Homework - Mineral Content in Bones
Due Date: Thursday, September 19, 2002
Bring Report To Class

Use the Minearal Content in Bones data given in Table 1.8, page 44, of our textbook (Johnson, R.A. and D.W. Wichern. 2002. Applied Multivariate Statistical Analysis, 5th Edition. Prentice Hall, Upper Saddle River, New Jersey) and shown below to perform the following analyses. I would encourage you to write a SAS program to perform the computations. Do your own work. The SAS examples that I have given should be sufficient to help you write your program, though you may ask a colleague for help with SAS syntax problems and errors. If you get stuck, my research associate, Ms. Rebecca Frederick, Room 149 Ag Admin, can help you with your SAS program.
  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?
     
  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.
     
  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].
     
  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?
     
  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.
     
  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. 
GOptions NoPrompt;
Title1 "Bone Mineral Analysis By Your Name Goes Here";

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;

/*
 * Create a stacked or univariate view data set
 * for plotting with PROC BOXPLOT. Could also
 * use PROC TRANSPOSE to do this.
 */
Data Dominant;
 Set Bones;
 Length Bone $7;
 MineralContent=DominantRadius;  Bone="Radius";  Output;
 MineralContent=DominantHumerus; Bone="Humerus"; Output;
 MineralContent=DominantUlna;    Bone="Ulna";    Output;
 Drop DominantRadius DominantHumerus DominantUlna Radius Humerus Ulna;
Run;

Proc Sort Data=Dominant;
 By Bone;
Run;

Proc Print Data=Dominant;
Run;

Proc BoxPlot Data=Dominant;
 Plot MineralContent*Bone / BoxStyle=Schematic VAxis=Axis1;
 Axis1 Label=(A=90 "Mineral Content");
 Symbol1 C=Black V=Dot;
Run; 
GOptions Reset=Symbol;