Quantitative Business Analysis for Decision Making

Quantitative Business Analysis for Decision MakingSimple Linear Regression403.72Lecture OutlinesScatter Plots Correlation AnalysisSimple Linear Regression ModelEstimation and Significance TestingCoefficient of DeterminationConfidence and Prediction IntervalsAnalysis of Residuals403.73Regression Analysis ?Regression analysis is used for modeling the mean of “response” variable Y as afunction of “predictor” variables X1, X2,.., Xk. When K = 1, it is called simple regression analysis.403.74Random SampleY: Response Variable, X: Predictor VariableFor each unit in a random sample of n, the pair (X, Y) is observed resulting a random sample: (x1, y1), (x2, y2),... (xn, yn) 403.75Scatter PlotScatter Plotis a graphical displays of the sample (x1, y1), (x2, y2),... (xn, yn) by n points in 2-dimension. It will suggest if there is a relationship between X and Y403.76A Scatter Plot Showing Linear Trend162126152025NielsenPeopleMA Scatter Plot Showing Linear Trendof Peoples Ratings and Nielsen Ratings403.77A Scatter Plot Showing No Linear Trend-101-101TodayYesterdaA Scatter Plot Showing No Linear Trendof Today's With Yesterday's DJIA403.78Modeling linear Trend A perfect linear relationship between Y and X exists if . Coefficient is the slope--quantifying the amount of change in y corresponding to one unit change in x.There are no perfect linear relationships in practical world. X of XY403.79Simple Linear Regression ModelModel: is linear function (nonrandom)is random error. It is assumed to be normallydistributed mean 0 and standard deviation . So are parameters of the modelXY  and ,XXy403.710EstimationSimple linear regression analysis estimates the meanof Y (linear trend) by and X y      bxayˆxbya2)())((xxyyxxb403.711Standard deviationStandard deviation(s) of the sample of n points in the scatter plot around the estimated regression line is:bxayˆ2ˆ2nyys403.712Testing the Slope of Linear TrendFor Testingcompute t-statistic and its p value:0a00:H vs.:Hbsb0-statistic-t403.713Coefficient of Determination: R2A quantification of the significance of estimated model is denoted by R2.R2> 85%= significant modelR2< 85%= model is perceived as inadequateLow R2will suggest a need for additional predictors for modeling the mean of Ybxayˆ403.714Correlation Coefficient: rThe correlation coefficient r is the square root of R2. It is a number between -1 and 1.–Closer ris to -1 or 1, the stronger is the linear trend –Its sign is positivefor increasing trend (slope b is positive)–Its sign is negativefor decreasing trend (slope b is negative)403.715Confidence and Prediction IntervalsTo estimate by a confidence interval, or to predict response Y corresponding to its predictor value x = x0–1. Compute:–2. compute:xy0ˆbxayyesyˆ..ˆ403.716What is ?i.e. Standard Error ofyesˆ..For estimating ,y220)()(1)ˆ.(.xxxxnsyesFor Predicting Y,220)()(11)ˆ.(.xxxxnsyesyˆ403.717Analysis of ResidualsResiduals are defined:Residual analysis is used to check the normality and homogeneity of variance assumptions of random errors .Histogram or box plot of residuals will help to ascertain if errors are normally distributed.  2,....n 1,i ,ˆiiiyye403.718Analysis of Residuals (con’t)Plot of residual against observed predictor values xi will help ascertain homogeneity assumption. –random appearance= homogeneity of variance assumption is valid.–non-random appearance=homogeneity assumption is not valid and variance is dependent on predictor values. ie