Determining and interpreting associations among variables
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Determining and Interpreting Associations Among Variables. 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?

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Associative analyses
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?

Relationships between two variables
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…

Relationships between two variables1
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.

  • 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.

Relationships between two variables2
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.

Relationships between two variables3
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)

Relationships between two variables4
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.

Characterizing relationships between variables
Characterizing Relationships Between Variables

  • 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 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

Cross tabulations1


  • You will find a detailed discussion of cross-tabulation tables in your text, pages 528-531.

Cross tabulations3

  • 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 analysis
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:

Cross tabulations4

  • 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…

Interpreting a significant cross tabulation finding
Interpreting a Significant Cross-Tabulation Finding

  • 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

Cross tabulations5

  • 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:

Cross tabulations6

  • 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.

Cross tabulations7

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

Cross tabulations8

  • 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.

Cross tabulations9


Chi square analysis1
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

Chi square analysis2
Chi-Square Analysis

  • Computed Chi-Square values:

Chi square analysis3
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

Chi square analysis4
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.

Chi square analysis5
Chi-Square Analysis

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

Chi square analysis6
Chi-Square Analysis

  • 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.

Chi square analysis7
Chi-Square Analysis

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

Presence direction and strength
Presence, Direction and Strength

  • 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.

Presence direction and strength1
Presence, Direction and Strength

  • 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 can you use crosstabs and chi square test
When can you use Crosstabs and Chi-Square test?

  • When you want to know if there is an association between two variables and…

  • Both of those variables have nominal (or ordinal) scales

Correlation coefficients and covariation
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.

Correlation coefficients and covariation1
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.

Measuring the association between interval or ratio scaled variables
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:


Pearson Product Moment Correlation

Correlation coefficients and covariation2
Correlation Coefficients and Covariation Variables

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

Pearson product moment correlation coefficient r
Pearson Product Moment Correlation Coefficient (r) Variables

  • 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?

Pearson product moment correlation coefficient r1
Pearson Product Moment Correlation Coefficient (r) Variables

  • 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”…

Rules of thumb determining strength of association
Rules of Thumb Determining Strength of Association Variables

  • 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 coefficient r2
Pearson Product Moment Correlation Coefficient (r) Variables

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

Pearson product moment correlation coefficient r3
Pearson Product Moment Correlation Coefficient (r) 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.

Example will be < or =0.05.

  • 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.

Concluding remarks on associative analyses
Concluding Remarks on Associative Analyses association.

  • 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.