Independent Samples T-test

Comparing Two Groups with Independent Samples T-test

• Research commonly aims to compare groups of people
• A t-test is a statistical test used to compare groups when the data supports its use
• A random sample is split into two groups, G1 and G2 in a t-test
• A numeric variable X, assumed to be normally distributed, is measured like systolic blood pressure
• Group membership (G1 vs. G2) is tested for association with different values of X
• A scientific question is whether the mean value of X differs between groups G1 and G2
• μG1 represents the population-level mean of X for G1, and μG2 for G2
• A key question is whether μG1 equals μG2

Statistical Hypotheses

• Null hypothesis (H0): μG1 = μG2 (the means are equal)
• Alternate hypothesis (HA): μG1 ≠ μG2 (the means are not equal)
• μG1 = μG2 is equivalent to μG1 - μG2 = 0
• H0 : μG1 − μG2 = 0
• HA : μG1 − μG2 ≠ 0
• The independent sample t-test assesses the probability of observed data assuming H0 is true

Logic of the t-test

• Statistical tests assume the null hypothesis is true
• If the null hypothesis is true (μG1 - μG2 = 0 at the population level), the difference in group means of X from samples of G1 and G2 is most likely 0
• μG1 and μG2 represent population-level means
• 𝑥¯𝐺1 and 𝑥¯𝐺2 represent mean values of sample groups

Sampling Variation

• Sampling rarely results in a perfect representation of populations
• Differences in sample group means (𝑥¯𝐺1 - 𝑥¯𝐺2) are likely to deviate from 0
• If the null hypothesis is true, values of 𝑥¯𝐺1 - 𝑥¯𝐺2 close to 0 are expected
• Values of 𝑥¯𝐺1 - 𝑥¯𝐺2 further from 0 are less likely
• Negative and positive values are equally likely, resembling a normal distribution

Normal Distribution Characteristics

• 0 is the most likely value
• Values closer to 0 are more likely
• Positive and negative values are equally likely (symmetry)

Student’s t-Distribution

• A normal distribution is defined by mean (μ) and standard deviation (σ)
• The population-level standard deviation of X may be unknown
• The t-distribution is a variation of the standard normal distribution (Z-distribution)
• The t-distribution is used when the population standard deviation is unknown
• Normal distribution (N(μ, σ)) can be transformed to the Z-distribution (N(0, 1))
• The t-distribution can be understood as a standardized distribution
• The t-distribution is a more conservative version of the Z-distribution and assumes wider variability
• With less information, observations are less certain to be near the mean

Degrees of Freedom

• Degrees of freedom indicate parameters able to "vary freely" given a defined outcome
• If 100 participants have a mean age of 60, there are many age possibilities with the average remaining 60
• If the exact age of 99 of 100 individuals is known, the age of the final person cannot "vary freely"
• There is only one value that can make the average 60
• Calculating a mean "spends" one degree of freedom
• With n observations, calculating the sample mean 𝑥¯ spends one degree of freedom
• There are n-1 degrees of freedom to calculate the standard deviation s
• Fewer observations (smaller n) mean less data to estimate variation in observed variable X

Capturing Uncertainty

• The t-distribution is intended to capture uncertainty in the measurement of standard deviation from a small sample
• Fewer degrees of freedom (smaller sample) make measured standard deviation s less representative of population-level standard deviation σ
• The t-distribution is shorter and wider than the normal distribution
• Values farther from 0 are more likely under the t-distribution than under the Z-distribution
• As data increases (n gets larger), the t-distribution's shape approaches that of the Z-distribution