In SOC345/lehmansociology: Tools for Sociology Students Using R

#suppressPackageStartupMessages()
library(lehmansociology)
education_and_poverty<-create_educ_poverty_data()
library(ggplot2)

Making even smaller sums of squares

Previously we saw that there is no statistic that will give you a total squared deviation (total sum of squared) that is smaller than the mean.

That is

Sum(x - mean(x))^2

will be at least as small as

Sum(x - something(x))^2

The only time they will be equal is when something(x) = mean(x) such as when the mean and median are the same value or the mean and the mode are the same value.

Let's do some calculations with our variable PCTPOVALL_2013

n <- length(education_and_poverty$PCTPOVALL_2013) 
meany <- mean(education_and_poverty$PCTPOVALL_2013)
vary <- var(education_and_poverty$PCTPOVALL_2013)
sumsquaresy <- vary * (n-1)

meanx <- mean(education_and_poverty$Percent.of.adults.with.less.than.a.high.school.diploma..2009.2013)
varx <- var(education_and_poverty$Percent.of.adults.with.less.than.a.high.school.diploma..2009.2013)
sumsquaresx <- varx * (n-1)

Mean poverty rate: r meanx Total sum of squares for poverty: r sumsquaresx.

Mean less than high school diploma: r meany Total summ of squares less than high school diploma: sumsquaresy

So far this looks at each as a single variable, but in sociology we are almost always interested in the relationship between variables.

baseplot <- ggplot(education_and_poverty,
       aes(x=Percent.of.adults.with.less.than.a.high.school.diploma..2009.2013,
           y=PCTPOVALL_2013)) +
  geom_point() +
  ggtitle("Fig # : Poverty Rate and High School Completion (for States)") +
  labs(x = "Percent of adults with less than highschool diploma",  
       y="Percent of population in poverty") 
baseplot

Let's add a horizontal line at the mean for

   plot2 <- baseplot + geom_hline(aes(yintercept=mean(PCTPOVALL_2013), color = "blue")) 
   plot2

What would be the equation for that line?

    regression0 <- lm(PCTPOVALL_2013 ~ 1, data = education_and_poverty)
    coefficients(regression0)

What is the slope of a horizontal line? Write out the equation for the line:

Let's also add a line representing the mean for the x variable: Percent of adults with less than high school diploma.

 plot3 <- plot2 +
     geom_vline(
            aes(xintercept =
              mean(Percent.of.adults.with.less.than.a.high.school.diploma..2009.2013),
              color='red'
              )
            ) 
 plot3

What would the equation for the vertical line be? (Kind of a tricky question.)

If you look at the four areads of the graph, what do you notice about where the points are? What pattern do you see?

If you drew a straight line that went through the point where the vertical and horizontal lines cross (the point representing the mean of the x and the mean of the y) what would the slope be?

Write your answer here:

Instead of drawing this line, we want to come up with a general formula or strategy for making this line.

Statisticans came up with the "least squares" line as an approach. We already know that the horizontal line will create the smallest total sum of squares of any possible single value.

But what if instead of using the mean to calculate the deviation, we could use different values depending on the value of the x variable. So instead of

Sum(y - mean(y))^2
We would have something like Sum(y - value(y from the line) )^2
and make that as small as possible.

In our data the values of Percent of Population in poverty from the line are going to be below the mean(Percent of population in poverty) when Percent of adults with less than a high school diploma is below the mean (Percent of adults with less than a high school diploma).

Likewise, in our data the values of Percent of Population in poverty from the line are going to be above the mean(Percent of population in poverty) when Percent of adults with less than a high school diploma is above the mean(Percent of adults with less than a high school diploma).

Let's add the least squares line

 plot3 + stat_smooth(method = lm, se=FALSE)

What would be the equation for that line?

  regression1 <- lm(PCTPOVALL_2013 ~
                Percent.of.adults.with.less.than.a.high.school.diploma..2009.2013, 
                data = education_and_poverty)
 coefficients(regression1)

What is the intercept for the line?

Remember from algebra the y intercept is the value of Y when X is 0. According to this line what percent of people would be expected to live in poverty even if 0% of the adult population had less than a high school diploma (that is if all adults had at least a high school diploma)?

What is the slope of the line?

Remember the slope is the change in y (Percent of population in poverty) associated with a 1 unit change in x (Percent of adults with less than a high school diploma).

Write the equation for the line here:

Write what the equation means in a sentence:

What is the the value of y on the line when x is 12.4?

SOC345/lehmansociology documentation built on May 9, 2019, 11:41 a.m.