FILTERING THE DATA

jumps<- filter(JUMP_DATA,SPORT_NAME %in% c("Football"))
jumps_s<-filter(jumps, POSITION %in% c("OL","WR"))
jumps_s2<-select(jumps_s,contains(c("POSITION","bodymass")))

We only want the observations that were performed on Football Offensive Linemen and Wide Receivers

We only want the variables that are expressed per unit of bodymass, which will get us the following variables: Bodymass Relative Eccentric Peak Power (per trial), Eccentric De-acceleration Impulse, Mean Eccentric Power(per trial), Mean Concentric Power(per trial), Mean Takeoff Force(per trial), Takeoff Power.

IS EXPLORATORY FACTOR ANALYSIS APPROPRIATE FOR THIS DATASET?

We will conduct a Keiser Meyer Olkin Test to see if Factor Analysis is appropriate for this dataset.

## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = jumps_s3)
## Overall MSA =  0.64
## MSA for each item = 
##  BODYMASS_RELATIVE_ECCENTRIC_PEAK_POWER_TRIAL 
##                                          0.62 
##     BODYMASS_RELATIVE_ECC_DECEL_IMPULSE_TRIAL 
##                                          0.55 
## BODYMASS_RELATIVE_MEAN_CONCENTRIC_POWER_TRIAL 
##                                          0.64 
##  BODYMASS_RELATIVE_MEAN_ECCENTRIC_POWER_TRIAL 
##                                          0.63 
##    BODYMASS_RELATIVE_MEAN_TAKEOFF_FORCE_TRIAL 
##                                          0.67 
##         BODYMASS_RELATIVE_TAKEOFF_POWER_TRIAL 
##                                          0.78

The MSA is greater than 0.6, so the correlation structure of the data is suitable for Exploratory Factor Analysis

PARALLEL ANALYSIS

Parallel Analysis will provide us with a suggestion about the number of underlying factors that explain the variation between Offensive Linemen and Wide Receivers in the observed variables

## Parallel analysis suggests that the number of factors =  2  and the number of components =  2

Based on this parallel analysis plot, there are two factors that explain the observed variables in our dataset.

FACTOR ANALYSIS DIAGRAM

We are going to construct a Factor Analysis Diagram to see which observed variables are explained by the first factor and which observed variables are explained by the second factor

## 
## Loadings:
##                                               MR1    MR2   
## BODYMASS_RELATIVE_ECCENTRIC_PEAK_POWER_TRIAL   0.223  0.848
## BODYMASS_RELATIVE_ECC_DECEL_IMPULSE_TRIAL      0.027  1.013
## BODYMASS_RELATIVE_MEAN_CONCENTRIC_POWER_TRIAL  0.986  0.186
## BODYMASS_RELATIVE_MEAN_ECCENTRIC_POWER_TRIAL  -0.031  0.885
## BODYMASS_RELATIVE_MEAN_TAKEOFF_FORCE_TRIAL     0.956  0.039
## BODYMASS_RELATIVE_TAKEOFF_POWER_TRIAL          0.892  0.022
## 
##                  MR1   MR2
## SS loadings    2.733 2.566
## Proportion Var 0.455 0.428
## Cumulative Var 0.455 0.883

With this, we see that Factor MR1 explains 99% of the variation in Mean Concentric Power, 96% of the variation in Mean Takeoff Force, and 89% of the variation in Takeoff Power.

Then, Factor MR2 explains 100% of the variation in Eccentric De-acceleraiton Impulse, Mean Eccentric Power, and Eccentric Peak Power.

COMPARING THE FACTOR SCORES OF WIDE RECEIVERS AND OFFENSIVE LINEMEN

## 
##  Welch Two Sample t-test
## 
## data:  MR1[POSITION == "OL"] and MR1[POSITION == "WR"]
## t = -96.546, df = 12863, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.298877 -1.247185
## sample estimates:
##  mean of x  mean of y 
## -0.6167923  0.6562383
## 
##  Welch Two Sample t-test
## 
## data:  MR2[POSITION == "OL"] and MR1[POSITION == "WR"]
## t = -43.121, df = 14121, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.7137581 -0.6516892
## sample estimates:
##   mean of x   mean of y 
## -0.02648534  0.65623835

Based on the Welch Two Sample T-Tests, it is clear that there is a statistically significant difference between the MR1 and MR2 scores/ratings of Wide Receivers and Offensive Linemen, with the difference favoring Wide Receivers.

WHY DO THE FACTOR SCORES OF THE RECEIVERS AND OFFENSIVE LINEMEN DIFFER?

Factor MR1 measures Explosiveness/Rate of Force Development (the generation of force per second), which explains the variations between Wide Receivers and Offensive Linemen in Eccentric Peak Power, Eccentric De-acceleration Impulse, and Mean Eccentric Power.

Given that Wide Receivers are more explosive than Offensive Linemen, they will generate more force and acceleration during their Eccentric descent. Consequently, the Receivers’ hamstrings, quadriceps, and glutes of will have to exert more force/power and do more work over the eccentric period to de-accelerate their bodies in preparation for the Concentric Phase.

Factor MR2 measures Elastic Energy Storage, which explains the variations between Wide Receivers and Offensive Linemen in Mean Concentric Power, Mean Takeoff Force(per trial), and Takeoff Power.

During the Eccentric Phase, since the Wide Receivers’ hamstrings, quadriceps, and glutes have to bear a heavier load/force, they will be less able to withstand said load and will be stretched to a greater extent than the counterparts of the offensive linemen.

This increased stretch will cause the Receivers hamstrings, quadricepts, and glutes to store more elastic energy, which can be transferred into kinetic energy and velocity during the Concentric phase. Since the wide receivers will generate higher concentric velocities than the offensive linemen and both positions start their concentric phase motionless, the wide receivers being able to generate greater Concentric Velocity means that they have a greater Concentric acceleration, and thus a greater concentric/takeoff power and force applied.