*Relationships Among VariablesCorrelation and RegressionKNES 510Research Methods in Kinesiology

*CorrelationCorrelation is a statistical technique used to determine the relationship between two or more variablesWe use two different techniques to determine score relationships:graphing techniquemathematical technique called correlation

Graphs of the RelationshipBetween Variables*

*Types of RelationshipsThe scattergram can indicate a positive relationship, a negative relationship, or a zero relationshipWhat are the characteristics of positive, negative, and zero relationships?

*Mathematical Technique: The Correlation Coefficient (r)The correlation coefficient, r,* represents the relationship between the z-scores of the subjects on two different variables (usually designated X and Y)This can be stated mathematically as the mean of the z-score products for all subjects*A more complete name for this statistic is Pearsons product-moment correlation coefficient

*Formula for the Correlation CoefficientThe correlation coefficient can be calculated as follows:

*The values of the coefficient will always range from +1.00 to -1.00A correlation coefficient near 0.00 indicates no relationship

*SPSS Bivariate Correlation Output

*Interpreting the Correlation CoefficientBecause the relationship between two sets of data is seldom perfect, the majority of correlation coefficients are fractions (0.92, -0.80, and the like)When interpreting correlation coefficients it is sometimes difficult to determine what is high, low, and average

*The Correlation Coefficient and Cause-and-EffectThere is a high correlation between a person's shoe size and their math skills in grades K through 6Is this an example of cause-and-effect?Can we predict math skill based on shoe size in grade K through 6 students?

*Coefficient of DeterminationThe coefficient of determination is the amount of variability in one measure that is explained by the other measureThe coefficient of determination is the square of the correlation coefficient (r2).For example, if the correlation coefficient between two variables is r = 0.90, the coefficient of determination is (0.90)2 = 0.81

*RegressionWhen two variables are related (correlated), it is possible to predict a persons score on one variable (Y) by knowing their score on the second variable (X)

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*This scatterplot illustrates that there is a strong, positive relationship between fat-free body mass and daily energy expenditure

*Regression Line (Line of Best Fit)The regression line is a line that best describes the trend in the dataThis line is as close as possible to the data pointsThe equation for this line is:

Y' = bX = C

Fitting a Regression Line*

*Simple PredictionTests have been developed to predict VO2 max from the time it takes a person to run 1.5 milesA person's VO2 max can thus be predicted from their 1.5 mile run time because a prediction or regression equation has been developed

*The simple linear prediction or regression equation takes the following form:Y' = a + bXY' = predicted valuea = intercept of the regression line (Y intercept) b = slope of the regression line (change in Y with each change in X)X = score on the predictor variable

*Determining Error in PredictionUnless two variables are perfectly related (-1.00 or +1.00) there will always be error associated with a prediction equationWe find the standard deviation of this error, the standard error of prediction (syx), using the following formula:

*Prediction and Residuals

*A predicted score (Y) syx yields a range of scores within which a persons true score on the predicted variable liesIf the standard error of prediction may be interpreted as the standard deviation of residuals, what are the odds that a persons true score lies between Y syx?

*The standard error of prediction for percent body fat estimated using the skinfold method is 3.5%If a person has their percent body fat estimated at 12%, between what two values does their true body fat lie (95% probability)?

*Which of the following will more precisely predict job performance?A: r = 0.168B: r = 0.686

*Sample SPSS OutputHere is the SPSS output for regressing Work Simulation Job Performance (Dependent Variable) against Supervisor Ratings (Independent Variable)

*This information can be used to create a prediction (regression) equation for predicting work performance of future applicants from supervisor ratings

Y = 1.156 + 0.033 X

*Work Simulation Job Performance may also be predicted from Arm StrengthHere is the SPSS output:

*This information can be used to create a prediction (regression) equation for predicting work performance of future applicants from supervisor ratings

Y = 4.095 + 0.055 X

*We now have two regression equations for predicting Work Simulation Job PerformanceWhich is the better equation for accurate prediction?To determine this, we must examine the standard error of prediction for each equation

*Standard error of prediction using Supervisor Ratings:

Standard error of prediction using Arm Strength:

Which is the better equation?

*Multiple PredictionA prediction formula using a single measure X is usually not very accurate for predicting a person's score on measure YMultiple correlation-regression techniques allow us to predict score Y using several X scores

*The general form of a two predictor multiple regression equation is:Y' = a + b1X1 + b2X2

*An example of multiple correlation-regression is the prediction of percent body fat from multiple skinfold measurementsDB (g/cc) = 1.0994921 - 0.0009929 (3SKF) + 0.0000023 (3SKF)2 0.0001392 (age)

Next ClassChapters 9 & 11Mock Proposals in class!*