T Tests

A t-test is a parametric statistical significance test utilized to assess and evaluate hypotheses regarding population means.
The procedure calculates a t-statistic, which represents the ratio of the difference between the group means to the variability (or standard error) within the groups.
A larger t-statistic implies that the difference between the groups is substantially greater than the difference within the groups, yielding stronger evidence against the null hypothesis.
To determine statistical significance, the calculated t-statistic is compared against a critical value derived from the t-distribution.
If the t-statistic falls into the rejection region (i.e., it is greater than the critical value), the null hypothesis ( $H_{0}$ ) is rejected in favour of the alternative hypothesis ( $H_{a}$ ).

The Student's t-distribution is a continuous probability distribution utilized for estimating the mean of a normally distributed population, specifically when the sample size is small and the true population standard deviation remains unknown.
Similar to the standard normal (z) distribution, the t-distribution is symmetrical and bell-shaped.
However, the t-distribution is slightly wider and features fatter, higher tails.
This specific shape accounts for the greater degree of uncertainty inherent in estimating a population mean from a small, limited sample size.
The exact shape of the t-distribution is dependent on the degrees of freedom (df); as the sample size and degrees of freedom increase, the t-distribution closely approximates the standard normal distribution.

The dependent variable being analyzed must be a continuous numerical variable.
The individual measurements within the populations must follow an approximately normal distribution.
When analyzing independent groups, the variances (or standard deviations) of the two populations should be nearly equal, an assumption known as homogeneity of variance.
If the assumption of equal variances is violated, an adjusted test, such as Welch's t-test, must be employed by adjusting the degrees of freedom.

In pediatric research and general medical statistics, the selection of the specific t-test depends entirely on the study design and the relationship between the comparison groups.
For example, an independent t-test is ideal for comparing the mean haematocrit levels between a control group of children with Tetralogy of Fallot and a separate treatment group.

Type of t-Test	Clinical Indication / Usage	Null Hypothesis ( $H_{0}$ )	Degrees of Freedom (df)	Non-Parametric Equivalent
One-Sample t-Test	Used to compare the mean of a single sample to a fixed, known population value or a established "gold standard".	$μ = μ_{0}$ (The population mean represented by the sample equals the known hypothesized value).	$n - 1$ (where n = sample size).	Sign test or Wilcoxon signed rank-sum test.
Independent-Samples (Two-Sample) t-Test	Used to compare the means of a particular variable between two completely independent and unrelated groups.	$μ_{1} = μ_{2}$ (There is no difference between the means of the two independent populations).	$n_{1} + n_{2} - 2$ (where $n_{1}$ and $n_{2}$ are the sample sizes of the two groups).	Mann-Whitney U test (also known as Wilcoxon rank sum test).
Paired-Samples (Dependent) t-Test	Used to compare two related or matched samples, such as measurements taken from the same individual before and after a specific treatment.	$μ_{d} = 0$ (The mean of the paired differences between the two conditions is zero).	$n - 1$ (where n = total number of matched pairs).	Wilcoxon matched-pairs signed-rank test.