What is T-Test

T-tests are a type of inferential statistics that allow us to compare the means of two groups to determine if they are significantly different from each other. The 't' in t-test stands for William Sealy Gosset's pseudonym, "Student," who developed the Student's t-test. Since the development of the Student's t-test, various adaptations, such as the Welch's t-test and Paired t-test, have been developed to accommodate different types of data and experimental designs.

The t-test works on the principle of null hypothesis testing. In a nutshell, this means that we start by assuming that there is no significant difference between the groups (this is our null hypothesis). We then perform the t-test, which gives us a p-value. The p-value tells us the probability of observing our data (or data more extreme) if the null hypothesis is true. If this probability is low (typically less than 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between the groups.

One critical aspect of the t-test is that it assumes the data follows a normal distribution, a bell-shaped curve that is symmetric about its mean. It also uses the standard deviation of the groups to estimate the standard error of the sampling distribution.

Student's T-Test

The Student's t-test, also known as an independent t-test, is a statistical method used to determine if there is a significant difference between the means of two independent groups. Named after William Sealy Gosset, who published under the pseudonym "Student," this t-test is widely applied in various fields. For instance, it can be used to compare the performance results of two different treatment groups in a clinical trial or to compare the average scores of two groups of students taught by different methods.

Assumptions

The Student's t-test operates under a few key assumptions:

• Independence of Observations
  The data collected in the two groups are independent of each other. This means that the measurement of one participant does not influence the measurement of another participant.
• Normal Distribution
  The data in both groups are normally distributed. Although the t-test is fairly robust to this assumption, extreme violations can alter the validity of the test results.
• Equal Variances
  Also known as the assumption of homogeneity of variances, this assumes that the standard deviations of the data in the two groups are approximately equal. If this assumption is violated, the results of the Student's t-test may not be valid, and the Welch's t-test is usually recommended instead.

Step-by-step Procedure

The Student's t-test follows a systematic procedure to ascertain whether the means of two independent groups are significantly different. Here are the steps involved:

1. Formulate the Null and Alternative Hypotheses

The null hypothesis (H_0) often states that there is no difference in the means of the two groups, while the alternative hypothesis (H_1 or H_a) states that there is a difference. The hypotheses can be formally written as:

• H_0: \mu_1 = \mu_2 (There is no difference between the group means)
• H_1: \mu_1 \neq \mu_2 (There is a difference between the group means)

2. Calculate the t-Statistic

The formula for the t-statistic in a Student's t-test is:

where \bar{X}_1 and \bar{X}_2 are the means of the two groups, s_1 and s_2 are the standard deviations of the two groups, and n_1 and n_2 are the sample sizes of the two groups.

3. Determine the Degrees of Freedom

For the Student's t-test, the degrees of freedom is calculated as the total number of observations in both groups minus 2.

4. Compare the t-Statistic with the Critical t-Value

The critical t-value can be found in a t-distribution table or calculated using statistical software. If the absolute value of the t-statistic is greater than the critical t-value, we reject the null hypothesis.

5. Calculate the p-Value

The p-value is the probability of observing a t-statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis.

Welch's T-Test

The Welch's t-test, is a variation of the Student's t-test designed to accommodate scenarios where the two samples have unequal variances and/or unequal sample sizes. The Welch's t-test is a more reliable alternative when the assumption of equal variances, critical to the Student's t-test, is not met.

Like the Student's t-test, the Welch's t-test is used to determine if there is a significant difference between the means of two independent groups. It finds its application in a wide range of scenarios, especially where the assumption of equal variances is not tenable.

Assumptions

The assumptions for the Welch's t-test are similar to those of the Student's t-test, with a key exception:

• Independence of Observations
  The data collected in the two groups are independent of each other.

• Normal Distribution
  The data in both groups should be approximately normally distributed.

• Unequal Variances
  Unlike the Student's t-test, the Welch's t-test does not require the assumption of equal variances. It is designed to be used even when the variances are unequal, a condition known as heteroscedasticity.

Step-by-step Procedure

The steps to perform a Welch's t-test are similar to those of a Student's t-test, with modifications in the calculation of the t-statistic and degrees of freedom:

1. Formulate the Null and Alternative Hypotheses

Similar to the Student's t-test, the null hypothesis often states that there is no difference in the means of the two groups, while the alternative hypothesis states that there is a difference.

2. Calculate the t-Statistic

The formula for the t-statistic in a Welch's t-test is the same as in the Student's t-test, but the standard error calculation in the denominator considers the variances separately for each group:

3. Determine the Degrees of Freedom

The degrees of freedom for the Welch's t-test are calculated using the Welch–Satterthwaite equation, which is more complex than the calculation for the Student's t-test:

4. Compare the t-Statistic with the Critical t-Value

As in the Student's t-test, if the absolute value of the t-statistic is greater than the critical t-value, we reject the null hypothesis.

5. Calculate the p-Value

If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis.

Paired T-Test

The Paired t-test, also known as the dependent or matched t-test, is a statistical procedure used to determine whether the mean difference between two sets of observations is zero. Each subject or entity in the paired t-test is measured twice, resulting in pairs of observations. These measurements could be from different times (e.g., before and after a treatment) or from matched pairs (e.g., twins).

The paired t-test is often used in case-control studies, crossover study designs, or repeated-measures designs, where the same subject is tested more than once.

Assumptions

The assumptions for the paired t-test include:

• Dependent Samples
  The samples or groups should be related or matched in some way or the observations should be taken in pairs.

• Normal Distribution
  The differences between the paired observations should be approximately normally distributed.

• Independence of Observations
  The pairs of observations should be independent of each other.

Step-by-step Procedure

Here are the steps to conduct a paired t-test:

1. Formulate the Null and Alternative Hypotheses

The null hypothesis typically states that there is no difference in the means of the paired observations, while the alternative hypothesis states that there is a difference.

2. Calculate the Differences between Pairs

For each pair, calculate the difference between the two observations.

3. Calculate the Mean and Standard Deviation of Differences

Compute the mean (denoted as d) and standard deviation (denoted as s_d) of these differences.

4. Calculate the t-Statistic

The formula for the t-statistic in a paired t-test is:

where n is the number of pairs.

5. Determine the Degrees of Freedom

The degrees of freedom for the paired t-test is the number of pairs minus 1 (df = n - 1).

6. Compare the t-Statistic with the Critical t-Value

If the absolute value of the t-statistic is greater than the critical t-value, we reject the null hypothesis.

7. Calculate the p-Value

If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis.

Python Implementation of Welch's T-Test

In Python, Welch's t-test can be performed using the scipy.stats.ttest_ind() function with the argument equal_var=False. Here's an example of how to do it:

import numpy as np
from scipy import stats

# Create two groups of data
group1 = np.random.normal(0, 1, 30)
group2 = np.random.normal(0.5, 1.5, 50)

# Perform Welch's t-test
t_stat, p_value = stats.ttest_ind(group1, group2, equal_var=False)

print("t statistic: ", t_stat)
print("p-value: ", p_value)

t statistic:  -1.544992743798216
p-value:  0.1264084891010019

A t-score of -1.5450 suggests that the mean of group1 is lower than the mean of group2, as a negative t-score indicates that the first group's mean is lower than the second group's mean.

A p-value of 0.1264 means that there is a 12.64% chance of obtaining a difference in means as large as (or larger than) the one observed, assuming that the null hypothesis is true (i.e., the true difference in population means is zero).

Typically, a p-value threshold (alpha level) of 0.05 is used in hypothesis testing. If the p-value is less than 0.05, the result is considered statistically significant, and the null hypothesis is rejected. In this case, your p-value is greater than 0.05, so we fail to reject the null hypothesis. This means that there's not enough evidence to say that there's a significant difference between the means of the two groups.