Thickness of SkullHao Chai

Dereck Shen

Skull dataset138 skulls from 10 regionsThickness was measured at 219 locations on

each skullOther variables in dataset:

Age of person (at time of death)Sex of personDimensions of the skull (length/width/height)

Goals of the studyTo determine what factors affect the

thickness of the skull. If such effects exist, where on the skull do we see it?

Hannah is particularly interested in: (1) Sex effect (2) Climate effect (warm/cold)

Challenges/issues Unbalanced dataCorrelation structureMultiple comparisons using Bonferroni

correction

Changes to the datasetTurned Age into a categorical variable:

Young (30 or younger)Mid-age (between 30 and 45)Old (45 or older)

Added a Climate factor derived from the regions:Warm (Australia, Egypt, etc.)Cold (Scandinavia, Northern Russia, etc.)China (can’t be classified as either warm or

cold)

Classical analysis: Model 1At each location, we fitted an additive

model with the following variables:Three factors: Sex, Age, RegionThree covariates: Length/Width/Height of skull

Obtained the p-value of each factor using the Anova() function from ‘car’ package

Determine significance by comparing p-value to the Bonferroni-adjusted threshold: .05/219

Anova() from ‘car’‘car’ stands for companion to applied

regression; it’s a package written as a companion to a textbook

Anova() gives us the anova table with Type III tests

We don’t use the anova() function in the base package because it only provides us with Type I tests

Sample output with Anova()> lm.out=lm(y~Sex + Age.fa + Region + Glabella.Opisthocranion + Vertex.Basion + Euryon.Euryon)> Anova(lm.out,type="III")Anova Table (Type III tests)

Response: y Sum Sq Df F value Pr(>F) (Intercept) 0.01384 1 0.9864 0.32262 Sex 0.03028 1 2.1591 0.14435 Age.fa 0.06297 2 2.2447 0.11039 Region 0.24197 9 1.9168 0.05572 .Glabella.Opisthocranion 0.00759 1 0.5410 0.46347 Vertex.Basion 0.00005 1 0.0035 0.95262 Euryon.Euryon 0.00595 1 0.4241 0.51613 Residuals 1.68315 120 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Classical analysis: Model 2Same as the first model, the only change is

we replace the Region factor with the Climate factor

Results Sex is not significant at any location Age is significant at certain locations Region is significant at certain locationsThe number of locations where Climate is

significant is fewer than that of Region, which suggests loss of information when we replace Region with Climate

Using pixel plots to show age effectAt each location, fit the previous additive

model without the Age factorThe residuals from the above model now

hold the age signalUsing the pixel plot function from

‘spastat’ package, we can plot the average of the residuals at each location on a 2-D map of the skull

We do this for each of the three age groups

Next stepUse permutation test to test for the effects