# Independent samples t-test

Command: | Statistics T-tests Independent samples t-test |

## Description

The independent samples (or two-sample) t-test is used to compare the means of two independent samples.

## Required input

Select the variables for sample 1 and sample 2. Differences will be calculated as Sample2-Sample1.

### Options

- Logarithmic transformation: if the data require a logarithmic transformation (e.g. when the data are positively skewed), select the Logarithmic transformation option.
- Correction for unequal variances: allows to select the t-test (assuming equal variances) or the t-test corrected for unequal variances (Welch test, Armitage et al., 2002). With the option "Automatic" the software will select the appropriate test based on the F-test (comparison of variances).

## Results

The results windows for the Independent samples t-test displays the summary statistics of the two samples, followed by the statistical tests.

First an F-test is performed. If the P-value is low (P<0.05) the variances of the two samples cannot be assumed to be equal and it should be considered to use the t-test with a correction for unequal variances (Welch test) (see above).

The independent samples t-test is used to test the hypothesis that the difference between the means of two samples is equal to 0 (this hypothesis is therefore called the null hypothesis). The program displays the difference between the two means, and the 95% Confidence Interval (CI) of this difference. Next follow the test statistic t, the Degrees of Freedom (DF) and the two-tailed probability P. When the P-value is less than the conventional 0.05, the null hypothesis is rejected and the conclusion is that the two means do indeed differ significantly.

### Logarithmic transformation

If you selected the Logarithmic transformation option, the program performs the calculations on the logarithms of the observations, but reports the back-transformed summary statistics.

For the t-test, the difference and 95% confidence are given, and the test is performed, on the log-transformed scale.

Next, the results of the t-test are transformed back and the interpretation is as follows: the back-transformed difference of the means of the logs is the ratio of the geometric means of the two samples (see Bland, 2000).

## One-sided or two-sided tests

In MedCalc, P-values are always *two-sided* (as recommended by Fleiss, 1981, and Altman, 1991) and not* one-sided*.

A *two-sided* (or two-tailed) P-value is appropriate when the difference between the two means can occur in both directions: it may be either negative or positive, the mean of one sample may either be smaller or larger than that of the other sample.

A *one-sided* test should only be performed when, before the start of the study, it has already been established that a difference can only occur in one direction. E.g. when the mean of sample *A* must be more than the mean of sample *B* for reasons other than those connected with the sample(s).

## Interpretation of P-values

P-values should not be interpreted too strictly. Although a significance level of 5% is generally accepted as a cut-off point for a significant versus a non-significant result, it would be a mistake to interpret a shift of P-value from e.g. 0.045 to 0.055 as a change from significance to non-significance. Therefore the real P-values are preferably reported, P=0.045 or P=0.055, instead of P<0.05 or P>0.05, so the reader can make his own interpretation.

With regards to the interpretation of P-values as significant versus not-significant, is has been recommended to select a smaller significance level of for example 0.01 *when it is necessary to be quite certain that a difference exists before accepting it*. When a study is designed to *uncover a difference*, or when *a life-saving* drug is being studied, we should be willing to accept that there is a difference even when the P-value is as large as 0.10 or even 0.20 (Lentner, 1982). The latter authors state that *"The tendency in medical and biological investigations is to use too small a significance probability"*.

## Confidence intervals

Whereas the P-value may give information on the statistical significance of the result, the 95% confidence interval gives information to assess the clinical importance of the result.

When the number of cases included in the study is large, a biologically unimportant difference can be statistically highly significant. *A statistically significant result does not necessarily indicate a real biological difference*.

On the other hand, a high P-value can lead to the conclusion of statistically non-significant difference although the difference is clinically meaningful and relevant, especially when the number of cases is small. *A non-significant result does not mean that there is no real biological difference*.

Confidence intervals are therefore helpful in interpretation of a difference, whether or not it is statistically significant (Altman et al., 1983).

## Presentation of results

It is recommended to report the results of the t-test (and other tests) not by a simple statement such as P<0.05, but by giving full statistical information, as in the following example by Gardner & Altman (1986):

*The difference between the sample mean systolic blood pressure in diabetics and non-diabetics was 6.0 mm Hg, with a 95% confidence interval from 1.1 to 10.9 mm Hg; the t test statistic was 2.4, with 198 degrees of freedom and an associated P value of P=0.02.*

In short:

*Mean 6.0 mm Hg, 95% CI 1.1 to 10.9; t=2.4, df=198, P=0.02*

## Literature

- Altman DG, Gore SM, Gardner MJ, Pocock SJ (1983) Statistical guidelines for contributors to medical journals. British Medical Journal, 286, 1489-1493. [Abstract]
- Armitage P, Berry G, Matthews JNS (2002) Statistical methods in medical research. 4
^{th}ed. Blackwell Science. - Bland M (2000) An introduction to medical statistics, 3
^{rd}ed. Oxford: Oxford University Press. - Fleiss JL (1981) Statistical methods for rates and proportions, 2
^{nd}ed. New York: John Wiley & Sons. - Gardner MJ, Altman DG (1986) Confidence intervals rather than P values: estimation rather than hypothesis testing. British Medical Journal, 292, 746-750. [Abstract]
- Lentner C (ed) (1982) Geigy Scientific Tables, 8
^{th}edition, Volume 2. Basle: Ciba-Geigy Limited.

## See also

- To perform different tests in one single procedure, see Comparison of independent samples
- One sample t-test
- Paired samples t-test
- Variance ratio test (F-test)

## External links

- Student's t-test on Wikipedia.
- Welch's t test on Wikipedia.