Bone Minerals Analysis

SAS Program

%Let Path=%Str(C:\temp);
ODS Listing Close;
ODS HTML File="&Path\BoneMineralsOutput.html";

Options PS=55 LS=80 PageNo=1;
GOptions NoPrompt;
Title1 "Bone Mineral Analysis Homework";

Data Bones;
 Subject+1;
 Input DominantRadius Radius 
       DominantHumerus Humerus 
       DominantUlna Ulna;
DataLines;
  1.103  1.052  2.139  2.238  0.873  0.872
  0.842  0.859  1.873  1.741  0.590  0.744
  0.925  0.873  1.887  1.809  0.767  0.713
  0.857  0.744  1.739  1.547  0.706  0.674
  0.795  0.809  1.734  1.715  0.549  0.654
  0.787  0.779  1.509  1.474  0.782  0.571
  0.933  0.880  1.695  1.656  0.737  0.803
  0.799  0.851  1.740  1.777  0.618  0.682
  0.945  0.876  1.811  1.759  0.853  0.777
  0.921  0.906  1.954  2.009  0.823  0.765
  0.792  0.825  1.624  1.657  0.686  0.668
  0.815  0.751  2.204  1.846  0.678  0.546
  0.755  0.724  1.508  1.458  0.662  0.595
  0.880  0.866  1.786  1.811  0.810  0.819
  0.900  0.838  1.902  1.606  0.723  0.677
  0.764  0.757  1.743  1.794  0.586  0.541
  0.733  0.748  1.863  1.869  0.672  0.752
  0.932  0.898  2.028  2.032  0.836  0.805
  0.856  0.786  1.390  1.324  0.578  0.610
  0.890  0.950  2.187  2.087  0.758  0.718
  0.688  0.532  1.650  1.378  0.533  0.482
  0.940  0.850  2.334  2.225  0.757  0.731
  0.493  0.616  1.037  1.268  0.546  0.615
  0.835  0.752  1.509  1.422  0.618  0.664
  0.915  0.936  1.971  1.869  0.869  0.868
;

Proc Print Data=Bones;
Run;

/*
 * 1. Compute the sample mean vector and sample covariance and 
 *    correlation matrices. Provide an interpretation of the correlations 
 *    that you observe. What do they tell you about the mineral content 
 *    in the bones? 
 */

Title2 "Question 1.";
Proc Corr Data=Bones Cov;
 Var DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna;
Run;

/*
 * 2. Assume that these data come from a single population. Are these 
 *    data consistent with the multivariate normal distribution model? 
 *    Why or why not? Give evidence to support your claims. 
 */

Title2 "Question 2.";
%Include "&Path\MNorPlt.sas";
%MNorPlt(Data=Bones,
   Vars=DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna,
   Analyze=B);

Proc GPlot Data=_PCA_;
 Plot PC2*PC1=1 PC1*PC2=2 / Overlay NoLegend;
 Symbol1 C=Black V=Dot I=None;
 Symbol2 C=Black V=None I=None;
Run;
Quit;

/*
 * 3. It is believed that the population from which these bone minerals 
 *    were determined differs from a standard reference population. 
 *    Using only the dominant radius, dominant humerus, and dominant ulna, 
 *    test the null hypothesis that the population means are 0.88, 1.68, 
 *    and 0.73, respectively, using both individual univariate tests and 
 *    then the multivariate Hotelling T-square test. State your conclusions 
 *    from each approach and draw conclusions in terms of the original 
 *    problem. Are the two analyses consistent with each other? If not, 
 *    how could they differ? [For this question and the ones that follow, 
 *    assume that the multivariate normal distribution model is a reasonable 
 *    one to use for these data -- note that you may not have reached that 
 *    conclusion based upon your analysis above]. 
 */

Title2 "Question 3.";
Data Hypothesis;
 Set Bones;
 R=DominantRadius-0.88;
 H=DominantHumerus-1.68;
 U=DominantUlna-0.73;
Run;

Proc GLM Data=Hypothesis;
 Model R H U =;
 MANOVA H=Intercept / PrintE PrintH;
Run;
Quit;

/*
 * 4. Construct box plots (I've given you code for PROC BOXPLOT below) 
 *    for each of the responses used in the T-square test above and 
 *    draw a line through the box plots at the value of each of the 
 *    individual null hypotheses. Also create a new variable that 
 *    corresponds with the linear combination of the responses that 
 *    most easily rejects the null hypothesis (the L-max vector or 
 *    Fisher's linear discrimiant function) and construct a box plot 
 *    for this new variable as well. Locate the null hypothesis on 
 *    this box plot as you did for the original variables. Do your 
 *    graphical findings support the test results from above? 
 *    Why or why not? 
 */

Title2 "Question 4.";
Data Dominant;
 Set Hypothesis;
 Length Bone $7;
 MineralContent=R;    Bone="Radius";  Order=1; Output;
 MineralContent=H;    Bone="Humerus"; Order=2; Output;
 MineralContent=U;    Bone="Ulna";    Order=3; Output;
 MineralContent=-2.08004682*R+0.93759368*H-0.21891940*U; 
                      Bone="LMax";    Order=4; Output;
 Drop DominantRadius DominantHumerus DominantUlna Radius Humerus Ulna;
Run;

Proc Sort Data=Dominant;
 By Order;
Run;

Proc Print Data=Dominant;
Run;

Proc BoxPlot Data=Dominant;
 Plot MineralContent*Bone / BoxStyle=SchematicId Notches 
                            VRef=0 VAxis=Axis1;
 Id Subject;
 Axis1 Label=(A=90 R=0 "Centered Mineral Content");
 Symbol1 C=Black V=Dot;
Run; 
Quit;
GOptions Reset=Symbol;

/*
 * 5. Construct individual univariate, simultaneous Bonferroni, and 
 *    simultaneous T-square 95% confidence intervals for the population 
 *    means for the dominant bones from which this sample was selected. 
 */

Title2 "Question 5.";
Title3 "Univariate Intervals";
Proc Means Data=Bones Mean StdErr CLM Alpha=0.05;
 Var DominantRadius DominantHumerus DominantUlna;
 Output Out=CIInfo
        N=N 
        Mean=DominantRadiusMean DominantHumerusMean DominantUlnaMean
		StdErr=DominantRadiusStdErr DominantHumerusStdErr DominantUlnaStdErr;
Run;

Title3 "Simultaneous Bonferroni Intervals";
Proc Means Data=Bones Mean StdErr CLM Alpha=%SysEvalF(0.05/3);
 Var DominantRadius DominantHumerus DominantUlna;
Run;

Title3 "Simultaneous T-Square Intervals";
Data TSquareIntervals;
 Length Bone $16;
 Set CIInfo;
 Retain p 3;
 ndf=p;
 ddf=(n-p);
 FCrit=FInv((1-0.05),ndf,ddf);
 m=sqrt((n-1)*p/(n-p)*FCrit);

 Bone="Dominant Radius";
 Mean=DominantRadiusMean;
 StdErr=DominantRadiusStdErr;
 LCL=Mean-m*StdErr;
 UCL=Mean+m*StdErr;
 Output;

 Bone="Dominant Humerus";
 Mean=DominantHumerusMean;
 StdErr=DominantHumerusStdErr;
 LCL=Mean-m*StdErr;
 UCL=Mean+m*StdErr;
 Output;

 Bone="Dominant Ulna";
 Mean=DominantUlnaMean;
 StdErr=DominantUlnaStdErr;
 LCL=Mean-m*StdErr;
 UCL=Mean+m*StdErr;
 Output;
 Keep Bone Mean StdErr LCL UCL FCrit m;
 Label LCL="95% LCL" UCL="95% UCL";
Run;

Proc Print Data=TSquareIntervals;
 Var Bone Mean StdErr LCL UCL FCrit m;
Run;

/*
 * 6. It is thought that exercise modifies bone mineral content, 
 *    and therefore, that the bones on the dominant side of a 
 *    person may have a different content from the bones on the 
 *    non-dominant side. Perform a paired T-square analysis of the 
 *    mineral content comparing the dominant to the non-dominant 
 *    bones. The null hypothesis would be that there are no differences 
 *    between the two sides. Provide an interpretation of your results 
 *    and draw conclusions in terms of the bone mineral problem. 
 */

Title2 "Question 6.";
Proc GLM Data=Bones;
 Model DominantRadius Radius 
       DominantHumerus Humerus 
       DominantUlna Ulna = / NoUni;
 MANOVA H=Intercept
        M=DominantRadius-Radius, 
          DominantHumerus-Humerus,
          DominantUlna-Ulna / PrintE PrintH Summary;
Run;
Quit;

ODS HTML Close;
ODS Listing;

 

Bone Mineral Analysis Homework

Obs	Subject	DominantRadius	Radius	DominantHumerus	Humerus	DominantUlna	Ulna
1	1	1.103	1.052	2.139	2.238	0.873	0.872
2	2	0.842	0.859	1.873	1.741	0.590	0.744
3	3	0.925	0.873	1.887	1.809	0.767	0.713
4	4	0.857	0.744	1.739	1.547	0.706	0.674
5	5	0.795	0.809	1.734	1.715	0.549	0.654
6	6	0.787	0.779	1.509	1.474	0.782	0.571
7	7	0.933	0.880	1.695	1.656	0.737	0.803
8	8	0.799	0.851	1.740	1.777	0.618	0.682
9	9	0.945	0.876	1.811	1.759	0.853	0.777
10	10	0.921	0.906	1.954	2.009	0.823	0.765
11	11	0.792	0.825	1.624	1.657	0.686	0.668
12	12	0.815	0.751	2.204	1.846	0.678	0.546
13	13	0.755	0.724	1.508	1.458	0.662	0.595
14	14	0.880	0.866	1.786	1.811	0.810	0.819
15	15	0.900	0.838	1.902	1.606	0.723	0.677
16	16	0.764	0.757	1.743	1.794	0.586	0.541
17	17	0.733	0.748	1.863	1.869	0.672	0.752
18	18	0.932	0.898	2.028	2.032	0.836	0.805
19	19	0.856	0.786	1.390	1.324	0.578	0.610
20	20	0.890	0.950	2.187	2.087	0.758	0.718
21	21	0.688	0.532	1.650	1.378	0.533	0.482
22	22	0.940	0.850	2.334	2.225	0.757	0.731
23	23	0.493	0.616	1.037	1.268	0.546	0.615
24	24	0.835	0.752	1.509	1.422	0.618	0.664
25	25	0.915	0.936	1.971	1.869	0.869	0.868

 

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Bone Mineral Analysis Homework

Question 1.

The CORR Procedure

6 Variables:

DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

 

Covariance Matrix, DF = 24
	DominantRadius	Radius	DominantHumerus	Humerus	DominantUlna	Ulna
DominantRadius	0.0130015833	0.0103784417	0.0223499750	0.0200856750	0.0091207083	0.0079578417
Radius	0.0103784417	0.0114178933	0.0185351900	0.0210995117	0.0085297833	0.0089085117
DominantHumerus	0.0223499750	0.0185351900	0.0803572267	0.0667761967	0.0168369250	0.0128470300
Humerus	0.0200856750	0.0210995117	0.0667761967	0.0694844733	0.0177354833	0.0167935983
DominantUlna	0.0091207083	0.0085297833	0.0168369250	0.0177354833	0.0115684167	0.0080711500
Ulna	0.0079578417	0.0089085117	0.0128470300	0.0167935983	0.0080711500	0.0105991400

 

Simple Statistics
Variable	N	Mean	Std Dev	Sum	Minimum	Maximum
DominantRadius	25	0.84380	0.11402	21.09500	0.49300	1.10300
Radius	25	0.81832	0.10685	20.45800	0.53200	1.05200
DominantHumerus	25	1.79268	0.28347	44.81700	1.03700	2.33400
Humerus	25	1.73484	0.26360	43.37100	1.26800	2.23800
DominantUlna	25	0.70440	0.10756	17.61000	0.53300	0.87300
Ulna	25	0.69384	0.10295	17.34600	0.48200	0.87200

 

Pearson Correlation Coefficients, N = 25 Prob > |r| under H0: Rho=0
	DominantRadius	Radius	DominantHumerus	Humerus	DominantUlna	Ulna
DominantRadius	1.00000	0.85181 <.0001	0.69146 0.0001	0.66826 0.0003	0.74369 <.0001	0.67789 0.0002
Radius	0.85181 <.0001	1.00000	0.61192 0.0012	0.74909 <.0001	0.74218 <.0001	0.80980 <.0001
DominantHumerus	0.69146 0.0001	0.61192 0.0012	1.00000	0.89365 <.0001	0.55222 0.0042	0.44020 0.0277
Humerus	0.66826 0.0003	0.74909 <.0001	0.89365 <.0001	1.00000	0.62555 0.0008	0.61882 0.0010
DominantUlna	0.74369 <.0001	0.74218 <.0001	0.55222 0.0042	0.62555 0.0008	1.00000	0.72889 <.0001
Ulna	0.67789 0.0002	0.80980 <.0001	0.44020 0.0277	0.61882 0.0010	0.72889 <.0001	1.00000

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Variable: DominantRadius

Moments
N	25	Sum Weights	25
Mean	0.8438	Sum Observations	21.095
Std Deviation	0.11402449	Variance	0.01300158
Skewness	-0.8447647	Kurtosis	3.22282639
Uncorrected SS	18.111999	Corrected SS	0.312038
Coeff Variation	13.5132123	Std Error Mean	0.0228049

 

Basic Statistical Measures
Location		Variability
Mean	0.843800	Std Deviation	0.11402
Median	0.856000	Variance	0.01300
Mode	.	Range	0.61000
		Interquartile Range	0.12900

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	37.00082	Pr > |t|	<.0001
Sign	M	12.5	Pr >= |M|	<.0001
Signed Rank	S	162.5	Pr >= |S|	<.0001

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	1.103
99%	1.103
95%	0.945
90%	0.940
75% Q3	0.921
50% Median	0.856
25% Q1	0.792
10%	0.733
5%	0.688
1%	0.493
0% Min	0.493

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
0.493	23	0.932	18
0.688	21	0.933	7
0.733	17	0.940	22
0.755	13	0.945	9
0.764	16	1.103	1

 

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The UNIVARIATE Procedure

Variable: DominantRadius

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Fitted Distribution for DominantRadius

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	0.8438
Std Dev	Sigma	0.114024

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.14739717	Pr > D	>0.150
Cramer-von Mises	W-Sq	0.07731306	Pr > W-Sq	0.221
Anderson-Darling	A-Sq	0.60241137	Pr > A-Sq	0.106

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	0.49300	0.57854
5.0	0.68800	0.65625
10.0	0.73300	0.69767
25.0	0.79200	0.76689
50.0	0.85600	0.84380
75.0	0.92100	0.92071
90.0	0.94000	0.98993
95.0	0.94500	1.03135
99.0	1.10300	1.10906

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Variable: Radius

Moments
N	25	Sum Weights	25
Mean	0.81832	Sum Observations	20.458
Std Deviation	0.10685454	Variance	0.01141789
Skewness	-0.5783573	Kurtosis	1.59636576
Uncorrected SS	17.01522	Corrected SS	0.27402944
Coeff Variation	13.0577944	Std Error Mean	0.02137091

 

Basic Statistical Measures
Location		Variability
Mean	0.818320	Std Deviation	0.10685
Median	0.838000	Variance	0.01142
Mode	.	Range	0.52000
		Interquartile Range	0.12400

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	38.29131	Pr > |t|	<.0001
Sign	M	12.5	Pr >= |M|	<.0001
Signed Rank	S	162.5	Pr >= |S|	<.0001

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	1.052
99%	1.052
95%	0.950
90%	0.936
75% Q3	0.876
50% Median	0.838
25% Q1	0.752
10%	0.724
5%	0.616
1%	0.532
0% Min	0.532

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
0.532	21	0.898	18
0.616	23	0.906	10
0.724	13	0.936	25
0.744	4	0.950	20
0.748	17	1.052	1

 

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The UNIVARIATE Procedure

Variable: Radius

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Fitted Distribution for Radius

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	0.81832
Std Dev	Sigma	0.106855

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.12336319	Pr > D	>0.150
Cramer-von Mises	W-Sq	0.06674410	Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.46193411	Pr > A-Sq	0.242

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	0.53200	0.56974
5.0	0.61600	0.64256
10.0	0.72400	0.68138
25.0	0.75200	0.74625
50.0	0.83800	0.81832
75.0	0.87600	0.89039
90.0	0.93600	0.95526
95.0	0.95000	0.99408
99.0	1.05200	1.06690

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Variable: DominantHumerus

Moments
N	25	Sum Weights	25
Mean	1.79268	Sum Observations	44.817
Std Deviation	0.2834735	Variance	0.08035723
Skewness	-0.4331181	Kurtosis	0.98938356
Uncorrected SS	82.271113	Corrected SS	1.92857344
Coeff Variation	15.8128334	Std Error Mean	0.0566947

 

Basic Statistical Measures
Location		Variability
Mean	1.792680	Std Deviation	0.28347
Median	1.786000	Variance	0.08036
Mode	1.509000	Range	1.29700
		Interquartile Range	0.30400

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	31.61989	Pr > |t|	<.0001
Sign	M	12.5	Pr >= |M|	<.0001
Signed Rank	S	162.5	Pr >= |S|	<.0001

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	2.334
99%	2.334
95%	2.204
90%	2.187
75% Q3	1.954
50% Median	1.786
25% Q1	1.650
10%	1.508
5%	1.390
1%	1.037
0% Min	1.037

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
1.037	23	2.028	18
1.390	19	2.139	1
1.508	13	2.187	20
1.509	24	2.204	12
1.509	6	2.334	22

 

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The UNIVARIATE Procedure

Variable: DominantHumerus

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Fitted Distribution for DominantHumerus

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	1.79268
Std Dev	Sigma	0.283474

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.09800357	Pr > D	>0.150
Cramer-von Mises	W-Sq	0.03527524	Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.24570882	Pr > A-Sq	>0.250

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	1.03700	1.13322
5.0	1.39000	1.32641
10.0	1.50800	1.42939
25.0	1.65000	1.60148
50.0	1.78600	1.79268
75.0	1.95400	1.98388
90.0	2.18700	2.15597
95.0	2.20400	2.25895
99.0	2.33400	2.45214

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Variable: Humerus

Moments
N	25	Sum Weights	25
Mean	1.73484	Sum Observations	43.371
Std Deviation	0.26359908	Variance	0.06948447
Skewness	0.11665025	Kurtosis	-0.4727338
Uncorrected SS	76.909373	Corrected SS	1.66762736
Coeff Variation	15.1944316	Std Error Mean	0.05271982

 

Basic Statistical Measures
Location		Variability
Mean	1.734840	Std Deviation	0.26360
Median	1.759000	Variance	0.06948
Mode	1.869000	Range	0.97000
		Interquartile Range	0.32200

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	32.90679	Pr > |t|	<.0001
Sign	M	12.5	Pr >= |M|	<.0001
Signed Rank	S	162.5	Pr >= |S|	<.0001

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	2.238
99%	2.238
95%	2.225
90%	2.087
75% Q3	1.869
50% Median	1.759
25% Q1	1.547
10%	1.378
5%	1.324
1%	1.268
0% Min	1.268

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
1.268	23	2.009	10
1.324	19	2.032	18
1.378	21	2.087	20
1.422	24	2.225	22
1.458	13	2.238	1

 

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The UNIVARIATE Procedure

Variable: Humerus

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Fitted Distribution for Humerus

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	1.73484
Std Dev	Sigma	0.263599

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.10539197	Pr > D	>0.150
Cramer-von Mises	W-Sq	0.03316777	Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.21982215	Pr > A-Sq	>0.250

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	1.26800	1.12162
5.0	1.32400	1.30126
10.0	1.37800	1.39702
25.0	1.54700	1.55705
50.0	1.75900	1.73484
75.0	1.86900	1.91263
90.0	2.08700	2.07266
95.0	2.22500	2.16842
99.0	2.23800	2.34806

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Variable: DominantUlna

Moments
N	25	Sum Weights	25
Mean	0.7044	Sum Observations	17.61
Std Deviation	0.10755657	Variance	0.01156842
Skewness	-0.0237368	Kurtosis	-1.2009741
Uncorrected SS	12.682126	Corrected SS	0.277642
Coeff Variation	15.2692468	Std Error Mean	0.02151131

 

Basic Statistical Measures
Location		Variability
Mean	0.704400	Std Deviation	0.10756
Median	0.706000	Variance	0.01157
Mode	0.618000	Range	0.34000
		Interquartile Range	0.16400

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	32.74556	Pr > |t|	<.0001
Sign	M	12.5	Pr >= |M|	<.0001
Signed Rank	S	162.5	Pr >= |S|	<.0001

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	0.873
99%	0.873
95%	0.869
90%	0.853
75% Q3	0.782
50% Median	0.706
25% Q1	0.618
10%	0.549
5%	0.546
1%	0.533
0% Min	0.533

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
0.533	21	0.823	10
0.546	23	0.836	18
0.549	5	0.853	9
0.578	19	0.869	25
0.586	16	0.873	1

 

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The UNIVARIATE Procedure

Variable: DominantUlna

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Fitted Distribution for DominantUlna

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	0.7044
Std Dev	Sigma	0.107557

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.10909879	Pr > D	>0.150
Cramer-von Mises	W-Sq	0.04181521	Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.32678370	Pr > A-Sq	>0.250

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	0.53300	0.45419
5.0	0.54600	0.52749
10.0	0.54900	0.56656
25.0	0.61800	0.63185
50.0	0.70600	0.70440
75.0	0.78200	0.77695
90.0	0.85300	0.84224
95.0	0.86900	0.88131
99.0	0.87300	0.95461

 

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Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Variable: Ulna

Moments
N	25	Sum Weights	25
Mean	0.69384	Sum Observations	17.346
Std Deviation	0.10295212	Variance	0.01059914
Skewness	-0.1421451	Kurtosis	-0.5498557
Uncorrected SS	12.289728	Corrected SS	0.25437936
Coeff Variation	14.838021	Std Error Mean	0.02059042

 

Basic Statistical Measures
Location		Variability
Mean	0.693840	Std Deviation	0.10295
Median	0.682000	Variance	0.01060
Mode	.	Range	0.39000
		Interquartile Range	0.15000

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	33.69722	Pr > |t|	<.0001
Sign	M	12.5	Pr >= |M|	<.0001
Signed Rank	S	162.5	Pr >= |S|	<.0001

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	0.872
99%	0.872
95%	0.868
90%	0.819
75% Q3	0.765
50% Median	0.682
25% Q1	0.615
10%	0.546
5%	0.541
1%	0.482
0% Min	0.482

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
0.482	21	0.803	7
0.541	16	0.805	18
0.546	12	0.819	14
0.571	6	0.868	25
0.595	13	0.872	1

 

---

 

The UNIVARIATE Procedure

Variable: Ulna

 

---

 

Bone Mineral Analysis Homework

Assessing Marginal Normality of DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

The UNIVARIATE Procedure

Fitted Distribution for Ulna

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	0.69384
Std Dev	Sigma	0.102952

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.06938699	Pr > D	>0.150
Cramer-von Mises	W-Sq	0.01912417	Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.13734321	Pr > A-Sq	>0.250

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	0.48200	0.45434
5.0	0.54100	0.52450
10.0	0.54600	0.56190
25.0	0.61500	0.62440
50.0	0.68200	0.69384
75.0	0.76500	0.76328
90.0	0.81900	0.82578
95.0	0.86800	0.86318
99.0	0.87200	0.93334

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Generalized Distance Information

XBAR
0.8438
0.81832
1.79268
1.73484
0.7044
0.69384

 

S
0.0130016	0.0103784	0.02235	0.0200857	0.0091207	0.0079578
0.0103784	0.0114179	0.0185352	0.0210995	0.0085298	0.0089085
0.02235	0.0185352	0.0803572	0.0667762	0.0168369	0.012847
0.0200857	0.0210995	0.0667762	0.0694845	0.0177355	0.0167936
0.0091207	0.0085298	0.0168369	0.0177355	0.0115684	0.0080712
0.0079578	0.0089085	0.012847	0.0167936	0.0080712	0.0105991

 

SINV
500.78034	-400.8874	-142.4433	143.66134	-99.36841	-18.34306
-400.8874	704.25683	118.25787	-173.2867	3.785007	-162.5968
-142.4433	118.25787	112.33741	-116.3159	2.3126965	53.922747
143.66134	-173.2867	-116.3159	154.11518	-14.5927	-54.30267
-99.36841	3.785007	2.3126965	-14.5927	249.6035	-98.32829
-18.34306	-162.5968	53.922747	-54.30267	-98.32829	340.33697

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Chi-squared QQ Plot Data

Obs	Squared Distance	Probability	Degrees of Freedom	Chi-squared Quantile
1	0.8477	0.02	6	1.1344
2	1.5703	0.06	6	1.7649
3	1.8969	0.10	6	2.2041
4	2.5604	0.14	6	2.5748
5	2.5998	0.18	6	2.9104
6	2.7417	0.22	6	3.2260
7	3.1587	0.26	6	3.5298
8	3.2081	0.30	6	3.8276
9	3.2543	0.34	6	4.1233
10	3.5693	0.38	6	4.4203
11	3.6075	0.42	6	4.7215
12	4.3740	0.46	6	5.0298
13	5.5291	0.50	6	5.3481
14	6.9308	0.54	6	5.6798
15	7.1699	0.58	6	6.0285
16	7.3446	0.62	6	6.3991
17	7.4222	0.66	6	6.7973
18	7.4592	0.70	6	7.2311
19	7.5112	0.74	6	7.7116
20	7.7145	0.78	6	8.2552
21	8.8661	0.82	6	8.8876
22	9.4023	0.86	6	9.6540
23	10.2828	0.90	6	10.6446
24	11.0572	0.94	6	12.0896
25	13.9215	0.98	6	15.0332

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Goodness of Fit of Chi-squared Distribution To Squared Distances

The UNIVARIATE Procedure

Variable: D2 (Squared Distance)

Moments
N	25	Sum Weights	25
Mean	5.76	Sum Observations	144
Std Deviation	3.37487326	Variance	11.3897695
Skewness	0.56454978	Kurtosis	-0.3038693
Uncorrected SS	1102.79447	Corrected SS	273.354469
Coeff Variation	58.5915497	Std Error Mean	0.67497465

 

Basic Statistical Measures
Location		Variability
Mean	5.760000	Std Deviation	3.37487
Median	5.529073	Variance	11.38977
Mode	.	Range	13.07387
		Interquartile Range	4.35254

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	8.533654	Pr > |t|	<.0001
Sign	M	12.5	Pr >= |M|	<.0001
Signed Rank	S	162.5	Pr >= |S|	<.0001

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	13.921544
99%	13.921544
95%	11.057204
90%	10.282787
75% Q3	7.511204
50% Median	5.529073
25% Q1	3.158665
10%	1.896947
5%	1.570318
1%	0.847677
0% Min	0.847677

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
0.847677	1	8.86614	21
1.570318	2	9.40230	22
1.896947	3	10.28279	23
2.560384	4	11.05720	24
2.599778	5	13.92154	25

 

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The UNIVARIATE Procedure

Variable: D2 (Squared Distance)

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Goodness of Fit of Chi-squared Distribution To Squared Distances

The UNIVARIATE Procedure

Fitted Distribution for D2

Parameters for Gamma Distribution
Parameter	Symbol	Estimate
Threshold	Theta	0
Scale	Sigma	2
Shape	Alpha	3
Mean		6
Std Dev		3.464102

 

Goodness-of-Fit Tests for Gamma Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.16962051	Pr > D	>0.250
Cramer-von Mises	W-Sq	0.12105067	Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.63315742	Pr > A-Sq	>0.250

 

Quantiles for Gamma Distribution
Percent	Quantile
	Observed	Estimated
1.0	0.84768	0.87209
5.0	1.57032	1.63538
10.0	1.89695	2.20413
25.0	3.15867	3.45460
50.0	5.52907	5.34812
75.0	7.51120	7.84080
90.0	10.28279	10.64464
95.0	11.05720	12.59159
99.0	13.92154	16.81189

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Principal Components with Largest and Smallest Eigenvalues

The UNIVARIATE Procedure

Variable: PC1

Moments
N	25	Sum Weights	25
Mean	0	Sum Observations	0
Std Deviation	0.40440634	Variance	0.16354449
Skewness	-0.234514	Kurtosis	0.14764425
Uncorrected SS	3.92506773	Corrected SS	3.92506773
Coeff Variation	.	Std Error Mean	0.08088127

 

Basic Statistical Measures
Location		Variability
Mean	0.000000	Std Deviation	0.40441
Median	0.009020	Variance	0.16354
Mode	.	Range	1.67983
		Interquartile Range	0.47892

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	0	Pr > |t|	1.0000
Sign	M	0.5	Pr >= |M|	1.0000
Signed Rank	S	1.5	Pr >= |S|	0.9688

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	0.71713982
99%	0.71713982
95%	0.71426771
90%	0.53856893
75% Q3	0.29882826
50% Median	0.00902024
25% Q1	-0.18009561
10%	-0.47876763
5%	-0.57099609
1%	-0.96268614
0% Min	-0.96268614

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
-0.962686	23	0.349464	10
-0.570996	19	0.423277	18
-0.478768	21	0.538569	20
-0.428628	13	0.714268	1
-0.424618	24	0.717140	22

 

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The UNIVARIATE Procedure

Variable: PC1

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Principal Components with Largest and Smallest Eigenvalues

The UNIVARIATE Procedure

Fitted Distribution for PC1

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	0
Std Dev	Sigma	0.404406

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.08803946	Pr > D	>0.150
Cramer-von Mises	W-Sq	0.02994813	Pr > W-Sq	>0.250
Anderson-Darling	A-Sq	0.19797707	Pr > A-Sq	>0.250

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	-0.96269	-0.940790
5.0	-0.57100	-0.665189
10.0	-0.47877	-0.518268
25.0	-0.18010	-0.272768
50.0	0.00902	-0.000000
75.0	0.29883	0.272768
90.0	0.53857	0.518268
95.0	0.71427	0.665189
99.0	0.71714	0.940790

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Principal Components with Largest and Smallest Eigenvalues

The UNIVARIATE Procedure

Variable: PC6

Moments
N	25	Sum Weights	25
Mean	0	Sum Observations	0
Std Deviation	0.02975919	Variance	0.00088561
Skewness	0.24773516	Kurtosis	-0.5433817
Uncorrected SS	0.02125463	Corrected SS	0.02125463
Coeff Variation	.	Std Error Mean	0.00595184

 

Basic Statistical Measures
Location		Variability
Mean	0.00000	Std Deviation	0.02976
Median	-0.01074	Variance	0.0008856
Mode	.	Range	0.11888
		Interquartile Range	0.04067

 

Tests for Location: Mu0=0
Test	Statistic		p Value
Student's t	t	0	Pr > |t|	1.0000
Sign	M	-1.5	Pr >= |M|	0.6900
Signed Rank	S	0.5	Pr >= |S|	0.9896

 

Quantiles (Definition 5)
Quantile	Estimate
100% Max	0.0643279
99%	0.0643279
95%	0.0413061
90%	0.0364342
75% Q3	0.0243986
50% Median	-0.0107412
25% Q1	-0.0162677
10%	-0.0366287
5%	-0.0452527
1%	-0.0545506
0% Min	-0.0545506

 

Extreme Observations
Lowest		Highest
Value	Obs	Value	Obs
-0.0545506	21	0.0331407	6
-0.0452527	22	0.0337184	12
-0.0366287	1	0.0364342	15
-0.0280514	24	0.0413061	25
-0.0263570	4	0.0643279	20

 

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The UNIVARIATE Procedure

Variable: PC6

 

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Bone Mineral Analysis Homework

Assessing Multivariate Normality of Bones for DominantRadius Radius DominantHumerus Humerus DominantUlna Ulna

Principal Components with Largest and Smallest Eigenvalues

The UNIVARIATE Procedure

Fitted Distribution for PC6

Parameters for Normal Distribution
Parameter	Symbol	Estimate
Mean	Mu	0
Std Dev	Sigma	0.029759

 

Goodness-of-Fit Tests for Normal Distribution
Test	Statistic		p Value
Kolmogorov-Smirnov	D	0.16092671	Pr > D	0.093
Cramer-von Mises	W-Sq	0.07950196	Pr > W-Sq	0.208
Anderson-Darling	A-Sq	0.40610100	Pr > A-Sq	>0.250

 

Quantiles for Normal Distribution
Percent	Quantile
	Observed	Estimated
1.0	-0.05455	-0.069230
5.0	-0.04525	-0.048950
10.0	-0.03663	-0.038138
25.0	-0.01627	-0.020072
50.0	-0.01074	-0.000000
75.0	0.02440	0.020072
90.0	0.03643	0.038138
95.0	0.04131	0.048950
99.0	0.06433	0.069230

 

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Bone Mineral Analysis Homework

Question 3.

The GLM Procedure

Number of observations

25

 

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Bone Mineral Analysis Homework

Question 3.

The GLM Procedure

Dependent Variable: R

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	0.03276100	0.03276100	2.52	0.1255
Error	24	0.31203800	0.01300158
Uncorrected Total	25	0.34479900

 

R-Square	Coeff Var	Root MSE	R Mean
0.000000	-314.9848	0.114024	-0.036200

 

Source	DF	Type I SS	Mean Square	F Value	Pr > F
Intercept	1	0.03276100	0.03276100	2.52	0.1255

 

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Intercept	1	0.03276100	0.03276100	2.52	0.1255

 

Parameter	Estimate	Standard Error	t Value	Pr > |t|
Intercept	-.0362000000	0.02280490	-1.59	0.1255

 

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Bone Mineral Analysis Homework

Question 3.

The GLM Procedure

Dependent Variable: H

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	0.31741956	0.31741956	3.95	0.0584
Error	24	1.92857344	0.08035723
Uncorrected Total	25	2.24599300

 

R-Square	Coeff Var	Root MSE	H Mean
0.000000	251.5739	0.283474	0.112680

 

Source	DF	Type I SS	Mean Square	F Value	Pr > F
Intercept	1	0.31741956	0.31741956	3.95	0.0584

 

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Intercept	1	0.31741956	0.31741956	3.95	0.0584

 

Parameter	Estimate	Standard Error	t Value	Pr > |t|
Intercept	0.1126800000	0.05669470	1.99	0.0584

 

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Bone Mineral Analysis Homework

Question 3.

The GLM Procedure

Dependent Variable: U

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	0.01638400	0.01638400	1.42	0.2457
Error	24	0.27764200	0.01156842
Uncorrected Total	25	0.29402600

 

R-Square	Coeff Var	Root MSE	U Mean
0.000000	-420.1429	0.107557	-0.025600

 

Source	DF	Type I SS	Mean Square	F Value	Pr > F
Intercept	1	0.01638400	0.01638400	1.42	0.2457

 

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Intercept	1	0.01638400	0.01638400	1.42	0.2457

 

Parameter	Estimate	Standard Error	t Value	Pr > |t|
Intercept	-.0256000000	0.02151131	-1.19	0.2457

 

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Bone Mineral Analysis Homework

Question 3.

The GLM Procedure

Multivariate Analysis of Variance

E = Error SSCP Matrix
	R	H	U
R	0.312038	0.5363994	0.218897
H	0.5363994	1.92857344	0.4040862
U	0.218897	0.4040862	0.277642

 

Partial Correlation Coefficients from the Error SSCP Matrix / Prob > |r|
DF = 24	R	H	U
R	1.000000	0.691459 0.0001	0.743693 <.0001
H	0.691459 0.0001	1.000000	0.552222 0.0042
U	0.743693 <.0001	0.552222 0.0042	1.000000

 

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Bone Mineral Analysis Homework

Question 3.

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Intercept
	R	H	U
R	0.032761	-0.1019754	0.023168
H	-0.1019754	0.31741956	-0.0721152
U	0.023168	-0.0721152	0.016384

 

Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix
Characteristic Root	Percent	Characteristic Vector V'EV=1
		R	H	U
0.87002337	100.00	-2.08004682	0.93759368	-0.21891940
0.00000000	0.00	-2.02837266	-0.00658930	2.83924250
0.00000000	0.00	1.08100957	0.34728919	0.00000000

 

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix S=1 M=0.5 N=10
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.53475268	6.38	3	22	0.0028
Pillai's Trace	0.46524732	6.38	3	22	0.0028
Hotelling-Lawley Trace	0.87002337	6.38	3	22	0.0028
Roy's Greatest Root	0.87002337	6.38	3	22	0.0028

 

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Bone Mineral Analysis Homework

Question 4.

Obs	Subject	R	H	U	Bone	MineralContent	Order
1	1	0.223	0.459	0.143	Radius	0.22300	1
2	2	-0.038	0.193	-0.140	Radius	-0.03800	1
3	3	0.045	0.207	0.037	Radius	0.04500	1
4	4	-0.023	0.059	-0.024	Radius	-0.02300	1
5	5	-0.085	0.054	-0.181	Radius	-0.08500	1
6	6	-0.093	-0.171	0.052	Radius	-0.09300	1
7	7	0.053	0.015	0.007	Radius	0.05300	1
8	8	-0.081	0.060	-0.112	Radius	-0.08100	1
9	9	0.065	0.131	0.123	Radius	0.06500	1
10	10	0.041	0.274	0.093	Radius	0.04100	1
11	11	-0.088	-0.056	-0.044	Radius	-0.08800	1
12	12	-0.065	0.524	-0.052	Radius	-0.06500	1
13	13	-0.125	-0.172	-0.068	Radius	-0.12500	1
14	14	0.000	0.106	0.080	Radius	0.00000	1
15	15	0.020	0.222	-0.007	Radius	0.02000	1
16	16	-0.116	0.063	-0.144	Radius	-0.11600	1
17	17	-0.147	0.183	-0.058	Radius	-0.14700	1
18	18	0.052	0.348	0.106	Radius	0.05200	1
19	19	-0.024	-0.290	-0.152	Radius	-0.02400	1
20	20	0.010	0.507	0.028	Radius	0.01000	1
21	21	-0.192	-0.030	-0.197	Radius	-0.19200	1
22	22	0.060	0.654	0.027	Radius	0.06000	1
23	23	-0.387	-0.643	-0.184	Radius	-0.38700	1
24	24	-0.045	-0.171	-0.112	Radius	-0.04500	1
25	25	0.035	0.291	0.139	Radius	0.03500	1
26	1	0.223	0.459	0.143	Humerus	0.45900	2
27	2	-0.038	0.193	-0.140	Humerus	0.19300	2
28	3	0.045	0.207	0.037	Humerus	0.20700	2
29	4	-0.023	0.059	-0.024	Humerus	0.05900	2
30	5	-0.085	0.054	-0.181	Humerus	0.05400	2
31	6	-0.093	-0.171	0.052	Humerus	-0.17100	2
32	7	0.053	0.015	0.007	Humerus	0.01500	2
33	8	-0.081	0.060	-0.112	Humerus	0.06000	2
34	9	0.065	0.131	0.123	Humerus	0.13100	2
35	10	0.041	0.274	0.093	Humerus	0.27400	2
36	11	-0.088	-0.056	-0.044	Humerus	-0.05600	2
37	12	-0.065	0.524	-0.052	Humerus	0.52400	2
38	13	-0.125	-0.172	-0.068	Humerus	-0.17200	2
39	14	0.000	0.106	0.080	Humerus	0.10600	2
40	15	0.020	0.222	-0.007	Humerus	0.22200	2
41	16	-0.116	0.063	-0.144	Humerus	0.06300	2
42	17	-0.147	0.183	-0.058	Humerus	0.18300	2
43	18	0.052	0.348	0.106	Humerus	0.34800	2
44	19	-0.024	-0.290	-0.152	Humerus	-0.29000	2
45	20	0.010	0.507	0.028	Humerus	0.50700	2
46	21	-0.192	-0.030	-0.197	Humerus	-0.03000	2
47	22	0.060	0.654	0.027	Humerus	0.65400	2
48	23	-0.387	-0.643	-0.184	Humerus	-0.64300	2
49	24	-0.045	-0.171	-0.112	Humerus	-0.17100	2
50	25	0.035	0.291	0.139	Humerus	0.29100	2
51	1	0.223	0.459	0.143	Ulna	0.14300	3
52	2	-0.038	0.193	-0.140	Ulna	-0.14000	3
53	3	0.045	0.207	0.037	Ulna	0.03700	3
54	4	-0.023	0.059	-0.024	Ulna	-0.02400	3
55	5	-0.085	0.054	-0.181	Ulna	-0.18100	3
56	6	-0.093	-0.171	0.052	Ulna	0.05200	3
57	7	0.053	0.015	0.007	Ulna	0.00700	3
58	8	-0.081	0.060	-0.112	Ulna	-0.11200	3
59	9	0.065	0.131	0.123	Ulna	0.12300	3
60	10	0.041	0.274	0.093	Ulna	0.09300	3
61	11	-0.088	-0.056	-0.044	Ulna	-0.04400	3
62	12	-0.065	0.524	-0.052	Ulna	-0.05200	3
63	13	-0.125	-0.172	-0.068	Ulna	-0.06800	3
64	14	0.000	0.106	0.080	Ulna	0.08000	3
65	15	0.020	0.222	-0.007	Ulna	-0.00700	3
66	16	-0.116	0.063	-0.144	Ulna	-0.14400	3
67	17	-0.147	0.183	-0.058	Ulna	-0.05800	3
68	18	0.052	0.348	0.106	Ulna	0.10600	3
69	19	-0.024	-0.290	-0.152	Ulna	-0.15200	3
70	20	0.010	0.507	0.028	Ulna	0.02800	3
71	21	-0.192	-0.030	-0.197	Ulna	-0.19700	3
72	22	0.060	0.654	0.027	Ulna	0.02700	3
73	23	-0.387	-0.643	-0.184	Ulna	-0.18400	3
74	24	-0.045	-0.171	-0.112	Ulna	-0.11200	3
75	25	0.035	0.291	0.139	Ulna	0.13900	3
76	1	0.223	0.459	0.143	LMax	-0.06480	4
77	2	-0.038	0.193	-0.140	LMax	0.29065	4
78	3	0.045	0.207	0.037	LMax	0.09238	4
79	4	-0.023	0.059	-0.024	LMax	0.10841	4
80	5	-0.085	0.054	-0.181	LMax	0.26706	4
81	6	-0.093	-0.171	0.052	LMax	0.02173	4
82	7	0.053	0.015	0.007	LMax	-0.09771	4
83	8	-0.081	0.060	-0.112	LMax	0.24926	4
84	9	0.065	0.131	0.123	LMax	-0.03931	4
85	10	0.041	0.274	0.093	LMax	0.15126	4
86	11	-0.088	-0.056	-0.044	LMax	0.14017	4
87	12	-0.065	0.524	-0.052	LMax	0.63789	4
88	13	-0.125	-0.172	-0.068	LMax	0.11363	4
89	14	0.000	0.106	0.080	LMax	0.08187	4
90	15	0.020	0.222	-0.007	LMax	0.16808	4
91	16	-0.116	0.063	-0.144	LMax	0.33188	4
92	17	-0.147	0.183	-0.058	LMax	0.49004	4
93	18	0.052	0.348	0.106	LMax	0.19491	4
94	19	-0.024	-0.290	-0.152	LMax	-0.18871	4
95	20	0.010	0.507	0.028	LMax	0.44843	4
96	21	-0.192	-0.030	-0.197	LMax	0.41437	4
97	22	0.060	0.654	0.027	LMax	0.48247	4
98	23	-0.387	-0.643	-0.184	LMax	0.24239	4
99	24	-0.045	-0.171	-0.112	LMax	-0.04221	4
100	25	0.035	0.291	0.139	LMax	0.16961	4

 

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Bone Mineral Analysis Homework

Question 5.

Univariate Intervals

The MEANS Procedure

Variable	Mean	Std Error	Lower 95% CL for Mean	Upper 95% CL for Mean
DominantRadius DominantHumerus DominantUlna	0.8438000 1.7926800 0.7044000	0.0228049 0.0566947 0.0215113	0.7967330 1.6756679 0.6600028	0.8908670 1.9096921 0.7487972

 

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Bone Mineral Analysis Homework

Question 5.

Simultaneous Bonferroni Intervals

The MEANS Procedure

Variable	Mean	Std Error	Lower 98.33333% CL for Mean	Upper 98.33333% CL for Mean
DominantRadius DominantHumerus DominantUlna	0.8438000 1.7926800 0.7044000	0.0228049 0.0566947 0.0215113	0.7851084 1.6467682 0.6490376	0.9024916 1.9385918 0.7597624

 

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Bone Mineral Analysis Homework

Question 5.

Simultaneous T-Square Intervals

Obs	Bone	Mean	StdErr	LCL	UCL	FCrit	m
1	Dominant Radius	0.84380	0.022805	0.77176	0.91584	3.04912	3.15895
2	Dominant Humerus	1.79268	0.056695	1.61358	1.97178	3.04912	3.15895
3	Dominant Ulna	0.70440	0.021511	0.63645	0.77235	3.04912	3.15895

 

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Bone Mineral Analysis Homework

Question 6.

The GLM Procedure

Number of observations

25

 

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Bone Mineral Analysis Homework

Question 6.

The GLM Procedure

Multivariate Analysis of Variance

M Matrix Describing Transformed Variables
	DominantRadius	Radius	DominantHumerus	Humerus	DominantUlna	Ulna
MVAR1	1	-1	0	0	0	0
MVAR2	0	0	1	-1	0	0
MVAR3	0	0	0	0	1	-1

 

E = Error SSCP Matrix
	MVAR1	MVAR2	MVAR3
MVAR1	0.08790224	0.11588692	0.03699828
MVAR2	0.11588692	0.39094336	0.07315224
MVAR3	0.03699828	0.07315224	0.14460616

 

Partial Correlation Coefficients from the Error SSCP Matrix of the Variables Defined by the Specified Transformation / Prob > |r|
DF = 24	MVAR1	MVAR2	MVAR3
MVAR1	1.000000	0.625140 0.0008	0.328162 0.1093
MVAR2	0.625140 0.0008	1.000000	0.307664 0.1346
MVAR3	0.328162 0.1093	0.307664 0.1346	1.000000

 

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Bone Mineral Analysis Homework

Question 6.

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Intercept
	MVAR1	MVAR2	MVAR3
MVAR1	0.01623076	0.03684408	0.00672672
MVAR2	0.03684408	0.08363664	0.01526976
MVAR3	0.00672672	0.01526976	0.00278784

 

Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix Variables have been transformed by the M Matrix
Characteristic Root	Percent	Characteristic Vector V'EV=1
		MVAR1	MVAR2	MVAR3
0.24774885	100.00	1.62353713	1.04425065	-0.21007873
0.00000000	0.00	-0.52236426	-0.28128175	2.80105848
0.00000000	0.00	-4.05113374	1.76409023	0.11249139

 

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect on the Variables Defined by the M Matrix Transformation H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix S=1 M=0.5 N=10
Statistic	Value