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Slide 1AssociationsKey characteristics of associationsAssociationsScatterplotsRelational ExistenceAssociation DirectionCorrelationCorrelation CoefficientCalculating the Correlation CoefficientExample 1: Correlation between education and incomeHypothesis Testing for Pearson’s rHypothesis Testing for Pearson’s rHypothesis Testing for Pearson’s rHypothesis Testing for Pearson’s rHypothesis Testing for Pearson’s rHypothesis Testing for Pearson’s rHypothesis Testing for Pearson’s rCorrelationsCorrelationsRelational ExistenceCorrelation and Associations: Interval Ratio DVs & IVs***Please take a handout at front of classAssociationsMeasures of association: Statistical tools used to indicate the degree of correspondence between two (or more) variables.•In other words, particular values of one variable tend to coincide with particular value of another▫E.g., if X and Y are associated:Particular values of Y tend to coincide with values of XThe values of Y are different across values of XThe average value of Y depends on the value of XKey characteristics of associations•Strength▫How strong is the tendency for values of Y to go with values of X?▫Maximum (perfect) association:All cases with a particular value of X have the same value on YKnowing the value of X allows for perfect prediction of Y▫Minimum (no) association:High conditional variation (value of Y varies even among those cases with the same value of X)Knowing the value of X does not improve ability to predict Y•Direction▫Positive: High values on Y tend to coincide with high values on X▫Negative: High values on Y tend to coincide with low values on X•Statistical Significance▫Ability to generalize association in a sample to the broader populationAssociations•T-tests, ANOVA, Chi square: Does a relationship exist? ▫Decide about direction/strength by looking at data▫Post-estimation tests (Cramer’s V) can help•Correlation (2 interval ratio variables)▫Does a relationship exist?▫How strong is it?▫In what direction is it?0 5 10 15 20 250102030405060Educational AttainmentIncome (in 1000s)ScatterplotsEducationIncomeEducationIncomeShirley 12 40,000 Hanna 16 56,000Bob 16 44,000 Karen 16 31,000Dwayne 13 14,600 Ron 11 35,600Phil 14 36,200 May 20 42,000Lenora 8 19,500 Steve 17 28,400Roberta 15 29,000 Bo 2 11,000and linearand linearNon-deterministicNon-deterministicRelational ExistenceA linear relationship is in contrast to a “curvilinear” relationship.0 5 10 15 20 25050100150Educational AttainmentIncome (in 1000s)Association Direction•Three types of direction in relationships:3. Zero: values of dependent variable do not differ by values on independent variableExamples: ???Correlation•The correlation coefficients simply formalizes what is contained in a scatter plot.Pearson Correlation Coefficient (r): A measure of association reflecting both the strength and direction of the relationship between two interval-ratio variables.Correlation Coefficient•Symbolized by r (sample) and ρ (population)•Ranges from -1.0 to +1.0•Rough Guideline (in absolute values):▫.00 to .09 = none▫.10 to .29 = weak▫.30 to .59 = moderate▫.60 to 1.00 = strongCalculating the Correlation Coefficient�=�����������(� , �)√(��������� �)(��������� �)G¿∑(� −´�)(� −´�)√∑(� −´�)�∑(� −´�)�G•Calculating the degree to which two variables vary together (i.e., covary) relative to how much each variable varies by themselves.Example 1: Correlation between education and incomeName (X) (Y) (X-X) (X-X)2(Y-Y) (Y-Y)2(X-X)(Y-Y)Shirley 12 20.00 -1.33 1.77 -2.92 8.53 3.88Bob 16 31.00 2.67 7.13 8.08 65.29 21.57Dwayne 13 19.00 -0.33 0.11 -3.92 15.37 1.29Phil 14 17.00 0.67 0.45 -5.92 35.05 -3.97Lenora 8 10.00 -5.33 28.41 -12.92 166.93 68.86Roberta 15 22.00 1.67 2.79 -0.92 0.85 -1.54Hanna 16 26.00 2.67 7.13 3.08 9.49 8.22Karen 16 35.00 2.67 7.13 12.08 145.93 32.25Ron 11 17.00 -2.33 5.46 -5.92 35.05 13.79May 20 34.00 6.67 44.49 11.08 122.77 73.90Steve 17 33.00 3.67 13.47 10.08 101.61 36.99Bo 2 11.00 -11.33 128.37 -11.92 142.09 135.05SUM160 275.00 0.00246.680.00848.96 390.29Mean13.33 22.92�=∑(�−´�)(� −´�)√∑(� −´�)�∑(� −´�)�GCovariation of X and YVariation of YVariation of XHypothesis Testing for Pearson’s r▫Like before, we want to know if the relationship revealed in our sample data reflects:1. A real relationship between two variables in the population.2. Chance sampling error when in reality the two variables are unrelated in the population.Hypothesis Testing for Pearson’s rFollow same five steps:1. Check assumptionsRandom sampleBivariate normal distributionLinear relationshipHomoscedasticity (“same scatter”)Hypothesis Testing for Pearson’s rFollow same five steps:1. Check assumptions2. State the research and null hypothesesH0: = 0H1: 0Hypothesis Testing for Pearson’s rFollow same five steps:1. Check assumptions2. State the research and null hypotheses3. Identify the sampling distribution and test statistic (identify critical region)Assuming above are met, the sampling distribution of r will follow the t-distributionSo use t-statisticHypothesis Testing for Pearson’s rFollow same five steps:1. Check assumptions2. State the research and null hypotheses3. Identify the sampling distribution and test statistic (identify critical region)Find the critical value of t with df = N -2Hypothesis Testing for Pearson’s rFollow same five steps:1. Check assumptions2. State the research and null hypotheses3. Identify the sampling distribution and test statistic (identify critical region)Find the critical value of t with df = N -2Education and Income Example: N=12 so, df = (12-2) = 10use α = .05 so, t(critical) = 2.228Education and Income Example: N=12 so, df = (12-2) = 10use α = .05 so, t(critical) = 2.228Hypothesis Testing for Pearson’s rFollow same five steps:1. Check assumptions2. State the research and null hypotheses3. Identify the sampling distribution and test statistic (identify critical region)4. Calculate the test statistic•GCorrelations•Correlations have some key strengths▫Allows us to compare two interval ratio variables (Chi square, ANOVA, T-tests do not let us do this)▫Summarizes strength, direction, and significance all in a single statistic▫Fundamentally very simple (based on a

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