Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more. 
Home Statistical Tests Means Paired Experiments  
See also: hypothesis testing, survey on statistical tests, comparing means, distribution calculator, Wilcoxon Test for Paired Differences  
Paired Experiments
When we try to compare methods, treatments, etc. by applying each to the same population, the resulting values are no longer independent. Some examples may contribute to a clarification: (1) comparison of the temperature measurement of a device to the calibrated temperature (both measurements origin from the same "real" temperature; (2) comparison of the yield of an orchard in two consecutive years (normally the trees stay the same); (3) the influence of alcohol on the reaction time; each client has to perform a rection test both in sober state and after drinking a 1/4 l of wine (the persons stay the same).
Think e.g. of comparing analytical methods applied to environmental samples.
The variation among the different samples will probably be larger than
the difference between the individual methods. Due to the large pooled
variance s_{p}^{2}, we cannot use the independent sample
ttest to distinguish between the two methods. We can calculate the pairwise
differences, d_{i}, and consider the d_{i} as a new variable
that follows a tdistribution. The mean and standard deviation of the d_{i}
are ,
and s_{d}, respectively. n_{D} is the number of pairs.
Depending on the sample size n_{D} we use the onesample tests
based on t scores.
Assumption: the distribution of the differences is normal^{(1)} and the values are measured at least at the interval level. The paired difference experiment is often more powerful, since it can eliminate differences in the samples that increase the total variance, σ^{2}. When the comparison is made between groups (of similar experimental units), it is called blocking. The paired difference experiment is a simple example of a randomized block experiment. When the normality assumption is not fulfilled, one can use the nonparametric Wilcoxon sign rank test for paired difference designs.


Home Statistical Tests Means Paired Experiments 
Last Update: 20121008