Module 3: Univariate Analysis

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Univariate Analysis

What is univariate analysis ?

The idea of univariate analysis is to first understand the variables individually. It is typically the first step in understanding a data set. A variable in UA is a condition or subset that your data falls into. You can think of it as a “category” such as “age”, “weight” or “length”. However, UA does not look at > than 1 variable at a time (this would be a bivariate analysis)

Learning Objectives:

Summarising Data
Frequency Tables
Univariate Hypothesis Testing
Visualising Univariate Data
Correlation
Simple Regression analysis

# Installation of packages (usually needed)
# install.packages("ggplot2")
# install.packages("dplyr")
# install.packages("ggpubr")
# install.packages("corrplot")

# Loading of packages
library(ggplot2)
library(dplyr)


Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

library(ggpubr)
library(corrplot)

corrplot 0.92 loaded

library(stats)

1. Summarising Data

# Using the data set stored in Rstudio called "cars"

# We need to create an array of our single variable for UA:
x <- cars$speed

Looking at the CENTRAL TENDENCY of the data:

mean(x)

[1] 15.4

median(x)

[1] 15

mode(x)

[1] "numeric"

Looking at the DISPERSION of the data:

min(x)

[1] 4

max(x)

[1] 25

# Range of the data:
range(x)

[1]  4 25

# Inter-quantile range:
IQR(x)

[1] 7

# Variance -->
var(x)

[1] 27.95918

# Standard Deviation:
sd(x)

[1] 5.287644

TIP: you can use the function summary to produce result summaries of the results of various model fitting functions.

summary(x)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    4.0    12.0    15.0    15.4    19.0    25.0

2. Frequency Tables:

The frequency of an observation tells you the number of times the observation occurs in the data.
A frequency table is a collection of these observations and their frequencies.
A frequency table can be shown either graphically (bar chart/histogram) or as a frequency distribution table.
These tables can show qualitative (categorical) or quantitative (numeric) variables.

Example Data

We will use a data frame with a categorical variable and a numerical variable to demonstrate each type of table.

# Create example data
set.seed(123)  # For reproducibility
data <- data.frame(
  category = sample(c("A", "B", "C", "D"), 100, replace = TRUE),
  value = rnorm(100, mean = 50, sd = 10)
)
head(data)

  category    value
1        C 52.53319
2        C 49.71453
3        C 49.57130
4        B 63.68602
5        C 47.74229
6        B 65.16471

# Frequency table for the categorical variable
freq_table <- table(data$category)
freq_table


 A  B  C  D 
28 26 29 17

# Qualitative Variables:
freq_table_numeric <- table(data$value)
freq_table_numeric


26.9083112435919 29.4675277845948 33.3205806341186 33.8211729171084 
               1                1                1                1 
33.9846382642541 34.2785584085451 34.5124719576978 34.8533234621825 
               1                1                1                1 
 35.382444150041 35.3935992907518  35.561068390282 37.1296952396482 
               1                1                1                1 
37.7928228774546 39.2820877352442 39.7357909969322 39.7587120939509 
               1                1                1                1 
39.8142461689291 40.3814336586987 40.4838143273498  40.525253858152 
               1                1                1                1 
41.5029565396642 42.1509553054292  42.895934363007 42.9079923741761 
               1                1                1                1 
43.1199138353264 43.4805009830454 43.5929399169462 43.7209392396063 
               1                1                1                1 
43.9974041285287 44.2465303739161  44.690934778297  44.976765468907 
               1                1                1                1 
45.0896883394346 45.0944255629933 45.7750316766038 46.1977347971224 
               1                1                1                1 
46.2933996820759 46.5245740060227 46.6679261633058 46.7406841446877 
               1                1                1                1 
47.1522699294899 47.3780251059753 47.4390780780175 47.5330812153763 
               1                1                1                1 
47.6429964089952 47.7422901434073 47.7951343818125 48.6110863756096 
               1                1                1                1 
 49.286919138764 49.4443803447546 49.5497227519108 49.5712954270868 
               1                1                1                1 
 49.714532446513 50.0576418589989 50.4123292199294  50.530042267305 
               1                1                1                1 
50.7796084956371 51.0567619414894 51.1764659710013 51.2385424384461 
               1                1                1                1 
51.8130347974915 52.1594156874397 52.3538657228486 52.3873173511144 
               1                1                1                1 
52.5331851399475 52.5688370915653 53.0115336216671 53.0352864140426 
               1                1                1                1 
 53.317819639157 53.7963948275988 53.8528040112633  54.351814908338 
               1                1                1                1 
54.4820977862943 54.5150405307921 55.1940720394346 55.4839695950807 
               1                1                1                1 
55.8461374963607 56.0796432222503 56.4437654851883 56.8791677297583 
               1                1                1                1 
57.0178433537471 57.3994751087733 59.1899660906077 59.2226746787974 
               1                1                1                1 
59.9350385596212 60.0573852446226  60.255713696967 60.9683901314935 
               1                1                1                1 
61.3133721341418 61.4880761845109 63.6065244853001 63.6860228401446 
               1                1                1                1 
64.4455085842335 65.1647060442954 65.3261062618519 68.4386200523221 
               1                1                1                1 
69.0910356921748 70.5008468562714 71.0010894052567 71.8733299301658 
               1                1                1                1

Note: the frequency table is CASE-SENSITIVE so the frequencies of the variables corresponds to how many times that specific number of string appears.

Grouped Tables:

Grouped tables aggregate the data into groups or bins.

# 1st Step: Create BINS for the numerical data
bins <- cut(x, breaks = 5)
freq_table_numeric <- table(bins)
freq_table_numeric

bins
 (3.98,8.2]  (8.2,12.4] (12.4,16.6] (16.6,20.8]   (20.8,25] 
          5          10          13          15           7

# Group data into bins and create a grouped table:
grouped_table <- table(cut(x, breaks = 5))
grouped_table


 (3.98,8.2]  (8.2,12.4] (12.4,16.6] (16.6,20.8]   (20.8,25] 
          5          10          13          15           7

Percentage (Proportion) Tables

Percentage tables show the proportion of each unique value or group in the data.

# Percentage table for the categorical variable
percentage_table <- prop.table(table(x)) * 100
percentage_table

x
 4  7  8  9 10 11 12 13 14 15 16 17 18 19 20 22 23 24 25 
 4  4  2  2  6  4  8  8  8  6  4  6  8  6 10  2  2  8  2

# Percentage table for the grouped numerical data
percentage_table_numeric <- prop.table(table(cut(x, breaks = 5))) * 100
percentage_table_numeric


 (3.98,8.2]  (8.2,12.4] (12.4,16.6] (16.6,20.8]   (20.8,25] 
         10          20          26          30          14

Cumulative Proportion Tables

Cumulative proportion tables show the cumulative proportion of each unique value or group.

# Cumulative proportion table for the categorical variable
cumulative_prop <- cumsum(prop.table(table(data$category)))
cumulative_prop <- cumulative_prop * 100
cumulative_prop

  A   B   C   D 
 28  54  83 100

# Cumulative proportion table for the grouped numerical data
cumulative_prop_numeric <- cumsum(prop.table(table(cut(x, breaks = 5))))
cumulative_prop_numeric <- cumulative_prop_numeric * 100
cumulative_prop_numeric

 (3.98,8.2]  (8.2,12.4] (12.4,16.6] (16.6,20.8]   (20.8,25] 
         10          30          56          86         100

Question 1:

Using the cars datset:

Calculate the mean, median, and standard deviation of variable “speed”.
Interpret what these statistics tell you about the speed data.
Compute the range and interquartile range (IQR) of speed.
What do these measures reveal about the dispersion of the speed data?
Use the summary function to get a summary of x.
Describe the central tendency and dispersion metrics provided by the summary output.

Question 2:

Using the below:

xy <- data.frame(
  category = sample(c("A", "B", "C", "D"), 100, replace = TRUE)
)
head(xy)

Create a frequency table for the category variable.
What is the frequency of each category?

Using the below:

data <- data.frame(
  value = rnorm(100, mean = 50, sd = 10)
)

Create a frequency table for the value variable.
How many observations fall into each unique value?

Using the below:

x <- data$value
bins <- cut(x, breaks = 5)

Create a grouped frequency table for the value variable using 5 bins.
What are the frequencies for each bin?

Using the below:

x <- data$value
bins <- cut(x, breaks = 5)

Create a percentage (proportion) table for the grouped value data.
What percentage of the observations fall into each bin?

Answers:

# Question 1:
# a. Calculate the mean, median, and standard deviation of variable "speed"
mean_speed <- mean(x)
median_speed <- median(x)
sd_speed <- sd(x)

# c. Compute the range and interquartile range (IQR) of speed
range_speed <- range(x)
iqr_speed <- IQR(x)

# e. Use the summary function to get a summary of x
summary_speed <- summary(x)

# Question 2:
# a. Create a frequency table for the category variable
freq_table_category <- table(xy$category)

# c. Create a frequency table for the value variable
freq_table_value <- table(data$value)

# e. Create a grouped frequency table for the value variable using 5 bins
grouped_table <- table(bins)

# g. Create a percentage (proportion) table for the grouped value data
percentage_table <- prop.table(grouped_table) * 100

3. Univariate Hypothesis Testing:

Often, the data you are dealing with is a subset (sample) of the complete data (population). Thus, the common question here is:

Can the findings of the sample be extrapolated to the population? i.e., Is the sample representative of the population, or has the population changed?

Such questions are answered using specific hypothesis tests designed to deal with such univariate data-based problems.

Example Dataframe:

set.seed(42)  # For reproducibility

# Generate numerical data
sample_data_large <- rnorm(50, mean = 100, sd = 15)  # Sample size > 30
sample_data_small <- rnorm(20, mean = 100, sd = 15)  # Sample size < 30

# Known population parameters
population_mean <- 100
population_sd <- 15

# Generate categorical data
category_data <- sample(c("A", "B", "C"), 100, replace = TRUE)
ordinal_data <- sample(c("Low", "Medium", "High"), 100, replace = TRUE)

Z Test: Used for numerical (quantitative) data where the sample size is greater than 30 and the population’s standard deviation is known.

# Z Test: Test if sample mean is significantly different from population mean
library(stats)

# Perform Z Test
z_score <- (mean(sample_data_large) - population_mean) / (population_sd / sqrt(length(sample_data_large)))
z_score

[1] -0.2522376

p_value_z <- 2 * pnorm(-abs(z_score))  # Two-tailed test
p_value_z

[1] 0.8008574

Interpretation: If the p-value is less than the significance level (commonly 0.05), the sample mean is significantly different from the population mean.

One-Sample t-Test: Used for numerical (quantitative) data where the sample size is less than 30 or the population’s standard deviation is unknown.

# One-Sample t-Test: Test if sample mean is significantly different from population mean
t_test_result <- t.test(sample_data_small, mu = population_mean)
t_test_result


    One Sample t-test

data:  sample_data_small
t = 1.2497, df = 19, p-value = 0.2266
alternative hypothesis: true mean is not equal to 100
95 percent confidence interval:
  97.17831 111.18375
sample estimates:
mean of x 
  104.181

Interpretation: The t-test result provides a p-value and confidence interval for the sample mean. A p-value less than 0.05 indicates a significant difference from the population mean.

Chi-Square Test: Used with ordinal categorical data

# Chi-Square Test: Test the distribution of categorical data
observed_counts <- table(category_data)
expected_counts <- rep(length(category_data) / length(observed_counts), length(observed_counts))

chi_square_result <- chisq.test(observed_counts, p = expected_counts / sum(expected_counts))
chi_square_result


    Chi-squared test for given probabilities

data:  observed_counts
X-squared = 2.18, df = 2, p-value = 0.3362

Interpretation: The Chi-Square test assesses whether the observed frequencies differ from the expected frequencies. A p-value less than 0.05 suggests a significant difference.

Kolmogorov-Smirnov Test: Used with nominal categorical data

# Kolmogorov-Smirnov Test: Compare sample distribution to a normal distribution
ks_test_result <- ks.test(sample_data_large, "pnorm", mean = population_mean, sd = population_sd)
ks_test_result


    Exact one-sample Kolmogorov-Smirnov test

data:  sample_data_large
D = 0.077011, p-value = 0.906
alternative hypothesis: two-sided

Interpretation: The KS test assesses whether the sample follows the specified distribution. A p-value less than 0.05 indicates a significant deviation from the normal distribution.

4. Visualising Univariate Data:

Visualizing univariate data helps us understand the distribution and patterns within a single variable. Below, we’ll cover visualization techniques for both categorical and numeric data.

Example Data:

set.seed(42)  # For reproducibility

# Numeric data
numeric_data <- rnorm(100, mean = 50, sd = 10)

# Categorical data
categorical_data <- sample(c("Category A", "Category B", "Category C"), 100, replace = TRUE)

Visualising Numeric Data:

A histogram shows the distribution of numeric data by dividing it into bins and counting the frequency of data points in each bin.

library(ggplot2)
# Histogram
ggplot(data.frame(numeric_data), aes(x = numeric_data)) +
  geom_histogram(binwidth = 2.5, aes(y = ..density..), fill = "skyblue", color = "white", alpha = 0.7) +
  geom_density(color = "darkblue", size = 1) +  # Added a density line
  labs(title = "Histogram",x = "Value", y = "Density") +
  theme_classic(base_size = 15) +
  theme(plot.title = element_text(hjust = 0.5, face = "bold"))

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.
This warning is displayed once every 8 hours.
Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
generated.

Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(density)` instead.
This warning is displayed once every 8 hours.
Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
generated.

A boxplot provides a summary of the numeric data distribution, highlighting the median, quartiles, and potential outliers.

# Boxplot
ggplot(data.frame(numeric_data), aes(x = "", y = numeric_data)) +
  geom_boxplot(fill = "lightgreen", color = "darkgreen", outlier.shape = NA) +
  geom_jitter(width = 0.2, alpha = 0.5, color = "darkgreen", aes(x = "")) +  
  labs(title = "Boxplot",y = "Value") +
  theme_classic(base_size = 15) +
  theme(plot.title = element_text(hjust = 0.5, face = "bold"))

A density plot shows the distribution of numeric data as a continuous probability density function.

# Density Plot
ggplot(data.frame(numeric_data), aes(x = numeric_data)) +
  geom_density(fill = "purple", color = "black", alpha = 0.5, size = 1) +  
  labs( title = "Density Plot",x = "Value", y = "Density") +
  theme_minimal(base_size = 15) +  # Increase base font size
  theme(
    plot.title = element_text(hjust = 0.5, face = "bold", size = 16),
    axis.title.x = element_text(size = 14),
    axis.title.y = element_text(size = 14),
    axis.text = element_text(size = 12),
    panel.border = element_rect(color = "black", fill = NA, linewidth = 1)
  )

Visualizing Categorical Data:

For categorical data, common visualizations include bar charts and pie charts.

A bar chart displays the frequency or count of each category in a categorical variable.

# Bar Chart
ggplot(data.frame(categorical_data), aes(x = categorical_data)) +
  geom_bar(fill = "coral", color = "black", width = 0.7) +
  geom_text(stat = 'count', aes(label = ..count..), vjust = -0.5, color = "black", size = 5) +  # Add labels on bars
  labs( x = "Category", y = "Count") +
  theme_classic(base_size = 15) +
  theme(plot.title = element_text(hjust = 0.5, face = "bold"))

A pie chart represents the proportion of each category in the dataset.

# Pie Chart
library(dplyr)
library(ggplot2)

pie_data <- as.data.frame(table(categorical_data)) %>%
  rename(Category = categorical_data, Count = Freq) %>%
  mutate(Proportion = Count / sum(Count),
         label = scales::percent(Proportion)) 

ggplot(pie_data, aes(x = "", y = Proportion, fill = Category)) +
  geom_bar(stat = "identity", width = 1) +
  coord_polar(theta = "y") +
  labs(title = "Pie Chart") +
  theme_void() +
  scale_fill_brewer(palette = "Pastel1") +  
  theme(legend.title = element_blank(),
        plot.title = element_text(hjust = 0.5, face = "bold")) +
  geom_text(aes(label = label), position = position_stack(vjust = 0.5))

5. Correlation:

Correlation analysis is used to investigate the association between two or more variables.

Step 1: Choose a Correlation Method

Pearson Correlation measures the linear relationship between two continuous variables. It assumes both variables follow a normal distribution.

Spearman and Kendall Correlation are non-parametric and measure the strength and direction of the association between two ranked variables.

Step 2: Preliminary Checks

Before applying Pearson’s correlation, check the assumptions:

Normality Check –> Use the Shapiro-Wilk test to assess if the variables are normally distributed.

Null hypothesis: the data = normally distributed
Alternative hypothesis: the data = not normally distributed
If the p-value is less than 0.05, the null hypothesis is rejected

# Shapiro-Wilk test for normality
shapiro.test(mtcars$mpg)


    Shapiro-Wilk normality test

data:  mtcars$mpg
W = 0.94756, p-value = 0.1229

shapiro.test(mtcars$wt)


    Shapiro-Wilk normality test

data:  mtcars$wt
W = 0.94326, p-value = 0.09265

Does from data of each of the 2 variables (mpg, wt) follow a normal distribution?:

mpg: The p-value is 0.1229, which is greater than 0.05. Therefore, we do not reject the null hypothesis. This suggests that the mpg variable does not significantly deviate from a normal distribution.

wt: The p-value is 0.09265, which is also greater than 0.05. Thus, we do not reject the null hypothesis. This indicates that the wt variable does not significantly deviate from a normal distribution.

Visualize Normality –> Use Q-Q plots to visually inspect normality.

library(ggpubr)
par(mfrow=c(1,2))
ggqqplot(mtcars$mpg, ylab = "MPG")

ggqqplot(mtcars$wt, ylab = "Weight")

Is the covariation linear?:

Yes, form the plot above, the relationship is linear.

Step 3: Calculate Correlation

Pearson Correlation

# Pearson correlation test
pearson_res <- cor.test(mtcars$mpg, mtcars$wt, method = "pearson")
pearson_res


    Pearson's product-moment correlation

data:  mtcars$mpg and mtcars$wt
t = -9.559, df = 30, p-value = 1.294e-10
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.9338264 -0.7440872
sample estimates:
       cor 
-0.8676594

Spearman and Kendall Correlation

# Spearman correlation test
spearman_res <- cor.test(mtcars$mpg, mtcars$wt, method = "spearman")

Warning in cor.test.default(mtcars$mpg, mtcars$wt, method = "spearman"): Cannot
compute exact p-value with ties

spearman_res


    Spearman's rank correlation rho

data:  mtcars$mpg and mtcars$wt
S = 10292, p-value = 1.488e-11
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
-0.886422

# Kendall correlation test
kendall_res <- cor.test(mtcars$mpg, mtcars$wt, method = "kendall")

Warning in cor.test.default(mtcars$mpg, mtcars$wt, method = "kendall"): Cannot
compute exact p-value with ties

kendall_res


    Kendall's rank correlation tau

data:  mtcars$mpg and mtcars$wt
z = -5.7981, p-value = 6.706e-09
alternative hypothesis: true tau is not equal to 0
sample estimates:
       tau 
-0.7278321

Step 4: Visualize the Correlation

# Scatter plot with Pearson correlation
ggscatter(mtcars, x = "mpg", y = "wt", 
          add = "reg.line", conf.int = TRUE, 
          cor.coef = TRUE, cor.method = "pearson",
          xlab = "Miles per Gallon", ylab = "Weight (1000 lbs)")

Step 5: Interpretation

Correlation Coefficient:

-1: Strong negative correlation (as one variable increases, the other decreases).
0: No correlation.
1: Strong positive correlation (both variables increase together).

P-Value:

p-value < 0.05 indicates a statistically significant correlation.

Exercise:

Perform a correlation analysis using the mpg and qsec variables from the mtcars to investigate the extent of correlation between the two variables. Provide an interpretation of the correlation coefficient and its p-value.

Example interpretation:

The Pearson correlation coefficient is -0.8677, which points to a strong negative linear relationship between the variables.
The p-value is significantly low (p < 0.001), indicating that the correlation is statistically significant.
The 95% confidence interval suggests that the true correlation lies between -0.9338 and -0.7441.

Compute correlation matrix in R

The R function cor() can be used to compute a correlation matrix.

# We start by loading the mtcars dataset and selecting a subset of columns for our analysis.
data("mtcars")
my_data <- mtcars[, c(1, 3, 4, 5, 6, 7)]
# Display first few rows
head(my_data)

                   mpg disp  hp drat    wt  qsec
Mazda RX4         21.0  160 110 3.90 2.620 16.46
Mazda RX4 Wag     21.0  160 110 3.90 2.875 17.02
Datsun 710        22.8  108  93 3.85 2.320 18.61
Hornet 4 Drive    21.4  258 110 3.08 3.215 19.44
Hornet Sportabout 18.7  360 175 3.15 3.440 17.02
Valiant           18.1  225 105 2.76 3.460 20.22

We then compute the correlation matrix:

rescm <- cor(my_data)
# Round the results to 2 decimal places for easier interpretation
round(rescm, 2)

       mpg  disp    hp  drat    wt  qsec
mpg   1.00 -0.85 -0.78  0.68 -0.87  0.42
disp -0.85  1.00  0.79 -0.71  0.89 -0.43
hp   -0.78  0.79  1.00 -0.45  0.66 -0.71
drat  0.68 -0.71 -0.45  1.00 -0.71  0.09
wt   -0.87  0.89  0.66 -0.71  1.00 -0.17
qsec  0.42 -0.43 -0.71  0.09 -0.17  1.00

Interpretation: - Values close to 1 or -1 indicate strong positive or negative correlations, respectively.

Values close to 0 suggest little to no linear relationship.

Visualising the Correlation Matrix:

library(corrplot)
corrplot(rescm,tl.col = "black", addCoef.col = "black")

6. Simple Linear Regression:

x <- mtcars$mpg
y <- mtcars$wt

model = lm(y ~ x)
summary(model)


Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.6516 -0.3490 -0.1381  0.3190  1.3684 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  6.04726    0.30869  19.590  < 2e-16 ***
x           -0.14086    0.01474  -9.559 1.29e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4945 on 30 degrees of freedom
Multiple R-squared:  0.7528,    Adjusted R-squared:  0.7446 
F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

par(mfrow = c(2, 2))
plot(model)

Checking Assumptions:

Assumption 1: Linearity –> Check if the relationship between variables is linear.

Plot of x vs y: This scatter plot displays the relationship between the predictor x and the response y.

Abline (Regression Line): The abline(model) adds the fitted regression line to the plot.

plot(x, y)
abline(model)

What to Look For:

Linear Relationship: The data points should roughly form a straight line if the linearity assumption is satisfied. The fitted regression line should capture the trend of the data points well.

Non-Linearity: If the data points show a clear curvature or systematic pattern not captured by the straight line, this suggests that the linearity assumption is violated. In such cases, consider polynomial regression or other non-linear models.

Assumption 2: Homoscedasticity –> Ensure that the residuals are evenly distributed.

Plot of Residuals: This plot shows the residuals from the model. Residuals are the differences between the observed values and the values predicted by the model.

plot(model$residuals)

What to Look For:

Even Spread: Ideally, the residuals should be randomly scattered around zero and should not display any clear pattern. This indicates homoscedasticity (constant variance of residuals).

Patterns: If you observe a pattern, such as a funnel shape (residuals increasing or decreasing as x increases), it suggests heteroscedasticity (non-constant variance). In such cases, consider transforming the dependent variable or using robust regression techniques.

Assumption 3: Normality of Residuals –> Use Q-Q plots to check the normality of residuals.

Q-Q Plot: The Q-Q plot (quantile-quantile plot) compares the quantiles of the residuals with the quantiles of a normal distribution.

qqnorm(model$residuals)
qqline(model$residuals)

What to Look For:

Straight Line: If the residuals are normally distributed, the points should closely follow the straight line (qqline). This suggests that the normality assumption is reasonable.

Deviations: Significant deviations from the line indicate that the residuals are not normally distributed. This could mean the presence of outliers or skewness in the residuals. If the normality assumption is violated, consider transforming the response variable or using non-parametric methods.

Summary of Interpretation:

Linearity: The plot of x vs. y with the regression line should show a clear linear relationship.
Homoscedasticity: The plot of residuals should display no obvious patterns or systematic structures.
Normality of Residuals: The Q-Q plot should show residuals following the diagonal line if they are normally distributed.

These plots help you validate the assumptions underlying your regression model, ensuring that your results are reliable and interpretable.

Simple Linear Regression Exercise:

You were asked to analyze the following dataset, mtcars, where mpg (miles per gallon) is used as the predictor variable and wt (weight) as the response variable. You have fitted a linear regression model and checked the assumptions.

Now, perform simple linear regression on two variables of your choosing from the mtcars data set and answer the following questions:

Describe the relationship between mpg and wt. Does the plot suggest a linear relationship?
Describe the spread of the residuals. Is there any noticeable pattern that might suggest a violation of the homoscedasticity assumption?
Assess whether the residuals appear to follow a normal distribution based on the Q-Q plot. Are there any significant deviations from the diagonal line?

sessionInfo()

R version 4.3.3 (2024-02-29)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.5

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: Africa/Johannesburg
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] corrplot_0.92   ggpubr_0.6.0    dplyr_1.1.4     ggplot2_3.5.1  
[5] workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] gtable_0.3.5       xfun_0.46          bslib_0.8.0        processx_3.8.4    
 [5] rstatix_0.7.2      lattice_0.22-6     callr_3.7.6        vctrs_0.6.5       
 [9] tools_4.3.3        ps_1.7.7           generics_0.1.3     tibble_3.2.1      
[13] fansi_1.0.6        highr_0.11         pkgconfig_2.0.3    Matrix_1.6-5      
[17] RColorBrewer_1.1-3 lifecycle_1.0.4    compiler_4.3.3     farver_2.1.2      
[21] stringr_1.5.1      git2r_0.33.0       munsell_0.5.1      getPass_0.2-4     
[25] carData_3.0-5      httpuv_1.6.15      htmltools_0.5.8.1  sass_0.4.9        
[29] yaml_2.3.10        later_1.3.2        pillar_1.9.0       car_3.1-2         
[33] crayon_1.5.3       jquerylib_0.1.4    whisker_0.4.1      tidyr_1.3.1       
[37] cachem_1.1.0       abind_1.4-5        nlme_3.1-165       tidyselect_1.2.1  
[41] digest_0.6.36      stringi_1.8.4      purrr_1.0.2        labeling_0.4.3    
[45] splines_4.3.3      rprojroot_2.0.4    fastmap_1.2.0      grid_4.3.3        
[49] colorspace_2.1-1   cli_3.6.3          magrittr_2.0.3     utf8_1.2.4        
[53] broom_1.0.6        withr_3.0.1        scales_1.3.0       promises_1.3.0    
[57] backports_1.5.0    rmarkdown_2.27     httr_1.4.7         ggsignif_0.6.4    
[61] evaluate_0.24.0    knitr_1.48         mgcv_1.9-1         rlang_1.1.4       
[65] Rcpp_1.0.13        glue_1.7.0         rstudioapi_0.16.0  jsonlite_1.8.8    
[69] R6_2.5.1           fs_1.6.4

Module 3: Univariate Analysis

Stefano Cacciatore

September 05, 2024

Univariate Analysis

Learning Objectives:

1. Summarising Data

2. Frequency Tables:

Example Data

Grouped Tables:

Percentage (Proportion) Tables

Cumulative Proportion Tables

3. Univariate Hypothesis Testing:

4. Visualising Univariate Data:

5. Correlation:

Step 1: Choose a Correlation Method

Step 2: Preliminary Checks

Step 3: Calculate Correlation

Step 4: Visualize the Correlation

Step 5: Interpretation

Compute correlation matrix in R

We then compute the correlation matrix:

Visualising the Correlation Matrix:

6. Simple Linear Regression:

Checking Assumptions:

Assumption 1: Linearity –> Check if the relationship between variables is linear.

Assumption 2: Homoscedasticity –> Ensure that the residuals are evenly distributed.

Assumption 3: Normality of Residuals –> Use Q-Q plots to check the normality of residuals.

Summary of Interpretation: