Homework
INSTRUCTION FOR DISCUSSION
1. Based on the SPSS/PSPP Bivariate Correlation Lecture complete the following: Is there a relationship between
Per Capita State and Local Govt. Spending for Elem. and Second. Education: 2007 (EDS140) and the percent of the population with a Bachelor’s Degree or More (EDS154)
.
Then do a correlation for percent of the population with a Bachelor’s Degree or More (EDS154) and State Minimum Wage Rates (EMS170). Use the STATES10 data to test this relationship.
Summarize and interpret your findings below:
Now perform the analysis and find:
Per Capita State and Local Govt. Spending for Elem. and Second. Education: 2007 (EDS140) and the percent of the population with a Bachelor’s Degree or More (EDS154)
Correlation __________________
Significance Level __________________
Is the relationship statistically significant? Yes No
Percent of the Population with a Bachelor’s Degree or More (EDS154) and State Minimum Wage Rates (EMS170)
Correlation __________________
Significance Level __________________
Is the relationship statistically significant? Yes No
How would you interpret the results? Do not just restate the statistics without any interpretation. What do you observe in the findings-be specific. Remember the null hypothesis is that there is no association. So, based on the level of significance, can we reject or confirm the null hypothesis? If we can reject the null hypothesis meaning that the correlation statistic is statistically significant, then what do the results tell us? Try to answer as though your audience has little, if any, statistical knowledge but also consider what might be the reasons for the findings (in other words, why?).
Chapter 5: Bivariate Correlation
Correlation and Causation
Before addressing bivariate correlation and Analysis of Variance, it is important to address the
assumptions of provide an overview of correlation and The essence of causality may be captured
by the notion of manipulation. If one could intervene without changing the surrounding
circumstances and make a change in the first thing, a change in the second thing would follow
from the original manipulation.
What are the criteria for causality?
There are several criteria necessary for causation. These include association, temporal order, and
spuriousness.
Association: The first criterion for causality is than an association must exist between
presumed cause and its effect. If two variables do not co-vary, meaning as one changes,
the other changes is a corresponding manner, then neither can be considered a candidate
to exert causal influence on the other.
Temporal Order: For variable A to be considered a causal candidate for the occurrence
of B, it must occur before B in time. Temporal order in the social and behavioral
sciences is often obvious, but because of the feedback nature of many of the things we
study, the order is not always easy to determine. Two things that are occurring at the
same time for example could be neither a cause nor effect for the other.
Spuriousness: The third criterion is that the relationship must not statistically disappear
when the influence of other variables is considered.
There should also be necessary and sufficient cause. A necessary cause or condition is one that
must be present for an effect to follow. A sufficient cause is a cause or condition that by itself is
able to produce an event.
Bivariate Correlation
The next procedure is bivariate correlation. Bivariate correlation is used to evaluate whether
two ratio or scale (in other words, continuous) variables are correlated, in other words, associated
with each other. This procedure should not be used with nominal or ordinal variables. If
necessary, review the discussion regarding levels of measurement in Chapter 2.
First, it is important to remember that correlation is not synonymous with causation. For
example, the number of firemen is positively correlated with fire damage but does it mean that
the firemen cause the fire damage? No! The fire itself, and more specifically the size of the fire,
is the cause of the damage not the number of firemen. Think of it terms of the assumption of
necessary or sufficient cause. Note: We can never completely satisfy the necessary and sufficient
criterion of causality and we never will (Walsh and Ollenburger 2001).
Bivariate correlation is often the first step before doing multivariate regression (Chapter 6). It
allows us to evaluate whether there is an association between two variables absent any other
influence. This means without controls. Multivariate regression allows one to evaluate whether
that association is still present while controlling for other variables (plausible alternative causes).
To illustrate how to do a bivariate correlation, consider the following: Is the percent of students
that have used marijuana negatively associated with high school graduation rates? Using the
STATES10 data set, we can examine this research question using Percent of High School
Students That Use Marijuana: 2007 (HTC255) and Percent of Population Graduated from High
School: 2008 (EDS131)
First, OPEN the STATES10 data set in PSPP
Graphic 5.1
Then, as illustrated in Graphic 5.2 go to
Analyze
Bivariate Correlation (left click)
Graphic 5.2
After left-clicking the following dialogue box appears:
Graphic 5.3
Left-click on the first variable so it is highlighted as shown in Graphic 5.3 and then begin typing
HTC255 (percent of high school students that use marijuana) and the proper variable should
become highlighted as shown in Graphic 5.4.
Graphic 5.4
Then click the arrow in the middle to move this variable HTC255 to the RIGHT so that it
appears as shown in
Graphic 5.5
Graphic 5.5
Following the same procedure but highlight the top variable and then begin typing EDS131 so
that Percent of Population Graduated from High School is highlighted as shown in Graphic 5.6.
Graphic 5.6
Move EDS131 over to the right as you did with HTC255
Graphic 5.7
Then click OK, and you will get an output that looks like what is displayed in Graphic 5.8.
Graphic 5.8
How should you read this output table? First, you have a 2 X 2 table. You need to make sure
you are reading the table properly. Make sure you are reading at the intersection of the two
variables. On the top row, you have the correlation for HTC255 and HTC255 (row 1, column 1)
and HTC255 and EDS131 (row 1, column 2) and in the second row, the correlation for EDS131
and HTC255 (row 2, column 1) and EDS131 and EDS131 (row 2, column 2). So the correlation
for the two variables is available in the cells at row 1, column 2 and row 2, column 1.
So what do the results for the bivariate correlation tell us?
The first line is the Pearson correlation: -.12
The second line is the significance level (Sig. (2-tailed)): .468
The third line is N, which is the number of cases: 51
The Pearson correlation is -.12, which is negative as expected and it indicates that there is a
negative weak correlation between the two variables. Before addressing the significance, the
strength of the association depends on the size of the Pearson correlation, which will be some
value between -1 and 1. The closer to 1 or -1 the Pearson correlation is, the stronger the
relationship. Conversely, the closer to 0 the Pearson correlation is, the weaker the relationship
As noted earlier there appears to be a negative weak correlation. In the social sciences, a
correlation of .40 (or -.40) is considered “strong.”
Graphic 5.9
Back to our example, -.12 is a weak relationship (at best). However, with a level of significance
of .468, which is above the .05 threshold suggests that these two variables are not significantly
correlated and as such the weak association could be due to chance. A scatterplot will visually
illustrate the association or lack there of as shown in Graphic 5.9.
Next, looking at an example so that you can see what a significant bivariate correlation looks
like. Using Percent of Population Graduated from High School: 2008 (EDS131) from the first
example and for the second variable: Per Capita Income: (ECS100). Unlike the previous
example, we will begin with the scatterplot (Graphic 5.10).
Graphic 5.10
Note the upward sloping cluster of most of the observations but to know whether there is a
significant correlation between per capita income and graduation rates. Beginning with the
procedure discussed above, move EDS131 over to the RIGHT as shown in Graphic 5.11. (if you
have been following along, just move HTC255 back to the LEFT if it is still there).
Graphic 5.11
Next move ECS100 over to the right as shown below (Graphic 5.12),
Graphic 5.12
Then click OK and the following output will appear (Graphic 5.13).
Graphic 5.13
So what does the output tell us about the bivariate correlation? Does it confirm what appears in
the above scatterplot (Graphic 5.10)? Based on the output:
The first line is the Pearson correlation: .30
The second line is the significance level (Sig. (2-tailed)): .036
The third line is N, which is the number of cases: 51
The Pearson correlation is .30, which indicates a moderate correlation. The level of significance
is .036, which is less than .05 and so this would indicate that there is a moderate correlation and
that it is statistically significant. This suggests that state graduation rates are higher in states with
higher per capita income. More sophisticated analysis, such as multivariate regression would be
necessary to determine if this correlation is causation. Multivariate regression will be covered in
Chapter 6.
