Feiran Jiao andBarbara Monaco

DETERMINING EFFECTS OF HABITAT AND TIME ON THE DIETS OF ANCIENT NATIVE

AMERICANS

Specific Goals:• Identify significant spatial (geography, habitat) and

temporal variability in prey species use, as well as any association between time and space patterning

• Evaluate how well variability in data is explained by cultural (pop. changes, landuse practices, overhunting, new technology, etc.) and/or natural (climate change, vegetation diff erences) factors

We focused on the fi rst of Matt’s goals and looked at the variables Habitat (mountains, plains, alluvial valley) and Time Group (Paleoindian, Archaic, Woodlands, Late Prehistoric) to determine whether or not they significantly aff ect various responses regarding dietary habits.

CLIENT’S ORIGINAL GOALS

Simpson’s Index (D):[0,1]Measure of Evenness

where Margalef’s Diversity Index (0, ∞)

Simple species richness index that attempts to compensate forsampling eff ects

Large Body Size [0,1]Proportion of bones that were classifi ed as belonging to a large animal (mammoth, buff alo)

Our data initially contained measures on three types of Function Groups (Kil l , Camp, and Other). To match with Matt’s discipline, we eliminated the Kil l and Other sites, analyzing only the Camp sites.

RESPONSES

1

1

NN

nnD ii

)ln(

1

N

SDmg

S

DE D

)/1(/1

Our initial analysis was to perform a Two-Way ANOVA for the three responses.

However, as you can see from the residual plots, the assumptions for the ANOVA are violated.

INITIAL ANALYSIS

FINAL RESULTS

Since the assumptions for the Two-Way ANOVA using the original data were clearly violated we attempted to transform the data: Simpson’s Index:

Margalef’s Diversity Index:

We could find no transformations that improved the residual plots enough that we could trust the p-values and hypothesis performed by the Two-Way ANOVA for Large Body Size

)ln(D

Margalef

Anova Table (Type III tests)

Response: neg.ln.D Sum Sq Df F value Pr(>F) (Intercept) 4.452 1 18.3976 0.00003 as.factor(Time.Group) 4.956 3 6.8267 0.0002 as.factor(Habitat) 2.143 2 4.4283 0.0132 as.factor(Time.Group):as.factor(Habitat) 5.337 6 3.6757 0.0018 Residuals 43.799 181

We have a signifi cant interaction between Time Group and Habitat, thus our next step is to look at contrasts and interaction plots between these twelve group means.

TRANSFORMED SIMPSON’S INDEX

P a r a m e t e r E s t i m a t e E r r o r t V a l u e P r > | t |  v a l l e y P a l e o - m o u n t a i n P a l e o 0 . 2 7 8 1 4 8 4 1 0 . 1 7 5 3 2 3 8 8 1 . 5 9 0 . 1 1 4 4 v a l l e y P a l e o - p l a i n s P a l e o 0 . 8 2 2 0 7 4 1 9 0 . 2 1 9 9 9 2 5 6 3 . 7 4 0 . 0 0 0 2 m o u n t a i n P a l e o - p l a i n s P a l e o 0 . 5 4 3 9 2 5 7 8 0 . 1 7 5 3 2 3 8 8 3 . 1 0 0 . 0 0 2 2 v a l l e y A r c h a i c - m o u n t a i n A r c h a i c - 0 . 3 7 3 3 6 0 3 4 0 . 1 4 2 0 0 4 5 9 - 2 . 6 3 0 . 0 0 9 3 v a l l e y A r c h a i c - p l a i n s A r c h a i c - 0 . 6 5 4 7 6 4 6 6 0 . 3 6 0 0 4 7 3 4 - 1 . 8 2 0 . 0 7 0 6 m o u n t a i n A r c h a i c - p l a i n s A r c h a i c - 0 . 2 8 1 4 0 4 3 2 0 . 3 6 4 0 2 5 8 9 - 0 . 7 7 0 . 4 4 0 5 v a l l e y W o o d - m o u n t a i n W o o d - 0 . 3 5 7 3 5 9 1 3 0 . 3 8 4 5 5 0 2 4 - 0 . 9 3 0 . 3 5 4 0 v a l l e y W o o d - p l a i n s W o o d - 0 . 3 8 0 8 5 8 3 2 0 . 3 8 4 5 5 0 2 4 - 0 . 9 9 0 . 3 2 3 3 m o u n t a i n W o o d - p l a i n s W o o d - 0 . 0 2 3 4 9 9 1 9 0 . 4 9 1 9 1 8 3 2 - 0 . 0 5 0 . 9 6 2 0 v a l l e y L a t e - m o u n t a i n L a t e 0 . 1 8 4 9 0 3 4 1 0 . 1 5 0 9 4 4 2 7 1 . 2 2 0 . 2 2 2 2 v a l l e y L a t e - p l a i n s L a t e 0 . 2 9 3 5 5 6 8 0 0 . 1 5 0 9 4 4 2 7 1 . 9 4 0 . 0 5 3 3 m o u n t a i n L a t e - p l a i n s L a t e 0 . 1 0 8 6 5 3 3 9 0 . 1 8 5 9 2 7 6 5 0 . 5 8 0 . 5 5 9 7

Since we are making 12 comparisons, we need to adjust the cut-off alpha level using a Bonferroni correction. So we will take alpha and divide it by 12, so the new cut-off will be alpha=0.004167

Thus the only comparisons that are signifi cantly diff erent are Valley – Plains (Paleoindian), and Mountain-Plains(Paleoindian).

SAS CONTRASTS

Anova Table (Type III tests)

Response: sqrt.marg Sum Sq Df F value Pr(>F) (Intercept) 31.06 1 467.06 <0.001 as.factor(Time.Group) 0.24 3 1.19 0.314 as.factor(Habitat) 0.17 2 1.28 0.280 as.factor(Time.Group):as.factor(Habitat) 1.62 6 4.06 0.001Residuals 12.44 187

We have a signifi cant interaction between Time Group and Habitat, thus our next step is to look at contrasts and interaction plots between these twelve group means.

TRANSFORMED MARGALEF’S INDEX

S t a n d a r d P a r a m e t e r E s t i m a t e E r r o r t V a l u e P r > | t |  v a l l e y P a l e o - m o u n t a i n P a l e o 0 . 0 5 3 3 9 4 2 7 0 . 0 9 1 9 0 8 3 6 0 . 5 8 0 . 5 6 2 0 v a l l e y P a l e o - p l a i n s P a l e o 0 . 4 3 0 9 8 3 7 9 0 . 1 1 2 6 7 3 1 1 3 . 8 3 0 . 0 0 0 2 m o u n t a i n P a l e o - p l a i n s P a l e o 0 . 3 7 7 5 8 9 5 2 0 . 0 8 8 5 5 8 5 3 4 . 2 6 < . 0 0 0 1 v a l l e y A r c h a i c - m o u n t a i n A r c h a i c - 0 . 0 5 5 0 0 0 5 5 0 . 0 7 4 4 4 1 7 1 - 0 . 7 4 0 . 4 6 0 9 v a l l e y A r c h a i c - p l a i n s A r c h a i c - 0 . 2 8 7 6 0 7 7 2 0 . 1 8 8 7 4 4 1 8 - 1 . 5 2 0 . 1 2 9 2 m o u n t a i n A r c h a i c - p l a i n s A r c h a i c - 0 . 2 3 2 6 0 7 1 7 0 . 1 9 0 8 2 9 8 2 - 1 . 2 2 0 . 2 2 4 4 v a l l e y W o o d - m o u n t a i n W o o d - 0 . 4 7 3 8 0 0 3 3 0 . 1 9 8 2 2 9 0 5 - 2 . 3 9 0 . 0 1 7 8 v a l l e y W o o d - p l a i n s W o o d - 0 . 3 9 7 6 1 9 7 6 0 . 1 9 8 2 2 9 0 5 - 2 . 0 1 0 . 0 4 6 3 m o u n t a i n W o o d - p l a i n s W o o d 0 . 0 7 6 1 8 0 5 8 0 . 2 5 7 8 7 3 6 5 0 . 3 0 0 . 7 6 8 0 v a l l e y L a t e - m o u n t a i n L a t e - 0 . 0 9 6 5 1 6 5 1 0 . 0 7 8 7 1 1 7 7 - 1 . 2 3 0 . 2 2 1 7 v a l l e y L a t e - p l a i n s L a t e - 0 . 0 2 1 4 1 8 7 1 0 . 0 7 6 6 7 3 8 6 - 0 . 2 8 0 . 7 8 0 3 m o u n t a i n L a t e - p l a i n s L a t e 0 . 0 7 5 0 9 7 8 1 0 . 0 9 5 8 2 8 8 6 0 . 7 8 0 . 4 3 4 2 

Since we are making 12 comparisons, we need to adjust the cut-off alpha level using a Bonferroni correction. So we will take alpha and divide it by 12, so the new cut-off will be alpha=0.004167

Thus the only comparisons that are signifi cantly diff erent are Valley – Plains (Paleoindian), and Mountain-Plains(Paleoindian), which are the same signifi cant diff erence that we found for the transformed Simpson’s Index.

SAS CONTRASTS

Since the Large Body Size variable had such a large proportions of 0’s and 1’s, it appeared to follow a Beta Density.

We plotted the density histograms for each of the twelve group means (Habitat*Time Group) and fi tted a Beta density to the data.

We excluded the Woodlands since we had so few observations for each Habitat and felt that would not give an accurate fi t to the density. We also dropped Plains/Rolling Hills for a similar reason.

LARGE BODY SIZE

Based on the assumption that Y is Beta-distributed and the E(Y) is related to the covariates through a linear predictor with unknown coeffi cient and a link function.

Assume y’s are in the standard unit interval (0,1).For y [0,1],

Incorporates features such as skewness which is commonly observed in data taking values in the standard unit interval, such as rates or proportions.

R package ‘betareg’

BETA REGRESSION

Since Beta Regression is not a commonly used (if ever) in Matt’s discipline, if he decides to move on with the Large Body Size analysis we have proposed then this would require additional work and collaboration between him and Feiran.

NEXT STEPS

Hill, M. E. (2008). Variation in paleoindian fauna use on the great plains and rocky mountains of north america. Quaternary International, 191, 34-52.

Magurran, A. E. (2004). Measuring Biological Diversity . Blackwell Science Ltd.

REFERENCES

QUESTIONS?