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Determining and Interpreting Associations Among Variables

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Determining and Interpreting Associations Among Variables

- 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?

- 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…

- 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.
- At the presence of breakfast, we shall have the presence of orders for coffee.
- At the presence of lunch, we shall have the absence of orders for coffee.

- 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.

- 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)

- 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.

- Presence:whether any systematic relationship exists between two variables of interest
- Direction:whether the relationship is positive or negative
- Strength of association: how strong the relationship is: strong? moderate? weak?
- Assess relationships in the order shown above.

- 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

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

- 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.

- 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:

- Example: Let’s suppose we want to know if there is a relationship between studying and test performance and both of these variables are measured using nominal scales…

- If the chi-square analysis determines that you have a significant relationship (no support for the null hypothesis) you may use the following to determine the nature of the relationship:
- The column percentages table or
- The raw percentages table

- Did you study for the midterm test? __yes __no
- How did you perform on the midterm test? __pass __fail
- Now, let’s look at the data in a crosstabulation table:

- Do you “see” a relationship? Do you “see” the “presence” of studying with the “presence” of passing? Do you “see” the “absence” of passing with the presence of not studying?
- Congratulations! You have just “seen” a nonmonotonic relationship.

- Bar charts can be used to “see” nonmonotonic relationships.

- 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.

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

- 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

- Computed Chi-Square values:

- 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

- 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.

- Read the P value (Asympt. Sig) across from Pearson Chi-Square. Since the P value is <0.05, we have a SIGNIFICANT association.

- How do I interpret a Chi-square result?
- A significant chi-square result means the researcher should look at the cross-tabulation row and column percentages to “see” the association pattern
- SPSS will calculate row, column, (or both) percentages for you. See the CELLS box at the bottom of the CROSSTABS dialog box.

- Look at the ROW %’s: 92% of those who studied passed; almost 70% of those who didn’t study failed. “See” the relationship!

- Presence? Yes, our Chi-Square was significant. This means that the pattern we observe between studying/not studying and passing/failing is a systematic relationship if we ran our study many, many times.
- Direction? Nonmonotonic relationships do not have direction…only presence and absence.

- Strength? Since the Chi-Square only tells us presence, you must judge the strength by looking at the pattern. Don’t you think there is a “strong” relationship between study/not studying and passing/failing?

- When you want to know if there is an association between two variables and…
- Both of those variables have nominal (or ordinal) scales

- 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.

- 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.

- 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

- Covariation can be examined with use of a scatter diagram.

- 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?

- 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”…

- A correlation coefficient’s size indicates the strength of association between two variables.
- The sign (+ or -) indicates the direction of the association

- Pearson product moment correlation: measures the degree of linear association between the two variables.

- 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.

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

- 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.

- What items are associated with preference for a waterfront view among restaurant patrons?
- Are preferences for unusual entrées, simple décor, and unusual desserts associated with preference for waterfront view while dining?
- Since all of these variables are interval-scaled we can run a Pearson Correlation to determine the association between each variable with the preference for waterfront view.

- Using SPSS, commands are ANALYZE, CORRELATE, BIVARIATE.

- The output shows presence, direction and strength of the association.
- Do you see any managerial significance to these associations?

- 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.