Simple Correlation Scatterplots r Linearity strength direction scatterplots for continuous binary relationships Quantitative binary predictors H0 RH Non linearity Interpreting r Outcomes vs RH Supporting vs contrary results Outcomes vs Population Correct vs Error results A scatterplot a graphical depiction of the relationship between two quantitative or binary variables each participant s x y values depicted as a point in x y space Pearson s correlation coefficient r value summarizes the direction and strength of the linear relationship between two quantitative variables into a single number range from 1 00 to 1 00 you should always examine the scatterplot before considering the correlation between two variable NHST can be applied to test if the correlation in the sample is sufficiently large to reject H0 of no linear relationship between the variables in the population A linear regression formula allows us to take advantage of this relationship to estimate or predict the value of one variable the criterion from the other the predictor prediction should only be applied if the relationship between the variables is linear and substantial Puppy Age x Eats y Sam 8 2 Ding 20 4 Ralf 12 2 Pit 4 1 Seff 24 4 Toby 16 3 Amount Puppy Eats pounds Example of a scatterplot 5 4 3 2 1 0 4 8 12 16 20 24 Age of Puppy weeks When examining a scatterplot we look for three things relationship no relationship linear non linear direction if linear positive negative strength strong moderate weak No relationship nonlinear strong Hi Hi Lo Lo Lo linear positive weak Hi Lo linear positive strong linear negative moderate Hi Hi Hi Lo Lo Lo Lo Hi Lo Hi Hi Lo Hi We can use correlation to examine the relationship between a quantitative predictor variable and a quantitative criterion variable Y Y Y strong weak X 1 00 X X A positive r tells us those higher X values tend to have higher Y values Y Y strong Y weak X 00 X X A negative r tells us those with lower X values tend to have higher Y values A nonsignificant r tells us there is no linear relationship between X Y We can also use correlation to examine the relationship between a binary predictor variable and a quantitative criterion variable Y Y Y strong grp 1 weak grp 2 grp 1 1 00 grp 2 grp 1 grp 2 A positive r tells us the group with the higher X code as the higher mean Y Y Y strong grp 1 Y weak grp 2 grp 1 00 grp 2 grp 1 grp 2 A negative r tells us the group with the lower X code as the higher mean Y A nonsignificant r tells us the groups have equivalent means on Y People who have more depression before therapy will be those who have more depression after therapy Depression after For each of the following show the envelope for the H0 and the RH H0 RH Errors Depression before H0 Those who study more have fewer errors on the spelling test RH Study Practice Instructor isn t related to practice H0 RH 0 1 Instructor People who score better on the pretest will be those who tend to score worse on the posttest Post test For each of the following show the envelope for the H0 and the RH H0 RH Depression Pretest H0 You can t predict depression from the number of therapy sessions RH I predict that snapping turtles coded 1 will eat more crickets than painted turtles coded 0 Crickets Sessions H0 RH 0 1 The Pearson s correlation r summarizes the direction and strength of the linear relationship shown in the scatterplot r has a range from 1 00 to 1 00 1 00 a perfect positive linear relationship 0 00 no linear relationship at all 1 00 a perfect negative linear relationship r assumes that the relationship is linear if the relationship is not linear then the r value is an underestimate of the strength of the relationship at best and meaningless at worst For a non linear relationship r will be based on a rounded out envelope leading to a misrepresentative r Extreme Non linear relationship r value is misinformative actual scatterplot Scatterplot as correlation sees it notice there is an x y relationship regression line has 0 slope r 0 no linear relationship Moderate Non linear relationship r value is an underestimate of the strength of the nonlinear relationship actual scatterplot Scatterplot as correlation sees it notice there is an x y relationship regression line has non 0 slope r 0 but the regression line not a great representation of the bivariate relationship NHST testing with r H0 r 0 00 is the same as r2 0 00 get used to working with both r the correlation between the 2 vars and r2 the variance shared between the 2 vars Performing the significance test software will usually provide an exact p value use p 05 a general formula is r F 1 r N 2 N sample size Find F critical using df 1 N 2 What retaining H0 and Rejecting H0 means When you retain H0 you re concluding The linear relationship between these variables in the sample is not strong enough to allow me to conclude there is a linear relationship between them in the population represented by the sample When you reject H0 you re concluding The linear relationship between these variables in the sample is strong enough to allow me to conclude there is a linear relationship between them in the population represented by the sample effect significance vs effect size vs shared variance The p value value range 1 0 0 tells the probability of making a Type I error if you reject the H0 based on the sample data e g p 10 means if we reject H0 based on these data there is a 10 chance that there really is no relationship between the variables in the population represented by the sample The usual acceptable risk is less than 5 or p 05 r range 1 0 1 0 tells strength and direction of the bivariate relationship between Y X large enough to be interesting value vary across research areas but a common guideline is 10 small 30 medium and 50 large r2 range 0 1 0 tells how much of the Y variability is accounted for predicted from or caused by X e g r 30 means that 302 9 of the Y variability is accounted for by X large enough to be interesting will vary across research areas but a common guideline is 1 small 10 medium and 25 large Interpret each of the following significance strength direction For age social skills r 25 p 043 Sig medium positive Older adolescents tend to have higher social skills scores For practice and performance errors r 52 p 015 Sig large negative Those who practiced more tended to have fewer errors For age and performance r 33 p 231 Nonsig medium negative There is no linear relationship between age and performance For gender m