1 G Lect 2M Examples of Correlation Random variables and manipulated variables Thinking about joint distributions Thinking about marginal distributions: Expectations Covariance as a statistical concept and tool G Multiple Regression in Psychology

2 G Lect 2M Three examples of correlation All from bar exam study discussed last week »Anxiety and Depression from POMS on day 29 (two days before bar exam) »Anger and Vigor from POMS on day 29 (two days before bar exam) »Anxiety and day to exam during week prior to start of exam.

3 G Lect 2M Anxious and Depressed Mood 2 Days Before Exam What do you notice about joint distribution? What is correlation? r = 0.64

4 G Lect 2M Anger and Vigor 2 Days Before Exam What do you notice about joint distribution? What is correlation? r = -.19

5 G Lect 2M Anxious Mood in Days Before the Exam What do you notice about joint distribution? What is correlation? r =.25

6 G Lect 2M Random Variables vs. Manipulated Variables A random variable is a quantity that is not known exactly prior to data collection. »E.g. anxiety and depression on any given day for a randomly selected subject A manipulated variable is a quantity that is determined by a sampling plan or an experimental design. »E.g. Day to exam, level of exposure, gender This distinction will have implications on statistical analysis of bivariate association.

7 G Lect 2M Thinking about bivariate (Joint) distributions Suppose we sample persons and measure two behaviors. »Both are random »The variables might be related or independent »The joint distribution contains information about each variable and the relation among them. When we ignore one of the two variables, and study the other, we say we are studying the Marginal distribution »This term simply reminds us that another variable is in the background

8 G Lect 2M Suppose we measure X, and Y, but choose to study only X (ignoring Y). We can describe the marginal distribution of X using the mean, the variance, and other moments such as coefficient of skewness and kurtosis. The population moments of the variable are described with Expectation Operators. Expectation operators can be used to study means and variances. Expectations and Moments for Marginal Distributions

9 G Lect 2M Expectation operators defined The population mean,  = E(X), is the average of all elements in the population. It can be derived knowing only the form of the population distribution. »Let f(X) be the density function describing the likelihood of different values of X in the population. »The population mean is the average of all values of X weighted by the likelihood of each value. If X has finite discrete values, each with probability f(X)=P(X), E(X)=  P(x i )x i If X has continuous values, we write E(X)=  x f(x) dx

10 G Lect 2M Rules for Expectation operators E(X)=  x is the first moment, the mean Let k represent some constant number (not random) »E(k*X) = k*E(X) = k*  x »E(X+k) = E(X)+k =  x +k Let Y represent another random variable (perhaps related to X) »E(X+Y) = E(X)+E(Y) =  x +  y »E(X-Y) = E(X)-E(Y) =  x -  y Putting these together »E( ) = E[(X 1 +X 2 )/2] =(  1 +  2 )/2 =  The expected value of the average of two random variables is the average of their means.

11 G Lect 2M Variance Operators Analogous to E(Y)= , is V(Y)=E(Y  ) 2 =  (y  ) 2 f(y) dy E[(X-  x ) 2 ] = V(X) =  x 2 Let k represent some constant »V(k*X) = k 2 *V(X) = k 2 *  x 2 »V(X+k) = V(X) =  x 2 Let Y represent another random variable that is independent of X »V(X+Y) = V(X)+V(Y) =  x 2 +  y 2 »V(X-Y) = V(X)+V(Y) =  x 2 +  y 2 A more general form of these formulas requires the concept of covariance

12 G Lect 2M Covariance: A Bivariate Moment E[(X-  x )(Y-  y )] = Cov(X,Y) =  XY is called the population covariance. »It is the average product of deviations from means »It is zero when the variables are linearly independent Formally it depends on the joint bivariate density of X and Y, f(X,Y). »f(X,Y) says how likely are any pair of values of X and Y »Cov(X,Y)=  (X-  x )(Y-  y )f(X,Y)dXdY

13 G Lect 2M Cov (X,Y) as an expectation operator »For k 1 and k 2 as constants, there are facts closely parallel to facts for variances: Cov(k 1 +X, k 2 +Y) = Cov(X,Y) =  XY Cov(k 1 X, k 2 Y) = k 1 *k 2 *Cov(X,Y) = k 1 *k 2 *  XY »Important special case: Let Y * = (1/  Y )Y and X * = (1/  X )X V(X * ) = V(Y * ) = 1.0 Cov(X *,Y * ) = (1/  Y ) (1/  X )  XY =  XY Cov (X *,Y * ) is the population correlation for the variables X and Y,  XY »Since  XY = (1/  Y ) (1/  X )  XY,  XY = (  Y ) (  X )  XY

14 G Lect 2M An important use of correlation and covariance We are often interested in linear functions of two random variables: aX+bY »a=1, b=1 gives sum »a=.5, b=.5 gives average »a=1, b=-1 gives difference What is the expected variance of W=aX+bY in general? »Var(W) = V(aX+bY) = a 2 V(X)+b 2 V(Y) + 2ab Cov(X,Y) = a 2  x 2 +b 2  y 2 + 2ab  x  y  xy »This can be used to compute expected standard error of contrasts of sample statistics.

15 G Lect 2M Example Suppose we want to average the POMS anxious and depressed moods. What is the expected variance? In the sample on day 29, »Var(Anx)=1.129, Var(Dep)=0.420 Corr(A,D)= 0.64 Cov(A,D)=.64*(1.129*.420) 1/2 = »Var(.5*A+.5*D) =.(25)(1.129)+(.25)(.420) +(2)(.25)(.441) = 0.648

16 G Lect 2M Final Comment Standard deviations and variances are particularly useful when variables are normally distributed Expectation operators assume that f(X), f(Y) and f(X,Y) can be known, but they do not assume that these describe bell shape or normal distributions Covariances and correlations can be estimated with non- normal variables, but be careful about statistical tests.