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Manual for psimpoll and pscomb

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Menu M: data analyses

Menu Mg: statistical description

• calculation of mean, standard deviation, skewness, and kurtosis for the values presented for each pollen type, with results written to the data analysis file (menu Ec). Values of skewness and kurtosis significantly different from zero (the value for a perfect normal distribution) are marked;
• calculation of a weighted average, or `optimum' location of each pollen type in time (or depth), together with a standard deviation, or `tolerance'. This is a measure that combines the extent of occurrence of a taxon over time, weighted by its abundance. It is included here, experimentally, as a potentially better statistic for the location of a taxon in a sequence than such subjective measures as `rational' or `empirical' limits etc. The idea is borrowed from the statistics used to define optima of diatoms along pH gradients: I have substituted time (depth) for the environmental gradient (Birks et al. 1990). Output in the data analysis file gives, for each taxon, the optimum, the tolerance, and the location (depth or age) of the taxon's maximum value, and the value of that maximum;
• calculation of a `runs' test. If abundances of a pollen type along a sequence were randomly distributed with time, then the length of `runs' (series of samples all successively greater (or lower) than the previous sample) will follow a predictable pattern. We can count the lengths of observed runs (how many runs of a single sample, runs of two samples, etc), and compare the resulting distribution with the distribution for random samples. The result is a runs test, and gives a measure of whether the overall pattern of change is, or is not, significantly different from random. The output gives the runs statistic, udr, and the goodness-of-fit, q. q greater than 0.05 indicates that the sequence of runs for the taxon is not significantly different from random.
• calculation of a Linear Correlation Coefficient. This is a statistic that can be used for detecting trends in the values for taxa. If a taxon's values increase continuously along the sequence, its correlation coefficient (r) will be high and significant. If there are no trends in the data, r will be low (near zero). The sign of r indicates the direction of the trend. In the normal situation where depth (time) values increase numerically down a sequence, r will be positive for taxa whose values are highest towards the base (i.e., decrease upwards). Conversely, r will be negative for taxa whose values increase up the sequence. Results (in the data analysis file) include for each taxon r, the significance level for the difference between r and zero, and Fisher's z, which can be used further to investigate differences between values of r. For 95% confidence values, look for values of 'signif' that are less than 0.05. See Press et al. (1992) for further details.
• calculation of Spearman's Rank Correlation Coefficient for each taxon. This is a non-parametric statistic, making no assumptions about the underlying distribution of the data. It is a means of detecting 'trend' in the dataset. The coefficient is based on the rank order of the values for the taxon concerned and assessing the degree to which the rank order is the same as the ordering of the samples (by depth or time). If the taxon values increase continuously along the sequence, the correlation coefficient will be high and significant. If there are no trends in the data, the coefficient will be low (near zero). The sign of the correlation coefficient indicates the direction of the trend. In the normal situation where depth (time) values increase numerically down a sequence, the coefficient will be positive for taxa whose values are highest towards the base (i.e., decrease upwards). Conversely, the coefficient will be negative for taxa whose values increase up the sequence. Results (in the data analysis file) include for each taxon the Correlation Coefficient 'r(s)', its significance 'signif.', the sum-squared difference of ranks 'D', the number of standard deviations by which D differs from zero 'num of sd', and the significance level for the difference from zero. For 95% confidence values, look for values of 'signif' that are less than 0.05. See Press et al. (1992) for further details, and Gauthier (2001) for some simple examples of use in environmental research.
• calculation of Kendall's Tau. This is another statistic that can be used for detecting trends in the values for taxa. It is even more non-parametric than Spearman's, being based only on the relative ordering of values: higher, lower, or the same. If the taxon values increase continuously along the sequence, Tau will be high and significant. If there are no trends in the data, Tau will be low (near zero). The sign of Tau indicates the direction of the trend. In the normal situation where depth (time) values increase numerically down a sequence, Tau will be positive for taxa whose values are highest towards the base (i.e., decrease upwards). Conversely, Tau will be negative for taxa whose values increase up the sequence. Results (in the data analysis file) include for each taxon Kendall's Tau, the number of standard deviations by which Tau differs from zero 'num of sd', and the significance level for the difference from zero. For 95% confidence values, look for values of 'signif' that are less than 0.05. See Press et al. (1992) for further details.

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