Chapter 9Correlational Research Designs What are correlational research designs, and why are they used in behavioral research. What patterns of association can occur between two quantitative variables? What is the Pearson product-moment correlation coefficient? What are its uses and limitations? How does the chi-square statistic assess association? What is multiple regression, and what are its uses in correlational research design? How can correlational data be used to make inferences about the causal relationships among measured variables? What are the limitation of correlational designs in doing so? What are the best uses for correlational design?

Statistical Assessment of RelationshipsDataAre the data quantitative or nominal?quantitativenominalCorrelation Analysis: rChi-Square Analysis: 2Do you have more than two predictor variables?Do you have more than two predictor variables?NoYesNoYesRegression Analysis: RLog-Linear AnalysisLogistic Regression

The Correlation Coefficientfor Association among Quantitative VariablesScatterplotRegression LineHigh SchoolGPACollegeGPA4.0

3.0

2.0

1.01.0 2.0 3.0 4.0A graph in which the x axis indicates the scores on the predictor variable and the y axis represents the scores on the outcome variable. A point is plotted for each individual at the intersection of their scores.A line in which the squared distances of the points from the line are minimized. (least square methods)

Linear Relationships and Nonliniar RelationshipsYY YY YXXXX XPositive LinearNegative LinearCurvilinearCurvilinearIndependent

The Pearson Correlation CoefficientCalculation r = Esteem 1 Esteem 214 424 333 242 252 1

Mean 3 2.4[(4-3)(4-2.4)]2 + ...(4-3)2 + ...(4-2.4)2 + ...Sesteem1 = 0.8 Sesteem2=1.04 (4-3)/0.8 =1.67==4 x 4 + 4 x 3 + ...54+4+3+2+24+3+2+2+14 x 4 + 4 x 4 + 3 x 3 ...Task 1: compute r4 x 4 + 3 x 3 + 2 x 2 ...

Interpretation of rIf the relationship between X and Y are positive:If the relationship between X and Y are negative: -1< r . 05There is no significant correlation between X and Y

Reporting Correlations As predicted by the research hypothesis, the variable of optimismand reported health behavior were (significantly) positively correlated in the sample (the data), r(20) = .52, p < .01r(Number of Participants) = Correlation Coefficient r, p < p value.

Limitation1. Cases in which the correlation between X and Y that have curvilinier relationships r = 02. Cases in which the range of variables is restricted.Restriction of RangeExample. SAT scores and college GPA3. Cases in which the data have outliers r > |.99|

Limitation (visual)Curviliniar Small Range Outlier

The Chi-square Statisticfor Association among nominal variablesNorthernerSouthernerYesNo30 (.15)70 (.35)60 (.30)40 (.20)

100 (.50)100 (.50)90 (.45)110 (.55)200 (1.00)45 (.225)55 (.275)45 (.225)55 (.275)X =2 =Row marginal X Column marginalNTask 2 computation 2

Interpretation of 2 Go to Table E in Appendix E.

Degree of Freedom (df): (Level of variable 1 - 1) X (Level of variable 2 -1)Number of ParticipantsSee the value at the intersection between Alpha p < .05 and dfIf 2 is greater than the value in Table E, the contingency tableis significantly differ from the expectation.If 2 is greater than the value in Table E, the contingency tableis not significantly differ from the expectation.

Reporting Chi-Square Statistic2(degree of freedom (df), Number of Participants(N)) = Chi value, p < p valueAs predicted by the research hypothesis, the southerners were more likely to approve of a policeman striking an adult male citizen who was being questioned as a suspect in a murder case, 2(1, N =30) = 34.23, p < .01