# Paired t Test

Menu location: **Analysis_Parametric_Paired t**.

This function gives a paired Student t test, confidence intervals for the difference between a pair of means and, optionally, limits of agreement for a pair of samples (Armitage and Berry, 1994; Altman, 1991).

The paired t test provides an hypothesis test of the difference between population means for a pair of random samples whose differences are approximately normally distributed. Please note that a pair of samples, each of which are not from normal a distribution, often yields differences that are normally distributed.

The test statistic is calculated as:

- where d bar is the mean difference, s² is the sample variance, n is the sample size and t is a Student t quantile with n-1 degrees of freedom.

Power is calculated as the power achieved with the given sample size and variance for detecting the observed mean difference with a two-sided type I error probability of (100-CI%)% (Dupont, 1990).

Limits of agreement

If the main purpose in studying a pair of samples is to see how closely the samples agree, rather than looking for evidence of difference, then limits of agreement are useful (Bland and Altman 1986, 1996a, 1996b). StatsDirect displays these limits with an agreement plot if you check the agreement box before a paired t test runs. For more detailed analysis of this type, see agreement analysis.

__Agreement plot__

When two methods of measurement are compared it is almost always wrong to present a scatter plot with correlation as a measure of agreement between the paired data. Highly correlated results often agree poorly, indeed large shifts in measurement scales may leave the correlation coefficient unaltered. It is therefore necessary to provide a measure of agreement. StatsDirect provides a plot of the difference against the mean for each pair of measurements. This plot also displays the overall mean difference bounded by the limits of agreement. A good review of this subject has been provided by Bland and Altman (Bland and Altman, 1986; Altman, 1991).

__Example__

Test workbook (Parametric worksheet: PEFR Before, PEFR After).

Comparison of peak expiratory flow rate (PEFR) before and after a walk on a cold winter's day for a random sample of 9 asthmatics. Enter two columns in the workbook, one of PEFR's before the walk and the other of PEFR's after the walk. In this example each row must represent the same subject:

Subject | Before | After |

1 | 312 | 300 |

2 | 242 | 201 |

3 | 340 | 232 |

4 | 388 | 312 |

5 | 296 | 220 |

6 | 254 | 256 |

7 | 391 | 328 |

8 | 402 | 330 |

9 | 290 | 231 |

If you were to plot these pairs using a ladder plot you would see that all but one pair decreases. You might also wish to test the assumption that the differences are from a normal distribution, this can be done with the Shapiro-Wilk test.

To run this example, open the test workbook using the open file function of the file menu then choose paired t test from the parametric methods section of the analysis menu. Select the columns marked "before" and "after" when prompted.

For this example:

__Paired t test__

For differences between PEFR Before and PEFR After:

Mean of differences = 56.111111 (n = 9)

Standard deviation = 34.173983

Standard error = 11.391328

95% CI = 29.842662 to 82.37956

df = 8

t = 4.925774

One sided P = .0006

Two sided P = .0012

Power (for 5% significance) = 98.47%

A null hypothesis of no difference between the means is clearly rejected; the confidence interval is a long way from including zero.