Using Univariate/ Multivariate analysis To Study Body Strength

Univariate simply means that your data has only a single variable. It is one of the simplest methods of analyzing data and its purpose is to describe, summarize, and find patterns in data. The univariate analysis differs from the multivariate analysis in that the latter is more complex and it is utilized when a data set has more than two variables. However, both are great statistical techniques and have been used in analyzing data for decades.

Multivariate analysis

Theoretically, as age increases, the injury index in a woman increases. Also, as medical difficulty increases, the injury index increases.
Coefficientsa
Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B
B Std. Error Beta Lower Bound Upper Bound
1 (Constant) -244.230 104.141   -2.345 .021 -450.922 -37.538
Age 3.628 1.541 .225 2.353 .021 .568 6.687
Medical difficulty index 284.142 95.017 .286 2.990 .004 95.559 472.725
a. Dependent Variable: Overall injury index
As the p-value for both the demographic variables is <0.05, we can say that both variables are statistically significant. And, as R2 is 0.162, it means that 16.2% variation in body strength is explained by the demographic variables.
Overall injury index (injury) = β0 + β1 age +β2 Medical difficulty index (medindex) + ei 

Regression Analysis

One of the team members, Tom, selected a model retaining four independent variables. His SPSS output is as follows:
The coefficient of 5.668 means that for each extra unit of age (1 year), the overall injury index increases by 5.668. Or, as age increases by 1 unit, the overall injury index increases by 5.668.
The coefficient of -3.981 means that for each extra unit of gluts, the overall injury index decreases by 3.981. Or, as strength in gluteus/hamstrings increases by 1 unit, the overall injury index decreases by 3.981.  
For the predicted injury index of -107.321, profile of independent variables has to be
i) 5.668*age + 0.300*quads - 3.981*gluts - 0.620*arms = 0 OR
ii) age = quads = gluts = arms = 0
H0 = βarms = 0
     H1 = βarms ≠ 0
Coefficientsa
Model Unstandardized Coefficients Standardized Coefficients t Sig. 95.0% Confidence Interval for B
B Std. Error Beta Lower Bound Upper Bound
1 (Constant) -107.321 96.170   -1.116 .267 -298.242 83.601
Age 5.668 1.420 .351 3.990 .000 2.848 8.488
Strength in quadriceps .300 .553 .056 .543 .588 -.797 1.398
Strength in gluteus/hamstrings -3.981 .918 -.441 -4.336 .000 -5.804 -2.158
Strength in arms/shoulders -.620 .583 -.101 -1.063 .291 -1.778 .538

a. Dependent Variable: Overall injury index
As the above table shows, the p-value for βarms is 0.291 (>0.05), i.e. we can reject the null hypothesis. Hence, we can say that βarms are not statistically significantly different than zero. Thus, the partial regression coefficient of -0.62 may differ from zero because of sampling error.
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Although Tom’s model may not be the one you picked, for some theoretical reasons, the team agreed that Tom’s model is the ultimate model to pick anyway. Recognizing that the correct model is never known, nonetheless, before reporting the result, you would like to check the aptness of Tom’s model. Using numbers and/or graphs, show the following diagnostics for Tom’s model. Specify your cut-off criterion when necessary. You do not need to re-run the analysis after checking the diagnostics.

Probability

We check the Variance Inflation Factor (VIF) to check multicollinearity.
           Coefficientsa
Model Collinearity Statistics
Tolerance VIF
1 Strength in quadriceps .717 1.395
Strength in gluteus/hamstrings .737 1.356
Strength in arms/shoulders .829 1.206
a. Dependent Variable: Age

As the above table shows, none of the VIF value is greater than 5, hence, we can say that there is no problem of multicollinearity in the model.
The normality assumption of residuals can be checked using normal probability plots for both standardized and unstandardized residuals.
As the above graph shows, residuals are scattered randomly, hence there is no problem of heteroscedasticity in the model.
To check residual outliers, tests such as Mahalanobis distance, Cook’s distance, etc. can be calculated:
 
Residuals Statistics
  Minimum Maximum Mean Std. Deviation N
Predicted Value 56.25 209.17 145.80 28.151 100
Std. Predicted Value -3.181 2.251 .000 1.000 100
Standard Error of Predicted Value 5.778 15.924 9.752 2.371 100
Adjusted Predicted Value 54.56 209.42 145.77 28.055 100
Residual -103.802 87.387 .000 43.953 100
Std. Residual -2.313 1.948 .000 .980 100
Stud. Residual -2.342 2.023 .000 1.004 100
Deleted Residual -106.349 94.677 .031 46.224 100
Stud. Deleted Residual -2.400 2.057 .000 1.014 100
Mahal. Distance .652 11.480 3.960 2.414 100
Cook's Distance .000 .091 .010 .015 100
Centered Leverage Value .007 .116 .040 .024 100
a. Dependent Variable: Overall injury index

As the above table shows, the maximum Mahalanobis distance in our model is 11.48 and the critical value for 4 degrees of freedom using the Chi-Sq test is 9.49. Hence, outliers are present in the dataset (Row 1,7,12 and 79 are the outliers).
Model Summary
Model R R Square Adjusted R Square Std. The error of the Estimate
1 .539a .291 .261 44.869
a. Predictors: (Constant), Strength in arms/shoulders, Age, Strength in gluteus/hamstrings, Strength in quadriceps
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ANOVA

Model Sum of Squares df Mean Square F Sig.
1 Regression 78456.806 4 19614.202 9.743 .000b
Residual 191257.194 95 2013.234    
Total 269714.000 99      
a. Dependent Variable: Overall injury index
b. Predictors: (Constant), Strength in arms/shoulders, Age, Strength in gluteus/hamstrings, Strength in quadriceps
Although the model R2 is quite low (0.291). But the model F-statistic is 9.743 and the significance F is 0.000. i.e. H0 (the model doesn’t fit the data)) is rejected. Hence, the model fits the data very well.