Methods for analysis

4.2.1 Methods for analysis o f tooth crown dimensions.

The metric data collected using the methods described above were transferred from Access version 2 to SPSS for Windows version 6.1. Most o f the statistical analysis o f the metric data was carried out using SPSS, although Minitab version 11.21 and Excel version 5C were also used. The data were subdivided before the statistical analysis. The Hakel sample was subdivided by sex and by age, using the four age groupings 4-6

months, 7-9 months, 10-12 months and 13-15 months; thus 8 subgroups were created. The modem domestic sample, which was not sexed, was divided according to breed, that is Tamworth, Berkshire and Middle White, thus generating three subgroups. In both samples the right and left sides were considered separately.

Before the data were examined in terms o f variation between and within subgroups (described above), tests were carried out in order to assess the statistical normality o f the data. This was important because a significant deviation from a normal distribution might indicate error in data collection or that the sample chosen was not representative o f a biological population. Dental measurements would be expected to have a normal distribution in such a population. Some clarification o f what is meant by the term population is required at this point. The biological definition o f a population is: “A breeding group o f organisms, all o f the same species, that occupies a particular area” (Allaby 1991:373). The statistical definition o f a population is: any group o f like

individuals (ibid.).

The term sample refers to a group o f individuals taken from the statistical population, it is assumed that each individual within the population has an equal chance o f being included in the sample (i.e. that the sample is random). In this project however, the choice o f sample was not completely random: the modem domestic material was collected fi'om breeders willing to donate material aged around six months. Within the collection only the three well represented breeds were included and only individuals that had both sides o f the jaw were included. Damaged material was not included. In the sampling o f the Hakel material age was the main selection criteria, the age groups listed above were chosen as the most comparable with the modem domestic material. In terms o f a biological definition o f population the Hakel material was drawn fi'om a breeding group which inhabited a particular geographical area, the modem domestic material however came several different farms which bred the particular breeds under study in this project.

The methods chosen to examine the normality o f the samples were firstly, stem and leaf plots o f the distribution secondly, normality probability plots (Q-Q plots), and thirdly the Kolmorgorov-Smimov (Lilliefors) test for normality. O f these the first two are means o f illustrating the distribution o f the sample, while the Kolmogorov-Smimov (Lilliefors) test assesses whether there is a significant deviation fi'om a normal

The stem and leaf plot can be considered to be a histogram on its side. If the tops o f the bars on a histogram or the ends o f the leaves on the stem and leaf plots are joined and smoothed to form a curve this will reveal some information concerning the form o f the distribution o f that measurement within the parent population (see Figure 4.10 for examples o f curve forms).

In a sample which is normally distributed within the population the shape o f the plot will reflect the bell-shaped curve o f the normal distribution; the curve is bilaterally symmetrical with equal tails to the right and left o f the mean, the mean, mode and median are equal. The plot is unimodal, and can be described as mesiokurtic (Figure 4 .10a). There are a number o f ways in which the shape o f the distribution can vary from this normal bell shaped curve. A curve with a longer tail on the right side is described as positively skewed (Figure 4.10b), this is due to a small number o f individuals within the sample being larger than expected. If the tail is greater to the left (the top in a stem and leaf) this is a negative skew (Figure 4.10c), due to a small number o f individuals being smaller than expected for that population. If there are more individuals concentrated about the mean than expected then the curve will be taller but narrower than the normal distribution (Figure 4. lOd), this is called a leptokurtic distribution resulting from a concentration o f individuals within the middle quartile. Conversely in a distribution where the points are spread out more widely than expected a lower wider shape than the normal distribution is seen which is called a platykurtic distribution (Figure 4. lOe). Finally, the distribution may be bimodal, that is having two peaks and can indicate that the variation is too great to be reasonably expected within a single population (Figure 4. lOf). In the stem and leaf plot the digit(s) before the decimal point form the stem, whilst the leaf consists o f the part o f the value that follows the decimal point. The frequency colunm indicates how often each value occurs in the sample, points which fall at a great distance from the majority o f those in the distribution (referred to as outliers in this project) are listed separately. An example o f a stem and leaf plot is shown in Figure 4.11.

Figure 4.10. Q-Q normality plot and histogram forms

Form of stem and leaf (histogram) Form of Q-Q plot Description

normal distribution mesiokurtic

positive skew a few individuals larger than

expected

negative skew a few individuals smaller than

expected

leptokurtic

more individuals concentrated around the mean

platykurtic more spread than expected

f bimodal

could indicate that sample was drawn from two

populations

Figure 4.11 Example of stem and leaf diagram.

Taken from the Hakel sample, first permanent molar anterior width for males 13-15 months (left side). Frequency Stem & L eaf

4 00 10 . 6789 6 00 11 * 013444 8 00 11 . 55677999 2 00 12 * 01 1 00 12 . 9 Stem width: Each leaf: 1.00 1 case(s)

The second method, the Q-Q plot also illustrates how the distribution deviates from a normal distribution. The plot is constructed in the following manner: for every measurement recorded the value actually obtained (i.e. the observed value) is plotted against the value that would be expected for a data set that came from a population with a normal distribution for that measurement. If the expected and observed values are identical, the points plotted create a diagonal line, the degree to which the points on the plot deviate from this line is an indication o f the deviation from the normal distribution within the data set. The Q-Q plots can indicate the range o f deviations described above for the stem and leaf plots, the range o f forms seen in Q-Q plots is shown in Figure 4.10, each form o f Q-Q plot is illustrated along with the stem and leaf or histogram curve that would be seen for the same data set. A positive skew is reflected in a plot which curves above the diagonal line (Figure 4 .10b). A negative skew is reflected in a curve below the diagonal line (Figure 4.10c). A leptokurtic distribution produces a plot where the points start above then line, join the line for the middle section and then fall below it towards the top o f the plot (Figure 4. lOd). The reverse is seen in a platykurtic distribution, the points initially fall below the line then join it for the middle section and then fall above the line in the upper section o f the plot (Figure 4. lOe). The bimodal distribution can be identified by a Q-Q plot which is stepped in appearance (Figure 4. lOf). An example o f a Q-Q plot is shown in Figure 4.12.

Figure 4.12 Example o f a Normal Q-Q plot

This is taken from data recorded for the Hakel sample.

Normal Q-Q Plot of first permanent molar anterior width