RPubs Link

Rpubs link comes here:

https://rpubs.com/iamashks/math1324_assign3

Introduction

According to Harvard Health,

“What’s a good breakfast? One that delivers some healthful protein, some slowly digested carbohydrates, and some fruit or vegetables.”

According to Better Health Channel (Victoria),

“Glucose is the body’s energy source… In the morning, after you have gone without food for as long as 12 hours, your glycogen stores are low… Without carbohydrate, … cause reduced energy levels. Eating breakfast restores your glycogen stores and boosts your energy levels.”

Then, sugar – a type of sweet-tasting, soluble carbohydrate – is a source of energy, which is mostly included in breakfast cereals (in one form or another) in Australia.

So, it brings us to the question:

Problem Statement

Does eating more sugar give enough energy?

In other words, as is the common belief in Australia:

• Does eating foods in the breakfast that are high in sugar provide enough energy for the day?

• Or, the other nutrients found in food like minerals, proteins, and vitamins are required as well?

So, it raises the question:

Is there exists a weak but positive relationship between sugar intake from breakfast cereals and energy levels?

Data

Dataset Origin

• AUSNUT 2011-13 Food Nutrient Database from Food Standards Australia & New Zealand, which was used for 2011–13 Australian Health Survey (AHS).

Dataset Contents

• Nutrient content of 5,740 foods.
• 53 nutrients reported for every food.
• Each value reported per 100 g portion.

Variables used in this Investigation

• Total Sugars (g) - numeric (double), and
• Energy With Dietary Fibre (kJ) - numeric (integer).

Data (Cont.)

Filter

• This dataset was filtered to comprise just breakfast cereals.
• The filtered dataset contains 168 observations over 57 variables.

Sampling Method

• Simple Random Sampling (SRS) was utilized to collect the data.
• The reason being every breakfast cereal has equal probability.
• Sampling result - The dataset now has 123 obersersations.

head(data)

Data (Cont.)

Subsetting Data

• Subset data to include just the two important variables:
  • Energy with dietary fibre (kJ), and
  • Total sugars (g).

data <- data[, c(1, 5, 13)]

head(data)

Descriptive Statistics and Visualisation

Summary and Missing Values

data %>% summarise(Variable = "Total Sugar",
                   Min = min(`Total sugars (g)`, na.rm = TRUE),
                   Max = max(`Total sugars (g)`, na.rm = TRUE),
                   Mean = mean(`Total sugars (g)`, na.rm = TRUE),
                   Missing = sum(is.na(`Total sugars (g)`))) -> table1

knitr::kable(table1)

data %>% summarise(Variable = "Energy with Dietary Fibre",
                   Min = min(`Energy, with dietary fibre (kJ)`, na.rm = TRUE),
                   Max = max(`Energy, with dietary fibre (kJ)`, na.rm = TRUE),
                   Mean = mean(`Energy, with dietary fibre (kJ)`, na.rm = TRUE),
                   Missing = sum(is.na(`Energy, with dietary fibre (kJ)`))) -> table2

knitr::kable(table2)

Descriptive Statistics and Visualisation (Cont.)

Multivariate Outlier Detection

data1 <- data[, 2:3]

results <- mvn(data1, 
               multivariateOutlierMethod = "quan", 
               showOutliers = TRUE, showNewData = TRUE)

Descriptive Statistics and Visualisation (Cont.)

Dataset without Outliers

data2 <- data.frame(energy = results$newData[,1],
                    sugars = results$newData[,2])

results <- mvn(data2, 
               multivariateOutlierMethod = "quan")

Hypothesis Testing

Assumptions for Linear Regression

1. Independence

Every breakfast cereal is the dataset is recorded independently.

2. Linearity

Residual vs. Fitted Plot

• The red almost flat line shows good indication of a linear relationship.

model1 <- lm(Energy..with.dietary.fibre..kJ. ~ Total.sugars..g., data = data2)

plot(model1, which = 1)

Hypothesis Testing (Cont.)

3. Normality of Residuals

Normal Q-Q Plot

• Since most of the residual points are close to or directly on the straight line except for the -1 and -2 quartiles.

plot(model1, which = 2)

Hypothesis Testing (Cont.)

4. Homoscedasticity

Scale-Location Plot

• Since the red line is slightly curved with small variance, the dataset is expected to be more homoscedastic.

plot(model1, which = 3)

Hypothesis Testing (Cont.)

Linear Regression

• Regression line - Energy = 1.8788 * TotalSugars + 1496.3033
• R-squared - Goodness of Fit
  • R-squared = 0.04532
  • TotalSugars evaluated 4.532% of variability for the energy levels.
  • However, low R-squared value suggests a weak goodness of fit.

summary(model1)

## 
## Call:
## lm(formula = Energy..with.dietary.fibre..kJ. ~ Total.sugars..g., 
##     data = data2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -238.043  -47.079    1.984   50.390  254.333 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      1496.3033    16.6862  89.673   <2e-16 ***
## Total.sugars..g.    1.8788     0.8076   2.326   0.0218 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 88.52 on 114 degrees of freedom
## Multiple R-squared:  0.04532,    Adjusted R-squared:  0.03695 
## F-statistic: 5.412 on 1 and 114 DF,  p-value: 0.02177

Hypothesis Testing (Cont.)

Overall Regression Model

• Using the f-statistic model.

Hypotheses

\[H_0: The\,data\,don't\,fit\,the\,linear\,regression\,model.\] \[H_A: The\,data\,do\,fit\,the\,linear\,regression\,model.\]

• F = 5.412 (df1 = 1, df2 = 114)
• p-value of f-statistic
  • p = 0.02177 < significance level (α) = 0.05, thus reject \(H_0\).

Hypothesis Testing (Cont.)

Intercept or Constant

• α = 1496.3033
  • So, When sugar levels = 0, average energy levels = 1496.3033.

Hypotheses

\[H_0: α = 0\] \[H_A: α ≠ 0\]

• Value of t-statistic = 89.673
• p < 0.001 < 0.05, thus reject \(H_0\).
• 95% Confidence Interval is [1463.2480701, 1529.358604].

confint(model1)

##                         2.5 %      97.5 %
## (Intercept)      1463.2480701 1529.358604
## Total.sugars..g.    0.2789205    3.478691

Hypothesis Testing (Cont.)

Slope

• β = 1.8788, which means it’s a positive slope.
  • That is, 1g increase in total sugar levels will result in average increase of 1.8788kJ in energy levels.

Hypotheses

\[H_0: β = 0\]

\[H_A: β ≠ 0\]

• Value of t-statistic = 2.326
• p < 0.002 < 0.05, thus reject \(H_0\).
• 95% Confidence Interval is [0.2789205, 3.478691].

confint(model1)

##                         2.5 %      97.5 %
## (Intercept)      1463.2480701 1529.358604
## Total.sugars..g.    0.2789205    3.478691

Discussion

Statistical Results

• There are enough evidence showing the dataset fits the linear regression model.
• The total sugar levels shown a small amount of variance (4.5%) in energy levels.
• The intercept – it’s being not equal to 0 – was statistically significant.
• The slope resulted positive in the tests, proving it significant as well.

Conclusion

• First of all, there were numerous evidencs proving the suggestion of a positive linear relationship among total sugar levels and energy levels in the breakfast cereals popular in Australia.

• Also, though the prediction’s accuracy was low, R-squared was 4.532%;

Discussion (Cont.)

Investigation Limitations

• The distribution of the total sugar levels was somewhat skewed.
  • The distribution became a little worse upon transformation attempts.
• It affected some of the assumptions’ plots shown in this test.
  • Q-Q plot for the residuals was almost identical to regular plots.
  • Scale-location showed a small curve, however, its variance < 0.5.

Investigation Strengths

• The outliers were removed, making the distrution lines more flat.

Data Limitations

• Data didn’t include any latest breakfast cereals, since it’s from 2011-2013.

Data Strengths

• There were sufficient, if not large, amount of observsations for this category.
• There was accurate representation of the population: dataset and sampled.

References

Data

• AUSNUT 2011-13 Food Nutrient Database

More info

• Harvard Health
• Better Health Channel (Victoria)