Paired sample t-test

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What is a paired sample t-test?

We use Paired sample t-test when we are determined to find the differences between two variables belonging to the same subject. The Paired sample t-test compares the mean of these two variables or measurements.

Like all the statistical procedures, Paired sample t-test formulates the hypothesis and there are two competing hypothesis: null hypothesis and alternative hypothesis.

Null hypothesis (\(H_0\)): assumes that the difference between the paired samples is equal to 0.
Alternative hypothesis: assumes that the difference between the paired samples is not equal to 0. Depending on the expected outcomes, alternative hypothesis can be one of the following forms –

Two-tailed alternative hypothesis (\(H_1\)): assumes that the direction of the difference does not matter and the difference between the paired sample is not equal to 0.
Upper-tailed alternative hypothesis (\(H_1\)): assumes that the difference between the paired samples is greater than 0.
Lower-tailed hypothesis (\(H_1\)): assumes that the difference between the paired samples is less than 0.

Point estimates : the sample mean difference

Test statistics:

where sd is the standard deviation of the differences.

Confidence intervals:

For a 95% confidence interval, the appropriate critical value of t for the two-sided α=0.05

Example of Paired sample t-test

Let’s say we wish to study the diabetic patients from 1952 to 1962. The Paired sample t-test would look like:

H0: The average difference in diabetic patients is 0 from 1952 to 1962
H1: The average difference in diabetic patients is NOT 0 from 1952 to 1962.
Our significance level is α = 0.05.

For α = 0.05, the critical value of t is 2.093. Since | -6.7| > 2.093, we will reject the null hypothesis (H0) and state that we have significant evidence that the average difference in diabetic patients from 1952 to 1962 is NOT 0. Specifically, there was an average decrease of 69.8 from 1952 to 1962.

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Uses of Paired sample t-test

Paired sample t-test is used to test the statistical difference between two time points.

It can also determine the difference between two conditions.

To determine the difference between the two measurements.

To draw a statistical difference between two matched pairs.

Paired sample t-test cannot be used:

To compare unpaired means between two independent groups on a continuous outcome that is normally distributed, you can use an independent sample t-test.

To compare unpaired means between more than two groups on a continuous outcome that is normally distributed, you can use ANOVA for this.

To compare paired means for continuous data that are not normally distributed, use the nonparametric Wilcoxon Signed-Ranks Test.
To compare paired means for ranked data, the nonparametric Wilcoxon Signed-Ranks test can also be used here.

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Assumptions in Paired sample t-test

The Paired sample t-test makes some assumptions and it is advised to derive the degree of deviation from these assumptions to make sure the quality of your results is preserved. Each of the assumptions refers to the observations which further define the differences in the sample measurements.

The dependent variable must be continuous (interval/ratio)
The observations are independent of one another
The dependent variable should be approximately normally distributed
The dependent variable should not contain any outliers
Level of measurement – as Paired sample t-test is based on nominal distribution, the sample data needs to be in a numeric and continuous form.
Independence – the independence of the observations cannot be measured but can be assumed if the sample is collected at random.
Normality – visual tools like histograms are used to test the normality. Although this can only be useful when the sample data is normal and can be plotted to look like a symmetrical shape.

Outliers – the values that appear far away from the sample data are called outliers. It is advised to handle these outliers as soon as possible for it to not cause any biases and discrepancies in the results. But as deleting values can also introduce some biases in the results, to minimize the significance of the outliers you can make use of the Wilcoxon Signed Rank Test. Boxplot can perfectly show how the outliers look:

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