Worked example 2
|Data obtained at perfusion
Our second worked example also uses data from De Bont et al. (2002)
but in this case the authors looked at the relationships between faecal and tissue egg counts. The rationale for doing this was not made clear, but since both measures are used as indicators of the intensity of infection, it seems reasonable to expect them to be correlated. A slope of one would indicate a simple proportional relationship between the two measures.
This is clearly a symmetrical regression. There are no dependent or independent variables - both variables are measures of egg numbers. One could estimate measurement error for these two variables, but since there will also be equation error, one would not be able to use these in the maximum likelihood equation. Since both variables are log10 transformed, reduced major axis regression is probably the best option.
Test for linearity
The effect of the log10 transformation can be seen in the figure below:
The relationship is more or less linear after the log10 transformation, although there is one outlier. Hence we proceed with the analysis.
Reduced major axis regression
The coefficients of the reduced major axis regression are estimated thus:
|| = +
|| = 0.6632
|| 0.6632 × 5.282
|| = −2.135
In R we initially ran the standard OLS regression using the linear model. This gave a P-value of 0.01341 with a slope of 0.569. We then ran the R package 'smatr', first for OLS regression as a check, and then for reduced major axis regression.
Coefficients for reduced major axis regression along with their 95% confidence intervals are given below:
slope = 0.6632 (95% CI: 0.3788 - 1.1611)
intercept = -2.135 (95% CI : -4.2101 - 0.0601)
Note that, as expected, the reduced major axis slope is greater than that estimated by simple OLS regression. Diagnostics for the reduced major axis regression are shown below:
The fit is less good than in the previous worked example, but still acceptable. The plot of residuals versus the predictor variable does not provide any evidence of heteroscedacticity. Again the normal QQ plot of residuals suggests some non-normality.
We assess whether the slope of the relationship differs significantly from 1 by testing whether Y − X is uncorrelated to Y + X. Pearson's product moment correlation coefficient comes out to -0.6355 which has a (non-significant) P-value of 0.1250. We conclude that the slope of the reduced major axis regression does not differ significantly from 1. The R package 'smatr' provides a test of slope against any specified value - results are the same as given here.