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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

Lecture aims

- To introduce correlation and regression techniques

The scattergram

- 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

example

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

Negative correlation

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

Positive correlation

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

Curvilinear monotonic relation

Again the further from source, the less quantity of artefacts

Arched relationship (non-monotonic)

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

scattergram

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

REGRESSION

- 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 = x y = 2x y = x2

General linear equations

- 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

Therefore:

- 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

1 correlation coefficient

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

- nominal
- ordinal
- interval
- ratio

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

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

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.

length (cm) width (cm)

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

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