BMDs_and_ABDs_fitxxxBin.Rmd

---
title: "Modelling Binomial Outcome Data using ABDs and BMDs"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Modelling Binomial Outcome Data using ABDs and BMDs}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE,comment = "|>")
library(fitODBOD)
library(ggplot2)
library(reshape2)
library(grid)
library(gridExtra)
library(bbmle)
library(tibble)
library(ggthemes)
library(flextable)
```

>IT WOULD BE CLEARLY BENEFICIAL FOR YOU BY USING THE RMD FILES IN THE GITHUB DIRECTORY FOR FURTHER EXPLANATION
OR UNDERSTANDING OF THE R CODE FOR THE RESULTS OBTAINED IN THE VIGNETTES. 

Fitting Alternate Binomial or Binomial Mixture distribution is the most crucial part of while 
handling Binomial Outcome Data. Finding the most suitable distribution which is similar to the 
given data is very important and essential. In order to compare distributions after fitted we can 
choose several measurements. They are namely

* Comparing actual frequency and estimated frequency.
* Using Chi-squared Test statistic and comparing p-values.
* Comparing actual variance with estimated variances of distributions.
* Comparing Negative Log Likelihood values of distributions.
* Comparing AIC values of distributions. 

The functions given below are used to fit respective distributions when BOD and estimated 
parameters are given. 

* `fitBin` - fitting the Binomial distribution.
* `fitTriBin`- fitting the Triangular Binomial distribution.
* `fitBetaBin` - fitting the Beta-Binomial distribution.
* `fitKumBin` - fitting the Kumaraswamy Binomial distribution.
* `fitGHGBB` - fitting the Gaussian Hyper-geometric Generalized Beta-Binomial distribution.
* `fitMcGBB` - fitting the McDonald Generalized Beta-Binomial distribution.
* `fitGammaBin` - fitting Gamma Binomial distribution.
* `fitGrassiaIIBin` - fitting Grassia II Binomial distribution.
* `fitAddBin` - fitting the Additive Binomial distribution.
* `fitBetaCorrBin` - fitting the Beta Correlated Binomial distribution.
* `fitCOMPBin` - fitting the COM Poisson Binomial distribution.
* `fitCorrBin` - fitting the Correlated Binomial distribution.
* `fitMultiBin` - fitting the Multiplicative Binomial distribution.
* `fitLMBin` - fitting the Lovinson Multiplicative Binomial distribution.

## Fitting Alternate Binomial Distributions

All six Alternate Binomial distributions will be fitted to the Alcohol data week 2 and their expected frequencies 
will be plotted with the actual frequency values. This is plot can be used to identify which distribution suits 
best for the Alcohol data week 2. 

```{r plotting Expected frequencies with actual frequency for Binomial Distribution, warning=FALSE}
Alcohol_data

BinRanVar <- Alcohol_data$Days
ActFreq <- Alcohol_data$week2

# Fitting Binomial Distribution
BinFreq <- fitBin(BinRanVar,ActFreq)

# printing the results of fitting Binomial distribution
print(BinFreq)
```

### Additive Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Additive Binomial Distribution, warning=FALSE}
# Estimating and fitting Additive Binomial distribution
Para_AddBin <- EstMLEAddBin(BinRanVar,ActFreq)

# printing the coefficients and using them
coef(Para_AddBin)

AddBin_p <- Para_AddBin$p
AddBin_alpha <- Para_AddBin$alpha

# Fitting Additive Binomial Distribution
AddBinFreq <- fitAddBin(BinRanVar, ActFreq, AddBin_p, AddBin_alpha)

# printing the results of fitting Additive Binomial Distribution
print(AddBinFreq)
```

### Beta-Correlated Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Beta-Correlated Binomial Distribution, warning=FALSE}
# Estimating and fitting Beta Correlated Binoial Distribution
Para_BetaCorrBin <- EstMLEBetaCorrBin(x=BinRanVar, freq=ActFreq,
                                    cov=0.001,a=10,b=10)

# printing the coefficients and using them
coef(Para_BetaCorrBin)

BetaCorrBin_cov <- coef(Para_BetaCorrBin)[1]
BetaCorrBin_a <- coef(Para_BetaCorrBin)[2]
BetaCorrBin_b <- coef(Para_BetaCorrBin)[3]

# Fitting Beta-Correlated Binomial Distribution
BetaCorrBinFreq <- fitBetaCorrBin(BinRanVar,ActFreq,BetaCorrBin_cov,
                                BetaCorrBin_a,BetaCorrBin_b)

# printing the results of fitting Beta-Correlated Binomial Distribution
print(BetaCorrBinFreq)
```

### COM-Poisson Binomial Distribution

```{r plotting Expected frequencies with actual frequency for COM-Poisson Binomial Distribution, warning=FALSE}
# Estimating and fitting COM Poisson Binomial Distribution
Para_COMPBin <- EstMLECOMPBin(x=BinRanVar, freq=ActFreq,
                            v=12.1,p=0.9)

# printing the coefficients and using them
coef(Para_COMPBin)

COMPBin_p <- coef(Para_COMPBin)[1]
COMPBin_v <- coef(Para_COMPBin)[2]

# Fitting COM-Poisson Binomial Distribution
COMPBinFreq <- fitCOMPBin(BinRanVar, ActFreq,COMPBin_p,COMPBin_v)

# printing the results of fitting COM-Poisson Binomial Distribution
print(COMPBinFreq)
```

### Correlated Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Correlated Binomial Distribution, warning=FALSE}
# Estimating and fitting Correlated Binomial Distribution
Para_CorrBin <- EstMLECorrBin(x=BinRanVar, freq=ActFreq,
                            cov=0.0021,p=0.19)

# printing the coefficients and using them
coef(Para_CorrBin)

CorrBin_p <- coef(Para_CorrBin)[1]
CorrBin_cov <- coef(Para_CorrBin)[2]

# Fitting Correlated Binomial Distribution
CorrBinFreq <- fitCorrBin(BinRanVar, ActFreq,CorrBin_p,CorrBin_cov)

# printing the results of fitting Correlated Binomial Distribution
print(CorrBinFreq)
```

### Multiplicative Binomial Distribution
 
```{r plotting Expected frequencies with actual frequency for Multiplicative Binomial Distribution, warning=FALSE}
# Estimating and fitting Multiplicative Binomial Distribution
Para_MultiBin <- EstMLEMultiBin(x=BinRanVar, freq=ActFreq,
                              theta=21,p=0.19)

# printing the coefficients and using them
coef(Para_MultiBin)

MultiBin_p <- coef(Para_MultiBin)[1]
MultiBin_theta <- coef(Para_MultiBin)[2]

# Fitting Multiplicative Binomial Distribution
MultiBinFreq <- fitMultiBin(BinRanVar, ActFreq,MultiBin_p,MultiBin_theta)

# printing the results of fitting Multiplicative Binomial Distribution
print(MultiBinFreq)
```

### Lovinson Multiplicative Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Lovinson Multiplicative Binomial Distribution, warning=FALSE}
# Estimating and fitting Lovinson Multiplicative Binomial Distribution
Para_LMBin <- EstMLELMBin(x=BinRanVar, freq=ActFreq,
                        phi=21,p=0.19)

# printing the coefficients and using them
coef(Para_LMBin)

LMBin_p <- coef(Para_LMBin)[1]
LMBin_phi <- coef(Para_LMBin)[2]

# Fitting Lovinson Multiplicative Binomial Distribution
LMBinFreq <- fitLMBin(BinRanVar, ActFreq,LMBin_p,LMBin_phi)

# printing the results of fitting Multiplicative Binomial Distribution
print(LMBinFreq)
```

### Conclusion

```{r plotting that dataset for ABD,fig.align='center',fig.height=7,fig.width=9,echo=FALSE}
# creating dataset for plotting
ABD_Data <- tibble(
                  w=BinRanVar,
                  x=ActFreq,
                  y=fitted(BinFreq),
                  z=fitted(AddBinFreq),
                  a=fitted(BetaCorrBinFreq),
                  b=fitted(COMPBinFreq),
                  c=fitted(CorrBinFreq),
                  d=fitted(MultiBinFreq),
                  e=fitted(LMBinFreq)
                )
names(ABD_Data) <- c("Bin_RV","Actual_Freq","EstFreq_BinD","EstFreq_AddBinD","EstFreq_BetaCorrBinD","EstFreq_COMPBinD",
                   "EstFreq_CorrBinD","EstFreq_MultiBinD","EstFreq_LMBinD")
```

```{r Flexxed Table ABD}
ABD_Total<-colSums(ABD_Data[,-1])
ABD_Variance<-c(var(rep(BinFreq$bin.ran.var,times=BinFreq$obs.freq)),
                var(rep(BinFreq$bin.ran.var,times=BinFreq$exp.freq)),
                var(rep(AddBinFreq$bin.ran.var,times=AddBinFreq$exp.freq)),
                var(rep(BetaCorrBinFreq$bin.ran.var,times=BetaCorrBinFreq$exp.freq)),
                var(rep(COMPBinFreq$bin.ran.var,times=COMPBinFreq$exp.freq)),
                var(rep(CorrBinFreq$bin.ran.var,times=CorrBinFreq$exp.freq)),
                var(rep(MultiBinFreq$bin.ran.var,times=MultiBinFreq$exp.freq)),
                var(rep(LMBinFreq$bin.ran.var,times=LMBinFreq$exp.freq))
                )
ABD_Variance<-round(ABD_Variance,4)

ABD_p_value<-c(BinFreq$p.value,
               AddBinFreq$p.value,
               BetaCorrBinFreq$p.value,
               COMPBinFreq$p.value,
               CorrBinFreq$p.value,
               MultiBinFreq$p.value,
               LMBinFreq$p.value)

ABD_NegLL<-c(AddBinFreq$NegLL,
             BetaCorrBinFreq$NegLL,
             COMPBinFreq$NegLL,
             CorrBinFreq$NegLL,
             MultiBinFreq$NegLL,
             LMBinFreq$NegLL)
ABD_NegLL<-round(ABD_NegLL,4)

ABD_AIC<-c(AIC(AddBinFreq),
           AIC(BetaCorrBinFreq),
           AIC(COMPBinFreq),
           AIC(CorrBinFreq),
           AIC(MultiBinFreq),
           AIC(LMBinFreq)
           )
ABD_AIC<-round(ABD_AIC,4)

C_of_diff_Values<-c(length(ABD_Data$Actual_Freq),
                    sum(abs(ABD_Data$Actual_Freq-ABD_Data$EstFreq_BinD) <= 5),
                    sum(abs(ABD_Data$Actual_Freq-ABD_Data$EstFreq_AddBinD) <= 5),
                    sum(abs(ABD_Data$Actual_Freq-ABD_Data$EstFreq_BetaCorrBinD) <= 5),
                    sum(abs(ABD_Data$Actual_Freq-ABD_Data$EstFreq_COMPBinD) <= 5),
                    sum(abs(ABD_Data$Actual_Freq-ABD_Data$EstFreq_CorrBinD) <= 5),
                    sum(abs(ABD_Data$Actual_Freq-ABD_Data$EstFreq_MultiBinD) <= 5),
                    sum(abs(ABD_Data$Actual_Freq-ABD_Data$EstFreq_LMBinD) <= 5)) 

rbind(ABD_Data,
      c("Total",ABD_Total),
      c("Count of difference Values",C_of_diff_Values),
      c("Variance",ABD_Variance),
      c("p-value","-",ABD_p_value),
      c("Negative Log Likelihood Value","-","-",ABD_NegLL),
      c("AIC","-","-",ABD_AIC))->ABD_flexed_Data

flextable(data=ABD_flexed_Data,
          col_keys = c("Bin_RV","Actual_Freq","EstFreq_BinD",
                       "EstFreq_AddBinD","EstFreq_BetaCorrBinD",
                       "EstFreq_COMPBinD","EstFreq_CorrBinD",
                       "EstFreq_MultiBinD","EstFreq_LMBinD")) %>%
  theme_box() %>% autofit()%>%
  bold(i=1,part = "header") %>%
  bold(i=c(9:14),j=1,part = "body") %>%
  align(i=c(1:14),j=c(1:9),align = "center") %>%
  set_header_labels(values = c(Bin_RV="Binomial Random Variable",
                               Actual_Freq="Observed Frequencies",
                               EstFreq_BinD="Binomial Distribution",
                               EstFreq_AddBinD="Additive Binomial Distribution",
                               EstFreq_BetaCorrBinD="Beta-Correlated Binomial Distribution",
                               EstFreq_COMPBinD="Composite Binomial Distribution",
                               EstFreq_CorrBinD="Correlated Binomial Distribution",
                               EstFreq_MultiBinD="Multiplicative Binomial Distribution",
                               EstFreq_LMBinD="Lovinson Multiplicative Binomial Distribution")) %>%
  align(i=1,part = "header",align = "center")

```

It is clearly visible that Additive and Correlated Binomial distributions behave very similarly, both 
generate same frequency values. Although they are close to actual frequencies. Comparing Binomial 
distribution generated expected frequencies with actual frequencies will lead to see that there is very 
much difference and Binomial distribution does not suit this Alcohol data of week 2. Multiplicative 
and COM Poisson Binomial distributions show close values to actual frequencies. Multiplicative and Lovinson
Multiplicative distributions are behaving similarly as a pair. 

Finally, the only distribution left is Beta Correlated Binomial distribution which shows more closeness to actual 
frequencies. Therefore it is clear that most suitable distribution for alcohol data week 2 is Beta Correlated Binomial 
distribution, second choice is Multiplicative and COM Poisson Binomial distributions and final choice is 
Correlated and Additive Binomial distributions.

## Fitting Binomial Mixture Distributions

In the eight BMD distributions except Uniform Binomial distribution others can be used for fitting the 
Alcohol data week 2. Here also as above a plot was generated to compare estimated frequencies 
with actual frequency. 

### Triangular Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Triangular Binomial Distribution,warning=FALSE}
# Estimating and fitting Triangular Binomial distribution
Para_TriBin <- EstMLETriBin(BinRanVar,ActFreq)

# printing the coefficients and using them
coef(Para_TriBin)

TriBin_c <- Para_TriBin$mode

# Fitting Triangular Binomial Distribution
TriBinFreq <- fitTriBin(BinRanVar, ActFreq, TriBin_c)

# printing the results of fitting Triangular Binomial Distribution
print(TriBinFreq)
```

### Beta-Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Beta-Binomial Distribution,warning=FALSE}
# Estimating and fitting Beta Correlated Binoial Distribution
Para_BetaBin <- EstMLEBetaBin(x=BinRanVar, freq=ActFreq,
                            a=10,b=10)
# printing the coefficients and using them
coef(Para_BetaBin)

BetaBin_a <- coef(Para_BetaBin)[1]
BetaBin_b <- coef(Para_BetaBin)[2]

# Fitting Beta-Binomial Distribution
BetaBinFreq <- fitBetaBin(BinRanVar, ActFreq,BetaBin_a,BetaBin_b)

# printing the results of fitting Beta-Binomial Distribution
print(BetaBinFreq)
```

### Kumaraswamy Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Kumaraswamy Binomial Distribution,warning=FALSE}
# Estimating and fitting Kumaraswamy Binomial Distribution
Para_KumBin <- EstMLEKumBin(x=BinRanVar, freq=ActFreq,
                          a=12.1,b=0.9,it=10000)

# printing the coefficients and using them
coef(Para_KumBin)

KumBin_a <- coef(Para_KumBin)[1]
KumBin_b <- coef(Para_KumBin)[2]
KumBin_it <- coef(Para_KumBin)[3]

# Fitting Kumaraswamy Binomial Distribution
KumBinFreq <- fitKumBin(BinRanVar, ActFreq,KumBin_a,KumBin_b,KumBin_it*10)

# printing the results of fitting Kumaraswamy Binomial Distribution
print(KumBinFreq)
```

### GHGBB Distribution

```{r plotting Expected frequencies with actual frequency for GHGBB Distribution,warning=FALSE}
# Estimating and fitting GHGBB Distribution
Para_GHGBB <- EstMLEGHGBB(x=BinRanVar, freq=ActFreq,
                        a=0.0021,b=0.19,c=0.3)

# printing the coefficients and using them
coef(Para_GHGBB)

GHGBB_a <- coef(Para_GHGBB)[1]
GHGBB_b <- coef(Para_GHGBB)[2]
GHGBB_c <- coef(Para_GHGBB)[3]

# Fitting GHGBB Distribution
GHGBBFreq <- fitGHGBB(BinRanVar, ActFreq,GHGBB_a,GHGBB_b,GHGBB_c)

# printing the results of fitting GHGBB Distribution
print(GHGBBFreq)
```

### McGBB Distribution

```{r plotting Expected frequencies with actual frequency for McGBB Distribution,warning=FALSE}
# Estimating and fitting McGBB Distribution
Para_McGBB <- EstMLEMcGBB(x=BinRanVar, freq=ActFreq,
                        a=21,b=0.19,c=0.1)

# printing the coefficients and using them
coef(Para_McGBB)

McGBB_a <- coef(Para_McGBB)[1]
McGBB_b <- coef(Para_McGBB)[2]
McGBB_c <- coef(Para_McGBB)[3]

# Fitting McGBB Distribution
McGBBFreq <- fitMcGBB(BinRanVar, ActFreq,McGBB_a,McGBB_b,McGBB_c)

# printing the results of fitting McGBB Distribution
print(McGBBFreq)
```

### Gamma Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Gamma Binomial Distribution,warning=FALSE}
# Estimating and fitting Gamma Binoial Distribution
Para_GammaBin <- EstMLEGammaBin(x=BinRanVar, freq=ActFreq,
                              c=10,l=10)

# printing the coefficients and using them
coef(Para_GammaBin)

GammaBin_c <- coef(Para_GammaBin)[1]
GammaBin_l <- coef(Para_GammaBin)[2]

# Fitting Gamma Binomial Distribution
GammaBinFreq <- fitGammaBin(BinRanVar, ActFreq,GammaBin_c,GammaBin_l)

# printing the results of fitting Beta-Binomial Distribution
print(GammaBinFreq)
```

### Grassia II Binomial Distribution

```{r plotting Expected frequencies with actual frequency for Grassia II Binomial Distribution,warning=FALSE}
# Estimating and fitting Grassia II Binoial Distribution
Para_GrassiaIIBin <- EstMLEGrassiaIIBin(x=BinRanVar, freq=ActFreq,
                                      a=10,b=10)
# printing the coefficients and using them
coef(Para_GrassiaIIBin)

GrassiaIIBin_a <- coef(Para_GrassiaIIBin)[1]
GrassiaIIBin_b <- coef(Para_GrassiaIIBin)[2]

# Fitting Grassia II Binomial Distribution
GrassiaIIBinFreq <- fitGrassiaIIBin(BinRanVar, ActFreq,GrassiaIIBin_a,GrassiaIIBin_b)

# printing the results of fitting Grassia II Binomial Distribution
print(GrassiaIIBinFreq)
```

### Conclusion

```{r plotting that dataset for BMD,fig.align='center',fig.height=7,fig.width=9,echo=FALSE}
BMD_Data <- tibble(
                  w=BinRanVar,
                  x=ActFreq,
                  y=fitted(BinFreq),
                  z=fitted(TriBinFreq),
                  a=fitted(BetaBinFreq),
                  b=fitted(KumBinFreq),
                  c=fitted(GHGBBFreq),
                  d=fitted(McGBBFreq),
                  e=fitted(GammaBinFreq),
                  f=fitted(GrassiaIIBinFreq)
                )
names(BMD_Data) <- c("Bin_RV","Actual_Freq","EstFreq_BinD","EstFreq_TriBinD","EstFreq_BetaBinD","EstFreq_KumBinD",
                   "EstFreq_GHGBBD","EstFreq_McGBBD","EstFreq_GammaBinD","EstFreq_GrassiaIIBinD")
```

```{r Flexxed Table BMD}
BMD_Total<-colSums(BMD_Data[,-1])
BMD_Variance<-c(var(rep(BinFreq$bin.ran.var,times=BinFreq$obs.freq)),
                var(rep(BinFreq$bin.ran.var,times=BinFreq$exp.freq)),
                var(rep(TriBinFreq$bin.ran.var,times=TriBinFreq$exp.freq)),
                var(rep(BetaBinFreq$bin.ran.var,times=BetaBinFreq$exp.freq)),
                var(rep(KumBinFreq$bin.ran.var,times=KumBinFreq$exp.freq)),
                var(rep(GHGBBFreq$bin.ran.var,times=GHGBBFreq$exp.freq)),
                var(rep(McGBBFreq$bin.ran.var,times=McGBBFreq$exp.freq)),
                var(rep(GammaBinFreq$bin.ran.var,times=GammaBinFreq$exp.freq)),
                var(rep(GrassiaIIBinFreq$bin.ran.var,times=GrassiaIIBinFreq$exp.freq))
                )
BMD_Variance<-round(BMD_Variance,4)

BMD_p_value<-c(BinFreq$p.value,
               TriBinFreq$p.value,
               BetaBinFreq$p.value,
               KumBinFreq$p.value,
               GHGBBFreq$p.value,
               McGBBFreq$p.value,
               GammaBinFreq$p.value,
               GrassiaIIBinFreq$p.value)

BMD_NegLL<-c(TriBinFreq$NegLL,
             BetaBinFreq$NegLL,
             KumBinFreq$NegLL,
             GHGBBFreq$NegLL,
             McGBBFreq$NegLL,
             GammaBinFreq$NegLL,
             GrassiaIIBinFreq$NegLL)

BMD_NegLL<-round(BMD_NegLL,4)

BMD_AIC<-c(AIC(TriBinFreq),
           AIC(BetaBinFreq),
           AIC(KumBinFreq),
           AIC(GHGBBFreq),
           AIC(McGBBFreq),
           AIC(GammaBinFreq),
           AIC(GrassiaIIBinFreq))

BMD_AIC<-round(BMD_AIC,4)

C_of_diff_Values<-c(length(BMD_Data$Actual_Freq),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_BinD) <= 5),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_TriBinD) <= 5),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_BetaBinD) <= 5),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_KumBinD) <= 5),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_GHGBBD) <= 5),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_McGBBD) <= 5),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_GammaBinD) <= 5),
                    sum(abs(BMD_Data$Actual_Freq-BMD_Data$EstFreq_GrassiaIIBinD) <= 5)) 

Overdispersion_BMD<-c(Overdispersion(TriBinFreq),
                      Overdispersion(BetaBinFreq),
                      Overdispersion(KumBinFreq),
                      Overdispersion(GHGBBFreq),
                      Overdispersion(McGBBFreq),
                      Overdispersion(GammaBinFreq),
                      Overdispersion(GrassiaIIBinFreq))

rbind(BMD_Data,
      c("Total",BMD_Total),
      c("Count of difference Values",C_of_diff_Values),
      c("Variance",BMD_Variance),
      c("p-value","-",BMD_p_value),
      c("Negative Log Likelihood Value","-","-",BMD_NegLL),
      c("AIC","-","-",BMD_AIC))->BMD_flexed_Data

flextable(data=BMD_flexed_Data,
          col_keys = c("Bin_RV","Actual_Freq","EstFreq_BinD",
                       "EstFreq_TriBinD","EstFreq_BetaBinD",
                       "EstFreq_KumBinD","EstFreq_GHGBBD",
                       "EstFreq_McGBBD","EstFreq_GammaBinD",
                       "EstFreq_GrassiaIIBinD")) %>%
  theme_box() %>% autofit()%>%
  bold(i=1,part = "header") %>%
  bold(i=c(9:14),j=1,part = "body") %>%
  align(i=c(1:14),j=c(1:10),align = "center") %>%
  set_header_labels(values = c(Bin_RV="Binomial Random Variable",
                               Actual_Freq="Observed Frequencies",
                               EstFreq_BinD="Binomial Distribution",
                               EstFreq_TriBinD="Triangular Binomial Distribution",
                               EstFreq_BetaBinD="Beta Binomial Distribution",
                               EstFreq_KumBinD="Kumaraswamy Binomial Distribution",
                               EstFreq_GHGBBD="Gaussian Hypergeometric Generalized Beta Binomial Distribution",
                               EstFreq_McGBBD="McDonald Generalized Beta Binomial Distribution",
                               EstFreq_GammaBinD="Gamma Binomial Distribution",
                               EstFreq_GrassiaIIBinD="Grassia II Binomial Distribution")) %>%
  align(i=1,part = "header",align = "center")

```

Estimated frequencies from Triangular distribution are far more better than estimated
frequencies from Binomial distribution. It is clear that Beta-Binomial, Kumaraswamy Binomial,
Gamma Binomial and Grassia II Binomial distributions behave identically for the alcohol data 
of week 2. Also McDonald Generalized Beta-Binomial distribution too behave equally for the alcohol
data of week 2. 

Finally,  Gaussian Hyper-geometric Generalized Beta-Binomial distribution is
best suited and generates more accurate frequencies. Therefore, first choice is GHGBB 
distribution and second choice would be McGBB distribution.