Chapter 2. General Methods And Materials
2.4. General Statistical Methods
2.4.1. Software and Common Procedures
All the statistics described in this thesis were performed on the JMP statistics package (versions 3.1.5 and 3.1.6, SAS Institute Inc.) except for the Mantel-Cox Log-Rank test, which was carried out upon BMDP software (BMDP Life Tables and Survival Functions, BMDP Software Ltd. (Dixon, 1988)).
Where appropriate, data sets were tested to see if they conformed to the assumptions of parametric analysis. Deviation from the normal
distribution was tested using the Shapiro-Wilk W test, and homogeneity of sample variances was verified using the O'Briens test. Any data violating these assumptions were either transformed using an appropriate procedure, so that parametric tests could be used, or analysed with a suitable non- parametric method. Unless otherwise mentioned, it may be assumed that data subjected to parametric analysis had homogeneous variances and were normally distributed. The main parametric method used was analysis of variance (ANOVA), but linear regression and Product-moment
correlations were also employed.
In the majority of cases, data that could not be analysed
parametrically were tested using the Kruskal Wallis one-way analysis of
variance by ranks. If such a test yielded a significant result, pairs of groups could then be tested using the Kruskal Wallis multiple comparisons test
(Siegel and Castellan, 1988). All Kruskal Wallis analyses were tested at the 5% significance level.
To avoid the possibility of obtaining spurious positive results, p values in multiple comparisons tests were corrected using the sequential Bonferroni method (Rice, 1989). This method involves reducing the significance level in a test by an amount that depends upon the number of seperate comparisons that are being made. Certain multiple comparisons procedures already account for the number of separate comparisons made; these include the Kruskal Wallis multiple comparisons test and the Tu key Kramer Honest Significant Difference (HSD). Therefore, Bonferroni correction was not used in these cases.
2.4.2. The Analysis of Survival and Mortality
A number of different analyses were carried out on the survival data collected during the experiments described in the following chapters.
• To assess lifetime survivorship, survival curves were analysed using the Mantel-Cox Log-Rank test. This compares cumulative mortality in each sampling interval with the expected mortality based on the number of live individuals in each group that enter the sampling interval. Accidental losses are taken into account. Cumulative survival probabilities for pairs of groups can be compared using an observed versus expected chi-squared statistic.
• The rate of increase in age-specific motality with age was estimating using the Gompertz survival model. The Gompertz model has been widely used to measure changes in rates of age-specific mortality (Curtsinger et al., 1992; Fukui et al., 1993; Tatar et al., 1993; Brooks et al.,
1994; Hughes and Charlesworth 1994). The Gompertz parameters were estimated using the following model fitted to survival data.
Sf = ((A/G)*(1-exp(G-day))
(Finch, 1990).
S, is the proportion of the population surviving to age t, A is the age- independent mortality rate, and G is the age-dependent rate of increase in mortality (Finch, 1990). G can be used as a measure of the rate of
senescence in a cohort, and was of particular interest. The model was fitted to the survival data for each experiment by least squares. The first step in testing G for variation between treatment groups was to estimate A for each treatment group and make a single estimate of G for all groups. Secondly, G was estimated individually for each group. The residual sums of squares error (SSE) for the two fits were then compared with an F test; an
improvement in fit was indicated by a reduction in the SSE. A significant difference between the two fits would indicate that one or more of the individual estimates of G differed from the original single value. If this was shown, then pairs of treatments could be compared in the same manner to establish exactly which ones differed.
For analysis of the age-dependent Gompertz parameter estimated for different treatment groups in experiments on populations selected for age at reproduction, described in chapters 4 and 6, an estimate of G was made for each replicate line within each selection regime. Comparisons between regimes were then made with either Kruskal Wallis or ANOVA, using the estimate of G for each replicate line within a regime to calculate a mean value of G for that regime.
• Age-specific mortality rate was also monitored in different way. In situations where it was desirable to measure mortality rates during a specific part, or interval, of the life span, the following formula was used.
Number dying during interval Age-Specific Mortality Rate = ______________________________
Number alive at the start of that interval
The Kruskal Wallis test or analysis of variance was then used to compare mean mortality rates of different cohorts within a particular interval.
• The point of 10% mortality in a cohort, when 10% of individuals have died, has been described previously as the start of a cohort's
senescent period (Arking, 1987), with this period continuing until 90% mortality. The point of 10% mortality was used as an additional way of monitoring the schedule of mortality in certain populations to assess the timing of the onset of the senescent period. Sample means were compared using the Kruskal Wallis test or ANOVA, depending on whether the data conformed to the assumptions of parametric analysis.