Topic IV: Correlating Two Continuous Variables

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IV. MEASURING RELATIONSHIPS: Two Continuous Variables

OCTOBER 12
THE PRODUCT-
MOMENT CORRELATION

SPSS Guide, Ch.11: "Bivariate and Partial Correlations," only 177-183.

If your variables are continuous, or if you can treat them as points along a conceptual continuum, relationships can be measured and expressed precisely and concisely through the twin techniques of product-moment correlation and regression. Correlational analysis is one of the most common techniques in social research. In essence, it tells us in what direction two variables are related and how strongly they are related.

OCTOBER 15
INTERPRETING THE PRODUCT-MOMENT CORRELATION

Also see

CORRELATION ARTIFACTS

SPSS Guide, Ch.11: "Bivariate and Partial Correlations," only 183-187. Read about the Spearman Rank Correlation, but you need not concentrate on this statistic. The section on "Partial Correlations" is more important.
Schmidt, Ch. 6: "Correlation: Measuring Relations," selections distributed in class on October 15.

The strength of association between two interval-level variables is actually expressed by the coefficient of determination, which is simply the square of the product moment correlation. Field tests several hypotheses about the policies of different nations toward abortion, including one holding that Catholic countries would have more conservative abortion policies than non-Catholic ones. Are these hypotheses supported or refuted by her data analysis?

Assignment: Formulate a hypothesis about the effect of socioeconomic characteristics on politics in American states and test it using data from the nustates2000 file. You must select a dependent variable (the political outcome you wish to explain) and an independent variable (the socioeconomic characteristic that is a likely explanation of the outcome).

From the Analyze Menu choose Correlate and then Bivariate

Transfer the variables you want to correlate into the right hand window.

Before clicking on OK to run the correlation, click on Options

Check off both boxes in the "Statistics" area

Then click the "Continue" button and then "OK" to run the correlation.

Print the output for this analysis and bring it to class on Monday. I will give a valuable PRIZE to the student who turns up the highest meaningful correlation in support of his or her chosen hypothesis.

Determine for yourself whether the hypothesis was supported by the data. Using the statistics generated from the sums of squares for the two variables and their cross-products try to compute the correlation value, r, as explained in the lecture notes. Try to work this out on your own before I show how in class. A QUESTION LIKE THIS WILL BE ON THE 2/3 EXAM.

OCTOBER 16
LINEAR REGRESSION

SPSS Guide, Ch.12: "Simple and Multiple Linear Regression," only 189-204.
Schmidt, "Ch. 7: Regression: Predicting Future Performance from Past Performance," pp. 183-198. Chapter distributed in class.

We visualized the value of the product-moment correlation, r, as indicating the degree to which the observations hugged "the best fitting straight line" without actually calculating that line. Regression analysis requires doing that. Whereas correlational analysis tells the strength of a relationship, regression analysis indicates the specific form of a relationship. Knowing the mathematical form of a relationship allows one to predict a value of a dependent variable (Y) with knowledge of an independent variable (X). A great deal of advanced statistical analysis is predicated on the ideas in simple linear regression. Learn this well.

OCTOBER 17

Optional Session: Review for 1/3 examination

Go here to view a "pre-test"--sample questions for the 1/3 exam. The discussion section will be based on this pre-test, which illustrates the type of questions that will appear on the 1/3 exam.

OCTOBER 18
RESEARCH APPLICATIONS

Marilyn J. Field, "Determinants of Abortion Policy in the Developed Nations," Policy Studies Journal, (Summer, 1979), 771-781. (On website)

Assignment: Earlier versions of SPSS contained a simple PLOT procedure that also produced regression statistics. SPSS 10 has dropped that simple procedure, but one can get a plot with regression statistics. This procedure is described by Ms. Wuyi Wang, a former TA in this class, who is now a technical consultant at SPSS, Inc. She writes:

Go to Graphs->Interactive->Scatterplot

drag and drop the variables to the vertical axis and horizontal axis.

Right-click on each variable to make sure that Scale is selected

(if the variable is tagged as Ordinal no regression line will be shown).

Click on the Fit tab. Select Regression from the drop-down list (the default is None).

Leave all other settings as default.

Click OK.

A scatterplot made this way has the regression line and R-squared. As you did when you formulated your hypothesis for Monday, choose a pair of variables that you'd like to correlate. But this time, follow Wuyi Wang's instructions. More instructions are given in the lecture notes for this day.

OCTOBER 19

1/3 EXAMINATION -- it will consist of 30 questions, mostly calling for short answer or selection of multiple responses. It will also ask for some elementary computations that require your understanding of key formulae in statistics.

OCTOBER 22

Discussion of examination results

IV. MEASURING RELATIONSHIPS: Two Continuous Variables
OCTOBER 12 THE PRODUCT- MOMENT CORRELATION		SPSS Guide, Ch.11: "Bivariate and Partial Correlations," only 177-183. If your variables are continuous, or if you can treat them as points along a conceptual continuum, relationships can be measured and expressed precisely and concisely through the twin techniques of product-moment correlation and regression. Correlational analysis is one of the most common techniques in social research. In essence, it tells us in what direction two variables are related and how strongly they are related.
OCTOBER 15 INTERPRETING THE PRODUCT-MOMENT CORRELATION Also see CORRELATION ARTIFACTS		SPSS Guide, Ch.11: "Bivariate and Partial Correlations," only 183-187. Read about the Spearman Rank Correlation, but you need not concentrate on this statistic. The section on "Partial Correlations" is more important. Schmidt, Ch. 6: "Correlation: Measuring Relations," selections distributed in class on October 15. The strength of association between two interval-level variables is actually expressed by the coefficient of determination, which is simply the square of the product moment correlation. Field tests several hypotheses about the policies of different nations toward abortion, including one holding that Catholic countries would have more conservative abortion policies than non-Catholic ones. Are these hypotheses supported or refuted by her data analysis? Assignment: Formulate a hypothesis about the effect of socioeconomic characteristics on politics in American states and test it using data from the nustates2000 file. You must select a dependent variable (the political outcome you wish to explain) and an independent variable (the socioeconomic characteristic that is a likely explanation of the outcome). From the Analyze Menu choose Correlate and then Bivariate Transfer the variables you want to correlate into the right hand window. Before clicking on OK to run the correlation, click on Options Check off both boxes in the "Statistics" area Then click the "Continue" button and then "OK" to run the correlation. Print the output for this analysis and bring it to class on Monday. I will give a valuable PRIZE to the student who turns up the highest meaningful correlation in support of his or her chosen hypothesis. Determine for yourself whether the hypothesis was supported by the data. Using the statistics generated from the sums of squares for the two variables and their cross-products try to compute the correlation value, r, as explained in the lecture notes. Try to work this out on your own before I show how in class. A QUESTION LIKE THIS WILL BE ON THE 2/3 EXAM.
OCTOBER 16 LINEAR REGRESSION		SPSS Guide, Ch.12: "Simple and Multiple Linear Regression," only 189-204. Schmidt, "Ch. 7: Regression: Predicting Future Performance from Past Performance," pp. 183-198. Chapter distributed in class. We visualized the value of the product-moment correlation, r, as indicating the degree to which the observations hugged "the best fitting straight line" without actually calculating that line. Regression analysis requires doing that. Whereas correlational analysis tells the strength of a relationship, regression analysis indicates the specific form of a relationship. Knowing the mathematical form of a relationship allows one to predict a value of a dependent variable (Y) with knowledge of an independent variable (X). A great deal of advanced statistical analysis is predicated on the ideas in simple linear regression. Learn this well.
OCTOBER 17		Optional Session: Review for 1/3 examination Go here to view a "pre-test"--sample questions for the 1/3 exam. The discussion section will be based on this pre-test, which illustrates the type of questions that will appear on the 1/3 exam.
OCTOBER 18 RESEARCH APPLICATIONS		Marilyn J. Field, "Determinants of Abortion Policy in the Developed Nations," Policy Studies Journal, (Summer, 1979), 771-781. (On website) Assignment: Earlier versions of SPSS contained a simple PLOT procedure that also produced regression statistics. SPSS 10 has dropped that simple procedure, but one can get a plot with regression statistics. This procedure is described by Ms. Wuyi Wang, a former TA in this class, who is now a technical consultant at SPSS, Inc. She writes: Go to Graphs->Interactive->Scatterplot drag and drop the variables to the vertical axis and horizontal axis. Right-click on each variable to make sure that Scale is selected (if the variable is tagged as Ordinal no regression line will be shown). Click on the Fit tab. Select Regression from the drop-down list (the default is None). Leave all other settings as default. Click OK. A scatterplot made this way has the regression line and R-squared. As you did when you formulated your hypothesis for Monday, choose a pair of variables that you'd like to correlate. But this time, follow Wuyi Wang's instructions. More instructions are given in the lecture notes for this day.
OCTOBER 19		1/3 EXAMINATION -- it will consist of 30 questions, mostly calling for short answer or selection of multiple responses. It will also ask for some elementary computations that require your understanding of key formulae in statistics.
OCTOBER 22		Discussion of examination results