RPubs link information

• http://rpubs.com/ThreeOfTheFive/Assignment4

Introduction

• The following link will help you with creating R Markdown Slidy Presentations
• http://rmarkdown.rstudio.com/slidy_presentation_format.html
• Don’t forget about Bootcamp 4
• A good introduction provides a brief background to the problem, defines important terms, and leads to a strong rationale.

Introduction Cont.

• Keep everything short and straight to the point.
• Use bullet points to help minimise text.
• Add relevant images to make your presentation more appealing
• Remember, you have a maximum of 20 slides to fit everything in.
• Ensure each slide fits on one screen. The reader should not have to scroll down.

Problem Statement

• State the overall problem/question driving the investigation
• Summarise how you will use statistics to solve the problem or answer your question.

Data

Dataset: https://www.kaggle.com/unsdsn/world-happiness

• You can use multiple slides to break up your sections.
• Explain how you collected your data.
• There should be enough detail here so that someone else could replicate your data collection.
• Explain the sampling method if known
• Ensure you reference the data source if you have used Open Data

# Read the data file from the project folder.
Dataset_for_Assignment_4 <- read_csv("Dataset for Assignment 4.csv")

Data Cont.

• List and explain the important variables.
• If you have factors, explain each level.
• Explain the scale of numeric variables.
• Justify everything that you do to preprocess the data.

Descriptive Statistics and Visualisation

• Summarise the important variables in your investigation.
• Use visualisation to highlight interesting features of the data and tell the overall story.
• Explain how you dealt with data issues (if any), e.g. missing data and outliers.
• Here are the examples of R chunks and outputs

Happiness_Score_Summary <- Dataset_for_Assignment_4 %>% 
            summarise(
             Min = min(Happiness.Score,na.rm = TRUE),
             Q1 = quantile(Happiness.Score,probs = .25,na.rm = TRUE),
             Median = median(Happiness.Score, na.rm = TRUE),
             Q3 = quantile(Happiness.Score,probs = .75,na.rm = TRUE),
             Max = max(Happiness.Score,na.rm = TRUE), 
             Mean = mean(Happiness.Score, na.rm = TRUE),
             SD = sd(Happiness.Score, na.rm = TRUE),
             n = n())
Happiness_Score_Summary

Happiness_Score_Outliers <- Dataset_for_Assignment_4 %>% summarise(
                LowerFence = (quantile(Happiness.Score,probs = .25)-1.5*IQR(Happiness.Score)) %>% round(1),
                UpperFence = (quantile(Happiness.Score,probs = .75)+1.5*IQR(Happiness.Score)) %>% round(1),
                LowerOutlierCount = sum(Happiness.Score < LowerFence) %>% round(0),
                UpperOutlierCount = sum(Happiness.Score > UpperFence) %>% round(0),
                SD_MinToMean = ((mean(Happiness.Score) - min(Happiness.Score)) / sd(Happiness.Score)) %>% round(1),
                SD_MeanToMax = ((max(Happiness.Score) - mean(Happiness.Score)) / sd(Happiness.Score)) %>% round(1))
Happiness_Score_Outliers

```

Dataset_for_Assignment_4$Happiness.Score  %>% qqPlot(dist="norm")

Wealth_Score_Summary <- Dataset_for_Assignment_4 %>% 
            summarise(
             Min = min(Economy..GDP.per.Capita.,na.rm = TRUE),
             Q1 = quantile(Economy..GDP.per.Capita.,probs = .25,na.rm = TRUE),
             Median = median(Economy..GDP.per.Capita., na.rm = TRUE),
             Q3 = quantile(Economy..GDP.per.Capita.,probs = .75,na.rm = TRUE),
             Max = max(Economy..GDP.per.Capita.,na.rm = TRUE), 
             Mean = mean(Economy..GDP.per.Capita., na.rm = TRUE),
             SD = sd(Economy..GDP.per.Capita., na.rm = TRUE),
             n = n())
Wealth_Score_Summary 

Wealth_Score_Outliers <- Dataset_for_Assignment_4 %>% summarise(
                LowerFence = (quantile(Economy..GDP.per.Capita.,probs = .25)-1.5*IQR(Economy..GDP.per.Capita.)) %>% round(1),
                LowerOutlierCount = sum(Economy..GDP.per.Capita. < LowerFence) %>% round(0),
                NumSD_MinToMean = ((mean(Economy..GDP.per.Capita.) - min(Economy..GDP.per.Capita.)) / sd(Economy..GDP.per.Capita.)) %>% round(1),
                UpperFence = (quantile(Economy..GDP.per.Capita.,probs = .75)+1.5*IQR(Economy..GDP.per.Capita.)) %>% round(1),
                UpperOutlierCount = sum(Economy..GDP.per.Capita. > UpperFence) %>% round(0),
                NumSD_MeanToMax = ((max(Economy..GDP.per.Capita.) - mean(Economy..GDP.per.Capita.)) / sd(Economy..GDP.per.Capita.)) %>% round(1))
Wealth_Score_Outliers

t.test(Dataset_for_Assignment_4$Economy..GDP.per.Capita., Dataset_for_Assignment_4$Happiness.Score,
       paired = TRUE,
       alternative = "two.sided")

## 
##  Paired t-test
## 
## data:  Dataset_for_Assignment_4$Economy..GDP.per.Capita. and Dataset_for_Assignment_4$Happiness.Score
## t = -65.809, df = 154, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -4.500461 -4.238141
## sample estimates:
## mean of the differences 
##               -4.369301

Dataset_for_Assignment_4$Economy..GDP.per.Capita.  %>% qqPlot(dist="norm")

boxplot(
  Dataset_for_Assignment_4$Economy..GDP.per.Capita.,
  Dataset_for_Assignment_4$Happiness.Score,
  ylab = "Scores Ratings",
  xlab = "Wealth and Happiness"
  )
axis(1, at = 1:2, labels = c("Wealth", "Happiness"))

matplot(t(data.frame(Dataset_for_Assignment_4$Economy..GDP.per.Capita., Dataset_for_Assignment_4$Happiness.Score)),
  type = "b",
  pch = 19,
  col = 1,
  lty = 1,
  xlab = "Comparison",
  ylab = "Score rating",
  xaxt = "n"
  )
axis(1, at = 1:2, labels = c("Wealth Score", "Happiness Score"))

From the matplot we can clearly visualise that it is a positive increase relationship between Economy and Happiness. If wealth socre or happiness score increase, another one will increase as well.

granova.ds(
  data.frame(Dataset_for_Assignment_4$Economy..GDP.per.Capita., Dataset_for_Assignment_4$Happiness.Score),
  xlab = "Wealth Score",
  ylab = "Happiness Score"
  )

##             Summary Stats
## n                 155.000
## mean(x)             0.985
## mean(y)             5.354
## mean(D=x-y)        -4.369
## SD(D)               0.827
## ES(D)              -5.286
## r(x,y)              0.812
## r(x+y,d)           -0.893
## LL 95%CI           -4.500
## UL 95%CI           -4.238
## t(D-bar)          -65.809
## df.t              154.000
## pval.t              0.000

A paired-samples tt-test was used to test for a significant mean difference between scores levels of economy and happiness. The mean difference following exercise was found to be -4.37 (SD = 0.827). Visual inspection of the Q-Q plot of the difference scores suggested that the Happiness.Score was approximately normally distributed, but Economy..GDP.per.Capita was not so clear. The paired-samples tt-test found a statistically significant mean difference between stress levels before and after exercise, t(df=154)=???65.8, p<0.05, 95% [ -4.500461 -4.238141]. Happiness scores were found to be significantly increased after wealth score increased.

Decsriptive Statistics Cont.

• You can use the knitr:kable function to print nice HTML tables. Here is an example R code:

y2 <- Dataset_for_Assignment_4$Happiness.Score^2
x2 <- Dataset_for_Assignment_4$Economy..GDP.per.Capita.^2
xy <- Dataset_for_Assignment_4$Happiness.Score*Dataset_for_Assignment_4$Economy..GDP.per.Capita.
sum_x <- sum(Dataset_for_Assignment_4$Economy..GDP.per.Capita.)
sum_y <- sum(Dataset_for_Assignment_4$Happiness.Score)
sum_x_sq <- sum(Dataset_for_Assignment_4$Economy..GDP.per.Capita.^2)
sum_y_sq <- sum(Dataset_for_Assignment_4$Happiness.Score^2)
sum_xy <- sum(Dataset_for_Assignment_4$Happiness.Score*Dataset_for_Assignment_4$Economy..GDP.per.Capita.)
n <- length(Dataset_for_Assignment_4$Economy..GDP.per.Capita.) #Sample size

Lxx <- sum_x_sq-((sum_x^2)/n)
Lyy <- sum_y_sq-((sum_y^2)/n)
Lxy = sum_xy - (((sum_x)*(sum_y))/n)
b = Lxy/Lxx
a = mean(Dataset_for_Assignment_4$Economy..GDP.per.Capita. - b*mean(Dataset_for_Assignment_4$Happiness.Score))

plot(Economy..GDP.per.Capita. ~ Happiness.Score, data = Dataset_for_Assignment_4, xlab = "Happiness Score", ylab = "Economy Score")
abline(a = a, b = b, col= "red")
abline(lm(Dataset_for_Assignment_4$Economy..GDP.per.Capita. ~ Dataset_for_Assignment_4$Happiness.Score))

HapEconmodel <- lm( Economy..GDP.per.Capita. ~ Happiness.Score, data = Dataset_for_Assignment_4)
HapEconmodel %>% summary()

## 
## Call:
## lm(formula = Economy..GDP.per.Capita. ~ Happiness.Score, data = Dataset_for_Assignment_4)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.90072 -0.16663  0.00354  0.16685  0.61731 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -0.63338    0.09593  -6.603 6.27e-10 ***
## Happiness.Score  0.30222    0.01753  17.238  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2461 on 153 degrees of freedom
## Multiple R-squared:  0.6601, Adjusted R-squared:  0.6579 
## F-statistic: 297.1 on 1 and 153 DF,  p-value: < 2.2e-16

R2 <- (b*Lxy)/Lyy
R2

## [1] 0.6601055

Hypothesis Testing

H0:The data do not fit the linear regression model

pf(q = 297.1,1,153,lower.tail = FALSE)

## [1] 1.117922e-37

(R2/(1-R2)*(153/1))

## [1] 297.1396

HapEconmodel %>% anova()

We confirm the pp-value reported in the summary to be p<.001. As p-value is less than the 0.05 level of significance, we reject H0. There was statistically significant evidence that the data fit a linear regression model.

Hypthesis Testing Cont.

HapEconmodel %>% summary() %>% coef()

##                   Estimate Std. Error   t value     Pr(>|t|)
## (Intercept)     -0.6333762 0.09592844 -6.602591 6.271247e-10
## Happiness.Score  0.3022205 0.01753249 17.237739 1.110391e-37

The intercept/constant is reported as a=-0.633.This value represents the average Economy..GDP.per.Capita score when happiness score is equal to 0. To test the statistical significance of the constant, we set the following statistical hypotheses: H0:=0 HA:<>0 This hypothesis is tested using a t statistic, reported as t=-6.6026, p<.001. The constant is statistically significant at the 0.05 level. This means that there is statistically significant evidence that the constant is not 0.

HapEconmodel %>% confint()

##                      2.5 %     97.5 %
## (Intercept)     -0.8228915 -0.4438609
## Happiness.Score  0.2675835  0.3368575

R reports the 95% CI for a to be [-0.8228915, -0.4438609]. H0:??=0 is clearly not captured by this interval, so was rejected.

H0:=0 HA:<>0 The slope of the regression line was reported as b=0.302.A one unit increase in Happiness Score was related to an average increase in Economy..GDP.per.Capita of .302 units. This is a positive change. We confirm that p<.001. As p<.05, we reject H0. There was statistically significant evidence that Happiness Score was positively related to Economy..GDP.per.Capita.

plot(HapEconmodel)

Discussion

• Discuss the major findings of your investigation
• Discuss any strengths and limitations.
• Propose directions for future investigations.
• This is a good place to re-state your findings as a final conclusion. What is the one take home message the reader should leave with?
• Your final conclusion needs to be very clear.

References

• Provide a list of any references you use in the presentation.