Analysis of Relationships among Variables Joel D. Sherman, Ph.D. Secretariat of Public Education – Mexico UNESCO Regional Office for Latin America and the Caribbean Nassau, Bahamas 9-10 December 2008

Why Analyze Relationships? Understand factors that may contribute to differences across countries. Identify potential “causes” of differences across countries. “Predict” future values on key measures of interest based on past and current values and factors that contribute to them. Determine consistency of indicators that reflect a similar concept.

Presenting Relationships among Variables: Cross-Tabulations Non-statistical relationships (presented in tables). Generally shows relationships between only two variables. Stratification on one variable, breakdown into groups such as quartiles or quintiles, averages for each variable calculated and compared. Upward progression on averages for both variables suggests a positive relationship between them. Upward progression on one variable with downward progression on the second suggests a negative relationship between them. No common pattern suggests “no relationship” between the variables.

Cross-Tabulations Suggesting a Positive Relationship Between Variables Cross-Tabulation Suggesting a Positive Relationships Between Variables Quartiles Variable A Variable B Quartile 1 (lowest) 25.3 16.8 Quartile 2 28.5 19.2 Quartile 3 33.7 21.4 Quartile 4 35.4

Cross-Tabulations Suggesting a Negative Relationship Between Variables Cross-Tabulation Suggesting a Negative Relationships Between Variables Quartiles Variable A Variable B Quartile 1 (lowest) 25.3 Quartile 2 28.5 21.4 Quartile 3 33.7 19.2 Quartile 4 35.4 16.8

Cross-Tabulations Suggesting “No Relationship” Between Variables Quartiles Variable A Variable B Quartile 1 (lowest) 25.3 21.4 Quartile 2 28.5 Quartile 3 33.7 16.8 Quartile 4 35.4 19.2

Presenting Relationships among Variables: Correlation Coefficients Measure s linear relationships between two variables. Range is from +1.0 to -1.0. Correlation of +1.0 shows a one-to-one positive correspondence between two variables; as one variable increases in value, there is a corresponding increase in the second variable. Correlation of -1.0 shows a one-to-one negative correspondence between two variables; as one variable increases in value, there is a corresponding decrease in the second variable. Correlation of 0.0 shows “no relationship” between two variables.

Strength of Correlations Correlation Coefficients Positive Negative No Relationship +0.01 to 0.24 -0.01 to -0.24 Weak +0.25 to +0.49 -0.25 to -0.49 Moderate +0.50 to +0.69 -0.50 to -0.69 Strong +0.70 to +1.00 -0.70 to -1.00

Correlation of +1.0 Between Variables Country Variable A Variable B Country A 43.0 21.2 Country B 57.3 23.5 Country C 62.5 24.8 Country D 72.5 27.9 Country E 73.2 29.2 Country F 86.4 30.7 Country G 89.1 38.5 Country H 95.1 40.2

Correlation of -1.0 Between Variables Country Variable A Variable B Country A 43.0 40.2 Country B 57.3 38.5 Country C 62.5 30.7 Country D 72.5 29.2 Country E 73.2 27.9 Country F 86.4 24.8 Country G 89.1 23.5 Country H 95.1 21.1

Graphing Relationships Between Two Variables: Positive Correlation

Graphing Relationships Between Two Variables: Negative Correlation

Limitations of Cross-Tabulations and Correlation Analysis Show relationships but not “causation”. Not always clear which is the independent variable. Doesn’t take into account other factors that may influence variable of interest. Results could be influenced by one or two units, which could be different if these units were removed. Doesn’t examine “how much” an increment in one unit relates to an increment in the other unit (percentage change).

Relationship Between Graduation Ratios and Educational Expectancy Upper Secondary Graduation Ratios and Educational Expectancy - 2005 Country Ratio Expectancy Argentina 43.0 17.5 Germany 99.7 17.4 Brazil 72.5 17.0 Hungary 84.3 17.8 Chile 73.2 15.2 Indonesia 43.5 12.1 China 57.3 12.5 Ireland 90.7 Czech Republic 89.1 17.2 Italy 81.6 17.1 Denmark 86.4 19.1 Jordan 72.8 14.0 Egypt 62.5 13.2 Malaysia 86.6 13.5 Finland 95.1 20.6 Mexico 40.4 13.8

First Steps in Organizing the Data Upper Secondary Graduation Ratios and Educational Expectancy - 2005 WEI Countries Ratio Expectancy OECD Countries Ratios Argentina 43.0 17.5 Czech Republic 89.1 17.2 Brazil 72.5 17.0 Denmark 86.4 19.1 Chile 73.2 15.2 Finland 95.1 20.6 China 57.3 12.5 Germany 99.7 17.4 Egypt 62.5 13.2 Hungary 84.3 17.8 Indonesia 43.5 12.1 Ireland 90.7 Jordan 72.8 14.0 Italy 81.6 17.1 Malaysia 86.6 13.5 Mexico 40.4 13.8 WEI Mean (All) 61.4 14.2 OECD Mean (All) 81.1 17.7

Making the Data More Useful Upper Secondary Graduation Ratios and Educational Expectancy - 2005 Country Ratio Expectancy Argentina 43.0 17.5 Mexico 40.4 13.8 Indonesia 43.5 12.1 Italy 81.6 17.1 China 57.3 12.5 OECD Mean (All) 81.1 17.7 WEI Mean (All) 61.4 14.2 Hungary 84.3 17.8 Egypt 62.5 13.2 Denmark 86.4 19.1 Brazil 72.5 17.0 Czech Republic 89.1 17.2 Jordan 72.8 14.0 Ireland 90.7 17.4 Chile 73.2 15.2 Finland 95.1 20.6 Malaysia 86.6 13.5 Germany 99.7

Final Steps in Organizing the Data Upper Secondary Graduation Ratios and Educational Expectancy – 2005 WEI Countries Ratio Expectancy OECD Countries Argentina 43.0 17.5 Mexico 40.4 13.8 Indonesia 43.4 12.1 OECD Mean (All) 81.6 17.7 China 57.3 12.5 Italy 17.1 WEI Mean (All) 61.4 14.2 Hungary 84.3 17.8 Egypt 62.5 13.2 Denmark 86.4 19.1 Brazil 72.5 17.0 Czech Republic 89.1 17.2 Jordan 72.8 14.0 Ireland 90.7 17.4 Chile 73.2 15.2 Finland 95.1 20.6 Malaysia 86.6 13.5 Germany 99.7

Graphing Relationship Between Ratios and Expectancy

Graphing Relationship Between Ratios and Expectancy (2)

Finding “Outliers” Upper Secondary Graduation Ratios and Educational Expectancy - 2005 Upper Secondary Ratios (Mean 72.7) Educational Expectancy (Mean 16.2) Below Mean Above Mean China Egypt Indonesia Mexico Argentina Brazil Chile Jordan Malaysia Czech Republic Denmark Finland Germany Hungary Ireland

What Can We Say?: General Findings WEI countries have a lower mean upper secondary graduation ratio and educational expectancy than OECD countries. Excluding Greece and India, the mean graduation ratio in WEI countries is 20 points lower than the OECD mean; educational expectancy is 3 years lower in WEI countries than in OECD countries. The ranges in WEI countries are narrower on both measures than in OECD countries. In WEI countries, the range in upper secondary graduation ratios is about 45 points; in OECD countries, it is almost 60 points. Similarly, the range in educational expectancy is about 5 years in WEI countries, but nearly 9 years in OECD countries.

What Can We Say?: General Findings (continued) Overall, there is a moderate correspondence between countries’ upper secondary graduation ratios and their total years of expected education. However, the correspondence is primarily in OECD countries (correlation +0.60) and not in WEI countries (correlation +0.06). In OECD countries with a mean upper secondary graduation ratio below 50, the mean educational expectancy is only 13 years; in countries with a mean ratio between 90 and 100, educational expectancy averages just over 18 years.

What Can We Say?: General Findings (continued) In WEI countries, countries with both low and high upper secondary graduation ratios have similar years of educational expectancy – 14.4 years in countries with a mean graduation ratio under 50 and 13.3 years in countries with a mean ratio between 80 and 89. A large majority of countries with available data demonstrate a consistency in their upper secondary graduation ratios and their levels of educational expectancy. Twelve countries fall below the median on both measures and another 12 are above the median on both measures.

What Can We Say?: General Findings (continued) Several countries show less consistency in their upper secondary graduation ratios and levels of educational expectancy. Slovakia, Malaysia, Switzerland and the Republic of Korea all are above the median in their graduation ratios, but below the median in their levels of educational expectancy. In contrast, Argentina, New Zealand and Spain fall above the median on the first measure, but below on the second.

Country Focus: Argentina In Argentina, there are significant differences between the upper secondary graduation ratio and the years of expected education. Argentina’s upper secondary graduation ratio (43.0) is the third-lowest among WEI and OECD countries with available data, while its level of educational expectancy (17.5 years) is more consistent with those found in OECD than WEI countries.

Country Focus: Malaysia In Malaysia, there are significant differences between the upper secondary graduation ratio and the years of expected education. Malaysia’s upper secondary graduation ratio 86.6 is well above the country mean for all countries and is more consistent with those found in OECD than WEI countries. However, its level of educational expectancy (13.5 years) is the sixth-lowest among countries with available data.

Country Focus: Thailand Thailand is typical of countries in their relationship between upper secondary graduation ratios and years of expected education. The upper secondary graduation ratio of 65.4 and the 16.1 years of expected education both fall below the median of WEI and OECD countries with available data.