Linear regression - elevation of coral clasts on Lanai, Hawaii.

Below is a subset of table data extracted from Table 1 of Rubin et al. 2000, which gives elevation and coral clast age for a unique surficial basalt-coral clast deposit on the Hawaiian Island of Lanai. There is distinct debate on the origin of this deposit. The age elevation data is one crucial part of this discussion. If the deposits were from gradual sea level changes a good correlation might be expected between the two and would yield an uplift history or rate(s). If not, other more complex explanations are required. Two alternate competing hypotheses are that the deposits are the result of one large wave, in which case one might not expect any relationship, or that they are the result of a more complex history including multiple depositional events during interglacial periods and associated high sea level stands.

The arguments are multiple, and if you want to decide which of the two hypotheses is correct you should read the relevant literature. For our purposes we will simply look at elevation vs. age data/ We will select elevation as the independent variable, because it is more precisely known, and because we are interested in how well elevation predicts age of clast, with an expectation that the higher clasts should be older. Note that one could reverse the variable roles here with a different perspective.

The table below has values. The error is associated with the age. We will not utilize the error information in our analysis. In most cases it is less than 1%. The sites are for different sites. The age is an isotopic date for coral clasts in the gravel bed.

site elevation age error
HB 0.5 2.01 0.07
HB 0.5 2.04 0.06
HB 0.5 3.22 0.07
C 3.8 134.6 0.9
C 3.8 249.6 1.6
C 2.3 249.6 1.6
c 8 196.2 1.3
C 4.2 250 1.7
AG 9.7 134.5 0.9
AG 35 211.4 1.5
KG 20 130.9 1.5
KG 19 134.8 0.9
KG 18 242.6 1.6
KG 23 136.4 0.9
KG 19 136.7 1
KG 28 230.2 3.9
KG 17 204.1 1.4
KG 58 222.4 1.5
KG 58.9 235.9 1.5
KG 47 250.2 5.2
KG 58 258.2 1

Excel scatter plots for one of the sites, and then for all of the sites is presented below. Note that the R-squared values are fairly low for a bivariate regression, and would generally not be considered usable. However, for an n of 12 for the data from site KG, the critical Pearson value is .5325 for a significance level of .025, while the observed r is .69. This means the r can be considered statistically significant, and there is at least a partial relationship between elevation and coral clast age. The lower graph including all the data but with a lower r-squared also turns out to be statistically significant. Note that the older clasts are at a greater range of elevations than younger. Note that the intercept is far from zero. What might be the geologic significance of this. What geologic processes might modify the age-elevation relationship? On the flip side, why might there be an elevation-age relationship for the coral fragments even if there was one big wave? Realize that the deposit from one big wave would be sorted by age. The argument is a complex one, and can't be resolved here, but this regression does give some insight.