Preview Correlation Experimental Psychology Arlo Clark-Foos Correlation Characteristics of Relationship to Cause-Effect Partial Correlations Standardized Scores (z scores) Psychometrics Reliability Validity What is a correlation? A general term to describe the association (or relation) between two variables co-relation Las Cucarachas How heavy are these cockroaches? 8.0 drachms(14.17 grams).25 pound (113.40 grams) 98 grams (98 grams) Need to put these on the same scale! Not always this easy Equal Footing Different variables, often different scales Need standardization z score (z is italicized) the number of standard deviations a particular score is from the mean z Distributions Mean = 0, SD = 1 (Always!) z distribution Scores vs. Distributions 1
Raw Score z Score Hypothetical Test Scores Mean = 70, SD = 10 Raw Score z Score NEED: Population parameters for the mean (μ) and standard deviation (σ) ( μ) z = X σ Correlation Coefficient A statistic that quantifies the relation between two variables Changing Gears: Correlations Pearson (r) correlation coefficient Measures the strength of a linear relationship Correlation Coefficient Positive Correlation Characteristics 1. Can be positive or negative 2. Ranges from -1.00 to 1.00 r > 0 3. Strength (magnitude) of correlation is indicated by size of resulting value, not the sign. 2
Positive Correlations Negative Correlation Positive correlation between smoking and lung cancer Positive correlation between SAT scores and high school grades r < 0 Positive correlation between the amount of diet soda you drink and your weight Negative Correlations No Correlation Negative correlation between education and years in jail Negative correlation between amount of time a baby is held and how much she cries r = 0 Negative correlation between amount of TV watched and grades Strength of Correlations Cohen (1988) How strong is your correlation? Strong, positive Weak/moderate, positive 3
Correlation & Causation Illusory Correlations True: After WWII, there was a strong positive correlation between the birth rate and the number of storks in Copenhagen. Why? Increasing population More buildings Increasing numbers of storks More baby makers Increasing numbers of babies Capricorn: Some rather boring and mundane tasks, perhaps involving paperwork, could take up much of your time today, Capricorn. You could get easily distracted and be tempted to set it aside and do something more interesting, but don't fall into this trap. You'll want your recent pattern of success to continue, and so it's best to get the boring stuff out of the way and then move on to what's exciting. Hang in there! Pisces: Some paperwork involving business enterprises could be executed today, Pisces, possibly in your home. This might be a new and unexpected development. It could have you feeling a bit disoriented, but try to pull yourself together and make the most of it. Whatever the opportunity, all signs say that it might prove quite successful, so don't let it pass you by. Go with the flow and see where it takes you. You could be pleasantly surprised! Third Variables Correlation is NOT Causation Break to Go Over Exams Karl Pearson r = Pearson correlation coefficient r (sample) or ρ (population) [ ( zx )( z y )] N or ( X M X )( Y MY) ( SS )( SS ) Σ r = Let s use attendance and exam grade as an example X Y 4
Correlation with z r = [ ( zx )( z y )] N Method 1: Correlation with Standard Scores 1. Plot the raw scores for each variable on a scatter plot to see if there might be a linear relationship If so, proceed with calculating the Pearson correlation coefficient. 2. Create z scores corresponding to each value Do this for both variables of interest for each person, keeping pairs of scores together ( μ) z = X σ Scatterplot Level the Field Mean Exam Grade 110 100 90 80 70 60 50 40 0 2 4 6 8 10 Absences 1. Plot the raw scores for each variable on a scatter plot to see if there might be a linear relationship If so, proceed with calculating the Pearson correlation coefficient. 2. Create z scores corresponding to each value Do this for both variables of interest for each person, keeping pairs of scores together ( μ) z = X σ 5
Cross-products 3. Pearson Correlation Coefficient Calculate the cross-products Multiple each pair of z scores together 4. Find the average of all of the crossproducts This is your Pearson correlation coefficient Cross-products 3. Pearson Correlation Coefficient Calculate the cross-products Multiple each pair of z scores together 4. Find the average of all of the crossproducts This is your Pearson correlation coefficient (Cross Products) / Number of Pairs = Pearson Correlation Coefficient r =.85 Same Data, New Method ( X M X )( Y MY) ( SS )( SS ) Σ r = X Y Method 2: Correlation with Deviations and Sums of Squares 6
Correlation with z 1. Plot the raw scores for each variable on a scatter plot to see if there might be a linear relationship If so, proceed with calculating the Pearson correlation coefficient. 2. Numerator: Create deviations (Score Mean) for each variable, keeping pairs together Mean Exam Grade 110 100 90 80 70 60 50 40 Scatterplot 0 2 4 6 8 10 Absences Correlation with z 1. Plot the raw scores for each variable on a scatter plot to see if there might be a linear relationship If so, proceed with calculating the Pearson correlation coefficient. 2. Numerator: Create deviations (Score Mean) for each variable, keeping pairs together Same Data, New Method Σ ( X M X )( Y MY) r = ( SS X )( SSY) Numerator: Deviations & Cross Products 3. Denominator: Create Sums of Squares (Sum of all Squared Deviations) for each variable 4. Denominator: Multiply your Sums of Squares 5. Take the Square Root of that product 6. Divide Numerator by Denominator to get your Correlation Coefficient 7
Denominator: Sums of Squares 3. Denominator: Create Sums of Squares (Sum of all Squared Deviations) for each variable 4. Denominator: Multiply your Sums of Squares 5. Take the Square Root of that product 6. Divide Numerator by Denominator to get your Correlation Coefficient 3. (X-M x2 ) = 56.4 (Y-M Y2 ) = 2262 56.4 * 2262 = 127576.8 4. Square Root of 127576.8 = 357.179 3. Denominator: Create Sums of Squares (Sum of all Squared Deviations) for each variable 4. Denominator: Multiply your Sums of Squares 5. Take the Square Root of that product 6. Divide Numerator by Denominator to get your Correlation Coefficient 304 / -357.179 = -.8511 r = -.85 110 Mean Exam Grade 100 90 80 70 60 Other Issues with Correlation 50 40 0 2 4 6 8 10 Absences 8
Correlation is NOT Causation Restriction of Range There appears to be a negative correlation between attendance and exam grades r =.05 What else could contribute to this? Restriction of Range Correlation is NOT Causation Outlier Effects Correlation is NOT Causation r =.56 r = -.135 Outlier Effects Correlation is NOT Causation r =.39 Uses of Correlation and Correlational Methods 9
Psychometrics Branch of statistics used in the development of tests and measures Examples Test for cultural biases in SAT Identify high-achieving employees critical shortage (NYT, Herszenhorn, 2006) Reliability Test-Retest Determines whether scale being used provides consistent information every time it is given Split-Half lf Reliability l Measures internal consistency of a test or scale by correlating the odd-numbered items with the even-numbered items Coefficient alpha (α) Calculated by taking the average of all possible splithalf corelations Validity Criterion-related The scale of measurement of interest (e.g., shyness) is correlated with a criterion, which is some external standard (e.g., shy behavior) Postdictive The scale or measurement of interest is correlated with a criterion measured in the past (e.g., home movies) Concurrent The scale or measurement of interest is correlated with a criterion measured at the same time (e.g., closeness) Partial Correlation A technique that quantifies the degree of association between two variables, after statistically removing the association of a third variable with both of those variables Review Correlation Characteristics of Relationship to Cause-Effect Relationship to Cause Effect Partial Correlations Standardized Scores (z scores) Psychometrics Reliability Validity 10