• Data collected from Australian Institute of Sports in 1994
• 102 males and 100 females analyzed
• Height, weight, body mass index, lean body mass, percentage ob body fat, sum of skin folds, red and white blood cell counts, hemoglobin, hematocrit, and ferritins
• Use the dimension reduction package in R Studio for analysis
• Objective is to determine number of reduced dimensions
• Use Sliced Inversion Regression, Principal Hessian Direction, Weights, and graphical representation

## 
## Call:
## dr(formula = LBM ~ Ht + Wt + log(RCC) + WCC, method = "sir", 
##     nslices = 8)
## 
## Method:
## sir with 8 slices, n = 202.
## 
## Slice Sizes:
## 25 25 25 25 27 27 30 18 
## 
## Estimated Basis Vectors for Central Subspace:
##              Dir1      Dir2    Dir3     Dir4
## Ht       -0.01053  0.001086  0.3068 -0.04158
## Wt       -0.02383  0.003352 -0.1896  0.01005
## log(RCC) -0.99960 -0.999976 -0.8965  0.51527
## WCC       0.01097  0.005932  0.2574  0.85596
## 
##               Dir1   Dir2    Dir3    Dir4
## Eigenvalues 0.8774 0.1614 0.04244 0.01313
## R^2(OLS|dr) 0.9987 0.9988 0.99997 1.00000
## 
## Large-sample Marginal Dimension Tests:
##                Stat df   p.value
## 0D vs >= 1D 221.057 28 0.0000000
## 1D vs >= 2D  43.822 18 0.0006117
## 2D vs >= 3D  11.224 10 0.3403385
## 3D vs >= 4D   2.651  4 0.6177515

## 
## Call:
## dr(formula = LBM ~ Ht + Wt + log(RCC) + WCC, method = "phdres", 
##     nslices = 8)
## 
## Method:
## phdres, n = 202.
## 
## Estimated Basis Vectors for Central Subspace:
##              Dir1       Dir2      Dir3     Dir4
## Ht       -0.12764  0.0003378 -0.005550  0.02549
## Wt        0.02163 -0.0326138  0.007342 -0.01343
## log(RCC)  0.74348 -0.9816463 -0.999930 -0.99909
## WCC      -0.65611  0.1879008  0.007408 -0.03157
## 
##               Dir1   Dir2    Dir3    Dir4
## Eigenvalues 1.4303 1.1750 -1.1244 -0.3999
## R^2(OLS|dr) 0.2781 0.9642  0.9642  1.0000
## 
## Large-sample Marginal Dimension Tests:
##               Stat df Normal theory Indep. test General theory
## 0D vs >= 1D 35.015 10     0.0001241    0.003676        0.01632
## 1D vs >= 2D 20.248  6     0.0025012          NA        0.03152
## 2D vs >= 3D 10.281  3     0.0163211          NA        0.05819
## 3D vs >= 4D  1.155  1     0.2825955          NA        0.26625

## $summary
##                   Stat p.value
## 0D vs >= 1D 221.056620   0.000
## 1D vs >= 2D  43.821844   0.002
## 2D vs >= 3D  11.223952   0.352
## 3D vs >= 4D   2.651365   0.536
## 
## $npermute
## [1] 499
## 
## attr(,"class")
## [1] "dr.permutation.test"

## 
## Call:
## dr(formula = LBM ~ Ht + Wt + RCC + WCC, weights = wts, method = "phdres")
## 
## Method:
## phdres, n = 191, using weights.
## 
## Estimated Basis Vectors for Central Subspace:
##         Dir1     Dir2     Dir3     Dir4
## Ht   0.02463 -0.02266  0.03279  0.05518
## Wt   0.00252  0.07276 -0.12843 -0.04936
## RCC -0.99635 -0.99397 -0.67622 -0.91361
## WCC  0.08166  0.07889  0.72468 -0.39979
## 
##                   Dir1    Dir2   Dir3   Dir4
## Eigenvalues -3.1348469 -1.3574 0.8266 0.3212
## R^2(OLS|dr)  0.0009775  0.7783 0.8404 0.8683
## 
## Large-sample Marginal Dimension Tests:
##                Stat df Normal theory Indep. test General theory
## 0D vs >= 1D 42.5277 10     6.039e-06    0.004499        0.01334
## 1D vs >= 2D  8.9755  6     1.750e-01          NA        0.37770
## 2D vs >= 3D  2.6848  3     4.428e-01          NA        0.54970
## 3D vs >= 4D  0.3521  1     5.529e-01          NA        0.60884

• AIS data 14 attributes and a complex dataset
• Dimension reduction package used to determine the number of dimensions, which decreased the amount of complexity
• SIR, pHd, and graphs showed two dimensions were best
• Permutation test from the original SIR model also showed two dimensions were best
• Weighted multipliers showed one dimension is achievable
• More analysis required to verify if the results from the dimension reduction makes logical sense