T Tests

Overview Of The T-Test

The t-test is a fundamental parametric statistical procedure. It is used to assess hypotheses about population means. It compares the difference of a continuous variable between groups. The test is only valid when performed on normally distributed data.

William S. Gosset invented the test under the pseudonym "Student".
It generates a t-statistic.
The t-statistic is a ratio of the difference between the group means and the variability within the groups.
A larger t-statistic provides greater evidence against the null hypothesis.

The T-Distribution

When sample sizes are small, the normal distribution is not a good indicator of data. In these cases, data follow the t-distribution.

The t-distribution is symmetric and bell-shaped like a normal distribution.
It has a larger standard deviation than the normal distribution.
It is lower and wider, and its tails are higher.
The shape depends on a single parameter called degrees of freedom.
The advantage of the t-distribution is that it accounts for sample size.
A smaller sample size yields a more prominent tail to account for greater uncertainty.
As sample size increases, the t-distribution approaches the standard normal distribution.

Core Assumptions

To perform a valid t-test, specific distributional assumptions must be met.

Normality: The dependent variables must be normally distributed within each group.
Independence: Subjects within the samples should be independent (for independent and one-sample tests).
Homogeneity Of Variance: The variances in the two populations must be nearly equal.

Evaluating Assumptions

Normality is checked using histograms, the Shapiro-Wilk test, or the Kolmogorov-Smirnov test.
Homogeneity of variance is tested using the F-test or Levene's test.
If variances are unequal, an adjusted t-test (like Welch's test) must be used.
If data are grossly skewed, non-parametric alternative tests must be utilized.

Types Of T-Tests

One-Sample T-Test

A one-sample t-test compares the mean of a single variable against a specified known value or gold standard. It determines if the sample comes from a population with a mean different from the hypothesized value.

Hypotheses

Null Hypothesis ( $H_{0}$ ): The population mean is equal to the hypothesized value ( $μ = μ_{0}$ ).
Alternative Hypothesis ( $H_{1}$ ): The population mean is not equal to the hypothesized value ( $μ \neq μ_{0}$ ).

Mathematical Formula

The t-statistic is calculated as follows: $$t = \frac{\bar{X} - \mu_0}{S / \sqrt{n}}$$

Where $\bar{X}$ is the sample mean.
$μ_{0}$ is the known population mean.
$S$ is the sample standard deviation.
$n$ is the sample size.
Degrees of freedom ( $d f$ ) = $n - 1$ .

Independent-Samples T-Test

An independent-samples t-test compares the means of two independent groups. It evaluates whether the unknown means of two populations differ from each other.

Characteristics

Subjects in the first group cannot be in the second group.
The groups can be from separate populations (e.g., males and females).
They can also be from a single population randomly divided into subgroups (e.g., treatment and control).

Hypotheses

Null Hypothesis ( $H_{0}$ ): There is no difference between the means of the two groups ( $μ_{1} = μ_{2}$ ).
Alternative Hypothesis ( $H_{1}$ ): There is a difference between the means ( $μ_{1} \neq μ_{2}$ ).

Mathematical Formula

The test calculates the difference between sample means divided by the standard error of the difference. $$t = \frac{\bar{X}_1 - \bar{X}2}{S{\bar{X}_1 - \bar{X}_2}}$$

Where ${\bar{X}}_{1}$ and ${\bar{X}}_{2}$ are the sample means.
$S_{{\bar{X}}_{1} - {\bar{X}}_{2}}$ is the standard error of the difference.
Degrees of freedom ( $d f$ ) = $n_{1} + n_{2} - 2$ .

Paired-Sample T-Test

A paired-sample t-test compares the means of two related variables. It is used when the data are paired or matched.

Characteristics

Used for pre-test/post-test measurements taken from the same individuals.
Used for subjects matched in pairs at baseline.
The test assesses whether the mean of the paired differences is zero.
Each subject serves as their own control.

Hypotheses

Null Hypothesis ( $H_{0}$ ): The mean of the paired differences is zero ( $μ_{d} = 0$ ).
Alternative Hypothesis ( $H_{1}$ ): The mean of the paired differences is not zero ( $μ_{d} \neq 0$ ).

Mathematical Formula

t = \frac{\bar{d}}{S_{d} / \sqrt{n}}

Where $\bar{d}$ is the mean of the differences.
$S_{d}$ is the standard deviation of the differences.
$n$ is the number of paired observations.
Degrees of freedom ( $d f$ ) = $n - 1$ .

Tabular Comparison Of T-Tests And Non-Parametric Equivalents

T-Test Type	Purpose	Degrees of Freedom	Non-Parametric Equivalent
One-Sample	Compares sample mean to a fixed known value.	$n - 1$	Wilcoxon signed rank-sum test or Sign test.
Independent-Samples	Compares means of two unrelated groups.	$n_{1} + n_{2} - 2$	Mann-Whitney U test.
Paired-Sample	Compares means of two related or matched observations.	$n - 1$ (pairs)	Wilcoxon matched-pair signed-rank test.

Statistical Decision Making

The objective of the t-test is to distinguish whether an observed difference suggests a real population difference or is due to chance.

Researchers calculate the t-statistic from the sample data.
They determine the degrees of freedom.
The t-statistic is compared against a critical value from the t-distribution table.
The critical value defines the rejection region based on a chosen significance level, usually $α = 0.05$ .
If the calculated t-statistic is equal to or greater than the critical value, the null hypothesis is rejected.
Rejecting the null hypothesis means the result is statistically significant and unlikely due to chance.
If the t-statistic is smaller than the critical value, researchers fail to reject the null hypothesis.
Failing to reject the null hypothesis implies there is insufficient evidence to conclude a difference exists.