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Computing in Archaeology

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Computing in Archaeology

Session 11. Correlation and regression analysis

© Richard Haddlesey www.medievalarchitecture.net

- To introduce correlation and regression techniques

- In correlation, we are always dealing with paired scores, and so values of the two variables taken together will be used to make a scattergram

- Quantities of New Forrest pottery recovered from sites at varying distances from the kilns

Here we can see that the quantity of pottery decreases as distance from the source increases

Here we see that the taller a pot, the wider the rim

Again the further from source, the less quantity of artefacts

Here we see the first molar increases with age and is then worn down as the animal gets older

- This shows us that scattergrams are the most important means of studying relationships between two variables

- Regression differs from other techniques we have looked at so far in that it is concerned not just with whether or not a relationship exists, or the strength of that relationship, but with its nature
- In regression analysis we use an independent variable to estimate (or predict) the values of a dependent variable

y = f(x)

- y = y axis (in this case the dependent
- f = function (of x)
- x = x axis

y = f(x)

y = x y = 2x y = x2

- y = a + bx
- Where y is the dependent variable, x is the independent variable, and the coefficients a and b are constants, i.e. they are fixed for a given data

- If x = 0 then the equation reduces to y = a, so a represents the point where the regression line crosses the y axis (the intercept)
- The b constant defines the slope of gradient of the regression line
- Thus for the pottery quantity in relation to distance from source, b represents the amount of decrease in pottery quantity from the source

y = a + bx

least-squares

least-squares

least-squares

least-squares

y = a + bx

y = a + bx

y = 102.64 – 1.8x

CORRELATION

CORRELATION

1 correlation coefficient

CORRELATION

1 correlation coefficient

2 significance

CORRELATION

- 1 correlation coefficient
- r

CORRELATION

- 1 correlation coefficient
- r
- -1 to +1

Levels of measurement:

- nominal – in name only
- ordinal – forming a sequence
- interval – a sequence with fixed distances
- ratio – fixed distances with a datum point

Levels of measurement:

- nominal
- ordinal
- interval
- ratio

Levels of measurement:

- nominal
- ordinal
- interval Product-Moment
- Correlation Coefficient
- ratio

Levels of measurement:

- nominal
- ordinal Spearman’s Rank
- Correlation Coefficient
- interval
- ratio

The Product-Moment

Correlation Coefficient

sample – 20 bronze spearheads

length (cm) width (cm)

n=20

r = nΣxy – (Σx)(Σy) g

√[nΣx2 – (Σx)2] [nΣy2 – (Σy)2]

length (cm) width (cm)

n=20

r = nΣxy – (Σx)(Σy) g

√[nΣx2 – (Σx)2] [nΣy2 – (Σy)2]

n=20

r = nΣxy – (Σx)(Σy) g

√[nΣx2 – (Σx)2] [nΣy2 – (Σy)2]

n=20

r = nΣxy – (Σx)(Σy) g= +0.67

√[nΣx2 – (Σx)2] [nΣy2 – (Σy)2]

n=20

Test of product moment correlation coefficient

Test of product moment correlation coefficient

H0 : true correlation coefficient = 0

Test of product moment correlation coefficient

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Test of product moment correlation coefficient

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables approximately random

Test of product moment correlation coefficient

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables approximately random

Sample statistics needed: n and r

Test of product moment correlation coefficient

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables approximately random

Sample statistics needed: n and r

Test statistic: TS = r

Test of product moment correlation coefficient

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables approximately random

Sample statistics needed: n and r

Test statistic: TS = r

Table: product moment correlation coefficient table.

n = 20

n = 20 r = 0.67 p<0.01

n = 20 r = 0.67 p<0.01

length (cm) width (cm)

Spearman’s Rank Correlation Coefficient (rs)

Spearman’s Rank Correlation Coefficient (rs)

H0 : true correlation coefficient = 0

Spearman’s Rank Correlation Coefficient (rs)

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Spearman’s Rank Correlation Coefficient (rs)

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables at least ordinal

Spearman’s Rank Correlation Coefficient (rs)

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables at least ordinal

Sample statistics needed: n and rs

Spearman’s Rank Correlation Coefficient (rs)

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables at least ordinal

Sample statistics needed: n and rs

Test statistic: TS = rs

Spearman’s Rank Correlation Coefficient (rs)

H0 : true correlation coefficient = 0

H1 : true correlation coefficient ≠ 0

Assumptions: both variables at least ordinal

Sample statistics needed: n and rs

Test statistic: TS = rs

Table: Spearman’s rankcorrelation coefficient table