Tag: Line of best fit

Adv Quant: General Linear Regression Model in R

Introduction

A goal for this post is to convert the dataset to a dataframe for analysis and performing a regression on the state.x77 dataset.

Results

IP1.5F1.png

Figure 1: Scatter plot matrix of the dataframe state.x77.  The red box illustrates the relationship that is personally identified for further analysis.

IP1.5F2.PNG

IP1.5F3.png

Figure 2: Scatter plot of murder rates versus illiteracy rates across the united states, with the linear regression function of illiteracy = 0.11607 * Murder + 0.31362; with a correlation of 0.729752.

Discussion

This post analyzes the dataset state.x77 under the MASS R library, was converted into a data frame (see code section), and an analysis of the data was conducted.  To identify which variable relationship would be interesting to conduct a regression on this dataset, all the relationships within the data frame were plotted in a matrix (Figure 1).  The relationship that personally seemed interesting was the relationship between illiteracy and murder.  Thus, moving forward with these variables a simple linear regression was conducted on that data.  It was determined that there is a positive correlation on this data of 0.729752, and the relationship between the data is defined by

illiteracy = 0.11607 * Murder + 0.31362                                        (1)

From this equation that describes the relationship (Figure 2) between these variables, can explain, 53.25% of the variance between these variables. Both the intercept value and the regression weight are statistically significant at the 0.01 level, meaning that there is less than a 1% chance that this relationship could be developed from pure random chance (R output between Figure 1 & 2).  In conclusion, this data is stating that states with lower illiteracy rates will have the least amount of murder rates in their state, and vice versa. 

Code

#

## Converting a dataset to a dataframe for analysis.

#

library(MASS)             # Activate the MASS library

library(nutshell)         # Activate the nutshell library to access the plot function

data()                    # Lists all data and datasets within the Mass Library

data(state)               # Data in question is located in state

head(state.x77)           # Print out the top five entries of state.x77

df= data.frame(state.x77) # Convert the state.x77 data into a dataframe

#

## Regression formulation

#

plot(df)                                           # Scatter plot matrix, of all relationships between the variables in the df

stateRegression = lm(Illiteracy~Murder, data= df)  # Selecting this relationship for further analysis

summary(stateRegression)                           # Plotting a summary of the regression data

# Plotting a scatterplot from a dataframe below

plot(df$Murder, df$Illiteracy, type=”p”, main=”Illiteracy rates vs Murder rates”, xlab=”Murder”, ylab=”Illiteracy”)           # Plotting a scatterplot from a dataframe

abline(lm(Illiteracy~Murder, data= df), col=”red”) # Plotting a red regression line

cor(df$Murder, df$Illiteracy)

References

Quant: Linear Regression in SPSS

Introduction

The aim of this analysis is to look at the relationship between a father’s education level (dependent variable) when you know the mother’s education level (independent variable). The variable names are “paeduc” and “maeduc.” Thus, the hope is to determine the linear regression equation for predicting the father’s education level from the mother’s education.

From the SPSS outputs the following questions will be addressed:

  • How much of the total variance have you accounted for with the equation?
  • Based upon your equation, what level of education would you predict for the father when the mother has 16 years of education?

Methodology

For this project, the gss.sav file is loaded into SPSS (GSS, n.d.).  The goal is to look at the relationships between the following variables: paeduc (HIGHEST YEAR SCHOOL COMPLETED, FATHER) and maeduc (HIGHEST YEAR SCHOOL COMPLETED, MOTHER). To conduct a linear regression analysis navigate through Analyze > Regression > Linear Regression.  The variable paeduc was placed in the “Dependent List” box, and maeduc was placed under “Independent(s)” box.  The procedures for this analysis are provided in video tutorial form by Miller (n.d.).  The following output was observed in the next four tables.

The relationship between paeduc and maeduc are plotted in a scatterplot by using the chart builder.  Code to run the chart builder code is shown in the code section, and the resulting image is shown in the results section.

Results

Table 1: Variables Entered/Removed

Model Variables Entered Variables Removed Method
1 HIGHEST YEAR SCHOOL COMPLETED, MOTHERb . Enter
a. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER
b. All requested variables entered.

Table 1, reports that for the linear regression analysis the dependent variable is the highest years of school completed for the father and the independent variable is the highest year of school completed by the mother.  No variables were removed.

Table 2: Model Summary

Model R R Square Adjusted R Square Std. Error of the Estimate
1 .639a .408 .407 3.162
a. Predictors: (Constant), HIGHEST YEAR SCHOOL COMPLETED, MOTHER
b. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER

For a linear regression trying to predict the father’s highest year of school completed based on his wife’s highest year of school completed, the correlation is positive with a value of 0.639, which can only 0.408 of the variance explained (Table 2) and 0.582 of the variance is unexplained.  The linear regression formula or line of best fit (Table 4) is: y = 0.76 x + (2.572 years) + e.  The line of best fit essentially explains in equation form the mathematical relationship between two variables and in this case the father’s and mother’s highest education level.  Thus, if the mother has completed her bachelors’ degree (16th year), then this equation would yield (y = 2.572 years + 0.76 (16 years) + e = 14.732 years + e).  The e is the error in this prediction formula, and it exists because of the r2 value is not exactly -1.0 or +1.0.  The ANOVA table (Table 3) describes that this relationship between these two variables is statistically significant at the 0.05 level.

Table 3: ANOVA Table

Model Sum of Squares df Mean Square F Sig.
1 Regression 6231.521 1 6231.521 623.457 .000b
Residual 9045.579 905 9.995
Total 15277.100 906
a. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER
b. Predictors: (Constant), HIGHEST YEAR SCHOOL COMPLETED, MOTHER

Table 4: Coefficients

Model Unstandardized Coefficients Standardized Coefficients t Sig.
B Std. Error Beta
1 (Constant) 2.572 .367 7.009 .000
HIGHEST YEAR SCHOOL COMPLETED, MOTHER .760 .030 .639 24.969 .000
a. Dependent Variable: HIGHEST YEAR SCHOOL COMPLETED, FATHER

The image below (Figure 1), is a scatter plot, which is plotting the highest year of school completed by the mother vs. the father along with the linear regression line (Table 4) and box plot images of each respective distribution.  There are more outliers in the husband’s education level compared to those of the wife’s education level, and the spread of the education level is more concentrated about the median for the husband’s education level.

u4db1f1.png

Figure 1: Highest year of school completed by the mother vs the father scatter plot with regression line and box plot images of each respective distribution.

Conclusion

There is a statistically significant relation between the husband’s and wife’s highest year of education completed.  The line of best-fit formula shows a moderately positive correlation and is defined as y = 0.76 x + (2.572 years) + e; which can only explain 40.8% of the variance, while 58.2% of the variance is unexplained.

SPSS Code

DATASET NAME DataSet1 WINDOW=FRONT.

REGRESSION

  /MISSING LISTWISE

  /STATISTICS COEFF OUTS R ANOVA

  /CRITERIA=PIN(.05) POUT(.10)

  /NOORIGIN

  /DEPENDENT paeduc

  /METHOD=ENTER maeduc

  /CASEWISE PLOT(ZRESID) OUTLIERS(3).

STATS REGRESS PLOT YVARS=paeduc XVARS=maeduc

/OPTIONS CATEGORICAL=BARS GROUP=1 BOXPLOTS INDENT=15 YSCALE=75

/FITLINES LINEAR APPLYTO=TOTAL.

References: