Slide 1

SOLVING THE HOMEWORK PROBLEMS

Pearson's r correlation coefficient measures the strength of the linear relationship between the distributions of two quantitative variables. If the relationship is not linear, the application of statistics that assume linearity may give questionable results. Determining whether a relationship should be characterized as linear or non-linear is challenging.

One indicator of non-linearity is the difference between the rank-order correlation correlation coefficient (Spearman's rho) and Pearson's r. When Spearman's rho is larger than Pearson's r, the relationship is likely to be non-linear, and Pearson's r may understate the strength of the relationship.

However, we can improve the linearity of the relationship and justify the use of statistics that assume linearity if one or both variables are badly skewed due to outliers, but can be corrected by re-expressing the data.

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The introductory statement in the question indicates:• The data set to use (2001WorldFactBook)• The task to accomplish (association between

variables)• The variables to use in the analysis: percent of the

total population who was literate [literacy] HIV-AIDS adult prevalence rate [hivaids]

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These problem also contains a second paragraph of instructions that provide the formulas to use if our examination of the association between quantitative variables requires us to re-express or transform one or both of the variables.

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The first statement concerns the number of valid cases. To answer this question, we produce the statistics using the SPSS Correlate procedure.

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To compute correlations, select Correlate > Bivariate from the Correlate menu.

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First, move the variables hivLive and literacy to the Variables list box.

Second, mark the check box for Spearman and leave the check box for Pearson marked.

Third, click on the OK button to produce the output.

Slide 7

The Correlations table shows us the number of cases available for computing the correlation – 131.

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The number of cases with valid data to analyze the relationship between "percent of the total population who was literate" and "number of people living with HIV-AIDS" was 131, out of the total of 218 cases in the data set.

Click on the check box to mark the statement as correct.

Slide 9

The next statement asks us to extract the correlation coefficients from the SPSS output, and compare the two.

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The Correlations table shows us that the Pearson r correlation is -0.203.

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The Nonparametric Correlations table shows us that the Spearman rho correlation is -0.548.

The comparison of the strength of the relationship indicated by each measure is based on the relative size of the absolute values of the coefficients. Since the absolute value of Spearman's rho (0.548) is larger than the absolute value of Pearson's r (0.203), Spearman's rho indicates a stronger relationship.

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Pearson's r was correctly identified as -0.203. Spearman's rho was correctly identified as -0.548.

The comparison of the strength of the relationship indicated by each measure is based on the relative size of the absolute values of the coefficients. Since the absolute value of Spearman's rho (0.548) is larger than the absolute value of Pearson's r (0.203), Spearman's rho indicates a stronger relationship.

Click on the check box to mark the statement as correct.

Slide 13

The next statement asks us to identify the outliers in the distribution. If there are outliers, we can re-express one or both variables to see if the linear correlation between the variables increases.

In these problems, outliers are defined as cases with scores that are three or more standard deviations from the mean. We use the SPSS Descriptives procedure to identify them.

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To compute the standard scores, select the Descriptive Statistics > Descriptives command from the Analyze menu.

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First, move the variable for the analysis hivlive and literacy to the Variable(s) list box.

Second, click on the Options button to request skewness in case we decide to re-express the variable.

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First, click the check box for Skewness so that we can decide which transformation to use should we decide to re-express the variable.

Second, click on Continue button to return to the prior dialog box.

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Finally, click on the OK button to produce the output.

Next, mark the check box Save standardized values as variables.

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If we scroll the Data View all the way to the right, we see that SPSS has created the standard scores. To name the standard score variables, SPSS prepends the letter “Z” to the variable name.

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Click the right mouse button on the column header for Zhivlive, and select Sort Ascending from the pop-up menu. This will show any negative outliers at the top of the column.

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After we scroll down past the cases with missing values, we do not see any negative values less than or equal to -3.0.

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Click the right mouse button again on the column header for Zpop, and select Sort Descending from the pop-up menu. This will show any positive outliers at the top of the column.

Click the right mouse button again on the column header for Zhivlive, and select Sort Descending from the pop-up menu. This will show any positive outliers at the top of the column.

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At the top of the column, we see four positive outliers with values greater than or equal to +3.0.

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If we scroll back to the country column, we see the names for the four outliers: Ethiopia, Kenya, Nigeria, and South Africa.

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Scroll right to the Zliterarcy column and click the right mouse button on the column header for Zliterarcy, and select Sort Ascending from the pop-up menu. This will show any negative outliers at the top of the column.

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After we scroll down past the cases with missing values, we see that we have one negative value less than or equal to -3.0.

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Click the right mouse button again on the column header for Zliteracy, and select Sort Descending from the pop-up menu. This will show any positive outliers at the top of the column.

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At the top of the column, we see that there are not any standard scores equal to or greater than +3.0.

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If we select the row with the negative outlier and scroll back to the country column, we see that Niger was an outlier on the variable literacy.

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The distribution of "number of people living with HIV-AIDS" [hivlive] contained four outliers that were three or more standard deviations from the mean: Ethiopia with a value of 3,000,000 (z=4.81), Kenya with a value of 2,100,000 (z=3.25), Nigeria with a value of 2,700,000 (z=4.29), and South Africa with a value of 4,200,000 (z=6.88). The distribution of "percent of the total population who was literate" [literacy] contained one outlier that was three or more standard deviations from the mean: Niger with a value of 13.6 (z=-3.02).

We mark the check box for correct.

Slide 30

Since Spearman’s rho indicated a stronger relationship than Pearson’s r in distributions that had outliers, we can re-express the variables to see if the strength of the linear relationship increases.

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In the table of Descriptive Statistics, we see that the skewness for number of people living with HIV-AIDS [hivlive] was 4.412. Since the variable was positively skewed, the data will be re-expressed as logarithms, transforming it to the log transformation of number of people living with HIV-AIDS [LG_hivlive].

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We type the formula identified in the second paragraph of the problem.

Click on the OK button to produce the output.

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The skewness for "percent of the total population who was literate" [literacy] was -1.112. Since the variable was negatively skewed, the data will be re-expressed as squares. The independent variable percent of the total population who was literate [literacy] was transformed to the square transformation of percent of the total population who was literate [SQ_literacy].

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We type the formula identified in the second paragraph of the problem.

Click on the OK button to produce the output.

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Next, we compute the correlation coefficients for the transformed variables.

To compute correlations, select Correlate > Bivariate from the Correlate menu.

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First, move the variables LG_hivLive and SQ_literacy to the Variables list box.

Second, click on the OK button to produce the output.

Slide 37

The linear fit of the relationship improved, and we report the Pearson's r for the relationship using the transformed variables.

Before re-expressing the data, Pearson's r was -0.203. After re-expressing both the dependent variable number of people living with HIV-AIDS [hivlive] as the log transformation of number of people living with HIV-AIDS [LG_hivlive] and the independent variable percent of the total population who was literate [literacy] as the square transformation of percent of the total population who was literate [SQ_literacy], Pearson's r increased to -0.487.

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Since the strength of the linear relationship increased when we used the transformed variables, we mark the check box for the question.

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The next statement asks us to interpret the Pearson correlation coefficient using the guidelines attributed to Tukey.

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Using the rule of thumb attributed to Tukey, a correlation between 0.0 and ±0.20 is very weak; ±0.20 to ±0.40 is weak; ±0.40 to ±0.60 is moderate; ±0.60 to ±0.80 is strong; and greater than ±0.80 is very strong, the relationship between the square transformation of "percent of the total population who was literate" and the log transformation of "number of people living with HIV-AIDS"was correctly characterized as a moderate relationship (Pearson's r = -.487).

Click on the check box to mark the statement as correct.

Slide 41

The next statement asks us to interpret the Pearson correlation coefficient using the guidelines attributed to Cohen.

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Applying Cohen's criteria for effect size (less than ±0.10 = trivial; ±0.10 up to ±0.30 = weak or small; ±0.30 up to ±0.50 = moderate; ±0.50 or greater = strong or large), the relationship was correctly characterized as a moderate relationship (Pearson's r = -.487).

Click on the check box to mark the statement as correct.

Slide 43

The next statement asks us to interpret the direction of the relationship between the variables. A direct or positive relationship means that the values of the variables change in the same direction, i.e. when one goes up or down, the other goes up or down. An inverse or negative relationship means that the values of the variables move in different directions.

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Since the sign of the correlation coefficient was negative (Pearson's r = -.487), the relationship is inverse and the values for the variables move in opposite directions. The statement that “higher scores on the variable the square transformation of percent of the total population who was literate were associated with lower scores on the log transformation of number of people living with HIV-AIDS” is correct.

Click on the check box to mark the statement as correct.