STAT 155 Principles of Statistics - Free Samples to Students

Body Mass Index (BMI) is a widely used measure assess whether the weight of an individual is in the healthy range. According to the Australian Government Department of Health, a BMI between 18.5 and 24.9 for adults is considered to be within the healthy weight range. A person with a BMI below 18.5 is considered to be underweight, a person with a BMI above 30 is considered to be obese and a person with a BMI between 25 and 30 is considered to be overweight. There are exceptions, for example, the healthy weight BMI range tends to be higher for older people.

In this study therefore, we sought to investigate whether the variables Age, Smoker, BMI, maxBMI, and minBMI influence spBMD. We present the results of the analysis in the next section.

Scatterplots

SPBMD versus Age

The scatter plot below shows that a negative relationship exists between spBMD and Age (Emerson, Green, Schoerke, & Crowley, 2013). That is to mean that an increase in the age of the respondent would be expected to result to a decrease in the spBMD while a decrease in the age of the respondent would be expected to result in an increase in the spBMD.

SPBMD versus BMI

A scatter plot of spBMD versus BMI was also plotted in order to visualize the relationship that exists between the two variables. For this case (spBMD versus BMI), we observe that a positive relationship exists between the two variables (Friendly & Denis, 2005). That is to mean that an increase in the BMI of the respondent would be expected to result to an increase in the spBMD while a decrease in the age of the respondent would be expected to result in a decrease in the spBMD.

SPBMD versus minBMI

The scatter plot below shows that a negative relationship exists between spBMD and minBMI. That is to mean that an increase in the minBMI of the respondent would be expected to result to a decrease in the spBMD while a decrease in the minBMI of the respondent would be expected to result in an increase in the spBMD.

SPBMD versus maxBMI

For the spBMD versus maxBMI we observe a positive relationship exists between spBMD and maxBMI. That is to mean that an increase in the age of the respondent would be expected to result to an increase in the spBMD while a decrease in the maxBMI of the respondent would be expected to result in an increase in the spBMD

Box Plot

In this section, we present the box plots for the different variables. Considering spBMD, we can see that the distribution of the data for spBMD seems to be normally distributed. However, it has some outliers both on the top and bottom of the graph.

The boxplot for the BMI also shows the data to be normally distributed though it has numerous outliers. The one for minBMI however does not show normality but a rather skewed distribution with few outliers both at the bottom and at the top. Lastly, for the maxBMI, the distribution is normally distributed with no evidence of outliers in the data. 

Results

In order to investigate the relationship between spBMD and the five independent variables (Age, Smoker, BMI, maxBMI, and minBMI), we performed a regression analysis (Armstrong, 2012). Regression analysis is a mathematical technique that allows one to estimate dependent variable based on one or more independent variables (Rouaud, 2013). The results are presented in the table below;

As can be seen, the value of R-squared is 0.2897; this means that only 28.97% of the variation in the dependent variable (spBMD) is explained by the five independent variables in the model.

The p-value of the F-Statistic is 0.000 (a value less than 5% level of significance), this leads to rejection of the null hypothesis and concluding that the model is different from zero hence it is fit and appropriate to estimate the spBMD at 5% level of significance (Tofallis, 2009).

Looking at the individual independent variables, we observed that out of the five independent variables only 2 were significant in the model. This means that only the two independent variables are significantly related with the dependent variable. The two significant variables are Age and BMI. The other three variables (Smoker, minBMI and mxBMI) were found to be insignificant in the model.

The coefficient of the Age was found to be -0.008137; this suggests that a unit increase in the age of the participant would result to a decrease in the spBMD by 0.0081. Conversely, a unit decrease in the age of the participant would result to an increase in the spBMD by 0.0081.

The coefficient of the BMI was found to be 0.014907; this suggests that a unit increase in the BMI of the respondent would result to an increase in the spBMD by 0.0149. Similarly, a unit decrease in the BMI of the participant would result to a decrease in the spBMD by 0.0149.

The coefficient of the Smoker was found to be -0.064940; this suggests that being a smoker would reduce the spBMD by 0.0649.

The coefficient of the minBMI was found to be -0.010276; this suggests that a unit increase in the minBMI of the participant would result to a decrease in the spBMD by 0.0103. Conversely, a unit decrease in the minBMI of the respondent would result to an increase in the spBMD by 0.0103.

The coefficient of the maxBMI was found to be 0.004401; this suggests that a unit increase in the maxBMI of the participant would result to an increase in the spBMD by 0.044. Similarly, a unit decrease in the maxBMI of the participant would result to a decrease in the spBMD by 0.0044.

Conclusion

The aim of this study was to investigate the influence that variables Age, Smoker, BMI, maxBMI, and minBMI have on spBMD. Results showed that only Age and BMI had significant influence on spBMD. Age for instance, had a negative relationship with spBMD while BMI had a positive relationship with spBMD.

References

Armstrong, J. S. (2012). Illusions in Regression Analysis. International Journal of Forecasting (forthcoming), 28(3), 689.

Emerson, J. W., Green, W. A., Schoerke, B., & Crowley, J. (2013). The Generalized Pairs Plot. Journal of Computational and Graphical Statistics, 22(1), 79–91.

Friendly, M., & Denis, D. (2005). The early origins and development of the scatterplot. Journal of the History of the Behavioral Sciences, 41(2), 103–130.

Rouaud, M. (2013). Probability, Statistics and Estimation. 60.

Tofallis, C. (2009). Least Squares Percentage Regression. Journal of Modern Applied Statistical Methods, 7(5), 526–534.