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correlation and percentages. association between variables can be explored using counts are high counts of bone needles associated with high counts of end scrapers? similar questions can be asked using percent-standardized data

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Correlation and percentages
correlation and percentages

  • association between variables can be explored using counts

    • are high counts of bone needles associated with high counts of end scrapers?

  • similar questions can be asked using percent-standardized data

    • are high proportions of decorated pottery associated with high proportions of copper bells?


but…

  • these are different questions with different implications for formal regression

  • percents will show some correlation even if underlying counts do not…

    • ‘spurious’ correlation (negative)

    • “closed-sum” effect


10 vars.

5 vars.

3 vars.

2 vars.

matrix(round(rnorm(100, 50, 15), nrow=10)))


original counts

 %s (10 vars.)

 %s (5 vars.)

 %s (3 vars.)

 %s (2 vars.)


original counts

%s 10 vars.

%s 5 vars.

%s 3 vars.

%s 2 vars.





Convex hull trimming
“convex hull trimming” idea…

> hull1 chull(x, y)

> plot(x, y)

> polygon(x[hull1], y[hull1])

> abline(lm(y[-hull1] ~ x[-hull1]))



Transformation1
transformation idea…

  • at least two major motivations in regression analysis:

    • create/improve a linear relationship

    • correct skewed distribution(s)



LG_DENS idea… log(DENSITY)

old.par  par(no.readonly = TRUE)

plot(DIST, DENSITY, log="y")

par(old.par)


> VAR1T idea… sqrt(VAR1)> plot(VAR1T, VAR2)


Transformation summary
transformation summary idea…

  • correcting left skew:

    x4 stronger

    x3 strong

    x2 mild

  • correcting right skew:

    x weak

    log(x) mild

    -1/x strong

    -1/x2 stronger



  • regression/correlation idea…

    • the strength of a relationship can be assessed by seeing how knowledge of one variable improves the ability to predict the other


  • if you ignore idea…x, the best predictor of y will be the mean of all y values (y-bar)

    • if the y measurements are widely scattered, prediction errors will be greater than if they are close together

  • we can assess the dispersion of y values around their mean by:


r idea…2=

  • “coefficient of determination” (r2)

  • describes the proportion of variation that is “explained” or accounted for by the regression line…

  • r2=.5

     half of the variation is explained by the regression…

     half of the variation in y is explained by variation in x…


x idea…

“explaining variance”

range

vs.


vs. idea…



Residuals
residuals idea…

  • vertical deviations of points around the regression

    • for case i, residual = yi-ŷi [yi-(a+bxi)]

  • residuals in y should not show patterned variation either with x or y-hat

  • should be normally distributed around the regression line

  • residual error should not be autocorrelated (errors/residuals in y are independent…)


  • residuals idea…may show patterning with respect to other variables…

  • explore this with a residual scatterplot

    • ŷ vs. other variables…

  • are there suggestions of linear or other kinds of relationships?

  • if r2 < 1, some of the remaining variation may be explainable with reference to other variables


  • paying close attention to idea…outliers in a residual plot may lead to important insights

  • e.g.: outlying residuals from quantities of exotic flint ~ distance from quarries

    • sites with special access though transport routes, political alliances…

  • residuals from regressions are often the main payoff


Middle Formative, idea…

Basin of Mexico


Formative basin of mexico
Formative Basin of Mexico idea…

  • settlement survey

  • 3 variables recorded from sites:

    • site size (proxy for population)

    • amount of arable land in standard “catchment”

    • productivity index for soils


How are these variables related?

Do any make sense as dependent or independent variables?


SIZE ~ AGLAND idea…


(ha) idea…

(km2)

r2 = .75

y = 35.4 + .66x

SIZE = 35.38 + .66*AGLAND


residuals?? idea…


residual SIZE = SIZE – SIZE-hat idea…

> resSize  frmdat$size – (35.4 +.66 * frmdat$agland)


PROD & SIZE idea…

SIZE = -29 + 98 * PROD

r2 = .69


r idea…2 = .75

What have we “explained” about site size??

r2 = .69


r idea…2 = .55


X idea…0

X1

X2

multiple regression…


X idea…0

1

1 = total variance observed in independent variable (x0)


X idea…0

X1

variance in x0 explained by x1, by itself…

variance in x0 unexplained by x1…


X idea…0

X2

variance in x0 explained by x2, by itself…

variance in x0 unexplained by x2…


X idea…0

X1

(total variance in x0 explained by x1, that is not explained by x2…)

partial correlation coefficient:

proportion of variance in x0 explained by x1, that is not explained by x2…


multiple coefficient of determination idea…:

variance in x0 explained by x1 and x2, both separately, and together…


productivity idea…

agricultural land

SITE-SIZE


y idea… = -1.8 + .42x1 + 50x2

SIZE = -1.8 + .42*AGLAND + 50*PROD


Size 1 8 42 agland 50 prod
size = idea…-1.8 + .42*agland + 50*prod

  • various scales are involved:

    size  hectares

    agland  km2

    prod  productivity index

  • increasing available agricultural land by 1 km2 increases site-size by about .4 hectares

  • a 1-unit increase of soil productivity increases site-size by about 50 hectares

  • which of these two factors is more important??


  • calculate “ idea…beta” coefficients to eliminate the effect differing scales…

  • convert the variables to Z-scores

    • mean of 0

    • standard deviation of 1

  • repeat multiple correlation analysis…


with(frmdat, { idea…

Bsize  (size-mean(size))/sd(size)

Bagland  (agland-mean(agland))/sd(agland)

Bprod  (prod-mean(prod))/sd(prod) })

lmBeta  lm(Bsize ~ Bagland + Bprod)


should be zero… idea…

doesn’t change…

size = .55*agland + .43*prod


site size idea…

r2=.83

=.45

=.55

productivity

r2=.55

agricultural land


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