2.6 Isochron regression line fitting

When radiogenic isotope ratios for a cogenetic sample suite (e.g. ⁸⁷Sr/⁸⁶Sr) are plotted against the parent/daughter ratio in those samples (e.g. ⁸⁷Rb/⁸⁶Sr), the points should ideally define a perfect straight line or ‘isochron’ (section 3.2.2). Typically, the slope of this isochron line can then be used to determine the age of the system. However, since both quantities involved are measured experimentally, experimental errors are inevitable. Hence, these errors must be considered when calculating the slope of a best-fit line though the data.

A straight line fitted to an array of points is termed a ‘linear regression’. One of the best approaches involves minimising the sum of the squares of the distances that data points lie away from a line drawn through the points, and hence is called a least squares fit. This involves iteration, and is therefore done by calculator or computer.

2.6.1 Types of regression fit

In simple linear regression programs, one ordinate is defined as ‘free of error’ and the regression line is calculated to minimise the mis-fit of points in the other ordinate (Fig. 2.26a,b). If the data define a very gentle slope and are somewhat scattered, disastrous fits can be produced by regressing onto the wrong ordinate. Where errors are present in both ordinates, as in the case of the isochron fit, a two-error regression treatment must be used (Fig. 2.26c, d). Several methods of this type have been presented in the literature (e.g. McIntyre et al., 1966; York, 1966; York, 1967; Brooks et al., 1972; Ludwig, 1997), sometimes including a ready-to-run computer program.

Fig. 2.26. Schematic illustration of least squares regression analysis with different conditions of weighting. a): infinite weighting of X (all errors in y); b): infinite weighting of Y; c): fixed weighting of X versus Y; d): individual weighting of each point, inversely proportional to squares of standard deviations. After York (1967).

In cases where the actual deviations of the data points from the regression line are equal to or less than those expected from experimental error, all regression treatments effectively give the same isochron age and initial ratio. In such cases, the only matter of debate is the manner in which experimental errors are assigned.

Ideally, the analysed error in ⁸⁷Sr/⁸⁶Sr and ⁸⁷Rb/⁸⁶Sr (for example) would be determined by measuring the reproducibility of an almost infinite number of duplicates (Brooks et al., 1972). Since this is very time-consuming, the best empirical estimate is probably the long-term reproducibility of standard analyses. Within-run precision of sample analyses is almost certainly an under-estimate of error, since it is typically about 50% of the reproducibility error. For ⁸⁷Rb/⁸⁶Sr, quoted accuracies must include an estimate for sample weighing errors, spike calibration errors etc., as well as mass spectrometry errors (in the case of isotope dilution).

While some of the regression programs in use provide the facility for weighting each data point according to its precision of measurement, this may sometimes be detrimental, as it tends to ‘destabilise’ the fit. In practice, ⁸⁷Rb/⁸⁶Sr and ⁸⁷Sr/⁸⁶Sr errors are probably best assigned as a blanket percentage (e.g. 0.5% and 0.002% 1F respectively). If one point has a particularly bad precision, it is better to re-analyse it than give it less weight in the regression.

2.6.2 Regression fitting with correlated errors

In conventional isochron analysis (e.g. Rb)Sr), analytical errors in the two ordinates (isotope ratio and elemental abundance ratio) are effectively uncorrelated. However, in the lead isotope dating methods, this is far from the case. In common Pb)Pb dating, correlated errors are found between ²⁰⁷Pb/²⁰⁴Pb and ²⁰⁶Pb/²⁰⁴Pb, due to the greater analytical uncertainties on the small ²⁰⁴Pb peak (which is common to both ratios) and due to the uncertainty of mass fractionation. These two correlation lines have different slopes, and while the former may be important for very small Pb beam sizes, the latter is normally dominant. Data for the NBS 981 standard shown in Fig. 2.27 yield a correlation coefficient of 0.94 (Ludwig, 1980).

Fig. 2.27. Results of seven analytical runs on the NBS 981 Pb standard performed with large beam sizes at varying filament temperatures. Data cluster near the mass fractionation line. Solid square = ‘true value’. After Ludwig (1980).

In U)Pb zircon dating, errors may show a much stronger correlation. This is because errors in ²⁰⁶Pb/²³⁸U and ²⁰⁷Pb/²³⁵U are mainly attributable to the elemental U/Pb ratio, which may be five or more times less reproducible than the ²⁰⁶Pb/²⁰⁷Pb ratio (Davis, 1982). This difference arises from the analytical errors inherent in isotope dilution, and to uncertainties in common Pb correction (section 5.2). Regression treatments for correlated errors using the least squares technique were given by York (1969) and Ludwig (1980). Davis (1982) used the alternative ‘maximum likelihood’ method, and showed that the two approaches yielded similar estimates of error using test data (see also Titterington and Halliday, 1979).

2.6.3 Errorchrons

Brooks et al. (1972) argued that ‘a line fitted to a set of data that display a scatter about this line in excess of the experimental error is simply not an isochron’. They proposed that Rb)Sr regression fits with excess or ‘geological’ scatter (McIntyre et al., 1966) should be called ‘errorchrons’ and treated with a high degree of suspicion. This raises the problem of how to detect the presence of geological scatter, bearing in mind the fact that analytical errors are only probabilities.

The sum of the squares of the mis-fits of each point to the regression line (= squared residuals; York, 1969) or sum of chi² (Brooks et al., 1968, 1972), may be divided by the degrees of freedom (number of data points minus two) to yield Mean Squared Weighted Deviates (MSWD) which is the most convenient expression of scatter. If the scatter of data points is, on average, exactly equivalent to that predicted from the analytical errors, then the calculation will yield MSWD = 1. Excess scatter of data points yields MSWD > 1, while less scatter than predicted from experimental errors yields MSWD < 1.

Problems may arise with the interpretation of these MSWD values, since the analytical errors input into the program are only estimates of error. To address this problem, Brooks et al. (1972) constructed a table of probabilities (Table 2.1) to distinguish between errorchrons and isochrons from their MSWD values. They established a ‘rule of thumb’ that on average if MSWD <2.5 then the data define an isochron, whereas if MSWD >2.5 they define an errorchron. Unfortunately this rule of thumb has been much abused over subsequent years, because the original objectives of Brooks et al. (1972) have been misunderstood. They set up the MSWD = 2.5 cut-off in order to reject errorchrons with a 95% certainty of excess scatter over analytical error. This corresponds to only 5% confidence that a fit with MSWD = 2.5 is an isochron (eg. Wendt and Carl, 1991). However, many workers have wrongly assumed that if MSWD is less than 2.5 then there is a high degree of confidence that the suite is a true isochron (where analytical errors express most or all of the error on the age). In actual fact, MSWD must be near unity for one to have a high degree of confidence that the data represent a true isochron.

Table 2.1 MSWD values indicating 95% confidence of an errorchron

)))))))))))))))))))))))))))))))))))))))))))))))))))))))

Number of Number of samples regressed

duplicates 3 4 5 6 8 10 12 14 26

)))))))))))))))))))))))))))))))))))))))))))))))))))))))

10 4.96 4.10 3.71 3.48 3.22 3.07 2.98 2.91 2.74

20 4.35 3.49 3.10 2.87 2.60 2.45 2.35 2.28 2.08

30 4.17 3.32 2.92 2.69 2.42 2.27 2.16 2.09 1.89

40 4.08 3.23 2.84 2.61 2.34 2.18 2.08 2.00 1.79

60 4.00 3.15 2.76 2.53 2.25 2.10 1.99 1.92 1.70

120 3.92 3.07 2.68 2.45 2.18 2.02 1.91 1.83 1.61

)))))))))))))))))))))))))))))))))))))))))))))))))))))))

Numbers underlined just exceed MSWD=2.5 cut-off

2.6.4 Dealing with errorchrons

Because the number of errorchrons will continually increase as analytical errors decrease, the suggestion that errorchrons be rejected outright is unhelpful. Therefore, other workers have looked for ways of quantifying geological scatter in terms of an error on the age result. McIntyre et al. (1966) emphasised that statistical error estimation of errorchrons cannot be properly meaningful unless the geological reasons for the mis-fit are understood. Therefore, they suggested four alternative approaches for error handling. These are as follows:

1) No excess scatter above predicted analytical errors (= true isochron).

2) All excess scatter is attributed to Rb/Sr, equivalent to assuming small differences between the initial ages of the samples.

3) All excess scatter is attributed to ⁸⁷Sr/⁸⁶Sr, equivalent to assuming variation in the initial isotopic ratio of samples.

4) Excess scatter is attributed to some combination of models 2 and 3.

The program of York (1966) allows the analytical errors on X and Y ordinates (e.g. ⁸⁷Rb/⁸⁶Sr and ⁸⁷Sr/⁸⁶Sr) to be multiplied in equal and uniform proportion by an error factor (/MSWD) until the expanded errors equal geological scatter (MSWD = 1). The error on the calculated age will be magnified by this process to give a reasonable estimate of uncertainty which includes both geological and analytical scatter.

Some form of error expansion must always be performed if a meaningful geological error estimate is to be given for a data set with MSWD > 1, because this is a definite indicator that excess scatter of some form is present. The only uncertainty is whether the excess scatter is geological or analytical. The York (1966) procedure is the most common method of dealing with excess scatter, but it is an arbitrary procedure which takes no account of geological processes, and their resulting contribution to errors. Where initial ratio variability is suspected, option 3 of McIntyre (above) is preferable (amplification of isotope ratios only). However, this can lead to misinterpretation of a data set if all points are not identical in age.

A new DOS/Windows-based computer program incorporating these concepts was provided by Ludwig (1997). Amongst several types of isochron fit, this program (Isoplot version 2.95) provides York-type fits under three categories which are similar to those above:

1) Fits based on individually assigned analytical errors. Errors on the age are calculated: a) based on analytical errors only (applicable if MSWD < or = 1); and b) by equal expansion of assigned analytical errors. A warning was given that the latter approach can give rise to serious errors if the assigned analytical errors are significantly variable.

2) A fit based on expansion of assigned errors to encompass the scatter, but all points have equal weights. If the assigned errors are uniform, fit 1b is the same as fit 2.

3) A fit based on model 3 of McIntyre, with excess scatter absorbed by expanding initial ratios only. Sketches to show categories (1) and (3) are shown in Fig. 2.28.

Fig. 2.28. Sketches to show the effects of Ludwig’s Model 1 and Model 3 on a 4-point errorchron (MSWD = 6.9) with one aberrant point with larger error. (Input data: .06, .1, .5106, .002; .10, .1, .511, .002; .14, .1 .5114, .002, .13, .1, .51135, .005).

Misinterpretation of errorchrons is usually due to a failure to adequately visualise the distribution of data and attendant errors. This can be avoided by using a graphical presentation. Different methods of graphical assessment using isochron diagrams will be discussed for the Rb)Sr method (section 3.2.2). However, an alternative approach is the so-called ‘bootstrap method’ (Kalsbeek and Hansen, 1989). In this method a set of errorchron data is analysed by computer to see how stable the regression line is to the application of a greater weighting to different points. This test is achieved by successive random selection of a sample of points from the data set, such that this sample is equal in size to the data set. (This is not as strange as it sounds). What will normally happen is that a few points are selected more than once, while others are omitted. By repeating this process a few thousand times, a probability distribution is set up which portrays the stability of the best-fit line to the influence of certain sub-sets of the data suite (Fig. 2.29).

If geological errors are randomly distributed, the frequency histogram derived from the data set will have a symmetrical (Poisson) distribution. In this case the result is identical to expanding analytical errors by /MSWD. (Of course, a true isochron should always yield a Poisson distribution, because analytical errors are assumed to be random.) However if geological scatter is uneven, the probability histogram of an errorchron may be skewed or even bimodal (Fig. 2.29), and hence highly suspect in terms of age assignment. This diagram therefore represents an excellent visual test for isochron data quality, and could help to avoid the misinterpretation of problematical data sets.

Fig. 2.29. Frequency distribution of 10 000 selection permutations from three sets of errorchron data. 95% (2F) confidence limits of the ‘bootstrap’ age determination are indicated (P_2.5 and P_97.5). Arrows represent symmetrical 2F confidence limits resulting from expansion of analytical errors until MSWD=1. After Kalsbeek and Hansen (1989).

A more recent form of the bootstrap approach was described by Powell et al. (2002). The objective of their method was to downplay the significance of extreme data points that lie outside a Poisson distribution, by amplifying the errors on these data points more than the main data set (similar to Fig. 2.28). This approach was intended to generate more ‘robust’ ages and error estimates by avoiding the distortion of an isochron age that could occur by full weighing of ‘aberrant’ points, while also reducing the ‘temptation’ to completely exclude such points from the calculation, as many workers do. The ideal approach would be to collect more data in order to resolve the non-Gaussian data more clearly, but in the real world this may not be possible.

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