Multivariate Regression using Skull Structures

Predicting Age of Adolescent Remains through Occipital Condyle Measurements

By:Paul Perrin

Justin Pierce

Grand Valley State UniversityMarch 21, 2013

Table of Contents

Abstract ………………………………. 3

Introduction ……………………………… 4

Terms & Definitions ……………………………… 4

Materials & Methods ……………………………… 5

Results ……………………………… 6

Discussion ……………………………… 7

Conclusion ……………………………… 8

Appendix ……………………………… 9

A.1: Occipital Shape Data …………. 9

A.2: Subject Demographics …………. 10

A.3: Delimited Coordinates with distances ….. 11

A.4: SAS Code Program …………. 12

A.5: SAS Output …………. 14

2

Abstract

In order to determine a new method for predicting the age of a human based on

their remains, the remains of 68 juveniles from the Hamann-Todd Collection in

Cleveland, Ohio underwent extensive metric analysis and documentation including

recorded known age at time of death.

As humans age, they undergo massive skull metamorphisms including changes in

a pair of oblong boney structures called Occipital Condyles. These condyles are typically

located inside the skull, one on either side of where the spinal cord attaches.

A procedure recorded markers on each Occipital Condyle within their skull using

a Microscribe 3D Digitizer. Using algebraic formulas to calculate length, height, and

width of each condyle, a multiple linear regression was used to model the recorded age of

death. The first model indicates a juvenile’s predicted age in years is:

Upon further investigation, the right condyle width and right condyle height were

correlated according to Pearson’s Correlation Coefficient of 0.48 so the model was

amended to not include the height variable with its higher P-value of 0.0951. The final

model generated that passed all diagnostic tests predicts age in years as:

with a significance value of less than 0.0001 and an R2 value of 0.1503.

.

3

Introduction

Identifying the age of specimens is a very important part of forensic sciences.

That is, given a skeleton, bones or a skull, it is very useful to be able to identify the age of

the specimen at time of death. Traditional methods of predicting age are based off of

dental information such as calcification levels in teeth. However, oftentimes this system

of identification is not always valid. Sometimes there aren’t any teeth in the specimen at

all. The goal of the project was to find a way to predict the age of a specimen based on

the size and shape of oblong bone structures in the skull called occipital condyles. The

theory is that these condyles change in size and shape in a predictable manner over time

in all human beings.

Terms & Definitions

When recording coordinate markers using the Microscribe 3D Digitizer, skulls

were individually positioned in the stand such that the anterior side of the skull (where

the facial features would be located) face straight up skywards. When relating in terms of

a mathematical 3 dimensional axis, this position would have the subject facing the

positive y-axis.

Once a baseline for orientation has been established, distances of each condyle in

millimeters are defined as follows.

Length: Distance between Anterior and Posterior markers

Width: Distance between Medial and Lateral markers perpendicular to length

Height: Greatest distance between any two different markers along the superior-inferior axis.

4

Materials & Methods

The remains of 68 juveniles from the Humann-Todd cadaver collection in

Cleveland, Ohio were used as a sample population due extensive documentation

including bone measurements and the known recorded age at the time of death. A table

showing the complete set of condyle marker measurements recorded can be seen in

Appendix A.1. Demographic and age variables for each observation are listed in

Appendix A.2.

Using coordinate markers collected from these 68 observations, the length and

width of each condyle can be calculated using the Pythagorean Theorem: a2 + b2 = c2

where a and b is the changes in x and z coordinates respectively, and c is the distance to

be found.

For finding width in millimeters, the expanded equation would be:

C =

For finding length in millimeters, the expanded equation would be:

C =

For finding the height, take the absolute value of the difference in y coordinates for every

pair of different condyle markers. The greatest value is what would be called the height.

A logical expression for finding the height would be:

MAX( |Anterior y – Posterior y|, |Anterior y – Lateral y|, |Anterior y – Medial y|, |Posterior y – Lateral y|, |Posterior y – Medial y|, |Lateral y – Medial y|)

5

A complete list of height, length, and width condyle measurements is available in Table

A.3 of the Appendix.

Once distance measurements have been collected for each condyle, Statistical

Analysis Software (SAS 9.3) is used to determine a multiple linear regression model to

predict recorded age of death for an observation using their length, width, and height

measurements of the left and right condyles using backwards selection method. All six

variables were included in the model for predicting age and dropped one by one if their

significance level was greater than 0.10. Through this method, the left condyle width was

first to be eliminated (p value = 0.688), followed by left condyle length (p value =

0.2629) and finally left condyle height (p value = 0.2071).

The three remaining variables: right condyle length, height, and width had p-

values less than the 0.1 level of significance. A copy of the SAS program code used to

conduct the analysis is shown in Appendix Figure A.4 while output produced by the code

is available in Appendix Figure A.5.

Results

Using backward selection process with a significance level of 0.10, the predicted

recorded age of remains in years was:

with an overall significance level of 0.0098 and an R2 value of 0.195073.A secondary

model generated using the same backward selection method but without incorporating the

right condyle height predicted the recorded age of remains in years as:

6

with an overall significance level of less than 0.0001 and an R2 value of 0.1503.

Discussion

While performing diagnostic tests to verify the first model developed, an issue

arose regarding variables that correlated with each other. In regression, correlated

variables are redundant and provide no further insight when predicting a variable

accurately. In the first model, the right width and height measurements were shown to be

correlated by Pearson’s correlation coefficient of 0.482. In statistics, a coefficient of 1 or

-1 suggests a strong correlation between two variables. When deciding which of the

correlating variables to eliminate, the one with the higher p-value (Right condyle height

had a p-value of 0.0951) should be dropped from the model.

When trying to compare how accurate a model is, the R2 figure reported by SAS

represents the proportion of all predicted values that can be explained using the

regression model. The first model generated in this study has an R2 value of 0.195 and the

secondary model has an R2 value of about 0.15. In order for a method of identifying the

age of human remains to be recognized and admissible in a court of law, a method must

be at least 80% accurate. Both models fail to meet that standard based on the low R2

values of 0.195 and 0.15.

One possible reason why both models have such a low accuracy is due to the

small sample size from which data was obtained from. The Hamann-Todd collection has

over 3,000 individuals with extensive documentation and from that large group, 68 were

sampled. Out of the selected 68, 4 samples were discarded due to having both left and

right condyles damaged. From the sample remaining, 8 observations had no

measurements taken of the left occipital condyle. Once more samples are included, than

7

the overall model will become more accurate since the linear model will have more data

points to fit a line with.

Conclusion

The linear model failed to give significant evidence to suggest a relationship

between condyle shape/size and age of a particular specimen. The data tell us now that

15% of the variation in condyle shape/size is accounted for by age. Although this was

not the outcome that was hoped for, the model still shows valuable information about

condyle shape/size and age.

There were also several limitations worth addressing in this study. All of our

specimens were from a population with little variability. For example, most of the

specimens were all of the same race (African American), and relatively small age range

(about 0-18). It is possible that with a larger more diverse sample, a more effective

model could have been created. Also, to come up with condyle shape and size

approximations, all there was to work with were four 3D coordinates per condyle. The

problem with this is it is impossible to tell the true shape and size of condyles by just

these coordinates. If further research were to be done, things like total volume and

surface area of condyles would we essential to creating an effective model.

8

Appendix

Table A.1: Occipital Shape DataID number Rt Occip condyle A Rt Occip Condyle P Rt Occipital Condyle L Rt Occipital Condyle M Lft Occipital Condyle A Lft Occipital Condyle P Left Occipital Condyle L Lft Occipital Condyle MHTH 0710 168.90 184.10 128.17 183.48 140.09 93.33 176.88 187.00 123.70 174.69 183.10 120.62 189.15 210.20 130.29 175.01 166.97 103.72 183.90 221.05 123.53 183.90 171.20 117.40HTH 0624 159.37 184.87 122.88 168.60 132.09 96.19 147.97 177.35 117.11 152.77 176.81 114.90 144.63 170.07 120.75 150.88 150.23 108.65 152.24 166.53 120.88 153.75 164.55 120.98HTH 0645 140.17 171.04 110.30 159.08 188.04 125.18 161.27 190.37 125.49 159.72 190.76 125.55HTH 0526 152.15 169.32 116.78 163.21 160.63 121.14 164.51 165.60 126.75 159.03 166.60 124.54 244.67 180.26 126.47 220.44 144.16 129.30 240.77 178.23 124.84 206.04 166.65 132.04 HTH 0632 197.53 110.47 112.79 200.78 106.94 95.05 203.90 120.16 108.55 210.71 116.24 105.91 185.78 115.22 113.37 210.05 113.36 100.69 196.22 130.99 119.22 189.03 128.49 118.87HTH 0633 X_117.14 Y_208.32 Z_136.17 X_156.62 Y_217.81 Z_141.06 137.20 211.17 141.28 135.92 181.32 139.51 121.81 199.98 120.15 154.77 209.78 138.76 158.84 219.62 139.23 160.10 226.85 141.04HTH 0527 110.63 202.75 111.73 124.67 165.19 115.29 136.64 212.11 119.03 132.56 212.16 121.01 210.90 189.40 121.58 240.22 166.10 130.94 240.87 166.04 130.56 220.17 218.06 126.91HTH 0245 178.71 179.96 56.37 200.79 100.94 55.00 202.45 108.72 54.96 196.29 112.27 53.50 197.92 120.25 51.95 200.37 104.83 44.32 197.01 112.27 49.07 194.21 116.20 43.00HTH 0485 137.34 206.21 130.25 156.10 188.89 130.72 152.21 201.97 129.05 147.02 200.03 133.10 194.34 159.19 132.72 200.65 145.55 131.27 199.78 178.01 131.32 196.47 173.53 138.61 HTH 0404 194.17 179.60 128.84 205.98 135.57 11.55 200.58 177.06 129.32 202.05 175.88 129.24 234.11 180.90 122.51 220.88 141.47 102.00 220.00 172.84 112.43 230.37 175.12 115.64HTH 1385 225.48 155.63 39.17 234.87 124.41 33.08 224.17 126.46 37.99 229.45 142.57 35.68 194.45 134.87 34.33 201.86 121.62 36.98 196.10 122.82 42.00 201.42 122.65 42.11HTH 1583 176.44 145.31 65.72 174.50 114.19 58.34 177.20 104.71 54.21 181.89 104.14 50.94 HTH 1557 170.57 166.17 124.58 202.64 119.82 102.38 187.61 142.84 112.62 179.75 141.01 115.68 175.51 178.70 130.42 197.98 145.20 119.62 173.98 181.20 130.12 182.70 178.57 129.76HTH 1074 177.28 187.70 42.25 184.66 163.41 45.17 187.52 170.30 40.84 185.80 172.24 40.71 220.32 127.26 136.42 223.04 141.23 133.59 217.75 137.32 133.23 221.07 127.20 135.54HTH 1156 210.46 192.97 118.46 229.81 201.88 109.45 214.20 208.79 112.61 212.97 202.08 112.16 209.26 188.15 29.10 203.81 200.46 19.72 201.71 193.63 24.52 208.95 198.71 22.12HTH 1115 149.30 159.80 60.85 175.27 92.06 46.68 175.57 103.95 52.59 182.69 110.2451.22 198.87 134.78 108.23 198.60 123.66 99.46 189.75 132.68 112.83 196.53 127.09 109.52HTH 1098 179.58 212.03 125.93 184.37 185.61 117.78 179.33 202.37 118.22 182.74 201.43 122.04 284.67 157.91 126.15 299.43 120.66 126.96 298.03 115.74 131.81 293.61 127.46 130.54HTH 1240 147.62 200.33 114.36 163.03 110.12 130.49 164.65 142.30 129.09 171.08 130.35 134.34 201.67 175.52 119.23 213.88 150.09 98.01 213.51 163.59 102.17 215.87 159.87 104.51HTH 0872 272.71 42.99 120.60 238.97 71.70 102.36 254.01 69.25 93.61 269.33 57.80 85.89 263.31 35.73 91.00 256.89 39.34 93.82 261.11 39.76 90.90 250.02 46.21 93.47HTH 0816 247.69 -51.62 37.59 234.87 -42.54 33.37 235.70 -44.31 36.06 259.42 -63.64 35.21 256.96 -58.31 34.99 237.28 -44.99 32.20 243.00 -47.48 34.31 244.11 -55.49 31.85HTH 1768 245.62 19.96 111.29 239.51 27.75 108.64 228.40 19.41 118.25 233.83 10.51 129.44HTH 1772 270.15 29.72 126.79 224.34 96.87 135.38 256.50 65.73 132.40 249.74 62.00 136.39 234.40 -37.29 142.66 203.85 -18.55 140.57 201.55 -63.61 141.32 195.86 -66.52 141.43HTH 1784 236.81 -59.01 33.76 222.55 -61.08 52.07 223.15 -71.72 35.83 224.70 -81.98 34.62 219.76 -79.98 132.25 214.71 -57.79 127.39 226.84 -78.02 131.03 223.33 -86.95 133.03HTH 2141 243.91 263.76 122.61 227.30 249.70 113.22 232.65 267.94 110.20 233.10 268.21 109.05 239.27 251.43 123.36 212.19 261.30 113.74 220.84 254.89 119.31 230.11 252.47 117.53HTH 2075 289.41 -8.45 95.07 277.13 1.78 81.20 331.02 -48.41 98.17 307.91 15.02 59.94HTH 2118 298.89 -16.34 117.92 284.80 -0.88 115.79 284.57 -7.24 121.17 288.89 -1.67 121.11 193.88 -43.49 121.63 198.27 -28.43 118.82 210.98 -23.50 122.30 198.54 -19.02 125.09HTH 2370 319.23 -59.68 31.83 306.63 -46.29 34.63 314.59 -65.86 31.05 296.88 -41.38 37.25 297.07 -47.27 37.70 273.08 -25.07 37.32 282.89 -29.89 37.37 278.87 -34.17 34.20HTH 2144 292.30 -19.15 108.92 291.83 -18.53 107.84 278.07 -16.99 98.57 288.08 -14.38 99.70 305.89 -2.66 99.76 300.03 3.34 80.30 306.14 -10.47 97.30 300.20 -13.16 98.37HTH 2074 331.98 -17.77 132.13 306.58 -15.71 127.57 307.02 -1.32 133.10 306.19 2.34 132.68 261.39 -6.89 135.66 265.08 -13.93 123.34 272.38 -9.59 129.46 272.65 -19.75 128.82HTH 1379 239.98 2.13 38.22 228.48 -2.55 34.76 225.15 -1.44 38.21 228.38 0.15 35.94 227.12 4.27 24.59 230.28 -5.56 22.75 233.99 -9.87 28.48 229.71 -8.53 24.45 HTH 1509 294.10 11.26 112.70 284.72 -48.45 96.52 299.78 -40.47 108.14 255.04 -35.19 114.59HTH 1441 321.58 8.59 98.25 294.71 16.75 88.07 318.47 -5.70 102.73 308.30 6.43 96.07 315.85 -4.84 109.07 321.10 11.58 93.12 319.94 15.01 100.65 309.77 10.35 99.25HTH 1168 277.74 -39.71 64.83 213.24 -13.28 62.82 212.95 -17.16 63.17 243.21 -27.37 63.79 279.01 -38.97 53.83 208.42 -23.47 45.58 204.82 -21.57 46.53 245.27 -35.81 49.44HTH 1453 248.81 -2.32 108.33 240.16 7.44 106.07 242.40 0.45 108.57 247.56 2.28 106.56 271.56 -24.98 103.64 255.89 12.24 92.01 262.57 -14.25 104.00 264.28 -9.72 99.82HTH 1232 318.47 -3.15 113.69 277.97 -19.26 127.20 308.40 -11.28 126.28 289.88 -3.60 135.58 242.34 -67.76 132.21 265.32 -35.57 127.76 273.81 -39.67 131.31 258.34 -36.77 134.83HTH 1867 220.96 8.26 29.22 225.99 -2.24 25.48 322.64 -47.73 29.70 314.09 -49.93 60.21HTH 1886 226.12 -7.62 35.11 217.06 -11.63 30.55 277.22 -56.44 38.28 246.62 -27.44 41.54 259.47 -42.77 44.03 238.05 -14.35 34.50 239.49 -13.54 40.54 245.28 -25.83 34.33HTH 1861 naHTH 1894 283.00 66.89 85.16 237.12 24.72 100.81 258.42 14.31 113.03 256.42 16.63 109.07 292.19 8.06 100.59 294.38 4.26 95.54 298.24 2.11 100.98 289.72 4.52 98.29HTH 1845 273.13 37.36 46.38 256.83 63.09 40.26 280.97 -26.17 43.29 276.54 29.04 42.93 292.18 18.38 51.86 268.63 37.01 34.71 275.00 21.46 33.60 276.38 3.49 37.19HTH 1688 332.79 67.08 128.63 331.12 52.31 128.01 302.42 67.31 136.90 305.83 65.20 137.92 278.64 34.72 142.06 325.22 10.42 141.39 315.06 -11.62 139.99 321.97 55.01 143.97HTH 2135 270.48 78.68 132.87 254.47 70.44 126.14 266.88 48.99 131.84 271.77 54.93 133.40 224.16 -9.63 136.23 242.98 -13.96 123.16 241.10 -6.21 131.82 234.40 -9.60 133.37HTH 3112 332.69 60.58 138.31 316.30 58.18 133.84 321.62 54.09 136.12 328.09 57.94 135.09 328.19 57.64 135.09 308.77 52.32 139.19 310.44 51.54 129.97 306.22 51.73 135.93HTH 3455 352.40 51.43 134.50 322.13 59.11 131.11 331.69 53.97 135.34 341.39 57.65 135.58 328.07 25.99 132.97 323.37 41.97 127.44 328.02 38.61 132.56 319.67 33.09 133.39HTH 3470 306.18 23.33 133.11 275.88 49.20 130.49 290.47 47.14 137.61 295.07 49.99 135.62 284.96 34.86 134.46 276.01 51.04 132.94 277.32 48.96 138.25 270.46 45.66 139.84HTH 1140 269.75 -25.19 141.48 251.62 -28.99 129.59 253.26 -31.22 135.06 259.25 -28.98 138.11 259.43 -21.29 140.32 254.93 -20.04 132.88 268.42 -37.21 132.53 257.04 -40.20 134.80HTH 0098 382.06 44.28 114.79 367.56 52.91 106.14 354.43 25.83 114.27 361.08 28.94 113.95 344.21 -7.02 109.62 333.93 47.63 81.76 342.08 52.60 84.82 333.67 52.82 88.79HTH 0437 328.01 52.58 121.76 298.99 87.04 132.95 328.70 54.64 119.10 324.98 56.69 120.71 265.41 6.73 118.05 290.64 45.26 121.85 292.52 18.84 117.23 287.80 14.71 118.14HTH 1041 313.77 38.41 130.71 228,31 191.60 129.07 256.74 91.07 129.78 255.97 96.02 130.53 248.51 40.43 123.499 256.07 63.01 118.48 259.26 41.40 118.32 255.21 39.79 122.90HTH 0548 307.37 -5.99 133.78 214.56 83.26 137.21 234.55 69.29 127.43 235.12 72.97 129.79 223.16 24.54 136.15 232.35 19.71 116.51 241.86 11.66 120.72 239.25 7.64 122.83HTH 1590 302.30 17.45 131.08 302.83 33.78 122.81 302.03 38.11 131.93 308.64 38.62 131.91 268.49 -0.83 131.06 273.73 10.74 126.79 270.71 -2.05 129.98 264.08 -7.313 131.31HTH 1606 339.09 32.13 130.11 313.07 55.04 127.94 328.55 45.41 130.99 334.90 49.83 132.58 318.98 -1.97 123.50 319.22 36.83 125.64 312.91 4.91 126.30 312.75 3.62 127.06HTH 1589 not availableHTH 1097 248.83 182.76 136.42 214.98 178.54 132.65 222.31 179.10 136.66 220.02 185.77 139.94 217.80 185.14 137.47 223.37 191.42 130.47 238.99 179.20 130.22 233.96 174.78 134.45HTH 1012 240.02 162.36 114.51 205.15 168.05 115.02 216.33 169.01 117.16 214.62 174.04 119.45 231.53 164.66 118.95 224.76 172.92 113.43 236.51 170.30 116.31 234.26 162.24 116.48HTH 1836 265.75 74.28 36.02 255.48 78.32 34.17 255.72 76.58 32.36 259.50 76.67 32.55 285.82 49.15 30.01 262.39 83.40 34.03 268.61 71.53 31.27 267. 62 69.98 26.81 HTH 1834 288.27 42.29 125.30 294.53 50.23 116.86 277.99 35.37 125.40 283.81 37.29 124.03 273.02 36.06 131.21 281.87 12.07 122.79 292.78 11.36 125.48 288.23 7.27 125.35HTH 1950 284.07 47.50 106.35 279.04 47.86 96.65 278.66 52.17 98.37 285.10 44.12 98.89 283.07 23.93 105.02 284.16 33.06 91.38 297.99 24.21 96.28 306.44 22.00 90.37HTH 1878 286.15 69.46 48.98 249.82 95.06 48.775 249.31 94.52 48.26 251.90 95.15 48.26HTH 2548 315.00 -16.76 112.24 303.93 -17.42 108.39 305.07 -12.85 109.15 302.39 -13.56 110.79 243.00 -60.83 119.66 241.44 -49.58 115.66 244.77 -56.94 116.85 247.90 -45.25 114.94HTH 2714 naHTH 1974 292.66 176.47 116.86 247.17 189.67 126.78 272.07 179.66 122.04 276.59 186.44 123.53 273.55 170.75 117.70 257.65 192.05 122.48 269.66 188.60 122.49 264.27 180.66 125.00HTH 0721 na 232.36 207.56 138.19 245.87 242.59 135.60 252.34 223.50 135.60 248.02 216.69 137.92HTH 1711 222.17 181.61 90.62 228.42 166.93 78.32 222.30 169.03 86.34 227.14 175.34 87.08HTH 0576 289.86 220.35 112.37 242.33 223.27 119.52 264.49 215.96 120.52 268.01 221.64 120.49 252.99 198.99 120.97 228.32 228.08 122.09 249.55 205.62 123.32 245.13 200.10 127.11HTH 0695 279.67 172.31 139.13 243.33 182.10 138.37 259.66 168.22 139.41 252.47 158.16 141.78 241.44 155.13 143.29 239.36 168.72 135.23 258.28 162.21 135.68 250.03 156.68 138.60HTH 2310 226.73 183.38 105.63 245.58 178.56 99.00 235.75 175.33 100.33 235.46 182.45 104.22 187.89 123.49 101.95 204.06 110.89 90.40 200.57 118.10 96.57 196.50 113.30 99.31HTH 0410 309.82 200.00 129.48 294.45 194.53 126.37 291.33 196.53 133.86 289.42 201.30 134.85 277.34 162.59 129.30 262.24 196.77 132.86 284.90 169.19 130.14 280.85 164.00 132.40

9

Table A.2: Subject DemographicsID number Recorded Age Dental Age Sex EthnicityHTH 0710 10 10 M BHTH 0624 6 6 F BHTH 0645 12 12 F WHTH 0526 11 10 F BHTH 0632 10 7 F BHTH 0633 14 10 F BHTH 0527 16 18 F WHTH 0245 0 6 mon M WHTH 0485 16 15 F BHTH 0404 11 10 M BHTH 1385 1 18 mon M BHTH 1583 1 9 mon M WHTH 1557 3 3 M BHTH 1074 4 4 F BHTH 1156 8 7 F BHTH 1115 5 5 F BHTH 1098 5 6 F BHTH 1240 12 12 F WHTH 0872 8 11 F BHTH 0816 0 9 mon M WHTH 1768 1 1 M BHTH 1772 12 11 F WHTH 1784 6 7 M BHTH 2141 4 4 F BHTH 2075 1 1.5 M BHTH 2118 13 11 F BHTH 2370 1 1 M BHTH 2144 6 6 M BHTH 2074 8 9 F BHTH 1379 1 6 mon M BHTH 1509 3 3 F BHTH 1441 10 10 M BHTH 1168 1 9 mon M BHTH 1453 0 6 mon F BHTH 1232 16 18 F BHTH 1867 0 0 M WHTH 1886 0 0.5 M BHTH 1861 0 0.5 F WHTH 1894 1 1 M BHTH 1845 0 8 fetal mon F WHTH 1688 10 11 M BHTH 2135 14 F BHTH 3112 15 15 M BHTH 3455 18 18 M BHTH 3470 18 18 M BHTH 1140 18 18 M BHTH 0098 18 18 M WHTH 0437 18 18 F WHTH 1041 17 17 F BHTH 0548 17 M BHTH 1590 18 18 F BHTH 1606 17 17 F BHTH 1589 17 15 M BHTH 1097 18 18 M BHTH 1012 18 20 F BHTH 1836 0 9 fetal mon M WHTH 1834 8 8 M BHTH 1950 4 4 M BHTH 1878 0 7 fetal mon M WHTH 2548 0 9 mon M BHTH 2714 1 1 F BHTH 1974 18 18 M BHTH 0721 18 18 M BHTH 1711 17 17 M BHTH 0576 16 18 F BHTH 0695 18 18 M BHTH 2310 15 15 M BHTH 0410 18 16 M W

10

Table A.3: Delimited Coordinates with DistancesID rocax rocay rocaz rocpx rocpy rocpz roclx rocly roclz rocmx rocmy rocmz locax locay locaz locpx locpy locpz loclx locly loclz locmx locmy locmz recage Rwidth Rlength Rheight Lwidth Llength Lheight

710 168.9 184.1 128.17 183.48 140.09 93.33 176.88 187 123.7 174.69 183.1 120.62 189.15 210.2 130.29 175.01 166.97 103.72 183.9 221.05 123.53 183.9 171.2 117.4 10 3.78 37.77 46.91 6.13 30.10 54.08624 159.37 184.87 122.88 168.6 132.09 96.19 147.97 177.35 117.11 152.77 176.81 114.9 144.63 170.07 120.75 150.88 150.23 108.65 152.24 166.53 120.88 153.75 164.55 120.98 6 5.28 28.24 52.78 1.51 13.62 19.84645 140.17 171.04 110.3 159.08 188.04 125.18 161.27 190.37 125.49 159.72 190.76 125.55 12 1.55 24.06 19.33526 152.15 169.32 116.78 163.21 160.63 121.14 164.51 165.6 126.75 159.03 166.6 124.54 244.67 180.26 126.47 220.44 144.16 129.3 240.77 178.23 124.84 206.04 166.65 132.04 11 5.91 11.89 8.69 35.47 24.39 36.1632 197.53 110.47 112.79 200.78 106.94 95.05 203.9 120.16 108.55 210.71 116.24 105.91 185.78 115.22 113.37 210.05 113.36 100.69 196.22 130.99 119.22 189.03 128.49 118.87 10 7.30 18.04 13.22 7.20 27.38 17.63633 117.14 208.32 136.17 156.62 217.81 141.06 137.2 211.17 141.28 135.92 181.32 139.51 121.81 199.98 120.15 154.77 209.78 138.76 158.84 219.62 139.23 160.1 226.85 141.04 14 2.18 39.78 36.49 2.21 37.85 19.64527 110.63 202.75 111.73 124.67 165.19 115.29 136.64 212.11 119.03 132.56 212.16 121.01 210.9 189.4 121.58 240.22 166.1 130.94 240.87 166.04 130.56 220.17 218.06 126.91 16 4.54 14.48 46.97 21.02 30.78 52.02245 178.71 179.96 56.37 200.79 100.94 55 202.45 108.72 54.96 196.29 112.27 53.5 197.92 120.25 51.95 200.37 104.83 44.32 197.01 112.27 49.07 194.21 116.2 43 0 6.33 22.12 79.02 6.68 8.01 15.42485 137.34 206.21 130.25 156.1 188.89 130.72 152.21 201.97 129.05 147.02 200.03 133.1 194.34 159.19 132.72 200.65 145.55 131.27 199.78 178.01 131.32 196.47 173.53 138.61 16 6.58 18.77 17.32 8.01 6.47 32.46404 194.17 179.6 128.84 205.98 135.57 11.55 200.58 177.06 129.32 202.05 175.88 129.24 234.11 180.9 122.51 220.88 141.47 102 220 172.84 112.43 230.37 175.12 115.64 11 1.47 117.88 44.03 10.86 24.41 39.43

1385 225.48 155.63 39.17 234.87 124.41 33.08 224.17 126.46 37.99 229.45 142.57 35.68 194.45 134.87 34.33 201.86 121.62 36.98 196.1 122.82 42 201.42 122.65 42.11 1 5.76 11.19 31.22 5.32 7.87 13.251583 176.44 145.31 65.72 174.5 114.19 58.34 177.2 104.71 54.21 181.89 104.14 50.94 1 5.72 7.63 40.61557 170.57 166.17 124.58 202.64 119.82 102.38 187.61 142.84 112.62 179.75 141.01 115.68 175.51 178.7 130.42 197.98 145.2 119.62 173.98 181.2 130.12 182.7 178.57 129.76 3 8.43 39.00 46.35 8.73 24.93 361074 177.28 187.7 42.25 184.66 163.41 45.17 187.52 170.3 40.84 185.8 172.24 40.71 220.32 127.26 136.42 223.04 141.23 133.59 217.75 137.32 133.23 221.07 127.2 135.54 4 1.72 7.94 24.29 4.04 3.93 14.031156 210.46 192.97 118.46 229.81 201.88 109.45 214.2 208.79 112.61 212.97 202.08 112.16 209.26 188.15 29.1 203.81 200.46 19.72 201.71 193.63 24.52 208.95 198.71 22.12 8 1.31 21.34 15.82 7.63 10.85 12.311115 149.3 159.8 60.85 175.27 92.06 46.68 175.57 103.95 52.59 182.69 110.24 51.22 198.87 134.78 108.23 198.6 123.66 99.46 189.75 132.68 112.83 196.53 127.09 109.52 5 7.25 29.58 67.74 7.54 8.77 11.121098 179.58 212.03 125.93 184.37 185.61 117.78 179.33 202.37 118.22 182.74 201.43 122.04 284.67 157.91 126.15 299.43 120.66 126.96 298.03 115.74 131.81 293.61 127.46 130.54 5 5.12 9.45 26.42 4.60 14.78 42.171240 147.62 200.33 114.36 163.03 110.12 130.49 164.65 142.3 129.09 171.08 130.35 134.34 201.67 175.52 119.23 213.88 150.09 98.01 213.51 163.59 102.17 215.87 159.87 104.51 12 8.30 22.31 90.21 3.32 24.48 25.43872 272.71 42.99 120.6 238.97 71.7 102.36 254.01 69.25 93.61 269.33 57.8 85.89 263.31 35.73 91 256.89 39.34 93.82 261.11 39.76 90.9 250.02 46.21 93.47 8 17.16 38.35 28.71 11.38 7.01 6.87816 247.69 -51.62 37.59 234.87 -42.54 33.37 235.7 -44.31 36.06 259.42 -63.64 35.21 256.96 -58.31 34.99 237.28 -44.99 32.2 243 -47.48 34.31 244.11 -55.49 31.85 0 23.74 13.50 21.1 2.70 19.88 13.32

1768 245.62 19.96 111.29 239.51 27.75 108.64 228.4 19.41 118.25 233.83 10.51 129.44 1 12.44 6.66 17.241772 270.15 29.72 126.79 224.34 96.87 135.38 256.5 65.73 132.4 249.74 62 136.39 234.4 -37.29 142.66 203.85 -18.55 140.57 201.55 -63.61 141.32 195.86 -66.52 141.43 12 7.85 46.61 67.15 5.69 30.62 47.971784 236.81 -59.01 33.76 222.55 -61.08 52.07 223.15 -71.72 35.83 224.7 -81.98 34.62 219.76 -79.98 132.25 214.71 -57.79 127.39 226.84 -78.02 131.03 223.33 -86.95 133.03 6 1.97 23.21 22.97 4.04 7.01 29.162141 243.91 263.76 122.61 227.3 249.7 113.22 232.65 267.94 110.2 233.1 268.21 109.05 239.27 251.43 123.36 212.19 261.3 113.74 220.84 254.89 119.31 230.11 252.47 117.53 4 1.23 19.08 18.51 9.44 28.74 9.872075 289.41 -8.45 95.07 277.13 1.78 81.2 331.02 -48.41 98.17 307.91 15.02 59.94 1 44.67 18.52 63.432118 298.89 -16.34 117.92 284.8 -0.88 115.79 284.57 -7.24 121.17 288.89 -1.67 121.11 193.88 -43.49 121.63 198.27 -28.43 118.82 210.98 -23.5 122.3 198.54 -19.02 125.09 13 4.32 14.25 15.46 12.75 5.21 19.992370 319.23 -59.68 31.83 306.63 -46.29 34.63 314.59 -65.86 31.05 296.88 -41.38 37.25 297.07 -47.27 37.7 273.08 -25.07 37.32 282.89 -29.89 37.37 278.87 -34.17 34.2 1 18.76 12.91 24.48 5.12 23.99 22.22144 292.3 -19.15 108.92 291.83 -18.53 107.84 278.07 -16.99 98.57 288.08 -14.38 99.7 305.89 -2.66 99.76 300.03 3.34 80.3 306.14 -10.47 97.3 300.2 -13.16 98.37 6 10.07 1.18 4.77 6.04 20.32 16.52074 331.98 -17.77 132.13 306.58 -15.71 127.57 307.02 -1.32 133.1 306.19 2.34 132.68 261.39 -6.89 135.66 265.08 -13.93 123.34 272.38 -9.59 129.46 272.65 -19.75 128.82 8 0.93 25.81 20.11 0.69 12.86 10.161379 239.98 2.13 38.22 228.48 -2.55 34.76 225.15 -1.44 38.21 228.38 0.15 35.94 227.12 4.27 24.59 230.28 -5.56 22.75 233.99 -9.87 28.48 229.71 -8.53 24.45 1 3.95 12.01 4.68 5.88 3.66 14.141509 294.1 11.26 112.7 284.72 -48.45 96.52 299.78 -40.47 108.14 255.04 -35.19 114.59 3 45.20 18.70 59.711441 321.58 8.59 98.25 294.71 16.75 88.07 318.47 -5.7 102.73 308.3 6.43 96.07 315.85 -4.84 109.07 321.1 11.58 93.12 319.94 15.01 100.65 309.77 10.35 99.25 10 12.16 28.73 22.45 10.27 16.79 19.851168 277.74 -39.71 64.83 213.24 -13.28 62.82 212.95 -17.16 63.17 243.21 -27.37 63.79 279.01 -38.97 53.83 208.42 -23.47 45.58 204.82 -21.57 46.53 245.27 -35.81 49.44 1 30.27 64.53 26.43 40.55 71.07 17.41453 248.81 -2.32 108.33 240.16 7.44 106.07 242.4 0.45 108.57 247.56 2.28 106.56 271.56 -24.98 103.64 255.89 12.24 92.01 262.57 -14.25 104 264.28 -9.72 99.82 0 5.54 8.94 9.76 4.52 19.51 37.221232 318.47 -3.15 113.69 277.97 -19.26 127.2 308.4 -11.28 126.28 289.88 -3.6 135.58 242.34 -67.76 132.21 265.32 -35.57 127.76 273.81 -39.67 131.31 258.34 -36.77 134.83 16 20.72 42.69 16.11 15.87 23.41 32.191867 220.96 8.26 29.22 225.99 -2.24 25.48 322.64 -47.73 29.7 314.09 -49.93 60.21 0 31.69 6.27 55.991886 226.12 -7.62 35.11 217.06 -11.63 30.55 277.22 -56.44 38.28 246.62 -27.44 41.54 259.47 -42.77 44.03 238.05 -14.35 34.5 239.49 -13.54 40.54 245.28 -25.83 34.33 0 30.77 10.14 48.82 8.49 23.44 29.231894 283 66.89 85.16 237.12 24.72 100.81 258.42 14.31 113.03 256.42 16.63 109.07 292.19 8.06 100.59 294.38 4.26 95.54 298.24 2.11 100.98 289.72 4.52 98.29 1 4.44 48.48 52.58 8.93 5.50 5.951845 273.13 37.36 46.38 256.83 63.09 40.26 280.97 -26.17 43.29 276.54 29.04 42.93 292.18 18.38 51.86 268.63 37.01 34.71 275 21.46 33.6 276.38 3.49 37.19 0 4.44 17.41 89.26 3.85 29.13 33.521688 332.79 67.08 128.63 331.12 52.31 128.01 302.42 67.31 136.9 305.83 65.2 137.92 278.64 34.72 142.06 325.22 10.42 141.39 315.06 -11.62 139.99 321.97 55.01 143.97 10 3.56 1.78 15 7.97 46.58 66.632135 270.48 78.68 132.87 254.47 70.44 126.14 266.88 48.99 131.84 271.77 54.93 133.4 224.16 -9.63 136.23 242.98 -13.96 123.16 241.1 -6.21 131.82 234.4 -9.6 133.37 14 5.13 17.37 29.69 6.88 22.91 7.753112 332.69 60.58 138.31 316.3 58.18 133.84 321.62 54.09 136.12 328.09 57.94 135.09 328.19 57.64 135.09 308.77 52.32 139.19 310.44 51.54 129.97 306.22 51.73 135.93 15 6.55 16.99 6.49 7.30 19.85 6.13455 352.4 51.43 134.5 322.13 59.11 131.11 331.69 53.97 135.34 341.39 57.65 135.58 328.07 25.99 132.97 323.37 41.97 127.44 328.02 38.61 132.56 319.67 33.09 133.39 18 9.70 30.46 7.68 8.39 7.26 15.983470 306.18 23.33 133.11 275.88 49.2 130.49 290.47 47.14 137.61 295.07 49.99 135.62 284.96 34.86 134.46 276.01 51.04 132.94 277.32 48.96 138.25 270.46 45.66 139.84 18 5.01 30.41 26.66 7.04 9.08 16.181140 269.75 -25.19 141.48 251.62 -28.99 129.59 253.26 -31.22 135.06 259.25 -28.98 138.11 259.43 -21.29 140.32 254.93 -20.04 132.88 268.42 -37.21 132.53 257.04 -40.2 134.8 18 6.72 21.68 6.03 11.60 8.70 20.16

98 382.06 44.28 114.79 367.56 52.91 106.14 354.43 25.83 114.27 361.08 28.94 113.95 344.21 -7.02 109.62 333.93 47.63 81.76 342.08 52.6 84.82 333.67 52.82 88.79 18 6.66 16.88 27.08 9.30 29.70 59.62437 328.01 52.58 121.76 298.99 87.04 132.95 328.7 54.64 119.1 324.98 56.69 120.71 265.41 6.73 118.05 290.64 45.26 121.85 292.52 18.84 117.23 287.8 14.71 118.14 18 4.05 31.10 34.46 4.81 25.51 38.53

1041 313.77 38.41 130.71 228.31 191.6 129.07 256.74 91.07 129.78 255.97 96.02 130.53 248.51 40.43 123.499 256.07 63.01 118.48 259.26 41.4 118.32 255.21 39.79 122.9 17 1.07 85.48 153.19 6.11 9.07 23.22548 307.37 -5.99 133.78 214.56 83.26 137.21 234.55 69.29 127.43 235.12 72.97 129.79 223.16 24.54 136.15 232.35 19.71 116.51 241.86 11.66 120.72 239.25 7.64 122.83 17 2.43 92.87 89.25 3.36 21.68 12.88

1590 302.3 17.45 131.08 302.83 33.78 122.81 302.03 38.11 131.93 308.64 38.62 131.91 268.49 -0.83 131.06 273.73 10.74 126.79 270.71 -2.05 129.98 264.08 -7.313 131.31 18 6.61 8.29 21.17 6.76 6.76 18.0531606 339.09 32.13 130.11 313.07 55.04 127.94 328.55 45.41 130.99 334.9 49.83 132.58 318.98 -1.97 123.5 319.22 36.83 125.64 312.91 4.91 126.3 312.75 3.62 127.06 17 6.55 26.11 22.91 0.78 2.15 38.81097 248.83 182.76 136.42 214.98 178.54 132.65 222.31 179.1 136.66 220.02 185.77 139.94 217.8 185.14 137.47 223.37 191.42 130.47 238.99 179.2 130.22 233.96 174.78 134.45 18 4.00 34.06 7.23 6.57 8.95 16.641012 240.02 162.36 114.51 205.15 168.05 115.02 216.33 169.01 117.16 214.62 174.04 119.45 231.53 164.66 118.95 224.76 172.92 113.43 236.51 170.3 116.31 234.26 162.24 116.48 18 2.86 34.87 11.68 2.26 8.74 10.681836 265.75 74.28 36.02 255.48 78.32 34.17 255.72 76.58 32.36 259.5 76.67 32.55 285.82 49.15 30.01 262.39 83.4 34.03 268.61 71.53 31.27 267.62 69.98 26.81 0 3.78 10.44 4.04 4.57 23.77 34.251834 288.27 42.29 125.3 294.53 50.23 116.86 277.99 35.37 125.4 283.81 37.29 124.03 273.02 36.06 131.21 281.87 12.07 122.79 292.78 11.36 125.48 288.23 7.27 125.35 8 5.98 10.51 14.86 4.55 12.22 24.71950 284.07 47.5 106.35 279.04 47.86 96.65 278.66 52.17 98.37 285.1 44.12 98.89 283.07 23.93 105.02 284.16 33.06 91.38 297.99 24.21 96.28 306.44 22 90.37 4 6.46 10.93 8.05 10.31 13.68 11.061878 286.15 69.46 48.98 249.82 95.06 48.775 249.31 94.52 48.26 251.9 95.15 48.26 0 2.59 36.33 25.62548 315 -16.76 112.24 303.93 -17.42 108.39 305.07 -12.85 109.15 302.39 -13.56 110.79 243 -60.83 119.66 241.44 -49.58 115.66 244.77 -56.94 116.85 247.9 -45.25 114.94 0 3.14 11.72 4.57 3.67 4.29 11.691974 292.66 176.47 116.86 247.17 189.67 126.78 272.07 179.66 122.04 276.59 186.44 123.53 273.55 170.75 117.7 257.65 192.05 122.48 269.66 188.6 122.49 264.27 180.66 125 18 4.76 46.56 13.2 5.95 16.60 21.3721 232.36 207.56 138.19 245.87 242.59 135.6 252.34 223.5 135.6 248.02 216.69 137.92 18 4.90 13.76 35.03

1711 222.17 181.61 90.62 228.42 166.93 78.32 222.3 169.03 86.34 227.14 175.34 87.08 17 4.90 13.80 14.68576 289.86 220.35 112.37 242.33 223.27 119.52 264.49 215.96 120.52 268.01 221.64 120.49 252.99 198.99 120.97 228.32 228.08 122.09 249.55 205.62 123.32 245.13 200.1 127.11 16 3.52 48.06 7.31 5.82 24.70 29.09695 279.67 172.31 139.13 243.33 182.1 138.37 259.66 168.22 139.41 252.47 158.16 141.78 241.44 155.13 143.29 239.36 168.72 135.23 258.28 162.21 135.68 250.03 156.68 138.6 18 7.57 36.35 23.94 8.75 8.32 13.59

2310 226.73 183.38 105.63 245.58 178.56 99 235.75 175.33 100.33 235.46 182.45 104.22 187.89 123.49 101.95 204.06 110.89 90.4 200.57 118.1 96.57 196.5 113.3 99.31 15 3.90 19.98 8.05 4.91 19.87 12.6410 309.82 200 129.48 294.45 194.53 126.37 291.33 196.53 133.86 289.42 201.3 134.85 277.34 162.59 129.3 262.24 196.77 132.86 284.9 169.19 130.14 280.85 164 132.4 18 2.15 15.68 6.77 4.64 15.51 34.18

11

A.4: SAS Code/*

Occipital Condyle MeasurementsBy: Justin PierceSTA 319 Project2/25/2013

*/

options nodate nonumber;

*Reads in Excel spreadsheet containing all condyle measurements;proc import datafile="N:\MY DOCUMENTS\Winter 2013\STA 319\Client\Occypital_Condyle_Data_Final.xls" out=condylemeas dbms=EXCEL97 replace; getnames=yes;run;

/* Displays All variables imported from excel */proc print data=condylemeas;title 'Data Import Test';run;

/* Generate a Multiple Linear Regression model using measurements of each condyle to predict recorded age using backward selection (alpha=0.1) */ proc reg;model recage = Rlength Rwidth Rheight Llength Lwidth Lheight / selection=backward;title 'Recorded Age model';run;

/* Checking Model Adequacy */proc glm;model recage = Rlength Rwidth Rheight;title 'Model Adequacy';run;

/* Residual Tests and Diagnostic Tools */ods graphics on;

proc glm plots=all;title 'Diagnostic Tests';model recage = Rlength Rwidth Rheight/ P ;output out = stat P=pred R=residual RSTUDENT=r1 DFFITS=diffits COOKD=cookd

H=hatvalue PRESS=res_del ;run;

ods graphics off;

12

/* Checking for Multicollinearity, correlation coefficent is close to 1 or -1 */proc corr;title 'Checking for Multicollinearity';var Rlength Rwidth Rheight;run;

/* Model Attempt #2: RHeight and RLength are correlated, dropping Rheight due to highest p value */proc reg;model recage = Rlength Rwidth;title 'Model #2';run;

/* Checking Model #2 Adequacy */proc glm;model recage = Rlength Rwidth;title 'Model #2 Adequacy';run;

/* Residual Tests and Diagnostic Tools for Model #2 */ods graphics on;

proc glm plots=all;title 'Diagnostic Tests for Model #2';model recage = Rlength Rwidth/ P ;output out = stat P=pred R=residual RSTUDENT=r1 DFFITS=diffits COOKD=cookd

H=hatvalue PRESS=res_del ;run;

ods graphics off;

quit;

13

A.5: SAS Output

Summary of Backward Elimination

Step VariableRemoved

Label NumberVars In

PartialR-Square

ModelR-Square

C(p) F Value Pr > F

1 Lwidth Lwidth 5 0.0025 0.2395 5.1647 0.16 0.6866

2 Llength Llength 4 0.0195 0.2200 4.4253 1.28 0.2629

3 Lheight Lheight 3 0.0250 0.1951 4.0399 1.63 0.2071

Model Adequacy

The GLM Procedure

Dependent Variable: recage recage

Source DF Sum of Squares Mean Square F Value Pr > F

Model 3 481.873250 160.624417 4.20 0.0098

Error 52 1988.341036 38.237328

Corrected Total 55 2470.214286

R-Square Coeff Var Root MSE recage Mean

0.195073 62.96064 6.183634 9.821429

Source DF Type I SS Mean Square F Value Pr > F

Rlength 1 190.1489154 190.1489154 4.97 0.0301

Rwidth 1 181.2441469 181.2441469 4.74 0.0340

Rheight 1 110.4801875 110.4801875 2.89 0.0951

Source DF Type III SS Mean Square F Value Pr > F

Rlength 1 276.6832081 276.6832081 7.24 0.0096

Rwidth 1 187.6878330 187.6878330 4.91 0.0311

Rheight 1 110.4801875 110.4801875 2.89 0.0951

Parameter Estimate Standard Error t Value Pr > |t|

Intercept 10.33513414 1.69614503 6.09 <.0001

Rlength 0.11690006 0.04345774 2.69 0.0096

14


Rwidth -0.28477666 0.12853758 -2.22 0.0311

Rheight -0.05726150 0.03368715 -1.70 0.0951

Diagnostic Tests

The GLM Procedure



Model 3 481.873250 160.624417 4.20 0.0098

Error 52 1988.341036 38.237328



0.195073 62.96064 6.183634 9.821429


Rlength 1 190.1489154 190.1489154 4.97 0.0301

Rwidth 1 181.2441469 181.2441469 4.74 0.0340

Rheight 1 110.4801875 110.4801875 2.89 0.0951


Rlength 1 276.6832081 276.6832081 7.24 0.0096

Rwidth 1 187.6878330 187.6878330 4.91 0.0311

Rheight 1 110.4801875 110.4801875 2.89 0.0951


Intercept 10.33513414 1.69614503 6.09 <.0001

Rlength 0.11690006 0.04345774 2.69 0.0096

Rwidth -0.28477666 0.12853758 -2.22 0.0311

Rheight -0.05726150 0.03368715 -1.70 0.0951

15

Sum of Residuals -0.000000

Sum of Squared Residuals 1988.341036

Sum of Squared Residuals - Error SS -0.000000

First Order Autocorrelation 0.318036

Durbin-Watson D 1.339962

16

Checking for Multicollinearity

The CORR Procedure

3 Variables: Rlength Rwidth Rheight

Simple Statistics

Variable N Mean Std Dev Sum Minimum Maximum Label

Rlength 56 27.43220 21.91396 1536 1.17784 117.88308 Rlength

Rwidth 56 6.99607 6.49442 391.77995 0.93022 30.77316 Rwidth

Rheight 56 30.18107 28.26751 1690 4.04000 153.19000 Rheight

Pearson Correlation Coefficients, N = 56 Prob > |r| under H0: Rho=0

Rlength Rwidth Rheight

Rlength

Rlength

1.00000

-0.04260

0.7552

0.48260

0.0002

Rwidth

Rwidth

-0.04260

0.7552

1.00000

-0.04060

0.7664

Rheight

Rheight

0.48260

0.0002

-0.04060

0.7664

1.00000

Model #2

The REG ProcedureModel: MODEL1


Number of Observations Read 65

Number of Observations Used 56

Number of Observations with Missing Values 9

Analysis of Variance

Source DF Sum ofSquares

MeanSquare

F Value Pr > F

Model 2 371.39306 185.69653 4.69 0.0133

17

Analysis of Variance

Source DF Sum ofSquares

MeanSquare

F Value Pr > F

Error 53 2098.82122 39.60040


Root MSE 6.29288 R-Square 0.1503

Dependent Mean 9.82143 Adj R-Sq 0.1183

Coeff Var 64.07301

Parameter Estimates

Variable Label DF ParameterEstimate

StandardError

t Value Pr > |t|

Intercept Intercept 1 9.54804 1.66054 5.75 <.0001

Rlength Rlength 1 0.08132 0.03876 2.10 0.0407

Rwidth Rwidth 1 -0.27977 0.13077 -2.14 0.0370

Model #2 Adequacy

The GLM Procedure



Model 2 371.393062 185.696531 4.69 0.0133

Error 53 2098.821223 39.600400



0.150349 64.07301 6.292885 9.821429


Rlength 1 190.1489154 190.1489154 4.80 0.0328

Rwidth 1 181.2441469 181.2441469 4.58 0.0370

18


Rlength 1 174.3307066 174.3307066 4.40 0.0407

Rwidth 1 181.2441469 181.2441469 4.58 0.0370


Intercept 9.548040859 1.66054345 5.75 <.0001

Rlength 0.081316603 0.03875628 2.10 0.0407

Rwidth -0.279772082 0.13077423 -2.14 0.0370

Diagnostic Tests for Model #2

The GLM Procedure



Model 2 371.393062 185.696531 4.69 0.0133

Error 53 2098.821223 39.600400



0.150349 64.07301 6.292885 9.821429


Rlength 1 190.1489154 190.1489154 4.80 0.0328

Rwidth 1 181.2441469 181.2441469 4.58 0.0370


Rlength 1 174.3307066 174.3307066 4.40 0.0407

Rwidth 1 181.2441469 181.2441469 4.58 0.0370


Intercept 9.548040859 1.66054345 5.75 <.0001

Rlength 0.081316603 0.03875628 2.10 0.0407

Rwidth -0.279772082 0.13077423 -2.14 0.0370

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Sum of Residuals 0.000000

Sum of Squared Residuals 2098.821223

Sum of Squared Residuals - Error SS -0.000000

First Order Autocorrelation 0.328108

Durbin-Watson D 1.313792

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Documents

Multivariate Regression using Skull Structures