Lesson -8

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Assignments and Information

Notes
So far in this course we have been looking at describing data from experiments. What change do we see in our dependent variable when we manipulate the independent variable? Correlation asks a very different question. Correlation looks at the statistical relationship between two variables.
Histograms were a graphic representation of frequency distributions. They let us see the basic shape of our distribution, so we would know if it is normal or skewed, unimodal or multimodal, tightly compressed or spread out. For correlation, we use scatterplots to visually inspect our data.

Using Scatterplots

Pearson Correlation Coefficient (also called the Pearson r and the Pearson product-moment Correlation Coefficient) - a statistic that indicates the degree of linear relationship between two variables measured at the interval or ratio level.

Using the Pearson r

Before you can use Pearson r for your data, you must make sure these assumptions have been met.

Correlation and Causality

Correlation can show a relationship between two variables, but you can almost never claim a cause and effect relationship. You don’t want to be the researcher who says that there should be a ban on ice cream sales because you found that there is a strong positive relationship between ice cream sales and boating accidents. While it is true that ice cream sales and boating accidents both increase dramatically in the summer, you can’t really claim that eating ice cream causes boating accidents. There is a possible outside influence (summer) that could play a part in that high correlation.

Coefficient of Determination (r2)

Now lets try some calculations based on our BASC study.

Lets assume that while we are gathering data on the BASC to look for cultural bias, we become curious about how these scores can be used. Is there a relationship between BASC scores and the number of detentions a student will get in a semester? Is there a relationship between BASC scores and scores on a much easier to administer test of sociability? Is there a relationship between BASC scores and IQ? We gather the following data on our 25 Hispanic youth that are participating in our original study.

Using our SPSS program, we fill this data in on the spreadsheet in columns 1-4. Make sure to save the data since we will be using it again in lesson 9. Then go to the tool bar at the top, and click on “graphs”. Slide down the list to “Scatter...” and click again. You want to keep it on “simple” and click “define”. I put the BASC data on the X axis, and the other variables were placed on the Y axis, one at a time. Here are the graphs I got, see if they seem to match yours.

The information typed underneath each graph also came from SPSS. Go up to the tool bar and click “Analyze”, “correlate”, “bivariate” and select the two variables you are interested in.

a Correlations BASC DETENTION BASC Pearson Correlation 1 .855 Sig. (2-tailed) . .000 N 25 25 DETENTION Pearson Correlation .855 1 Sig. (2-tailed) .000 . N 25 25 ** Correlation is significant at the 0.01 level (2-tailed).				This, in English, states that there is a Pearson r of 0.855 which is significant at and alpha of less then 0.01. There are 25 subjects in each group.
b Correlations BASC SOCIABILITY BASC Pearson Correlation 1 -.892 Sig. (2-tailed) . .000 N 25 25 SOCIABILITY Pearson Correlation -.892 1 Sig. (2-tailed) .000 . N 25 25 ** Correlation is significant at the 0.01 level (2-tailed).				This, in English, states that there is a Pearson r of 0.896 which is significant at and alpha of less then 0.01. There are 25 subjects in each group.
c Correlations BASC IQ BASC Pearson Correlation 1 .089 Sig. (2-tailed) . .672 N 25 25 IQ Pearson Correlation .089 1 Sig. (2-tailed) .672 . N 25 25				This, in English, states that there is a Pearson r of 0..089 which is not significant. This is also obvious by the grouping on the graph. There are 25 subjects in each group.

SPSS Tips:

We can see from the graphs and the Pearson r scores that there is a positive relationship between our BASC scores and the number of detentions a student serves. The higher the BASC, the more detentions they are likely to do. We can also see that we have an r of 0.855, which is statistically significant at an alpha of 0.01. (On page 709 in the text you will find the table of critical values of r. We have 25 pairs of scores, so our degrees of freedom are 2 less, which is 23. We would need an r of at least 0.396 to have significance at the alpha of 0.05, or an r of 0.505 for significance at the 0.01 alpha level. We are well above both, and would report our significance at the 0.01 alpha level.)

We can see from the graphs and the Pearson r scores that there is a negative relationship between BASC scores and scores on the sociability scale. (This is because BASC scores get worse as they get higher, and sociability scores get better as they get higher.) The higher a student’s score on the BASC the lower their score is likely to be on the sociability measure. We can also see that we have an r of -0.892, which again is significant at the 0.01 alpha level.

Finally, there seems to be no relationship between BASC scores and IQ scores. The points are all over the graph. The Pearson r for this correlation is 0.089, which is not significant at either the 0.05 or the 0.01 alpha level.

For those of you who want to try the math by hand, to get a better feel for how the formula works, lets double check the computers r for BASC scores and Sociability. I have made a table of the scores, their squares, and the total of the BASC scores times the sociability scores.

We will use the computational formula, since it works directly with our raw scores.

a) We fill in the data from the table	a.
b) We do our multiplying and exponents	b.
c) We do the subtracting within the parentheses	c.
d) We multiply again	d.
e )We find the square root	e.
f) And finally, we divide.	f.

As you can see, we end up with the same r, -0.892, that the SPSS program gave us.

Vocabulary

Bivariate distribution – A distribution in which two scores are obtained from each subject.

Scatterplot – A graph of a bivariate distribution in which the X variable is plotted on the horizontal axis and the Y variable is plotted on the vertical axis.

Correlational studies – Studies in which two or more variables are measured to find the direction and degree to which they covary.

Covary – Two variables covary when a change in one variable is related to a change in the other variable.

Linear relationship – A relationship between two variables that can be described by a straight line.

Positive relationships – A linear relationship between two variables in which as the value of the first variable increases, the value of the second variable tends to increase as well.

Negative relationship – A linear relationship between two variables in which as the value of one variable increases, the value of the other variable tends to decrease.

Curvilinear relationship – A relationship between two variables that is not linear. It starts with a positive or negative trend, but at some midpoint changes direction forming a U-shaped curve.

Near- Zero relationship – A bivariate distribution that has no obvious relationship between variables.

Correlation coefficient – A descriptive statistic that expresses the degree of relationship between two variables.

Pearson correlation coefficient ( r ) – A statistic that indicates the degree of linear relationship between two variables that have been measured in either interval or ratio level.

Sums of Products of X and Y ( SPXY) - The value of SP= Sum(X-)(Y-) for variables X and Y.

Coefficient of determination ( r2 ) – The value of r2 indicating the common variance of variables X and Y.

Spearman rank-order correlation coefficient ( rs) – a correlation coefficient used with ordinal measurements.