Basic Statistics Correlation

Var Relationships Associations

The Need for a Measure of Relationship INDIVIDUAL DIFFERENCES (Variance) Describe Predict Control Explain

Information ? COvary In Research Dependent variable Independent variables X1X1 X2X2 X3X3 Y

The Concept of Correlation Association or relationship between two variables X Y Covary---Go together Co-relate?relationr

Patterns of Covariation

Y Positive correlation Negative correlation Correlation Covary Go together XY XY X Zero or no correlation

Scatter plots allow us to visualize the relationships Scatter Plots The chief purpose of the scatter diagram is to study the nature of the relationship between two variables  Linear/curvilinear relationship  Direction of relationship  Magnitude (size) of relationship

Represents both the X and Y scores Variable X Variable Y An illustration of a perfect positive correlation high low Scatter Plot A Exact value

Variable X Variable Y An illustration of a positive correlation high low Scatter Plot B Estimated Y value

Variable X Variable Y An illustration of a perfect negative correlation high low Scatter Plot C Exact value

Variable X Variable Y An illustration of a negative correlation high low Scatter Plot D Estimated Y value

Variable X Variable Y An illustration of a zero correlation high low Scatter Plot E

Variable X Variable Y An illustration of a curvilinear relationship high low Scatter Plot F

The Measurement of Correlation The degree of correlation between two variables can be described by such terms as “strong,” ”low,” ”positive,” or “moderate,” but these terms are not very precise. If a correlation coefficient is computed between two sets of scores, the relationship can be described more accurately. The Correlation Coefficient A statistical summary of the degree and direction of relationship or association between two variables can be computed

Pearson’s Product-Moment Correlation Coefficient r  Direction of relationship: Sign (+ or –)  Magnitude: 0 through +1 or 0 through -1 Negative correlationPositive correlation No Relationship

The Pearson Product-Moment Correlation Coefficient Recall that the formula for a variance is: If we replaced the second X that was squared with a second variable, Y, it would be: This is called a co-variance and is an index of the relationship between X and Y.

Conceptual Formula for Pearson r This formula may be rewritten to reflect the actual method of calculation

Calculation of Pearson r You should notice that this formula is merely the sum of squares for covariance divided by the square root of the product of the sum of squares for X and Y

Formulae for Sums of Squares Therefore, the formula for calculating r may be rewritten as:

Calculation of r Using Sums of Squares

An Example Suppose that a college statistics professor is interested in how the number of hours that a student spends studying is related to how many errors students make on the mid- term examination. To determine the relationship the professor collects the following data:

The Stats Professor’s Data Student Hours Studied (X) Errors (Y) X2X2 Y2Y2 XY Total  X = 70  Y = 73  X 2 =546  Y2=695  XY=429

The Data Needed to Calculate the Sum of Squares XYX2X2 Y2Y2 XY Total  X = 70  Y = 73  X 2 =546  Y2=695  XY=429 = /10 = = 56 = /10 = = = 429 – (70)(73)/10 = 429 – 511 = -82

Calculating the Correlation Coefficient = -82 / √(56)(162.1) = Thus, the correlation between hours studied and errors made on the mid-term examination is -0.86; indicating that more time spend studying is related to fewer errors on the mid-term examination. Hopefully an obvious, but now a statistical conclusion!

Pearson Product-Moment Correlation Coefficient r 0+1 Negative correlation Positive correlation perfect negative correlation Perfect positive correlation Zero correlation

Numerical values Negative correlation Zero correlation Positive correlation Perfect Strong Moderate

The Pearson r and Marginal Distribution The marginal distribution of X is simply the distribution of the X’s; the marginal distribution of Y is the frequency distribution of the Y’s. Y X Bivariate Normal Distribution Bivariate relationship

Marginal distribution of X and Y are precisely the same shape. X variable Y variable

Interpreting r, the Correlation Coefficient Recall that r includes two types of information:  The direction of the relationship (+ or -)  The magnitude of the relationship (0 to 1) However, there is a more precise way to use the correlation coefficient, r, to interpret the magnitude of a relationship. That is, the square of the correlation coefficient or r 2. The square of r tells us what proportion of the variance of Y can be explained by X or vice versa.

Variable X Variable Y An illustration of how the squared correlation accounts for variance in X, r =.7, r 2 =.49 high low How does correlation explain variance? Explained Suppose you wish to estimate Y for a given value of X. 49% of variance is explained Free to Vary

Now, lets look at some correlation coefficients and their corresponding scatter plots.

What is your estimate of r? r =.87r 2 =.76 = 76%

X Y r = -1.00r 2 = 1.00 = 100%

X Y r = +1.00r 2 = 1.00 = 100%

r =.04r 2 =.002 =.2%

r = -.44r 2 =.19 = 19%

Pearson r assumes that we are using interval or ratio data. What do we do if one or both of the variables we measured at the ordinal level? If we replace the scores with ranks, we can use the same formula. However, it can be simplified if we are using ordinal data. It is called a Spearman Rank-Order Correlation Coefficient.

Spearman’s Rank Order Correlation As noted, the Spearman r s is a special case of the Pearson r (when the data are ordinal). The formula, derived from the Pearson, is as follows: The characteristics and interpretation of a Spearman r s are exactly the same as a Pearson r. That is, r S ranges from -1 to +1, and the square provides an estimate of the shared variance.

Spearman Rank Order Correlation Coefficient One or both of the variables are in the form of ranks. Raw data may be converted to ranks, or ranks may be gathered as the original data. Example

Illustrated Calculation XY d= X – Yd2d N = 4

Choosing Between Pearson and Spearman If the data are ordinal, we have no choice, we have to use Spearman. If the data are interval or ratio, we do have a choice. –Pearson is more sensitive –Spearman easier to compute by hand

Summary of Measures of Relationship Spearman Rank Correlation Coefficient The Biserial Correlation Coefficient The Point-Biserial Correlation Coefficient The Phi Correlation Coefficient The Tetrachoric Correlation Coefficient The Rank-Biserial Correlation Coefficient rSrS There are other correlation coefficients for other levels of measurement. However, we will only study three, the two we have already reviewed and later, one more for nominal data.

Summarizing Correlations Pearson and Spearman Correlation Coefficients range from -1.0 to Pearson and Spearman Correlation Coefficients indicate both direction and magnitude of the relationship Correlation does NOT imply Causation