Enter your name :Your name is : Hypothesis Testing
t-tests & the Truth About p-values
A Simple Blast Off Countdown Thingy 🚀
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Blast Off!
A SUPER Plotting Function
Another SUPER-DUPER Plotting Function
# calculating sine of an angle in degrees
sine_degrees <- function(angle_degrees) {
# Convert degrees to radians
angle_radians <- angle_degrees * (pi / 180)
# Calculate sine
sine_value <- sin(angle_radians)
# Return the sine value
return(sine_value)
}
plot_sine <- function(start, end, step) {
# Generate a sequence of angles in degrees
angles <- seq(start, end, by = step)
# Calculate sine values for each angle
sine_values <- sapply(angles, sine_degrees)
# Create the plot
plot(angles, sine_values, type = "l", col = "blue", lwd = 2,
xlab = "Angle (degrees)", ylab = "Sine Value",
main = "Sine Function Plot")
grid() # Add grid lines for better visualization
}
plot_sine(1,400,1)
A Quick Word About Quantiles
Tip
- Quartiles: Splits the data into four equal parts (Q₁ at 25%, Q₂ at 50%, Q₃ at 75%).
- Common quantiles include quartiles (4 parts), quintiles (5 parts), and percentiles (100 parts).
- Useful when summarizing data spread, central tendency, and range
A Function to Check Normality of Data
If your data is normally distributed, the points in the Q-Q plot will fall approximately along the reference line. Deviations from this line indicate departures from normality.
library(tidyverse)
# Quartiles
myQuartilePlot <- function(data) {
# Create a data frame for ggplot
df <- data.frame(value = data)
# Create the quantile plot
ggplot(df, aes(sample = value)) +
stat_qq() +
stat_qq_line() +
labs(title = "Quantile-Quantile Plot",
x = "Theoretical Quantiles",
y = "Sample Quantiles") +
theme_minimal()
}
# Generate some sample data
set.seed(301)
data <- rnorm(100)
myQuartilePlot(data)
A Function to Check Normality of Data Plot
In normal distributions, the points will align with the reference line.
Correlation: A Relatedness Measure
Important
A correlation coefficient measures the strength and direction of the linear relationship between two continuous variables.
Correlation coefficients range from -1 (negative) to 1 (positive), with values closer to -1 or 1 indicating stronger relationships.
Positive Correlation
Positive correlation: as one variable increases, the other also increases.
Negative Correlation
Negative correlation: as one variable increases, the other decreases.
Neutral Correlation
Neutral correlation: the increase in one variable is not related to the increase in the other.
All Correlations
All correlations: positive, negative, and neutral correlations.
Correlations with Numerical Values
Different types of correlation coefficients exist, such as Pearson’s correlation coefficient (for linear relationships) and Spearman’s rank correlation coefficient (for monotonic relationships).
Correlations with Numerical Values
The correlation coefficient is a unitless measure; it does not depend on the units of the variables being measured.
Whats the Spread of the Data?
How spread out is the data? Does the data have a lot of variability about the mean?
Variance is a measure of how spread out the data is from the mean. It is calculated by taking the average of the squared differences from the mean.
Standard Deviation Visually Part 1
Standard Deviation Visually Part 2
Tip
- Large standard deviation → data is spread out over a wider range of values.
- Large and Lazy
- Small standard deviation → data is clustered closely around the mean.
- Small and Snug
Standard Deviation Visually Part 3
Tip
- How does the SD compare to your requirements or limits?
- How does the SD change over time?
Standard Deviation and Means
library(ggplot2)
# Define the distributions
params <- data.frame(
group = c("Low", "Moderate", "High", "Very high"),
mean = c(10, 20, 30, 40),
sd = c(2, 4, 6, 8)
)
# Generate density values for each distribution
set.seed(123)
data <- do.call(rbind, lapply(1:nrow(params), function(i) {
x <- seq(
params$mean[i] - 4 * params$sd[i],
params$mean[i] + 4 * params$sd[i],
length.out = 500
)
data.frame(
x = x,
density = dnorm(
x,
mean = params$mean[i],
sd = params$sd[i]
),
group = params$group[i]
)
}))
# Plot the distributions
ggplot(data, aes(x = x, y = density, color = group)) +
geom_line(linewidth = 1.2) +
geom_vline(
data = params,
aes(xintercept = mean, color = group),
linetype = "dashed",
linewidth = 0.7,
show.legend = FALSE
) +
labs(
title = "Increasing Means and Standard Deviations",
subtitle = "Dashed lines indicate the means",
x = "Value",
y = "Probability density",
color = "Distribution"
) +
theme_minimal(base_size = 13)
Standard Deviation and Means (Plot)
A SUPER Fast View of P-Values!
Tip
- If the p-value is small (< 0.05), we reject H₀. If the p-value is large, we fail to reject H₀.
- If the p-value is very small (< 0.01), we have stronger evidence for Ha.
A SUPER Fast Example with \(H_0\) and \(H_1\)! 🍹
Tip
Let’s say we’re testing whether a new drink increases alertness. We have 30 participants, and we measure their alertness before and after drinking the new drink.
- \(H_0\): The new drink has no effect on alertness.
- \(H_1\): The new drink increases alertness.
We collect the data and find a p-value of 0.03, which is less than our significance level of 0.05. So we reject H₀ and conclude that the new drink increases alertness.
Seeing the Null Distribution 👀
library(tidyverse)
set.seed(301)
# Simulate H0 being TRUE: two groups from the same population
null_ts <- replicate(5000, {
a <- rnorm(20, 100, 15)
b <- rnorm(20, 100, 15) # identical population!
t.test(a, b)$statistic
})
observed <- 2.3 # pretend this is our real study's t statistic
ggplot(tibble(t = null_ts), aes(t)) +
geom_histogram(bins = 60, fill = "#BDC3C7", color = "white") +
geom_vline(xintercept = c(-observed, observed), color = "#E74C3C", linewidth = 1) +
labs(title = "5,000 studies where H0 was true",
subtitle = paste0("p-value = fraction beyond the red lines = ",
round(mean(abs(null_ts) >= observed), 4)),
x = "t statistic", y = "Count") +
theme_minimal(base_size = 12)
A Green Drink, a Purple Drink, and a p-value! 🍹
Important
Imagine your company bottles two drinks: green and purple. You suspect that the Green drink bottles are filled to higher levels by the green machine than the Purple machine fills its purple bottles.
Green and Purple: Our Selection
Important
We randomly select 9 bottles of each drink and measure the fill levels. These nine bottles are used to represent the total population of bottles, and to test our hypotheses.
The data is as follows:
- Green drink: 102, 107, 111, 110, 112, 115, 118, 120, 122
- Purple drink: 98, 100, 102, 102, 107, 110, 113, 104, 111
A SUPER Fast View of P-Values (Again!)
Tip
- If the p-value is small (< 0.05), we reject H₀. If the p-value is large, we fail to reject H₀.
- If the p-value is very small (< 0.01), we have stronger evidence for Ha.
Interpretation of the p-value for Green and Purple Drinks: The p-value = 0.01293 and so we reject the null hypothesis and conclude that there is a statistically significant difference in fill levels between the green and purple drinks.
One-Sample t-test: Visualizing the Null Distribution
library(tidyverse)
# Create a histogram of the sample data
ggplot(mtcars, aes(x = mpg)) +
geom_histogram(bins = 10, fill = "lightblue", color = "black", alpha = 0.7) +
geom_vline(xintercept = 20, color = "red", linetype = "dashed", size = 1) +
geom_vline(xintercept = mean(mtcars$mpg), color = "blue", linetype = "dashed", size = 1) +
labs(title = "Null Distribution for One-Sample t-test",
x = "MPG",
y = "Frequency") +
theme_minimal()
As a plot, the difference is clear
library(tidyverse)
ggplot(mtcars,
# Note, factor(am) places transmission type
# on the x-axis as a category.
aes(factor(am), mpg,
fill = factor(am))) +
geom_boxplot(alpha = 0.7) +
# geo_jitter() adds the points to box plots
geom_jitter(width = 0.12, size = 2) +
scale_x_discrete(
labels = c("Automatic", "Manual")) +
scale_fill_manual(
values = c("#3498DB", "#E67E22")) +
labs(x = NULL, y = "MPG") +
theme_minimal(base_size = 12) +
theme(legend.position = "none")
Checking Assumptions Plot
