Testing Two Paired Samples

10.6.3 Testing Two Paired Samples

The previous two-sample tests assumed that the samples were independent. The two-paired-sample test can be reduced to a one-sample test using the same technique as in Chapter 4 (see section 4.4.3.1), i.e., employing the differences between pair members. If the distributions of the two samples are similar, we expect that the difference sample will be uniformly distributed. The function dirdif implemented in MATLAB (see Commands 10.3) computes the directional data of the difference set in standard format.

Example 10.19

Q: Consider the measurements M2 and M3 of the Soil Pollution dataset. Assess, at the 5% significance level, if one can accept that the two measurement methods yield similar distributions.

A: Let soil denote the data matrix containing all measurements of the Soil Pollution dataset. Measurements M2 and M3 correspond to the column pairs 3-4 and 5-6 of soil, respectively. We use the sequence of R commands shown below and do not reject the hypothesis of similar distributions at the 5% level of significance.

> m2<-soil[,3:4] > m3<-soil[,5:6] > d<-dirdif(m2,m3) > p<-rayleigh(d) >p [1] 0.1772144

10 Directional Data

Exercises

10.1 Compute the mean directions of the wind variable WD ( Weather dataset, Data 2) for the four seasons and perform the following analyses: a)

Assess the uniformity of the measurements both graphically and with the Rayleigh test. Comment on the relation between the uniform plot shape and the observed value of the test statistic. Which set(s) can be accepted as being uniformly distributed at a 1% level of significance?

b) Assess the von Misesness of the measurements.

10.2 Consider the three measurements sets, H, A and I, of the VCG dataset. Using a specific methodology, each of these measurement sets represents circular direction estimates of the maximum electrical heart vector in 97 patients. a)

Inspect the circular plots of the three sets. b)

Assess the uniformity of the measurements both graphically and with the Rayleigh test. Comment on the relation between the uniform plot shape and the observed value of the test statistic. Which set(s) can be accepted as being uniformly distributed at a 1% level of significance?

c) Assess the von Misesness of the measurements.

10.3 Which type of test is adequate for the comparison of any pair of measurement sets studied in the previous Exercise 10.2? Perform the respective pair-wise comparison of the distributions.

10.4 Assuming a von Mises distribution, compute the 95% confidence intervals of the mean directions of the measurement sets studied in the previous Exercise 10.2. Plot the data in order to graphically interpret the results.

10.5 In the von Misesness assessment of the WDB measurement set studied in Example 10.12, an estimate of the concentration parameter κ was used. Show that if instead of this estimate we had used the value employed in the data generation ( κ = 2), we still would not have rejected the null hypothesis.

10.6 Compare the wind directions during March on two streets in Porto, using the Weather dataset (Data 3) and assuming that the datasets are valid random samples.

10.7 Consider the Wave dataset containing angular measurements corresponding to minimal acoustic pressure in ultrasonic radiation fields. Perform the following analyses: a)

Determine the mean directions of the TRa and TRb measurement sets.

b) Show that both measurement sets support at a 5% significance level the hypothesis of a von Mises distribution.

c)

Compute the 95% confidence interval of the mean direction estimates.

d) Compute the concentration parameter for both measurement sets. e)

For the two transducers TRa and TRb, compute the angular sector spanning 95% of the measurements, according to a von Mises distribution.

Exercises 401

10.8 Compare the two measurement sets, TRa and TRb, studied in the previous Exercise

10.7, using appropriate parametric and non-parametric tests.

10.9 The Pleiades data of the Stars dataset contains measurements of the longitude ’ and co-latitude of the stars constituting the Pleiades’ constellation as well as their photo-visual magnitude. Perform the following analyses: a) Determine whether the Pleiades’ data can be modelled by a von Mises

distribution. b) Compute the mean direction of the Pleiades’ data with the 95% confidence

interval. c) Compare the mean direction of the Pleiades’ stars with photo-visual magnitude

above 12 with the mean direction of the remaining stars.

10.10 The Praesepe data of the Stars’ dataset contains measurements of the longitude and co-latitude of the stars constituting the Praesepe constellation obtained by two researchers (Gould and Hall). a)

Determine whether the Praesepe data can be modelled by a von Mises distribution.

b) Determine the mean direction of the Praesepe data with the 95% confidence interval.

c) Compare the mean directions of the Prasepe data obtained by the two researchers.