Hypothesis of Association: CorrelationChapter 11

Difference vs. Association

Hypothesis of Association

• relationship between two sets of variables are examined to determine whether they are associated or correlated

-IV is not manipulated but assigned

-can not determine cause and effect

-Correlation

Hypothesis of Difference

• A deliberate manipulation of a variable to see if a difference in behavior occurs

-IV is not assigned but manipulated

-can determine cause-and-effect relationships

-Experiment

Correlation: a measure of the relationship between two variables-correlation coefficient (r): a number calculated from the formula for measuring a correlation

*indicating strength & direction *ranges from +1 to -1 -strong relationship if the correlation coefficient is close to +1 or -1

*knowing the relationship between two variables allows us to make predictions -EX: if you study X amount of hours, a score of X is predicted

-positive correlation *represented by a positive # (0 to +1) *as the value of one variable increases, the other variable also increases

-EX: study time goes up, test scores go up OR as study time goes down, test scores go down-negative correlation

*represented by a negative # (0 to -1) *as the value of one variable increases, the other variable decreases

-EX: as study time goes up, party time goes down-zero correlation

*represented by 0 *no relationship between variables

-EX: study time & height

Correlation

Study Time

Exam

Sco

res

Exam

Sco

res

Party Time

r =-0.6321

Correlation: Scatter Plots

Exam

Sco

res

Height

r =0.8273

• Formula:

• Calculation

Step 1: Calculate the means

Step 2: Calculate the standard deviations

Step 3: Plug all values into the formula

Correlation: Pearson r

• Testing a hypothesis using correlation

-r refers to the sample correlation and ρ (rho) refers to the population correlation

Ho: ρ = 0 there is no correlation in the population

Ha: ρ ≠ 0 there is a correlation in the population

• Critical Values (Table E)

-df are N-2 (number of pairs of score minus 2)

-if the calculated r value is greater than or equal to the table r value then reject Ho

-NOTE: as sample size increases, really small correlations become significant

**EX: look at df=400

• Guilford’s Interpretation for significant r values

Correlation: Pearson r

Correlation: Pearson r

Requirements for using Pearson r• The sample has been randomly selected from the population

• Measurement for both variables must be in the form of interval and/or ratio data

• The variables being measured must not depart significantly from normality

-variable data should take the shape of the normal curve if you measured the whole population

• The assumption of homoscedasticity is reasonable

-points are fairly equally distributed above & below the regression line

• The association is between X & Y is linear (not curvilinear)

-plot your data & make sure it takes an oval shape

Correlation: Pearson r

• Use Spearman rs when you can’t meet the requirements to use the Pearson

-when both sets of data are not interval and/or ratio

-when the data are skewed/non-normal distributions

-note: the Spearman rs (unike the Pearson r) is considered a non-parametric test

**ie. it does not make assumptions about normality of the population including the parameter mean or the parameter standard deviation

• Calculating Spearman rs (when you have ordinal data)

Formula:

Step 1: determine the rank of each subject on both variables

**if you have interval data (on one set) convert it to ordinal by ranking it

Step 2: Obtain the absolute difference, d, between each subject’s pair of ranks

Step 3: Square each difference, d2

Step 4: Calculate Σd2 by adding the squared differences

Step 5: Plug the values into the formula

Correlation: Spearman rs

• Case of ties

-if you have the same score for two or more subjects (see worksheet for example):

*add the ranks (that the scores are tied for) and divide by the number of tied scores

*give all the subjects that same rank

• Testing a hypothesis using correlation

-r refers to the sample correlation and ρs (rho) refers to the population correlation

Ho: ρs= 0 there is no correlation in the population

Ha: ρs≠ 0 there is a correlation in the population

• Critical Values (Table F)

-Use N (not df)

-if the calculated rs value is greater than or equal to the table r value then reject Ho

Correlation: Spearman rs