Regression Modeling

Model for Predicting Tax Assessment Value

Scatter Plot with FloorArea and Assessment Value

Looking at the plot above, one can conclude that there is a linear relationship between Assessment Value and Floor Area. The Assessment Value increases as the Floor Area increases. Therefore, a linear relationship can be said to be present between the two variables.

Regression Analysis of FloorArea and AssessmentValue

The following is the output for the regression analysis of Floor Area and Assessment Value.”

The figures that would be key in determining whether the Floor Area is a significant predictor of Assessment Value are the R square and p-value. The R² value, which is 0.9377, indicates that the Floor Area influences 93.77% of the Assessment Value. The figure shows that the floor area has a great influence on the assessment value. Therefore, Floor Area can be deemed a strong factor that impacts the Assessment Value. The p-value of the regression analysis is 1.23E-19, which is less than 0.05. A p-value of less than 0.05 usually suggests statistical significance. Therefore, one can conclude that Floor Area is a significant predictor of Assessment Value.

Scatter Plot with Age and AssessmentValue

The following is the scatter plot with age as the independent variable and assessment value as the dependent variable.

From looking at the scatter plot, one can deduce that there exists no linear relationship between Age and Assessment Value. This is because the data in the plot is evenly distributed and not an indicator of one change affecting the other. The linear regression line is almost level indicating that one variable has little to no effect on the other variable.

Regression Analysis of Age and AssessmentValue

The following tables detail the output of the regression analysis of age and assessment value.

Similar to the regression analysis of floor area and assessment, the key areas of the analysis that would help in determining whether age is a significant predictor of assessment value are the R square and the p-value. In this regression analysis, the R square is 0.032. This value implies that age influences 3.2% of the Assessment Area. The impact that age has on the Assessment Area is very little, and thus, age cannot be considered a significant predictor of assessment value. Additionally, the p-value of 0.327, which is greater than 0.05, indicates that there is no statistical significance. Therefore, age is not a significant predictor of assessment value.

Multiple Regression Model

Regression Analysis with AssessmentValue, FloorArea, Offices, Entrances, and Age

The following is the output from the multiple regression analysis:

Based on the results of the multiple regression above, the R square figure is 0.953, while the adjusted R square is 0.946. The R square is the adjusted goodness of fit model for measuring linear models. It is used for identifying the variance percentage in the target fields explained by the inputs (Bar-Gera, 2017). Therefore, the figure obtained above means that 95.3% of the variation is explained by the change in the variables. On the other hand, the adjusted R square is the modified version of the R square that is used for adjusting the predictors that are not significant in the model (Bar-Gera, 2017). The current multiple linear regression model has several predictor variables, thus the need to evaluate and determine the adjusted R square. If the R square is less than the R square, it means that the additional variables in the model are not adding any value. This is the case with the current analysis since Adjusted R squared = 0.946 < R square =0.953.

Predictors Considered Significant

As mentioned initially, if the R square is less than the R square, it means that the additional variables in the model are not adding any value. The condition is true for the current case, and this means that there are predictors that are considered significant and those not considered significant. However, to determine the predictors that are significant, it is essential to check the p values of each of the variables, including floor area, offices, entrances, and age. A predictor is only significant if its p-value is less than 0.05 (Di Leo & Sardanelli, 2020). Looking at the output from the analysis, only one of the predictors has a p-value of less than 0.05. The only significant variable is Floor Area (p = 0.0313 < 0.05).

Final Model if FloorArea and Offices are Used as Predictors

When floor area and office are used as predictors, the following are the results of the regression analysis:

The results of the regression analysis can help in formulating the final model.

The final model can be derived based on the following formula:

y = mx + c

where c is the intercept and m the constant.

Y(AssessmentValue) = 115.91 + (0.26 x FloorArea) + (78.34 x Offices)

Assessed Value Based on Final Model

AssessmentValue = 115.91 + (0.26 x FloorArea) + (78.34 x Offices)

We are given FloorArea =1500, Offices = 2

Therefore, the assessed value will be calculated as follows:

Assessed Value = 115.9 + (0.26 x 3500) + (78.34 x 2)

= 1025.9 + 166.68

= 1192.58

The assessed value based on the final model will be 1192.58 thousand dollars.

The obtained Assessed Value is consistent with what is provided within the database because it falls between the minimum and maximum values.

References

Bar-Gera, H. (2017). The Target Parameter of Adjusted R-Squared in Fixed-Design Experiments. The American Statistician, 71(2), 112–119. https://doi.org/10.1080/00031305.2016.1200489

Di Leo, G., & Sardanelli, F. (2020). Statistical significance: p-value, 0.05 threshold, and applications to radionics—reasons for a conservative approach. European Radiology Experimental, 4(1). https://doi.org/10.1186/s41747-020-0145-y

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Instructions:

The Excel file for this assignment contains a database with information about the tax assessment value assigned to medical office buildings in a city. The following is a list of the variables in the database:

Regression Modeling

FloorArea: square feet of floor space
Offices: number of offices in the building
Entrances: number of customer entrances
Age: age of the building (years)
AssessedValue: tax assessment value (thousands of dollars)

Use the data to construct a model that predicts the tax assessment value assigned to medical office buildings with specific characteristics.

Construct a scatter plot in Excel with FloorArea as the independent variable and AssessmentValue as the dependent variable. Insert the bivariate linear regression equation and r^2 in your graph. Do you observe a linear relationship between the 2 variables?
Use Excel’s Analysis ToolPak to conduct a regression analysis of FloorArea and AssessmentValue. Is FloorArea a significant predictor of AssessmentValue?
Construct a scatter plot in Excel with Age as the independent variable and AssessmentValue as the dependent variable. Insert the bivariate linear regression equation and r^2 in your graph. Do you observe a linear relationship between the 2 variables?
Use Excel’s Analysis ToolPak to conduct a regression analysis of Age and Assessment Value. Is Age a significant predictor of AssessmentValue?

Construct a multiple regression model.

Use Excel’s Analysis ToolPak to conduct a regression analysis with AssessmentValue as the dependent variable and FloorArea, Offices, Entrances, and Age as independent variables. What is the overall fit r^2? What is the adjusted r^2?
Which predictors are considered significant if we work with ?=0.05? Which predictors can be eliminated?
What is the final model if we only use FloorArea and Offices as predictors?
Suppose our final model is:
AssessedValue = 115.9 + 0.26 x FloorArea + 78.34 x Offices
What would be the assessed value of a medical office building with a floor area of 3500 sq. ft., 2 offices, that was built 15 years ago? Is this assessed value consistent with what appears in the database?