Interpreting the Chi-square and A index for two Combined dates

[Erik]

I have combined two dates, but am a little sure if this is valid, given the chi-square confidence and A index.

The chi-square T value is 1.49 (1 degree of freedom), meaning I can only be 78% confident that the distributions are consistent.

This is lower than usual standard of 95%, suggesting I shouldn't be combining them.

BUT,

1. Stratigraphically they are closely associated (this seems to be strongest argument to combine them)

2. The A index is fairly robust (101). Why does this look so much more robust than the Chi-square test?

So: Can I combine these dates or not?

Thanks for any ideas.

Erik

 Plot()

 {

  Curve("ShCal13","ShCal13.14c");

  Combine()

  {

   R_Date(583,43);

   R_Date(470,80);

  };

 };

[MILLARD A.R.]

The Ward & Wilson test (it's not a chi-square test, though it uses a chi-square distribution) tests whether two measurements are consistent with the samples having the same 14C content. The A-index tests, in a rather different way, whether they have the same calendar date. In this case you are on a steep bit of calibration curve (try viewing this on a curve plot) so small changes in calendar age lead to large differences in 14C content. Thus these samples, which may differ in age by only a few decades, do differ in 14C content but after calibration the uncertainties are large enough that they cannot be distinguished. Using Difference on these two dates gives a 95% probability range of -30 to 290 years.


Best wishes

Andrew
--
 Dr. Andrew Millard 
e: A.R.M...@durham.ac.uk | t: +44 191 334 1147
 w: http://www.dur.ac.uk/archaeology/staff/?id=160
 Senior Lecturer in Archaeology, Durham University, UK

> --
> You received this message because you are subscribed to the Google
> Groups "OxCal" group.
> To unsubscribe from this group and stop receiving emails from it, send
> an email to oxcal+un...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.

[Erik]

Great, Andrew. This clarifies a lot.

So the Ward & Wilson test would be more relevant for R_Combine (pre-calibration), right?

Good point about the calibration curve, I hadn't noticed that.

"after calibration the uncertainties are large enough that they cannot be distinguished."

The high A index is confirming this, right?

And by extension, it is reasonable to combine the dates (despite the low chi-score score), right?

What are you getting out of the Difference command? I don't understand why you used that in this case.

Thanks,

Erik

[MILLARD A.R.]

> From: Erik
> Sent: 27 January 2015 14:23

> So the Ward & Wilson test would be more relevant for R_Combine (pre-
> calibration), right?

Yes.

> "after calibration the uncertainties are large enough that they cannot
> be distinguished."
> The high A index is confirming this, right?
> And by extension, it is reasonable to combine the dates (despite the
> low chi-score score), right?
>
> What are you getting out of the Difference command? I don't understand
> why you used that in this case.

The A-index is sort of indicating this. A low A-index indicates conflict between your data and your prior model and a high one lack of conflict. It is designed to be used with the 60% cut-off just like a p-value in the W&W test. But like the p-value from the W&W test it doesn't do anything else; it is just a Classical statistical decision criterion. The A-index is also an approximate calculation for the comparison of models with data, and is not a measure with a precise statistical meaning.

The Difference is an explicit calculation of the difference between the two dates, with a precise statistical meaning. Just as a confidence interval in a classical statistical framework is more informative than a p-value, the credible interval on the difference is more informative. So a difference of -30 to 290 years at 95% probability tells me that there is no strong evidence to reject the difference of zero (which is assumed by Combine), but the difference might be more than a couple of hundred years (in which case Combine might lead me astray if I am working with high-precision results). Difference also has the advantage that because it is a Bayesian posterior probability distribution it can be manipulated further. Usually we only want to combine dates which are contemporary, but a zero year difference is not the only possible definition of contemporaneity. For example, it might be good enough for my purposes to define contemporaneity as dates within 25 years of one another (perhaps on the grounds that this is less than a human generation or the resolution of an established periodisation I will compare the results to). From the Difference probability distribution I can calculate the probability that the difference is within 0±25 years, and base my judgement on that. This approach allows me to evaluate the relative strengths of two hypotheses: contemporary within acceptable limits versus different. In the case of your two dates there is only ~15% chance that the difference is 0±25, and ~85% chance that is it larger. So the odds are about 5.8:1 that the difference is more than 25 years. With an A-index or p-value you could only have said that the data was consistent with the null hypothesis of no difference at 95% confidence, and nothing about the alternative hypothesis. That's one of the big advantages of Bayesian statistics over Classical statistics :-)

[Erik]

Fascinating, Andrew. I am learning a lot today. Good explanation.

... the 60% cut-off just like a p-value in the W&W test.

I thought W&W (1981) used 95%, like in most classic statistics. Why chose 60% or 95%?

 

From the Difference probability distribution I can calculate the probability that the difference is within 0±25 years, and base my judgement on that. This approach allows me to evaluate the relative strengths of two hypotheses: contemporary within acceptable limits versus different. In the case of your two dates there is only ~15% chance that the difference is 0±25, and ~85% chance that is it larger.

What's the OxCal code for this?

 

That's one of the big advantages of Bayesian statistics over Classical statistics

Very interesting.

I have another example that I'll put in a separate thread.

Thanks

Erik

[Pam Groves]

Hello,
I have a series of replicate dates for the same samples. Some have two
dates, some up to 5. Does OxCal have a way to tell if the replicate
dates are statistically the same or different?

Many thanks,
Pam
--
Pamela Groves, Institute of Arctic Biology, University of Alaska,
Fairbanks, AK 99775, 907-474-7165

[Bayliss, Alex]

Hi Pam,

If these are replicates on the same single entity (eg the same bone), then you want the R_COMBINE function of OxCal (testing if they are of the same radiocarbon age as described by Ward & Wilson 1978). If these are replicates on different pieces of charcoal or grains from the same feature, then you probably want the COMBINE function. This will give you an index of agreement that will help you assess whether the fragment are of the same date.

I suspect yours is the former case.

Best wishes,

Alex

--
You received this message because you are subscribed to the Google Groups "OxCal" group.
To unsubscribe from this group and stop receiving emails from it, send an email to oxcal+un...@googlegroups.com.
For more options, visit https://groups.google.com/d/optout.

English Heritage is changing into two organisations.

From Spring 2015, we shall become Historic England, a government service championing England's heritage and giving expert, constructive advice, and the English Heritage Trust, a charity caring for the National Heritage Collection of more than 400 historic properties and their collections.

This e-mail (and any attachments) is confidential and may contain personal views which are not the views of English Heritage unless specifically stated. If you have received it in error, please delete it from your system and notify the sender immediately. Do not use, copy or disclose the information in any way nor act in reliance on it. Any information sent to English Heritage may become publicly available.