The analysis of the model began with examining the regression statistics (see APPENDIX I). The model’s multiple correlation statistic (Multiple R) was 0.90. Although this statistic does not directly equate to the Pearson’s Product Moment correlation, it is the square root o f R2 (R Square) and does suggest that there is a high degree o f correlation between the linear combination o f all the predictor variables and the
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response variable. The coefficient o f determination (R2) was 0.81, which meant that the predictor variables explain approximately 81% o f the total sum o f squares deviation about their mean. The adjusted R2 (Adjusted R Square) was 0.76, which is within approximately 6% o f the R2 value. The adjusted R2 tends to correct the R2 value based on the number o f predictor variables used to develop the model. Both the R2 and the adjusted R2 values were high enough to give the researcher confidence in the fit of the regression line to the data, and confidence that the model was useful in making cost predictions.
The standard error estimates the standard deviation o f the predicted cost o f a ground combat weapon system. Based on the 42 observations in this case, the typical error or deviation about the regression line was approximately $4,000.
The researcher also used analysis o f variance to determine if the model’s estimated multiple regression coefficients were all statistically equal to zero, or if at least one regression coefficient was statistically not equal to zero (see APPENDIX I). The F- distribution statistic is 30.88, which equates to a probability o f random occurrence of approximately zero (Significance F). This result meant that the researcher could be at least 99% certain that at least one multiple regression coefficient was statistically greater than zero. Further analysis o f the P-value for each regression coefficient showed that they were all statistically significant to the model and had less than a 0.1% chance of randomly occurring. This led the researcher to conclude that all the regression coefficients were important to predicting the response o f the cost variable.
The researcher analyzed the residuals to determine if the model exhibited signs of bias or had variance issues needing resolution. First, the researcher summed the residuals
from the original regression equation to ensure that they did add up to zero (see
APPENDIX H). This was significant because it verified that the assumption E(s) = 0 was valid. However, in this study the researcher chose to force the Y-intercept to occur at the origin. This constraint simply implied that an Army ground combat weapon system that has zero values for its predictor variables does not exist and therefore costs zero dollars. This constraint also introduced an upward shift in the location o f the Y-intercept and introduced a negative bias in the error (e). The sum o f the residuals was $-78.49, which is an average o f $-1.87 for the 42 observations. This evidence strongly suggested that the residuals were randomly distributed about the regression line, which satisfied one o f the verification requirements.
The researcher also examined the normality plot o f the residuals and the plot o f each predictor variable to the residuals. This procedure provided significant verification information because least-square model tests are more optimal if the residuals have a normal distribution (Birkes and Dodge, 1993). The normal plot o f the residuals,
APPENDIX I, showed that there is a strong linear relationship between the standardized residuals and the cost. This confirmed that the residuals were approximately normally distributed, which satisfied one o f the underlying regression assumptions. In addition, the plot o f all the residuals to the response variable showed that they visually appear to be randomly distributed. As an additional check for randomness in the residuals, an
autocorrelation analysis was performed on all the residual data points (Minitab 13, 2000). The autocorrelation plot showed that there was no significant correlation between the residuals, thus confirming that the residuals exhibited a random pattern (see APPENDIX I). The residuals were also plotted against each predictor variable to determine if
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problems o f unequal variance existed with any of the predictor variables. The Horsepower/WT ratio plot was the only graph that showed any evidence o f possible unequal variance. The predictor variables were transformed to their logarithm form in an attempt to eliminate this problem. The resulting plot o f Horsepower/WT to residuals, see APPENDIX I, showed little or no improvement in the variance. Therefore, the original form o f the predictor variables was maintained for the study.
Lastly, the researcher examined the plots o f each predictor variable to determine if the variables exhibited a linear relationship with the response variable (see APPENDIX I). The purpose o f this test was to ensure that the model was not incorrectly specified, i.e., no curvilinear trends. All the plots showed that the predictor variables do have a linear relationship with the response variable. As expected, Activity showed the strongest linear relationship with cost. Based on all the evidence, the researcher concluded that the issues o f the small error term (s) and the possible unequal variance were insignificant and that they have little effect on the model’s usefulness as a valid operating cost prediction tool for Army ground combat weapon systems.