I would like to compare the mean values of petrophysical properties for the same stratigraphic interval at different borehole locations. These properties are determined using empirical regression equations and are assumed to be valid and applicable. Usually, if the preconditions are met, I would apply, e.g., an independent t-test to compare the means of both groups. However, by doing so I am not able to consider the uncertainty in the determination of the single group values (resulting from the application of the regression equation that should allow prediction of the target properties with a mean error ranging between 5 to 15 %). I want to avoid having a false positive result in the comparing-mean test (that means, seeing a significant difference between both means although this results from the uncertainty of the prediction equation and not from real significant variations between both groups). How can I circumnavigate this problem? Does anyone have an idea? Raw data and all statistical parameter of the applied regression equations are available to me. Big thanks in advance.
If the dependant variable is petrophysical properties in the model, you could introduce a variable to differenciate the different borehole locations (a binary if you have only a comparaison of two data). You juste have to look for the p value of this variable to test for the difference between group.
@ David:
Well, do you mean the raw data of the regression equations? The applied regression equations have been originally developed on a synthetical data set which is assumed to cover the full range of the studied petrophysical property in sedimentary rocks. Now, they are applied on real borehole data. Therefore, unfortunately, I don't see how the ANOVA applied to the raw data set could help me with my borehole data set.
@Gregor:
Yes, how to perform the mean test is fully clear. But let me give you an example.
In a specific stratigraphic interval of borehole 1, I am able to predict the parameter with an mean error of 5 per cent, at borehole 2 with an mean error of 12%. In borehole 1 I have 150 observations, in borehole 2 280.
If I now compare the means of both groups (locations), this mean is not only determined by the variability in the groups, it is also affected by the prediction error. Is there any approach to consider these input prediction errors in the mean comparison test?
I want to look for the real variations between both boreholes which are not biased by the prediction uncertainty.
I meant your measured borehole data. Since you wanted to circumvent the extra uncertainty introduced by the prediction equation, directly comparing your measured input parameters from both boreholes seemed like an option.
Normally your prediction variability is compraised in the distribution of the observation. However, I think there is a possibility to calculate the distribution of your mean predicted by the two models using the bootstraps technic.
I hope I could help you.
Best regards
Gregor
@ David
Yes, I have thought about this idea. The problem is that properties, for which well-log measurements from the boreholes are available, are not simply correlated to the parameter of interest. Furthermore, I do not have direct measurements of the specific parameter of interest. Applying the prediction equations as approximation is the only chance to deduce data I need.
@ Gregor
Your are right, in general, prediction variability is comprised in the distribution of the observations. But I am sure that the distribution of the observations is broader due to the additional uncertainties resulting from the prediction procedure than it would be by using direct measurements of the parameter. Anyway, I will check if bootstrapping would be an approach.
Thanks to both of you.
Sven
Sven,
Do you have reason to believe the errors at different boreholes are correlated? If not, it seems like a random-effects t-test would be unbiased, if not necessarily highly efficient.
If they are correlated, that's a different problem that you might be able to address using simulations from your data as a sensitivity analysis.
Pat
@Patrick
With respect to the available well logs, different predictive equations are applied at different borehole locations. It is documented in the original study, that these equations have varying predictive capabilities and, thus, are afflicted with different mean errors to be expected. Therefore, I have reasons to believe that the errors at different boreholes are not correlated. Are you convinced, in this case, that a random-effects t-test would be unbiased?
Sven,
Yes, with adjustments for unequal group variance, I believe you're fine.
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