10/6/20081Multivariate MethodsLIR 832Multivariate Methods: Topics of the DayA Isolating Interventions in a multi-causalA. Isolating Interventions in a multicausal worldB. Multivariate probability DistributionsC. The Building Block: covarianceD.The Next Step: CorrelationD.The Next Step: Correlation10/6/20082A Multivariate WorldIsolating Interventions in a Multi-Causal WorldIsolating Interventions in a MultiCausal World A. Example of problem:  Evaluate a program to reduce absences from a plant? Is there age discrimination? B. Types of data ExperimentalQuasiexperimentalQuasi-experimental Non-experimental C. Need multivariate analysis to sort out causal relationships.Bi-Variate Relations: A First Run at Multivariate MethodsA. Many of the issues we are interested in areA. Many of the issues we are interested in are essentially about the relationship between two variables.B. Bi-variate can be generalized to multivariate relationshipsC. We learn bi-variate formally and make moreC. We learn bivariate formally and make more intuitive reference to multivariate.D. What do we mean by bi-variate relationship?10/6/20083Bi-Variate ExampleOur firm, has formed teams of engineers, accountants and ,g,general managers at all plants to work on several issues that are considered important in the firm. The firm has long been committed to gender diversity and we are interested in the distribution of gender among our managerial classifications. We are particularly concerned about the distribution of gender on these teams and particularly among engineers. Consider the distribution of two statistics about these three person teams.  a. gender of the team members (X: x = number of men) b. is the engineer a woman (Y: 0 = man, 1 = woman)Bi-Variate Example (cont.)10/6/20084Bi-Variate Example (cont.)Bi-Variate Example (cont.)10/6/20085Bi-Variate Example (cont.)Bi-Variate Example (cont.)We can also use this information to buildWe can also use this information to build conditional probabilities: What is the likelihood that the engineer is a woman, given that we have a man on the team?10/6/20086Bi-Variate Example (cont.)What is the likelihood that the engineer is a gwoman, given that we have a man on the team? P(Y = 1 & X = 1|X= 1)  = P(Y = 1 & X = 1)/P(X= 1)  = (2/8) / (3/8) = 2/3Note: P(Y= 1|X=2) is:“th b bilit th t Y i l t 1 i th t X 2"“the probability that Y is equal to 1 given that X = 2" or  “the probability that Y = 1 conditional on X = 2"Bi-Variate Example (cont.)What is the likelihood that there is only oneWhat is the likelihood that there is only one man, given the engineer is a woman? P(Y = 1 & X = 1|Y= 1)  = P(Y = 1 & X = 1)/P(Y= 1)  = (2/8)/(4/8) = 2/4 =1/210/6/20087Bi-Variate Example (cont.)What is the likelihood that the engineer is a gwoman?  P(Y= 1) = 1/2But if we know that there are two men, we can improve our estimate: P(Y=1 |X=2)P(Y1&X2|X2) = P(Y=1 & X=2|X=2) = P(Y=1 &X=2) / P(X=2) = 1/8 / 3/8 = 1/3What about calculating the likelihood of two men given the engineer is a woman? Example: Gender Distribution10/6/20088Example: Gender DistributionWorking with Conditional Probability:gy P(female) = 50.91% P(female| LRHR) = p(Female & LRHR)/P(LRHR) = 0.36/0.55 = 65% P(LRHR) = 0.55% P(LRHR|Female) = p(lrhr & female)/p(female) = .36/50.91 = .70%Independence DefinedNow that we know a bit about bi-variateNow that we know a bit about bivariate relationships, we can define what it means, in a statistical sense, for two events to be independent.If events are independent, then Their conditional probability is equal to theirTheir conditional probability is equal to their unconditional probability The probability of the two independent events occurring is P(X)*P(Y) = P(X,Y).10/6/20089Importance of IndependenceWhy is independence important?Why is independence important?  If events are independent, then we are getting unique information from each data point. If events are not independent, then A practical example on running a survey on employee satisfaction within an establishmentemployee satisfaction within an establishment. Example: Employee Satisfaction10/6/200810CovarianceCovariance: Building Block of Multi-Covariance: Building Block of Multivariate Analysis All very nice, but what we are looking for is a means of expressing and measuring the strength of association of two variables. How closely do they move together? Is variable A a good predictor of variable B? Move to a slightly more complex world, no more 2 and three category variablesExample: Age and Income Data10/6/200811Example:Age and Income DataExample: Age and Income Data10/6/200812Example: Age and Income Data__________________________________________________________________Descriptive Statistics: age, annual incomeVariable N Mean Median StDev SE Meanage 23 24.565 23.000 4.251 0.886annual I 23 17174 10000 15712 3276Variable Minimum Maximum Q1 Q3age 22.000 42.000 22.000 26.000annual I 0 65000 7000 25000_________________________________________________________________Example:Age and Income Data10/6/200813Example:Age and Income Data•Adding some info to the graph…Covariance and Correlation DefinedDefine Covariance and Correlation for aDefine Covariance and Correlation for a random sample of data: Let our data be composed of pairs of data (Xi,Yi) where X has mean μxand Y has mean μy. Then the covariance, the co-movement around their means is defined as:around their means, is defined as:10/6/200814Example: CovarianceWe observe the relationship between the number pof employees at work at a plant and the output for five days in a row:Attendance Output840328220639428What is the covariance of attendance and output?Example: Covariance (cont.)The covariance is positive. This suggests that when attendance is above its mean, output is also above its mean. Similarly, when attendance is below its mean, output is below its mean.10/6/200815Example: Overtime Hours and Productivity10/6/200816Example: Overtime Hours and ProductivityExample: Overtime Hours and ProductivityCovariances: prod-avg weekCovariances: prodavg, week prod-avg weekprod-avg 113.7292week-49 5667 22 6667week 49.5667 22.666710/6/200817Example: Overtime Hours and Productivity