## Regression skewed data
## Set working directory
setwd("C:/Users/Documents")

## Install (if not already) and load following libraries
library(readstata13)        ## to read STATA data
library(ggplot2)            ## for plots (ggplot graphs)
library("MASS")             ## To fit negative binomial model
library(AER)                ## For dispersion parameter test
library(pscl)               ## Testing between Poisson and negative binomial regression models, using Vuong test.
library(ggpubr)             ## For plotting multiple ggplots using `ggarrange' function


rm(list=ls())
## reading data from STATA data file and saving in an object `regdata'.
regdata = read.dta13("filename.dta")
                               
## Converting numerical variable, gender (0,1) to factor variable, sex (Female, Male)
regdata$sex = factor(regdata$gender)       
levels(regdata$sex) = c("Female","Male")

## Histograms of weekly minutes of habitual PA (Figure 1)
ggplot(regdata, aes(x=TotPA)) + geom_histogram(color="black", fill="white",bins=15) +
      labs(x="",y="")+ #theme_minimal() +
      theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
      panel.background = element_blank(), axis.line = element_line(colour = "black"))

## Descriptive measures
summary(regdata$age)   		## summary measures of age
var(regdata$age)       		## variance of age
summary(regdata$medCond)   	## summary measures of NCMC
var(regdata$medCond)      	## variance of NCMC
summary(regdata$TotPA)   	## summary measures of PA
var(regdata$TotPA)       	## variance of PA
summary(regdata$ave_mvpa)  	## summary measures of mvpa
var(regdata$ave_mvpa)       	## variance of mvpa

## Proportion of zeros in PA
length(which(regdata$TotPA==0))/dim(regdata)[1]

#########  fitting models to COUNT DATA  ########
## outcome variable (TotPA) to fit on age and NCMC (medCond)
## Standard linear regression model 
fitlm=lm(TotPA ~ age + sex + medCond , data=regdata)
## Poisson regression
fitp=glm(TotPA ~ age + sex + medCond ,family=poisson, data=regdata)
## quasi-Poisson regression
fitqp=glm(TotPA ~ age + sex + medCond ,family=quasipoisson, data=regdata)
## Negative binoial regression
fitnb=glm.nb(TotPA ~ age + sex + medCond , data=regdata)
## calculating 95% confidence interval of various regession models
confint(fitlm); confint(fitp); confint(fitqp); confint(fitnb)

## Test of dispersion, only for poisson glm
dispersiontest(fitp)

## Corresponding negative binomial model, testing significance of dispersion parameter
## of negative binomial regression. Quadratic formulation (trafo = function(x) x^2)
dispersiontest(fitp, trafo = 2) 

## Testing between Poisson and negative binomial regression models, using Vuong test.
## Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses.
## Econometrica. 57(2). 307–333.
vuong(fitp, fitnb)

## AIC's
## Calculating AIC of quasipoisson distribution
dof = length(fitqp$coeff)+1         	## model parameters, plus phi
phi.fit = summary(fitqp)$dispersion 	## fitted phi value 
mu.fit = fitqp$fitted.values        	## fitted values
aicqp = 2*dof - 2*sum(dnbinom(regdata$TotPA, mu=mu.fit, size = mu.fit/(phi.fit - 1), log=T))

## AIC  of other models
aic=t(AIC(fitlm,fitp,fitqp,fitnb))
## Replacing AIC of quasiPoisson
aic[2,"fitqp"]=aicqp
aic

##############################################################################
###############################################################################
########## CONTINUOUS OUTCOME #########
###  Plotting original (untransformed) and transformed MVPA  ###
cdata1=ggplot(regdata, aes(x=ave_mvpa)) + geom_histogram(color="black", fill="white",bins=15) + 
	labs(x="minutes of mvpa",y="Frequency") + ylim(0,90) +
      	theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
      	panel.background = element_blank(), axis.line = element_line(colour = "black"))
cdata2=ggplot(regdata, aes(x=log(ave_mvpa))) + geom_histogram(color="black", fill="white",bins=15) + 
	labs(x="log of minutes of mvpa",y="Frequency") + ylim(0,90) +
      	theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
      	panel.background = element_blank(), axis.line = element_line(colour = "black"))
cdata3=ggplot(regdata, aes(x=sqrt(ave_mvpa))) + geom_histogram(color="black", fill="white",bins=15) + 
	labs(x="square root of minutes of mvpa",y="Frequency") + ylim(0,90) +
      	theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
      	panel.background = element_blank(), axis.line = element_line(colour = "black"))
## plotting in one window
ggarrange(cdata1, cdata2, cdata3 , ncol = 1, nrow = 3)


#########################################################
########## MODEL FITTINGS TO CONTINUOUS OUTCOME #########
## outcome variable (MVPA) to fit on age and sex

# Standard Linear regression model
mod1 <- lm(ave_mvpa ~ age + sex + medCond, data=regdata)
# Linear regression on Square roort transformed MVPA
mod2 <- lm(sqrt(ave_mvpa) ~ age + sex + medCond, data=regdata)
# Linear regression on Log transformed MVPA
mod3 <- lm(log(ave_mvpa) ~ age + sex + medCond, data=regdata)
# GLM: Gamma regression with Log link
mod4 <- glm(ave_mvpa ~ age + sex + medCond, family=Gamma(link = "log"),data=regdata)
# GLM: Inverse guassian regression with log link
mod5 <- glm(ave_mvpa ~ age + sex + medCond, family=inverse.gaussian(link = "log"),data=regdata)
## 95% confidence interval of various models
confint(mod1); confint(mod2); confint(mod3); confint(mod4); confint(mod5)

## Calculating Predicted values from various models on original scale
predlm = predict(mod1)
predlmsqrt = predict(mod2)^2
predlmlog = exp(predict(mod3))
predglmg = predict(mod4,type="response")
predglmig = predict(mod5,type="response")

## Plotting predicted and actual values from various models on original scale against age
ggplot(regdata,aes(y=ave_mvpa,x=age,shape=sex))+ geom_point() +
       geom_point(aes(y = predlm,x=age,shape=sex,color="LM")) +
       geom_point(aes(y = predlmsqrt,x=age,shape=sex,color="LM sqrt")) +
       geom_point(aes(y = predlmlog,x=age,shape=sex,color="LM log")) +
       geom_point(aes(y = predglmg,x=age,shape=sex,color="GLM gamma")) +
       geom_point(aes(y = predglmig,x=age,shape=sex,color="GLM IG")) + 
       geom_line(aes(y = predlm,x=age,group=sex,color="LM")) +
       geom_line(aes(y = predlmsqrt,x=age,group=sex,color="LM sqrt")) +
       geom_line(aes(y = predlmlog,x=age,group=sex,color="LM log")) +
       geom_line(aes(y = predglmg,x=age,group=sex,color="GLM gamma")) +
       geom_line(aes(y = predglmig,x=age,group=sex,color="GLM IG")) + 
       	labs(shape="Original", colour="Predicted", x="Age",y="MVPA") + 
       	scale_color_discrete(breaks=c("LM","LM sqrt","LM log","GLM gamma","GLM IG"), 
	guide = guide_legend(override.aes = list(shape = c(rep(NA,5))))) +
       	theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
	panel.background = element_blank(), axis.line = element_line(colour = "black"))+
       	geom_hline(yintercept=0, linetype="dashed")

## Comparing GLM-gamma with GLM IG using residuals vs fitted values plot
rvfglmg=ggplot(regdata, aes(y=residuals(mod4),x=predict(mod4))) + geom_point(color="black", fill="white",shape=21) + 	geom_hline(yintercept=0, linetype="dashed") + ylim(-2.5,2.5) +  
      	labs(x="Linear predictor",y="Deviance residuals",title="GLM-gamma") + 
	theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
      	panel.background = element_blank(), axis.line = element_line(colour = "black"))
rvfglmig=ggplot(regdata, aes(y=residuals(mod5),x=predict(mod5))) + geom_point(color="black", fill="white",shape=21) + 	geom_hline(yintercept=0, linetype="dashed") + ylim(-2.5,2.5)+  
      	labs(x="Linear predictor",y="Deviance residuals",title="GLM-IG") + 
	theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(),
      	panel.background = element_blank(), axis.line = element_line(colour = "black"))

##residuals vs fitted values plot all together in one plot window
ggarrange(rvfglmg, rvfglmig , ncol = 1, nrow = 2)

###### AIC values of various models   ######
## Corrected/adjusted AIC values from transformed models
N=length(regdata$ave_mvpa)
## AIC of Square roort transformed MVPA
aic.sqrt=sum(log(regdata$ave_mvpa))+2*N*log(2) + AIC(mod2)
## AIC of Log transformed MVPA
aic.log=2*sum(log(regdata$ave_mvpa))+AIC(mod3)
## AIC values of all models
aic=c(AIC(mod1),aic.sqrt,aic.log,AIC(mod4),AIC(mod5))