Bivariate Correlation and Regression PSYCHOLOGY 3800 , LAB 002

In Today’s Lab

•  split plot ANOVA feedback link •  introduction to correlation and regression •  example analyses •  assignment #8 summary

Assignment #6: Feedback

•  the statistical component was typically well done (woot!) •  check lab blog for list of commonly made errors:

http://uwo3800g.tumblr.com/post/80090405911/assignment-6-commonly-made-errors

Correlation & Regression: Overview

Correlation indicates the nature and strength of a relationship

Nature

•  direction of the relationship between variables

positive negative zero

Correlation indicates the nature and strength of a relationship

Nature

•  direction of the relationship between variables

positive negative zero

as one variable increases/decreases, so does the other

Correlation indicates the nature and strength of a relationship

Nature

•  direction of the relationship between variables

positive negative zero

as one variable increases/decreases, the other does the opposite

Correlation indicates the nature and strength of a relationship

Nature

•  direction of the relationship between variables

positive negative zero

no relationship (in this case) but could also indicate non-linear relationship

indicates the strength and nature of a relationship

Strength

•  how far the plotted data points fall from one other •  closer together = stronger relationship (value closer to 1)

Correlation

strong moderate weak/none

1 r 0

•  significance of correlation is based on strength of the relationship between variable (stronger = more significant)

Example…

r = -.83

negative relationship strong relationship

Correlation

Simple Regression

•  a simple regression is similar to correlation: deals with the relationship between two variables

predictor (x): variable used to make a prediction criterion (y): variable being predicted

•  uses relationship data for the two variables to:

(1) assess whether x adds significantly to the prediction of y (significance of the model)

(2) calculate a predicted y-score given a specific value of x

Simple Regression

Significance of Relationship/Model

• F-value (outputted in ANOVA table) •  assesses overall model fit (i.e., if x adds significantly to the prediction of y) •  if significant (p < .05): slope of the regression line is significantly different than zero •  if slope of the regression line was 0, we wouldn’t be able to predict anything (no relation between two variables)

• t-value (outputted in Coefficients table) •  assesses each predictor in the model (only one this week) •  indicates whether each predictor adds significantly to the prediction of the criterion

Simple Regression

Other Indicators of Effectiveness of Prediction

(1) standard error of the estimate o  in general: the average distance between each actual score and its predicted score o  can indicate how closely (within how many points) we can predict a score on x (outcome variable) by knowing a score on y (predictor variable)

(2) r2

o  proportion of variance in y (outcome variable) that is accounted for by x (predictor variable)

Simple Regression

€

ˆ y = b0 + b1(x)

•  regression equation represents line of best fit that runs through the data b0 = intercept of line of best fit (constant) b1 = slope of line of best fit (unstandardized coefficient) x = value of predictor y = predicted criterion score

•  accuracy of the prediction will depend on the relationship between the two variables (x and y)

as x and y are more strongly related, x will do a better job at predicting y

Prediction

Level of Studying

Exam

Gra

de

Simple Regression

Prediction

€

ˆ y = b0 + b1(x)

y-intercept

Simple Regression

Prediction

€

ˆ y = b0 + b1(x)

Level of Studying

Exam

Gra

de

y-intercept

slope

Simple Regression

Prediction

€

ˆ y = b0 + b1(x)

Level of Studying

Exam

Gra

de

y-intercept

slope

studying score prior to exam

Simple Regression

Prediction

€

ˆ y = b0 + b1(x)

Level of Studying

Exam

Gra

de

y-intercept

slope

predicted score on exam

Simple Regression

Prediction

€

ˆ y = b0 + b1(x)

Level of Studying

Exam

Gra

de

studying score prior to exam

Connecting Correlation and Regression

•  neither correlation nor regression imply causation   phrase interpretations/conclusions correctly   consider alternative explanations/variables

Example Analysis

The Study

•  interested in which variables are associated with grades on the final exam for Psych 3800

Bivariate Correlation

Analyze Correlate Bivariate

Move all variables into the “Variables” box. Select the “Options” menu.

Bivariate Correlation

Options Menu

Request that descriptive statistics be outputted (means and standard deviations)

Bivariate Correlation

Output

average rating for each construct variability in ratings for each construct

number of participants scores analyzed for each

construct

Bivariate Correlation

Output

Bivariate Correlation

Output exact significance values given below each correlation coefficient

overall significance levels indicated using asterisk (*) markers

Bivariate Correlation

Conclusions

Correlation between exam grades in Psych 3800 and:

(a) enthusiasm toward pie r = -.182, ns (b) pre-exam shots r = -.341, p < .01 (c) tendency to sleepwalk r = .132, ns (d) level of studying r = .383, p < .001

“The results revealed a significant negative correlation between grades on the Psychology 3800 final exam and one’s tendency to consume pre-exam shots, r = -.341, p < .01.”

Bivariate Regression

Analyze Regression Linear

Enter your predictor variable as the independent variable, and your criterion variable as the dependent variable.

Bivariate Regression

Save Menu … doing this will create two new columns in your data file

F(1, 78) = 13.410, p < .001

Bivariate Regression

Output: Test of Significance

“The results revealed that studying adds significantly to the prediction of final exam grades in Psychology 3800.”

contains info about the bivariate

regression

contains info about the error

Bivariate Regression

Additional Information About Prediction

R correlation between variables (absolute value) r = .383

R Square proportion of variance in exam grades accounted for by studying r2 = .147 (14.7%)

Std. Error of Estimate average difference between actual and predicted scores sy.x = 8.898

Bivariate Regression

Additional Information About Prediction

t(78) = 3.662, p < .001

residual df from the “ANOVA” table

Significance of each predictor in the model (here, only one predictor)…

Bivariate Regression

Output: Regression Equation

€

ˆ y = b0 + b1(x)

€

ˆ y = 41.603+ 3.207(x)

€

ˆ y = 41.603+ 3.207(7)

€

ˆ y = 64.052

predicted exam score for someone who studies quite a bit (rating of 7 on 10-point Likert scale)

Bivariate Regression

Output: Prediction Equation

predicted score using prediction equation from previous slide

(some differences due to rounding)

differences between obtained score on predictor (56) and predicted score using equation (64.05426)

residual = 56 – 64.05426 residual = -8.05426 … we over-predicted by about 8 points

*these are the new columns that have been added to your data file

The Assignment

•  not a results section (number your responses) but adhere to APA formatting

•  run bivariate correlation and linear regression analyses in SPSS (report all statistics in APA style)

•  be sure to answer ALL parts of each question and to submit all output

Assignment: Overview

Note: Because this assignment is straightforward, you are being asked to work independently to complete it. I can help you to run the data, but I cannot provide direction in answering the questions. All needed information was covered in lab and lecture. Additional help is in your textbook.