Statistics Homework: Analysis of Osteocalcin and Bone Formation

This statistics assignment analyzes the relationship between Osteocalcin (OC) levels and bone formation (measured by VO+ units) to determine if OC can be used to predict bone formation, potentially avoiding more complex and expensive VO+ measurements. The assignment includes scatter plots ill...Read More

Running head: STATISTICS 1
Statistics
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Q 1.
The research is designed to determine whether there is a significant linear relationship
between Osteocalcin (OC) and bone formation (measured by VO+ units). If the relationship
is significant, and the fitted model works efficiently, then the cost and other inconveniences
associated with VO+ measurements can be sidestepped. Thus, the study is designed to assess
whether the level of OC is linked/associated to the VO+.
Q 2.
Figure 1: Scatter plot of VO+ against OC level
The scatter plot indicates that there is a positive relationship between the OC level and VO+
units (Chambers, 2017). That is, as the OC level increases, the bone formation is expected to
increase. The OC level below 35mg/mL shows a bit of consistency, but above that value, the

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bone formation is inconsistent. However, a confirmatory test is necessary to ascertain if the
relationship is significant.
Q 3.
The regression model assumptions include; linearity, homoscedasticity, independence, and
normality of the independent variable (Cohen, West, & Aiken, 2014). In this case, the
assumption is that the data are normal, and the readings of one person are independent of
each other. The hypothesis that the linear relationship is significant was tested, and the results
are as follows.
Table 1: Model 1 summary
Call:
glm(formula = Bone.Formation$voplus ~ Bone.Formation$oc)
Deviance Residuals:
Min 1Q Median 3Q Max
-727.45 -234.43 -85.08 43.66 1500.99
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 334.034 159.241 2.098 0.0448 *
Bone.Formation$oc 19.505 4.127 4.726 5.43e-05 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for gaussian family taken to be 196492.3)
Null deviance: 10087061 on 30 degrees of freedom
Residual deviance: 5698276 on 29 degrees of freedom
AIC: 469.75
Number of Fisher Scoring iterations: 2
The summary indicates that there is a significant relationship between OC and VO+ (bone
formation) ( β=19.505 , t ( 30 ) =4.127 , p−value<.05 ). Thus, the linearity assumption is met.
Also, it can be concluded that the OC can be used to predict bone formation. The residual
assessment was performed, and the results are as follows:

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Figure 2: Model 1 residual analysis
The plot deduces that the residuals evenly, with no discerning trend. This points out that the
residuals are normally distributed. Also, the QQ-plot suggests that the data are normally
distributed as the observations lie on a relatively straight line. Thus, the homoskedasticity
assumption is met. However, there are a few points that could be influencing the association
between the dependent and independent variable.
Q 4.

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Q 5.
Figure 3: Scatter plot of log VO+ against log OC level
The scatter plot indicates that there is a positive relationship between the log of OC level and
log of VO+ (Chambers, 2017). The plot has a consistent trend, which might suggest a
significant relationship between the two variables.
Q 6.
The linear regression model is fitted to determine whether there is a significant association
between the log of VO+ (bone formation) and log of CO. The test results are as follows.
Table 2: Model 2 summary
Call:
glm(formula = Bone.Formation$ln_voplus ~ Bone.Formation$ln_os)
Deviance Residuals:
Min 1Q Median 3Q Max
-0.53157 -0.30172 -0.03691 0.22209 0.91881
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.3852 0.3643 12.039 8.38e-13 ***
Bone.Formation$ln_os 0.7060 0.1074 6.573 3.35e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

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(Dispersion parameter for gaussian family taken to be 0.1281696)
Null deviance: 9.2541 on 30 degrees of freedom
Residual deviance: 3.7169 on 29 degrees of freedom
AIC: 28.22
Number of Fisher Scoring iterations: 2
The developed model is statistically significant ( β=0.7060 ,t (30 )=6.573 , p−value< .05 )
(Cohen, West, & Aiken, 2014). Thus, the log of OC can be used to predict the log of VO+
significantly. Thus, the linearity assumption is met. The residual analysis was carried out, and
the results are as follows.
Figure 4: Model 2 Residual analysis
The residual versus fitted values plot shows that there is no discerning trend which can
violate the homoscedastic assumption (Cohen, West, & Aiken, 2014). The QQ-plot supports
the same by indicating that the data are normally distributed, but have a few values that can
be influential to the model.
Lastly, Model 2 (AIC = 28.22) is better compared to model 1 (AIC = 469.75) since its AIC
value is smaller. Also, model 2 residual deviance is quite small.

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References
Chambers, J. M. (2017). Graphical Methods for Data Analysis: 0. Chapman and Hall/CRC.
Cohen, P., West, S. G., & Aiken, L. S. (2014). Applied multiple regression/correlation
analysis for the behavioral sciences (2nd ed.). Psychology Press.

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Appendix
R-code used:
#########Bone Formation########
## Loading the data
Bone.Formation <- read.csv("C:/Users/Administrator/Downloads/Bone.Formation.dat",
sep="")
View(Bone.Formation)
plot(Bone.Formation$oc,Bone.Formation$voplus,main = "Scatter plot of VO+ against OC
level", ylab = "VO+",
xlab = "OC level", col= "blue")
abline(glm(Bone.Formation$voplus~ Bone.Formation$oc), col="red")
mod1<-glm(Bone.Formation$voplus~ Bone.Formation$oc)
summary(mod1)
par(mfrow=c(1,2))
plot(mod1, which=1:2)
shapiro.test(Bone.Formation$voplus)

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##computation of the log of OC and log VO+
Bone.Formation$ln_voplus<-log(Bone.Formation$voplus)
Bone.Formation$ln_os<-log(Bone.Formation$oc)
##sactter plot for log of OC against log VO+
par(mfrow=c(1,1))
plot(Bone.Formation$ln_os,Bone.Formation$ln_voplus,main = "Scatter plot of log VO+
against log OC level",
ylab = "log of VO+",
xlab = "log of OC level", col= "blue")
abline(glm(Bone.Formation$ln_voplus~ Bone.Formation$ln_os), col="red")
#Fitting the second model
mod2<-glm(Bone.Formation$ln_voplus~Bone.Formation$ln_os)
summary(mod2)
#Performing residual analysis

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par(mfrow=c(1,2))
plot(mod1, which=1:2)