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Determining and Interpreting Associations Among Variables

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Ch 182 Associative Analyses Associative analyses: determine where stable relationships exist between two variables Examples –What methods of doing business are associated with level of customer satisfaction? –What demographic variables are associated with repeat buying of Brand A? –Is type of sales training associated with sales performance of sales representatives? –Are purchase intention scores of a new product associated with actual sales of the product?

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Ch 183 Relationships Between Two Variables Relationship: a consistent, systematic linkage between the levels or labels for two variables “Levels” refers to the characteristics of description for interval or ratio scales…the level of temperature, etc. “Labels” refers to the characteristics of description for nominal or ordinal scales, buyers v. non-buyers, etc. As we shall see, this concept is important in understanding the type of relationship…

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Ch 184 Relationships Between Two Variables Nonmonotonic: two variables are associated, but only in a very general sense; don’t know “direction” of relationship, but we do know that the presence (or absence) of one variable is associated with the presence (or absence) of another.

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Ch 185 Nonmonotonic Relationship

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Ch 186 Relationships Between Two Variables Monotonic: the general direction of a relationship between two variables is known –Increasing –Decreasing Shoe store managers know that there is an association between the age of a child and shoe size. The older a child, the larger the shoe size. The direction is increasing, though we only know general direction, not actual size.

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Ch 187 Monotonic Increasing Relationship

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Ch 188 Relationships Between Two Variables Linear: “straight-line” association between two variables Here knowledge of one variable will yield knowledge of another variable “100 customers produce $500 in revenue at Jack-in- the-Box” (p. 525)

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Ch 189 Relationships Between Two Variables Curvilinear: some smooth curve pattern describes the association Example: Research shows that job satisfaction is high when one first starts to work for a company but goes down after a few years and then back up after workers have been with the same company for many years. This would be a U-shaped relationship.

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Ch 1810 Characterizing Relationships Between Variables 1.Presence: whether any systematic relationship exists between two variables of interest 2.Direction: whether the relationship is positive or negative 3.Strength of association: how strong the relationship is: strong? moderate? weak? Assess relationships in the order shown above.

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Ch 1811 Cross-Tabulations Cross-tabulation: consists of rows and columns defined by the categories classifying each variable…used for nonmonotonic relationships Cross-tabulation table: four types of numbers in each cell –Frequency –Raw percentage –Column percentage –Row percentage

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Ch 1812 Cross-Tabulations Using SPSS, commands are ANALYZE, DESCRIPTIVE STATISTICS, CROSSTABS You will find a detailed discussion of cross- tabulation tables in your text, pages 528-531.

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Ch 1813 Cross-Tabulations When we have two nominal-scaled variables and we want to know if they are associated, we use cross-tabulations to examine the relationship and the Chi-Square test to test for presence of a systematic relationship. In this situation: two variables, both with nominal scales, we are testing for a nonmonotonic relationship.

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Ch 1814 Chi-Square Analysis Chi-square (X2) analysis: is the examination of frequencies for two nominal-scaled variables in a cross-tabulation table to determine whether the variables have a significant relationship. The null hypothesis is that the two variables are not related. Observed and expected frequencies:

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Ch 1815 Cross-Tabulations Undersøgelses- spørgsmål Relevante variabler AnalyseteknikForventet output Er der sammenhæng mellem køn og planer for at gå videre på universitet? s_16 uni_ind Krydstabel for de to variabler Procentberegning til vurdering af retning Contingency coefficient til vurdering af styrken Chi-square test til vurdering signifikans Procentvis fordeling af mænd og kvinder på planerne for at gå videre på universitet og vurdering af evt. forskels betydning

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Ch 1816 Cross-Tabulations

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Ch 1817 Cross-Tabulations But while we can “see” this association, how do we know there is the presence of a systematic association? In other words, is this association statistically significant? Would it likely appear again and again if we sampled other students? We use the Chi-Square test to tell us if nonmonotonic relationships are really present.

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Ch 1818 Cross-Tabulations Using SPSS, commands are ANALYZE, DESCRIPTIVE STATISTICS, CROSSTABS and within the CROSSTABS dialog box, STATISTICS, CHI-SQUARE.

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Ch 1819 Chi-Square Analysis Chi-square analysis: assesses nonmonotonic associations in cross-tabulation tables and is based upon differences between observed and expected frequencies Observed frequencies: counts for each cell found in the sample Expected frequencies: calculated on the null of “no association” between the two variables under examination

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Ch 1820 Chi-Square Analysis Computed Chi-Square values:

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Ch 1821 Chi-Square Analysis The chi-square distribution’s shape changes depending on the number of degrees of freedom The computed chi-square value is compared to a table value to determine statistical significance

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Ch 1822 Chi-Square Analysis How do I interpret a Chi-square result? –The chi-square analysis yields the probability that the researcher would find evidence in support of the null hypothesis if he or she repeated the study many, many times with independent samples. –If the P value is < or = to 0.05, this means there is little support for the null hypothesis (no association). Therefore, we have a significant association…we have the PRESENCE of a systematic relationship between the two variables.

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Ch 1823 Cross-Tabulations Read the P value (Asympt. Sig) across from Pearson Chi-Square. Since the P value is >0.05, we have a NON-SIGNIFICANT association.

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Ch 1824 Correlation Coefficients and Covariation The correlation coefficient: is an index number, constrained to fall between the range of −1.0 and +1.0. The correlation coefficient communicates both the strength and the direction of the linear relationship between two metric variables.

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Ch 1825 Correlation Coefficients and Covariation The amount of linear relationship between two variables is communicated by the absolute size of the correlation coefficient. The direction of the association is communicated by the sign (+, -) of the correlation coefficient. Covariation: is defined as the amount of change in one variable systematically associated with a change in another variable.

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Ch 1826 Measuring the Association Between Interval- or Ratio-Scaled Variables In this case, we are trying to assess presence, direction and strength of a monotonic relationship. We are aided in doing this by using: Using SPSS, commands are ANALYZE, CORRELATE, BIVARIATE. Pearson Product Moment Correlation

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Ch 1827 Correlation Coefficients and Covariation Covariation can be examined with use of a scatter diagram.

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Ch 1828 Pearson Product Moment Correlation Coefficient (r) Presence? Determine if there is a significant association. The P value should be examined FIRST! If it is significant, there is a significant association. If not, there is no association. Direction? Look at the coefficient. Is it positive or negative?

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Ch 1829 Pearson Product Moment Correlation Coefficient (r) Strength? The correlation coefficient (r) is a number ranging from -1.0 to +1.0. the closer to 1.00 (+ or -), the stronger the association. There are “rules of thumb”…

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Ch 1830 Rules of Thumb Determining Strength of Association A correlation coefficient’s size indicates the strength of association between two variables. The sign (+ or -) indicates the direction of the association

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Ch 1831 Pearson Product Moment Correlation Coefficient (r) Pearson product moment correlation: measures the degree of linear association between the two variables.

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Ch 1832 Pearson Product Moment Correlation Coefficient (r) Special considerations in linear procedures: –Correlation takes into account only the relationship between two variables, not interaction with other variables. –Correlation does not demonstrate cause and effect. –Correlations will not detect non-linear relationships between variables.

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Ch 1833 When there is NO association, the P value for the Pearson r will be >0.05.

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Ch 1834 When there IS association, the P value for the Pearson r will be < or =0.05. Examples: negative association between sales force rewards and turnover; positive association between length of sales force training and sales.

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Ch 1835 Example Undersøgelses- spørgsmål Relevante variabler AnalyseteknikForventet output Er der lineær sammenhæng mellem kriterierne ” socialt samvær med andre studerende” og ”Ligelig fordeling mellem drenge og piger” s_8_1 s_8_2 Begge antages intervalskalerede Pearson’s korrelationskoefficient Fortegn til vurdering retning Numerisk størrelse (0-1) til vurdering af styrken Test til vurdering signifikans Et enkelt mål for styrke, retning og statistisk signifikans

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Ch 1836 Undersøgelses- spørgsmål Relevante variabler AnalyseteknikForventet output Er der lineær sammenhæng mellem kriterierne ” socialt samvær med andre studerende” og ”Ligelig fordeling mellem drenge og piger” s_8_1 s_8_2 Begge antages intervalskalerede Pearson’s korrelationskoefficient Fortegn til vurdering retning Numerisk størrelse (0-1) til vurdering af styrken Test til vurdering signifikans Et enkelt mål for styrke, retning og statistisk signifikans Read the Pearson correlation coefficient Since the estimated value is flagged with **, we have a SIGNIFICANT association at the 1%-level The table is a (simple) example of a correlation matrix

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Ch 1837 Undersøgelses- spørgsmål Relevante variabler AnalyseteknikForventet output Er der lineær sammenhænge mellem kriterierne for valg af studium? s_8_1 … s_8_17 Begge antages intervalskalerede Pearson’s korrelationsmatrix Fortegn til vurdering retning Numerisk størrelse (0-1) til vurdering af styrken Test til vurdering signifikans Et enkelt mål for styrke, retning og statistisk signifikans for hvert par af variabler Uddrag:

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Ch 1838 Concluding Remarks on Associative Analyses Researchers will always test the null hypothesis of NO relationship or no correlation. When the null hypothesis is rejected, then the researcher may have a managerially important relationship to share with the manager.

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Examining Relationships in Quantitative Research

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