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# Statistic & Research Design II PSY 862

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This 5 page Class Notes was uploaded by Miss Mellie O'Conner on Sunday October 11, 2015. The Class Notes belongs to PSY 862 at Eastern Kentucky University taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/221434/psy-862-eastern-kentucky-university in Psychlogy at Eastern Kentucky University.

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Date Created: 10/11/15

Logic of Regression Discuss the logic of regression analysis Include discussion of the following in your answer scatterplot regression line actual Y predicted Y regression and residual least squares criterion lt X x Y 7 Total Mean afYY Figure 1 The scatterplot Figure 1 shows the scores of a group of people on two valiables X and Y Vaiiables X and Y are related and we wish to use X to predict Y If we don t know a persons score on X the best prediction we can make is to predict the mean on Y Y If we do know X then we can use the regression equation and predict that the person will fall on the regression line That is we predict the persons Y score will be Y The amount of error in predicting Y is this is called error or residual The amount of accuracy we gain in predicting Y when we know X is this is called regression The equation for the regression line is Y39 a b X where b is the slope b in Figure l and a is the n intercept We select the line using the least squares criterion That means we select the line for which the sum of the squared errors is a minimum 2 X e The total variance in the Y variable is broken down into two components residual and regression SSTofal SSreridua regression Note that Y39 is the same for all individuals with the same X Any differences among the individuals with a particular X score are residual differences What happens when the correlation between X and Y is small The residual sum of squares will be relatively large when the scatter of the points about the regression line is relatively large What happens when the correlation between X and Y is large or perfect Conversely the closer the points are to the regression line the smaller the residual sum of squares When all the points are on the regression line the residual sum of squares is zero and explanation or prediciton of Y using X is perfect the correlation is 100 What happens when the correlation between X and Y is zero If on the other hand the regression of Y on X is zero the regression line has no slope and will be drawn horizontally at the mean of Y and the correlation betweenX and Y is zero In that case the mean of Y Y will be the predicted value for everyone regardless of their X score Under such circumstances SS 2 Pi2 Z 17 Y 0 and all the deviations are due to error residual as follows S5 Zez ZltY W ZltY 1 02 SSW and in this situation knowledge of X does not enhance the prediction of Y Selecting the Appropriate Statistic Part 1 Used to test Number of Groups Question to be Statistic to Choose Null and hypothesis about Answered Verbal Alternative statement of the Hypothesis hypothesis Means One Is the mean of the Single Sample ttest H0 p number group significantly H1 p number different from some or number H1 p lt number or H1 p gt number example H0 p 30 Two independent Is the mean of one Between Subjects H0 pl p2 separate groups group significantly Independent H1 p it different from the Samples ttest or mean of the other H1 pltp or H1 in it Two groups of Is the mean of one Within Subjects HO pl p2 0 matched subjects or set of scores repeated measures H1411 p2 0 two measures on significantly or dependent the same subjects different from the groups ttest repeated measures mean of the other set of scores Ranks Two independent Are the ranks MannWhitney U HO The ranks are separate groups assigned by one Test no different group higher or lower than the H1 The ranks are ranks assigned by higher or lower the other group Two groups of Are the ranks Wilcoxon Matched HO The ranks are matched subjects or assigned in one Pairs Signed Ranks no different two measures on situation higher or Test the same subjects lower than the H1 The ranks are repeated measures ranks assigned in higher or lower another situation Shape of the One groupset of Are the scores Chi Square H0 The distribution Distribution scores norm ally goodness of fit is normal distributed Frequency H1 The distribution Distribution Or differs from normal Are the scores distributed in some specified way Frequency Count the number Is there a Chi Square for H0 There is no Distribution in each of several relationship Contingency Tables relationship the Counts categories on two between the dimensions dimensions dimensions variables are variables independent H1 There is a relationship DECISION TREE FOR SELECTION OF STATISTICAL TECHNIQUES Predictor Independe t0utcome Hypothesis Tested Criterion for Combining Vari bl ame of Techniques Variables Dependent Criterion of Optimality Variables 1 discrete group 1 Is the mean of group 1 significantly tstest membership 2 1eve1 different from the mean of group 2 1 discrete group 1 Are there any significant difference among Onesway Ana1ysis of membership more th the means of the groups Variance 2 2 or more discrete 1 ther gni 39 ant interaction b tween Higher Order ANOVA group membership the Independent Variab1es or a main ffect the dependent variabl s Effec s interpreted as differences among the means 1 continuous 1 Is the predictor variable correlated withimize correlation of pred ct arson r or for the outcome varia outcome values with actual prediction purposes outcome values bivariate regression 2 or more continuous 1 Is a significant amount of the variaicthximize the correlation of eMultiple correlation the outcome variable accounted for b thembined predictor variables imor prediction purpose predictors or Can you predict the o tcahe outcome varia multiple regression n the basis of the predictors analysis RA Mixture of discrete 1 Is there a significant difference amnthhdmize the correlation of eAnalysis of Covariance group membership ard groups holding the continuous variab escontinuous predictors with t e ANACOVA continuous variables constant ie statistically control ingutcome variable within leve s of for the continuous variab1es the discrete variab1es 1 discrete group 2 or more Is there a difference between the prtfiimimize the taratio on the llotelling39s 12 membership with 2 of outcome variab1es for these group meined outcome variab1es discriminant ana1ysis the groups differ on any one or any combination of the outcome variables 1 discrete grou or more Is the profile of outcome variables Maximize the Faratio for the onOnesway multivariate membership with mor significant1y different for any one f why ANOVA on the combined oulcomalysis of variance th groups Do the groups differ on any onvaaniab es MANOVA Discriminan any combination of the outcome varia les alysi 2 or more discrete 2 or more s there a significant interaction b thematio on the combined outcx me Higher Order MANOVA group membership the independent variab1es or a signil icmtiables is maximized for e ch main ef ect in terms of the combined main effect and each interac ion outcome variab es fect Mixture of discrete 2 or more Is there a significant interaction o thximize the correlation of Multivariate ANACOVA group membership d effect of the combined outcome a 39 predictors with MANACOVA continuous statistically controlling for the outcome variables within lev continuous predictor variables of the discrete variables 2 or more continuous 2 or more Is there a relationship between the etMa imize the correlation of Canonical Correlation predictor variab1es and the set of o tccmnbined predictor variab1es wi nnonical Ana1ysis variables combined outcome variables 2 or more continuous What is the underlying structure of hiMaximize the variance of 39 39 Components set of observations Dimension redux tidpredictor7 variab1es Factor Ana1ysis PCA 2 or more continuous What is the underlying structure of hiReproduce intercorrelations odgctor Analysis FA set of observations Dimension redu tioirigina1 variab1es as accura e1y as possib1e Combined variable refers to an optimal w ch variab1e is computed by a formu1a su W ased on iiarris R J 1985 A Primer of Multiva as W X w2X2 WJX te Sta kam s Academic Press Inc Orlando p 11 eighted combination of the set of variables usually outcome variables In most statistical techniques The weighting coefficients w are selected to maximize the criterion of optimality identified in the table above Shrinkage The choice of weights in a regression analysis is designed to yield the highest possible correlation between the independent variables and the dependent variable Rsquare That is the multiple correlation can be expressed as the correlation between the predicted scores based on the regression equation Y predicted and the observed criterion scores actual Y Since you pick the weights to produce the largest possible correlation the regression equation may be over tted to idiosyncrasies of the sample data If your development sample has some unusual or atypical individuals they may pull the regression equation off If you drew another sample that didn t contain these outliers the results could be quite different It can be demonstrated that when a regression equation developed on one sample is applied to a new sample the resulting R square is almost always smaller than the Rsquare obtained with the original development sample This phenomenon is called shrinkage Dealing with Shrinkage There are a couple of ways to deal with Shrinkage in a Multiple Regression Analysis Adjusted R Square Rsquare can be interpreted as the percent of criterion variance accounted for by the linear combination of the predictors The sample multiple correlation and the squared multiple correlation are biased estimates oftheir corresponding population values The sample Rsquare typically overestimates the population Rsquare and needs to be adjusted downward The adjusted Rsquare reported by SPSS makes the adjustment assuming a fixed effects model There are several other formulas for computing the adjusted Rsquare depending on which assumptions are made CrossValidation Probably the best method for estimating the degree of shrinkage is to perform a crossvalidation This is done by using two samples for which both the predictor variables and the criterion or outcome variable are known The first sample is the development sample the second sample is called the calibration sample The steps are as follows 1 Forthe first sample a regular regression analysis is performed and Rsquare and the regression equation are calculated 2 The regression equation is then applied to the predictor variables of the second sample yielding a Ypredicted for each subject 3 The correlation between the predicted scores and the observed criterion scores in the calibration sample is computed 4 The difference between Rsquared obtained from the development sample and the R squared obtained from the calibration sample is an estimate ofthe amount of shrinkage lfthe shrinkage is small the regression is considered valid Normally the two samples are then combined to produce one large development sample and the final regression equation is computed 01

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