) + c(-1, 1) * qt(0.975, df=67) * sqrt(var(x)/68)
# t-test to test the mean difference equals 0
t.test(x)
```

# Illustrated R functions 


## The butterfly function 

This function draws a butterfly as on the front cover of the book. This is a plot obtained as follows.  Initially a sequence of 
angles denoted by $\theta$ is chosen in the range 0 to 24$\pi$. Then, we specify two parameters $a$ and $b$ and evaluate the following equations. The illustrations show the effect of these two parameters on the shape of the resulting plot obtained by plotting $x-y$ pairs. Successive points on the plot are joined by  lines.   
\begin{equation}
r = \exp(\cos(\theta)) - a \cos(b \,  \theta) + \sin\left(\frac{\theta}{12}\right) \\
x = r \sin(\theta) \\
y = -r \cos(\theta)
\end{equation}
```{r butterfly, eval=TRUE}
butterfly(color = 6)
old.par <- par(no.readonly = TRUE)
par(mfrow=c(2, 2))
butterfly(a=10, b=1.5, color = "seagreen")
butterfly(color = 6)
butterfly(a=5, b=5, color=2)
butterfly(a=20, b=4, color = "blue")
par(old.par) # par(mfrow=c(1, 1))
```
  

## Simulation of the Monty Hall game


```{r Montyplot, eval=TRUE, echo=FALSE, fig.cap="Monty Hall game", fig.width =2, fig.height=1}
library(ipsRdbs)
figpath <- system.file("figures", package = "ipsRdbs") 
knitr::include_graphics(paste0(figpath, "/monty.png"))
```

The function `?monty` simulates the famous Monty Hall game. This function is written by [@Monty.Corey2012]. The function takes the arguments: `strat`: short form for strategy which can take one of the three choices: 
* "stay" : Do not change the initial door chosen
* "swap" : Swap the door chosen initially.
* "random" : Randomly decide to stay or swap.

The other parameters are $N$: How many games to play, defaults to 1000 and 
`print_games`, which is a logical argument that tells whether to print the results of each of the $N$ games. Here are three illustrations of the games. 
```{r}
monty("stay", print_games = FALSE)
monty("switch", print_games = FALSE)
monty("random", print_games = FALSE)
```


  
  
## Functions illustrating the Central Limit Theorem (CLT)

There are two functions illustrating the CLT. The first one 
'?see_the_clt_for_Bernoulli' simulates `nrep=10000` samples of size 
`nsize=10` with probability of success $p=0.8$ by default. It then finds 
the means of the `nrep` samples and standardises the means by subtracting 
the overall mean and dividing by the sample standard deviation. It then draws a 
histogram of the sample means and superimposes the theoretical density of the standard normal distribution. The histogram will closely resemble the density of the standard normal distribution if the CLT approximation is good. The quality of the approximation depends on the sample size `nsize`. This can be observed from the illustrations provided below. 
```{r cltforBernoulli, eval=TRUE}
a <- see_the_clt_for_Bernoulli()
old.par <- par(no.readonly = TRUE)
par(mfrow=c(2, 3))
a30 <- see_the_clt_for_Bernoulli(nsize=30)
a50 <- see_the_clt_for_Bernoulli(nsize=50)
a100 <- see_the_clt_for_Bernoulli(nsize=100)
a500 <- see_the_clt_for_Bernoulli(nsize=500)
a1000 <- see_the_clt_for_Bernoulli(nsize=1000)
a5000 <- see_the_clt_for_Bernoulli(nsize=5000)
par(old.par)
```

The second function `?see_the_clt_for_uniform` demonstrates the CLT for sampling from the uniform distribution in the interval (0,1). As in the previous function for the Bernoulli distribution, this also takes two similar arguments and behaves very similarly as demonstrated below. But the sampling is done from the uniform distribution. 
```{r, seethecltforuniform, eval=TRUE}
a <- see_the_clt_for_uniform()
old.par <- par(no.readonly = TRUE) 
par(mfrow=c(2, 3))
a1 <- see_the_clt_for_uniform(nsize=1)
a2 <- see_the_clt_for_uniform(nsize=2)
a3 <- see_the_clt_for_uniform(nsize=5)
a4 <- see_the_clt_for_uniform(nsize=10)
a5 <- see_the_clt_for_uniform(nsize=20)
a6 <- see_the_clt_for_uniform(nsize=50)
par(old.par)
ybars <- see_the_clt_for_uniform(nsize=12)
zbars <- (ybars - mean(ybars))/sd(ybars)
k <- 100
u <- seq(from=min(zbars), to= max(zbars), length=k)
ecdf <-  rep(NA, k)
for(i in 1:k) ecdf[i] <- length(zbars[zbars<u[i]])/length(zbars)
tcdf <- pnorm(u)
plot(u, tcdf, type="l", col="red", lwd=4, xlab="", ylab="cdf")
lines(u, ecdf, lty=2, col="darkgreen", lwd=4)
symb <- c("cdf of sample means", "cdf of N(0, 1)")
legend(x=-3.5, y=0.4, legend = symb, lty = c(2, 1), 
col = c("darkgreen","red"), bty="n")
```

This function also provides a plot of the estimated cumulative density function (cdf) of the standarised sample means and superimposes the cdf of the standard normal distribution. The CLT states that these cdf's will be very close to each other as the sample size $n \to \infty$.   

## Functions illustrating the Weak Law of Large Numbers (WLLN)

The function `?see_the_wlln_for_uniform` illustrates the WLLN for sampling from the uniform distribution in the interval (0, 1).  Similar to the two functions demonstrating the CLT, this function takes two arguments `nsize` for sample size and `nrep` for number of replications.   The function draws the histogram of the replicated sample means and draws an estimated density function of the samples. 
The below illustration shows that the estimated density gets more peaked as the sample size $n$ increases. The WLLN says that the estimated density will gradually become just a spike as $n \to \infty$.   

```{r seethewlln, eval=TRUE}
a1 <- see_the_wlln_for_uniform(nsize=1, nrep=50000)
a2 <- see_the_wlln_for_uniform(nsize=10, nrep=50000)
a3 <- see_the_wlln_for_uniform(nsize=50, nrep=50000)
a4 <- see_the_wlln_for_uniform(nsize=100, nrep=50000)
plot(a4, type="l", lwd=2, ylim=range(c(a1$y, a2$y, a3$y, a4$y)), col=1, 
lty=1, xlab="mean", ylab="density estimates")
lines(a3, type="l", lwd=2, col=2, lty=2)
lines(a2, type="l", lwd=2, col=3, lty=3)
lines(a1, type="l", lwd=2, col=4, lty=4)
symb <- c("n=1", "n=10", "n=50", "n=100")
legend(x=0.37, y=11.5, legend = symb, lty =4:1, col = 4:1)
```




# Discussion 
The data sets included in this package and discussed in this vignette can be used in teaching and learning of introductory probability, statistics, `R` for data-based sciences with or without reading the book by the same author, [@Sahu2023]. So that a beginner reader can understand the methods, and also being introductory in nature, this package has used very basic `R` commands and scripts throughout. But the package  has illustrated the use of the advanced graphics package  `r  BiocStyle::CRANpkg("ggplot2")` and function writing through the `butterfly` function.  However, `r  BiocStyle::CRANpkg("ipsRdbs")`  does not intentionally teach R programming and does not expect the users to be proficient in adopting those techniques in their own data analysis and processing.  It has been purposefully designed this way to interest the reader to further learn and use more advanced graphics and data manipulation and analysis capabilities of R.           

<!-- A fuller version of this document with additional graphical illustrations 
<a href="https://www.sujitsahu.com/ipsRdbs/vignette.html" > is also available.  </a> 
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# References 
