Editing Student's t-test (section)

=== The two-sample ''t''-test is a special case of simple linear regression ===
{{unsourced section|date=March 2025}}

The two-sample ''t''-test is a special case of simple [[linear regression]] as illustrated by the following example.

A clinical trial examines 6 patients given drug or placebo. Three (3) patients get 0 units of drug (the placebo group). Three (3) patients get 1 unit of drug (the active treatment group). At the end of treatment, the researchers measure the change from baseline in the number of words that each patient can recall in a memory test.

[[File:Graph_of_word_recall_vs_drug_dose.svg|300px|alt=Scatter plot with six point. Three points on the left and are aligned vertically at the drug dose of 0 units. And the other three points on the right and are aligned vertically at the drug dose of 1 unit.]]

A table of the patients' word recall and drug dose values are shown below. 

{| {{Table}}
! Patient !! drug.dose !! word.recall
|-
! 1
! 0
| 1
|-
! 2
! 0
| 2
|-
! 3
! 0
| 3
|-
! 4
! 1
| 5
|-
! 5
! 1
| 6
|-
! 6
! 1
| 7
|}

Data and code are given for the analysis using the [[R programming language]] with the <code>t.test</code> and <code>lm</code>functions for the t-test and linear regression. Here are the same (fictitious) data above generated in R.

<syntaxhighlight lang="R">
> word.recall.data=data.frame(drug.dose=c(0,0,0,1,1,1), word.recall=c(1,2,3,5,6,7)) 
</syntaxhighlight>

Perform the ''t''-test. Notice that the assumption of equal variance, <code>var.equal=T</code>, is required to make the analysis exactly equivalent to simple linear regression.

<syntaxhighlight lang="R">
> with(word.recall.data, t.test(word.recall~drug.dose, var.equal=T))
</syntaxhighlight>

Running the R code gives the following results.
*The mean word.recall in the 0 drug.dose group is 2. 
*The mean word.recall in the 1 drug.dose group is 6. 
*The difference between treatment groups in the mean word.recall is 6 – 2 = 4.
* The difference in word.recall between drug doses is significant (p=0.00805).

Perform a linear regression of the same data. Calculations may be performed using the R function <code>lm()</code> for a linear model. 
<syntaxhighlight lang="R">
> word.recall.data.lm =  lm(word.recall~drug.dose, data=word.recall.data)
> summary(word.recall.data.lm)
</syntaxhighlight>

The linear regression provides a table of coefficients and p-values.

{| {{Table}}
! Coefficient !! Estimate!! Std. Error !! t value !! P-value 
|-
! Intercept
! 2
! 0.5774
! 3.464
| 0.02572
|-
! drug.dose
! 4
! 0.8165
! 4.899
| 0.000805
|}

The table of coefficients gives the following results.
*The estimate value of 2 for the intercept is the mean value of the word recall when the drug dose is 0. 
*The estimate value of 4 for the drug dose indicates that for a 1-unit change in drug dose (from 0 to 1) there is a 4-unit change in mean word recall (from 2 to 6). This is the slope of the line joining the two group means.
*The p-value that the slope of 4 is different from 0 is p = 0.00805.

The coefficients for the linear regression specify the slope and intercept of the line that joins the two group means, as illustrated in the graph. The intercept is 2 and the slope is 4.

[[File:Regression_lines_with_slopes_4_and_0.jpg|400px|Regression lines]]

Compare the result from the linear regression to the result from the ''t''-test.
* From the ''t''-test, the difference between the group means is 6-2=4. 
*From the regression, the slope is also 4 indicating that a 1-unit change in drug dose (from 0 to 1) gives a 4-unit change in mean word recall (from 2 to 6).
* The ''t''-test ''p''-value for the difference in means, and the regression p-value for the slope, are both 0.00805. The methods give identical results.

This example shows that, for the special case of a simple linear regression where there is a single x-variable that has values 0 and 1, the ''t''-test gives the same results as the linear regression. The relationship can also be shown algebraically.

Recognizing this relationship between the ''t''-test and linear regression facilitates the use of multiple linear regression and multi-way [[analysis of variance]]. These alternatives to ''t''-tests allow for the inclusion of additional [[Dependent and independent variables|explanatory variables]]  that are associated with the response. Including such additional explanatory variables using regression or anova reduces the otherwise unexplained [[variance]], and commonly yields greater [[Power of a test|power]] to detect differences than do two-sample ''t''-tests.