determining and interpreting associations among variables

Determining and Interpreting Associations

Among Variables

Ch 18 2

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?

Ch 18 3

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…

Ch 18 4


• 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.

Ch 18 5

Nonmonotonic Relationship

Ch 18 6


• 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.

Ch 18 7

Monotonic Increasing Relationship

Ch 18 8


• 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)

Ch 18 9


• 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.

Ch 18 10

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.

Ch 18 11

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

Ch 18 12

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.

Ch 18 13

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.

Ch 18 14

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:

Ch 18 15

Cross-TabulationsUndersøgelses-spørgsmål

Relevante variabler

Analyseteknik Forventet output

Er der sammenhæng mellem køn og planer for at gå videre på universitet?

s_16uni_ind

Krydstabel for de to variablerProcentberegning til vurdering af retningContingency coefficient til vurdering af styrkenChi-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

Ch 18 16

Cross-Tabulations

Ch 18 17

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.

Ch 18 18

Cross-Tabulations

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

Ch 18 19

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

Ch 18 20

Chi-Square Analysis

• Computed Chi-Square values:

Ch 18 21

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

Ch 18 22

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.

Ch 18 23

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.

Ch 18 24

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.

Ch 18 25


• 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.

Ch 18 26

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

Ch 18 27


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

Ch 18 28

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?

Ch 18 29


• 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”…

Ch 18 30

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

Ch 18 31


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

Ch 18 32


• 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.

Ch 18 33

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

Ch 18 34

• 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.

Ch 18 35

ExampleUndersøgelses-spørgsmål

Relevante variabler


Er der lineær sammenhæng mellem kriterierne ” socialt samvær med andre studerende” og ”Ligelig fordeling mellem drenge og piger”

s_8_1s_8_2Begge antages intervalskalerede

Pearson’s korrelationskoefficientFortegn til vurdering retningNumerisk størrelse (0-1) til vurdering af styrkenTest til vurdering signifikans

Et enkelt mål for styrke, retning og statistisk signifikans

Ch 18 36

Undersøgelses-spørgsmål

Relevante variabler


Er der lineær sammenhæng mellem kriterierne ” socialt samvær med andre studerende” og ”Ligelig fordeling mellem drenge og piger”

s_8_1s_8_2Begge antages intervalskalerede

Pearson’s korrelationskoefficientFortegn til vurdering retningNumerisk størrelse (0-1) til vurdering af styrkenTest 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

Ch 18 37

Undersøgelses-spørgsmål

Relevante variabler


Er der lineær sammenhænge mellem kriterierne for valg af studium?

s_8_1 … s_8_17Begge antages intervalskalerede

Pearson’s korrelationsmatrixFortegn til vurdering retningNumerisk størrelse (0-1) til vurdering af styrkenTest til vurdering signifikans

Et enkelt mål for styrke, retning og statistisk signifikans for hvert par af variabler

Uddrag:

Ch 18 38

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.

determining and interpreting associations among variables

Documents

variables relationship

variables nonmonotonic

variables levels

variables monotonic

variables curvilinear

variables linear

variables examples

demographic variables