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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.

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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


Regression equation

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


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


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

  • 2 significance


  • CORRELATION

    • 1 correlation coefficient

      • r

      • -1 to +1

  • 2 significance


  • 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


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