START First steps Deutsch Guide : Repeated observations

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Empirical measures of dispersion, skewness and excess kurtosis Normal probability plot Explanations of the statistical tests Anderson-Darling-test for normality Repeated height determination of a point Try this computation tool using normal distributed random numbers Try this computation tool using different random numbers In the library
In geodesy and in other measuring professions a quantity is often measured multiple times under the same conditions, to increase accuracy and reliability. It is assumed that the measured values differ only due to the effect of random independent identically distributed errors. Such measurements are comprehensively evaluated, including all applicable statistical tests. Instead of geodetic measured values also other random samples of that kind may be evaluated.

START First steps Deutsch Empirical measures of dispersion, skewness and excess kurtosis

Histogram of the measured values
Histogram of the measured values

The inter quartile range (IQR) is the difference of the 1st and 3rd quartile. This interval includes 50% of all values of a quantity. For the computation of the inter quartile range at least 12 measured values are required.

The skewness indicates, how symmetrical or non-symmetrical the measurement distribution is.
The excess kurtosis indicates, how thin or thick the tails of the measurement distribution, in comparison to the Gaussian normal distribution.

Skewnessdistribution excess kurtosisdistribution
< 0left-skewed < 0thin-tailed (blunt peaked)
= 0symmetrical = 0normal shape (Gaussian bell)
> 0right-skewed > 0thick-tailed (sharp peaked)

START First steps Deutsch Normal probability plot

Deviations from normal distribution
Deviations from normal distribution (from left to right):
right-skewed, left-skewed, thick-tailed, thin-tailed

This plot yields a fast graphical method to assess the deviation of a sample distribution from normality. The measured values are plotted on the horizontal axis against the normal order statistic medians on the vertical axis. In case of a Gaussian normal distribution all points in the plot should be close to collinear. Deviations of this may indicate non-symmetric (skewed) or thick-tailed, i.e. peak-shaped, or thin-tailed distributions. Isolated points at the left-bottom or right-top of the graphic are outlier-suspected.

START First steps Deutsch Explanations of the statistical tests

The measured values are treated as independent, identically Gaussian normal distributes random variates. If the measured values only approximately follow a normal distribution then some tests still yield correct results, provided that the number of measurements is not too small. Correlations would falsify the result.

Did you know? Although for a set of measurements all possible tests are computed consecutively, it would not be correct to use the results of two or more tests simultaneously without adapting the probability of type I decision error α . If you want to perform a multiple test with the same measurements, e.g. an outlier test and thereafter a test of the expectation then it has to be taken into account that the total probability of type I decision error α is increasing. In the simplest case of nearly independent multiple test statistics, the desired total probability must by divided by the number of individual tests according to the Bonferroni equation, to arrive at the probability α to be set for each individual test.

All tests are computed, which are theoretically computable, even if a different test would less probably bring about a false decision. Example: If the standard deviation of the measurements is a priori correctly known then the w-Test after Baarda less probably brings about a false decision than the Pope-test and the Z-test less probably brings about a false decision than the t-test.

Acceptance region : The null hypothesis Ho is accepted if the test statistic

two-tailed test
of the two-tailed test falls between the critical values c1,c2 .
right-tailed test
of the right-tailed test falls below the critical value c2 .
left-tailed test
of the left-tailed test exceeds the critical value c1 .

 

Test distributions

N(μ,σ) Gaussian normal distribution with expectation μ and standard deviation σ
t(r) t-distribution r degrees of freedom
τ(r) τ-distribution r degrees of freedom
χ²(r) χ²-distribution r degrees of freedom
F(r,r') F-distribution r and r' degrees of freedom

START First steps Deutsch Anderson-Darling-test for normality

This test checks the hypothesis of Gaussian normal distribution. The test statistic measures the difference between the empirical distribution of the sample and the normal distribution, giving more weights to the tails of the distribution than similar tests, e.g. the Cramér–von-Mises criterion.

The Anderson-Darling test is computed in up to four variants:

The first three variants work only if the required values have been given.

START First steps Deutsch Repeated height determination of a point

Every year in the third semester of the curriculum Surveying/Geoinformatics of University of Applied Sciences, Dresden (Germany) every student has to measure tacheometric point positionings of the same points. These can be treated as independent repeated measurements. In the years 2010 and 2011 the following results have been obtained for a chosen point:

YearDetermination of the point height, unit=metre
2010 116.774 116.755 116.755 116.751 116.742 116.745 116.760 116.754 116.753 116.739 116.752 116.747 116.732 116.752 116.736 116.764 116.738 116.765 116.757 116.750 116.741 116.759 116.751 116.753 116.734 116.737 116.757 116.730 116.755
2011 116.764 116.748 116.758 116.743 116.757 116.659 116.744 116.754 116.761 116.762 116.769 116.741 116.747 116.738 116.744 116.750 116.746 116.736 116.760 116.762 116.760 116.756 116.739 116.754 116.728 116.745 116.737 116.750

In the exercise network this point has the nominal height of 116.767 m . Despite of lacking routine the students can be asked for a standard deviation of the determination of σo=0.01 m . The repeated measurements shall be tested statistically with a probability of type I decision error of α=0.05 .

and Compute

The normal probability plot shows the distribution of the measured values (dots) relative to a best fitting Gaussiannormal distribution (straight line). Immediately it is seen that at the left bottom there is an isolated red (=2011) dot. This clearly indicates an outlier.
The remaining dots scatter around the blue (=2010) line. In the middle zone of the dots (around the median) the blue dots are located right of the red (=2011) dots. Consequently, the median of the 2010 measurements is larger by 3 mm.

First of all, it can be tested if the required measurement precision has been realized, i.e. if Ho:σ≤σo can be assumed. This is done by the right-tailed global test, which for the measurement series Year 2011 is rejected. This means that the measurement precision has probably not been realized. Strictly speaking, the probability, that the measurement precision has been realized and the global test is rejected nonetheless, is α=0.05.

Also the w-test after Baarda detects an outlier for Year 2011 . This is the measured value 116.659, which is most departed from the mean. If you eliminate this value and repeat the computation, both the left-tailed global test as well as the w-test are accepted. However, it could be irritating that also the right-tailed global test Ho:σ≥σo is accepted. The reason for this phenomenon as follows: If α is chosen small enough, the null hypotheses of all statistical tests are always accepted. And for a small number of measurements, α=0.05 is practically small.

If σo=0.01 would have been unknown then the outlier could have been detected by the τ-test after Pope.

The posterior standard deviations of a single determination of the point height are estimated by 0.0107 m for Year 2010 and 0.0102 m for Year 2011 . The answer to the question if after elimination of the outlier the measurement precision of both years should be treated as identical, is given by the two-tailed F-test with Hoxy. This hypothesis is accepted. Therefore, the precision in the Year 2011 was not significantly higher.

The means amount to 116.7496 m for Year 2010 and 116.7501 m for Year 2011 . The answer of the question, if after elimination of the outlier the expected values of both series coincide, yields the double-sample Z-test with Hoxy . The hypothesis is accepted. Therefore, the expected values should be treated as identical.

The hypothesis that the nominal value of that point μo=116.767 is identical to the mean, is tested by the two-tailed single-sample Z-test with Ho:μ=μo . This hypothesis is rejected for both years. The conclusion could be that the nominal value is not correct or that one control height used in all determinations of instrument heights was not correct and this has not been detected during station setup.

Since both measurement series do not show significant differences, it is possible to merge them into one series and repeat the computation:

and Compute

As in the cases before, the nominal height value of μo=116.767 is rejected. This value is either incorrect or the heights are biased. Moreover, the hypothesis of normal distribution with the parameters μo=116.767 is rejected. Without this parameter the Anderson-Darling-test succeeds.

START First steps Deutsch Try this computation tool using normal distributed random numbers

Two series of Gaussian normal distributed pseudo random numbers N(53.06;16.10) from www.random.org/gaussian-distributions are investigated.

and Compute

START First steps Deutsch Try this computation tool using different random numbers

One series of Laplace distributed and one series of χ²(1) distributed pseudo random numbers, 100 values each, computed by GNU Octave are investigated. The true parameters of the distributions are

seriesexpectationmedianstddev.inter quartile rangeskewnessexcess kurtosis
Laplace 00 1.4141.3860 3
χ²(1)10.4551.4141.2222.82812
and Compute
Did you know? All numbers are always accepted with decimal point or with decimal comma.

START First steps Deutsch




Ausgleichungslehrbücher


START First steps Deutsch In the library

Link Author(s)Title Year Type Pages
MByte
PDF: open accessFoppe KFehlerlehre und Statistik201694
0.9
PDF: open accessVermeer MStatistical methods in geodesy2015157
0.1
PDF: open accessLehmann ROn the formulation of the alternative hypothesis for geodetic outlier detection201314
0.1
PDF: open accessLehmann ROn the formulation of the alternative hypothesis for geodetic outlier detection (postprint)201320
1.3
PDF: open accessFoppe KRepetorium zur Fehlerlehre und Statistik und Ausgleichungsrechnung200987
1.9
PDF: restricted accessLehmann RDas arithmetische Mittel von Mehrfachmessungen unter Wiederholbedingungen20085
0.1
PDF: restricted accessKoch KRParameterschätzung und Hypothesentests in linearen Modellen2004382
0.1
PDF: open accessTanizaki HComputational Methods in Statistics and Econometrics2004512
5.5