PSYCH 210 Lecture 26 Outline of Last Lecture I. Factorial ANOVAa. Interpretation of interactionb. AssumptionsII. CorrelationOutline of Current Lecture I. Finish Correlationa. Calculational example: Pearson rb. Test for significance of rc. Assumptions for inferential testd. Important aspects of correlationII. RegressionCurrent LectureI. Correlationa. Pearson ri. Calculational formula1. r=SP/√SSx-SSya. SP=Sums Productsi. Def= Σ(x – Mx)(y – My)1. Remember SS def = Σ(x – μ)2a. =Σ (x – μ)(x – μ)ii. Computational 1. r= {Σxy – (ΣxΣy)/n}/√[Σx2-(Σx)2/n]-[ Σy2-(Σy)2/n]2. Precalculations (Ex)These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.a. n = 6b. Σx = 30c. Σy = 498d. ΣxΣy = 30*498 = e. Σxy = 2458 = f. Σx2 = 168g. Σy2 = 41406i. Remember Orders of Operation!3. Formulaa. SP = 2458 – [(30)(498)]/6 = -32i. Ok to have negative numbershere! (Unlike for SS)b. SSx = 168 – (30)2/6 = 18c. SSy = 41406 – (498)2/6 = 724. r = (-32)/√18*72)a. r= -0.88b. Significance test of ri. r value is ‘r observed’ (Step 3 of hypothesis test)ii. Inferential1. Non-directionala. Is the r observed value significantly different from 0?2. Directionala. Is r obs. sig positive? (Or…)b. Is r obs. sig negative?iii. Other 4 steps of hypothesis test (using shyness example still)1. Hypothesesa. Non-directionali. H0: ρ = 0ii. H1: ρ ≠ 0b. Directionali. Negative1. H0: ρ ≥ 02. H1: ρ <02. Critical Value(s) for ra. Using the new tablei. df = n – 2b. r crit. = -0.729i. (Because Negative Directional)3. r obs.a. =-0.884. Decision: Reject H05. Conclusiona. The correlation between early socialization and shyness is significantly negativec. Criteria for Pearson r significance testi. Testing for a relationship between 2 DVs (measurements)d. A note on sample size and correlations:i. The larger the sample size, the smaller the correlation you need1. Correlations for huge data sets easy to get (always must be critical of data!)e. Assumptions for Pearson significance test of ri. Interval or ratio data1. Different test for rank data = Spearmana. Don’t need to know thisii. Random Samples of x and yiii. Distributions of x and y must be normaliv. The x/y relationship must be linearf. Important concepts on Correlationi. (Pearson) only for Linear relationships1. Make a scatterplot first! If appears curvilinear, use different testii. Restriction of range1. Performance and Motivationa. Researchers only getting part of the picture leads to incorrect conclusion from data observediii. Effects of outliers1. Could give you significantly positive (or negative) data even if there is no correlationa. Skew the datab. Make scatter plot firsti. Run test with outliers includedii. Run test with outliers excluded1. If vastly different, use excluded versioniv. Correlation and Causation1. Violent Video Games and Aggressiona. 3 possible scenarios (we don’t know which):i. x causes yii. y causes xiii. z causes x and yb. It’s not that causation doesn’t exist… we just cannot determine which one is occurringv. Coefficient of determination1. r squareda. Amount of variability in y that can be explained by variability in xi. Shyness Ex)1. (-0.88)2 = 0.7742. 77% of the variance in Shyness (y) can be explained by the variability in Early Socialization (x)a. Think Pie Graphi. 33% = unexplained varianceII. Regressiona. Scatterplots and regression linesi. Scatterplots: tell us about direction and degree of relationship1. Summarize this relationship using a regression lineii. Line of best fit: minimizes distance between actual y and predicted y (y-hat)1. Linear equation: y-hat = bx + aa. b = slopeb. a = y-interceptiii. What will regression line be able to tell us?1. Direction of relationship2. Degree of relationshipa. Amount of scatter around line3. Prediction (predict y for a given x)4. Measure of central tendency (average degree of relationship)iv. Finding the regression line1. Best fir – minimize distance between y and y-hat = “least squared error solution”2. Thing ok y – y-hat as a deviation scorea. Find average deviation, but square so deviations don’t sumv. Regression equation does NOT tell us about the accuracy of predictions!1. The same regression line could be linked to different levels of prediction accuracya. Ex) Varying degree of amounts of scatter but same line usedi. Standard error of estimate (SEE) is related to correlation value1. Measures amount of scatter between line and pointsa. The lower SEE, the more accurate the regression line (smaller amount of distance between point and line)2. Higher r = lower SEE = better fitting regression