Soundpost Questions, re: T&M

1. Equipment acceptable? Henri

2. Dated chronologies?Malcolm

3. t value? Paul

4. Instrument chronology? Paul

5. Anticipated date? Henri

6. t values acceptable? Paul

Dendro Dating ArtifactsSouthwestern Ruins (~violins)

Grand SW Houses

A.E. Douglass

• Collected living-tree samples– Dated

chronology• Collected ruins

samples– Undated

chronology• Connected the

two

Dating

• Living back to ~AD 1250

• 700 year “floating” chronology

• Separated by a gap

Gap

Piece

• Buried wood

The Bridging

• With skeleton plots

• No t values: who needs ‘em?

Raw width series not helpful for

dating

• Yes: Trends look similar

• But: Most tree series have that trend

• Geometric constraint of tree growth

0.0

0.5

1.0

1.5

2.0

2.5

0 20 40 60 80 100 120

Ring Number

Rin

g W

idth

(m

m)

Messiah Treble Messiah Bass

500

550

600

650

700

0 20 40 60 80 100 120

Ring Number

Are

a (m

m2 )

0

2

4

6

Wid

th (

mm

)

Area Incr. (mm2) Width Incr. (mm)

DetrendedSeries

• Standard series (blue) ok:

– No trend

– But has autocorrelation

0.6

0.8

1.0

1.2

1.4

1.6

0 20 40 60 80 100 120

Ring Number

Inde

xStandard: AR(1) = +0.24 Residual: AR(1) = +0.02

correlation = +0.96

t

tt fit

RWI

t1tt II

• Residual series (red) even better, best

– No trend

– No autocorrelation

Autocorrelation a Bad Thing for Dating• Undated tree sample with 290 rings • Dated chronology with almost 600 rings • “Significant” t values for 113 dating positions• With residual series, only one dated position

• ~ Standard deviation• Difference of two

rings divided by average

average

differenceySensitivitMean

n

2t 1tt

1tt

2)II(

IIySensitivitMean

n

2t 1tt

1tt

)II(

II2ySensitivitMean

Mean Sensitivity

• If two rings equal each other– Difference is 0– Sensitivity 0

• If one ring is 0: – Difference =

average– Sensitivity 2

Messiah MS• Quite low:

– Residual series: 0.148– SW chronologies 0.5

• Stats dating of Messiah will be difficult• Perhaps some other ring variable would

be better, e.g., density

0.10

0.15

0.20

0 20 40 60 80 100 120

Ring Number

Mea

n S

ensi

tivity

40-yr running ms

samplepooled

BsampleAsamples errordardtans

meanmeant

Question 3: What is the t value?

Sample BSample A

80

70

60

50

Sam

ple

A

Two Samples That Are Different

samplepooled

BsampleAsamples errordardtans

meanmeant

Question 3: What is the t value?

Sample BSample A

200

100

0

Sam

ple

A

Two Samples That Are Not Different

Question 3: What is the t value?

43210-1-2-3-4

500

400

300

200

100

0

t

Fre

quen

cyNormal Distribution

mean = 0, standard deviation = 1

2.5%2.5%

• t value is compared to Student’s distribution

• Large |t values| rare significant

• Adjusts correlation r values by sample size

• As r 0, t 0

• As r , t • As r 1, t skyrockets

• t varies with sqrt(n)

testinvaluesofnumbern

valuencorrelatiother

valuetfamousthet

:where

r1

2nrt

2

0

2

4

6

0 100 200 300 400 500

Number of Data

t va

lue

r = +0.25

Famous t value

• Messiah has 109 rings

• Not bad, not great either

• Multiple testing issue:– Do enough tests,

you’re bound to get significant results

• Alpha level should be adjusted

• Standard stats• As m , () , P

(penalty)

testedpositionsdatingofnumberm

chancebybigbeingtof

likelihoodthe,levelcesignificantable

levelcesignificancorrectedP

:where

)1(1P m

t Value Penalty

• Example:– 500-yr chron – 100-yr series– 50-yr minimum

overlap– 500 tests

• Significant t value (p = 0.0005) non-significance

• Worse with many chronologies

0.00

0.05

0.10

0.15

0.20

0.25

0 100 200 300 400 500

Number of Tests

P

alpha = 0.0005

t Value Penalty