Managing diabetes People with diabetes measure their fasting plasma glucose (FPG; measured in units of milligrams per milliliter) after fasting for at least 8 hours. Another measurement, made at regular medical checkups, is called HbA. This is roughly the percent of red blood cells that have a glucose

molecule attached. It measures average exposure to glucose over a period of several months. The table below gives data on both HbA and FPG for $18$ diabetics five months after they had completed a diabetes education class.

(a) Make a scatterplot with HbA as the explanatory variable. There is a positive linear relationship, but it is surprisingly weak.

(b) Subject $15$ is an outlier in the y-direction. Subject $18$ is an outlier in the x-direction. Find the correlation for all $18$ subjects, for all except Subject $15$ and

for all except Subject $18$ Are either or both of these subjects influential for the correlation? Explain in simple language why r changes in opposite directions when we remove each of these points.

(c) Add three regression lines for predicting FPG from HbA to your scatterplot: for all $18$ subjects, for all except Subject $15$ and for all except Subject $18$

Is either Subject $15$ or Subject $18$ strongly influential for the least-squares line? Explain in simple language what features of the scatterplot explain the degree of influence.

Linear regression is commonly used for predictive analysis and modeling.

Set the horizontal axis for HbA (the explanatory variable) and the vertical axis for FPG (the response variable).

The scatterplot for the supplied data is presented below using the MINITAB:

The general pattern moves from the bottom left to the higher right, as shown in the graph. That is, people with a higher HbA have a higher FPG. This is referred to as a positive relationship between the two variables. The relationship is linear in nature. That example, the general pattern runs from bottom left to higher right in a straight line. Because the points deviate greatly from the line and there are some outliers, the relationship is weak. Therefore, the required scatterplot is drawn.

The correlation $r$ with all $18$ individuals using the MINITAB is $r=0.482$

Without subject $15$ the correlation coefficient is $r=0.568$

Without subject $18$ the correlation coefficient is$r=0.384$

Without subject $15$ and without subject $18$ the correlation is $r=0.324$

The Correlation increases by $0.086$ after outlier subject $15$ is removed. However, removing subject $15$ from the equation has no influence on the association. Because of subject $15\text{'}s$ extreme position on the HbA scale, the position of the regression line is strongly influenced by this point. The Correlation drops by $0.098$ when the outlier subject$18$ is removed. One outlier can be wholly responsible for a high correlation value that would otherwise be quite low (without the outlier). Needless to note, major decisions should never be made solely on the basis of the correlation coefficient's value (i.e., examining the respective scatterplot is always recommended). These are known as 'good' outliers. Both subjects $15$ and $18$ have an impact since the linear correlation coefficient varies dramatically when they are combined.

Therefore,

The correlation $r$ with all $18$ subjects is $r=0.482$

The correlation $r$ without subject $15$ is $r=0.568$

The correlation $r$ without subject $18$ is $r=0.384$

The least-square lines with all $18$topics, without subject $15$ and without subject $18$ are shown in the diagram below.

The relevance of subject $18$ can be shown here. This point can be considered an excellent outlier because it spreads the pattern to the top right. When this point is removed, the correlation decreases since the remaining points exhibit no discernible pattern. Because this point is so distant from the regression line, Subject $15$ has a very large residual. Least-squares lines minimize the sum of squares of the vertical distances between the points. The line is pulled toward itself by a point that is extreme in the $X$ direction and has no other points nearby. It's known as influential spots. It lowers the line's incline. Therefore, subject $15$ and subject $18$ both are influential.