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Data Analytics for Management (Subject) / AppliedQ (Lesson)

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  • According to the estimated regression coefficients. is there a difference between the mean salaries earned by male and female employees at Datacom? If so, how large is the difference? 1. Repeat Regression to a Simple Linear Regression with only GENDER and Y.  2. The mean salary of male is... The mean salary of female is.. We can see that there is a difference in the salaries of male and female. The difference is large and the difference in amount is ..  According to the equation. the value is only dependent on the variable gender. If we use other explanatory variables, then the equation would be different. 
  • Types of Data Cross-Sectional Data Cross-sectional data is a random sample Each observation is a new individual, firm, etc. with information at a point in time If the data is not a random sample, we have a sample-selection problemv Time Series Data Time series data has a separate observation for each time period – e.g. stock prices Since not a random sample, different problems to consider  Trends and seasonality will be important
  • Are the Regression Assumptions satisfies? 1) Linearity Assumption: For each of the predictors X, the data points appear to be randomly distributed about the LSL (least squares line) on the scatter plot. Consequently, the assumption appears to be satisfied. EACH INDIVIDUAL SCATTERPLOT X on Y (2) Uniform Variance (Homoscedasticity) Assumption:  The variation in standardized residuals plotted against standardized predicted values looks reasonably unifrom around the horizontal line.  PLOT Regression Standardized Predicted Values (x) on Standardized Predicted Residuals (Y) (3) Normality Assumption:  The historgram of standardized residuals looks somewhat bell shaped and the point on the normal probability plot do not seem to depart too far from the diagonal  HISTOGRAM Standardized Residuals + Normal P-P Plot (Predicted Cum Prob, Observed Cum Prob.) (4) No Multicollinearity Assumption:  Since the correlation matrix does not contain any correlation greater than 0.8 for any pair of Xs, there is no indication that multicollinearity exists. CORRLEATION MATRIX
  • Write the results of the f test in the ANOVA table for the regression to predict Y The f test in the ANOVA for the regression to predict Y from X, X, X, is statistically significant at the 5% level.  We can safely conclude that the overall regression is significant, i.e. at least one coefficient in the regression is different from zero. 
  • What is the average effect of a variable with large p-vale? Strictly speaking, the coefficient of RUSH is not significantly different from 0 at the 5% level, i.e. RUSH has no effect. The probability of the corefficient being due to chance alone is moderate. However, if we are prepared to accept borderline evidence of an effect, then the best estimate is tht the time for oder processing is reduced by ...
  • Are there other assumptions of regression analysis you would like to check before accepting the above conclusions? We did not check whether the assumption of a linear relationship between the variables holds true. Hence, we should check whether the data points for each of the predictors X appear to be randomly distributed about the LSL (least squares line) on the scatter plot. Further, we have to check the residuals. The residuals should be normally distributed, homoscedastic and have no autocorrelation. Thus, we should check whether the standardized residuals plotted against standardized predicted values looks reasonably unifrom around the horizontal line. Further, we should whether the historgram of standardized residuals looks somewhat bell shaped and the point on the normal probability plot do not seem to depart too far from the diagonal. It would also be useful to obtain a correlation matrix for the independent variables to check on possible multicollinearity. If the correlation matrix does not contain any correlation greater than 0.8 for any pair of Xs, there is no indication that multicollinearity. Consequently, the assumption appears to be satisfied. There is still sizeable unexplained variation in the model. Is there a relevant variable that is not yet included? 
  • Should any variables be deleted from the model? Possible RUSH. It appeard to be marginally significant 5.7% For all other variables, all variables have a significant t-values, i.e. >2. Thus, we have enough reason to believe that the slope is actually different from zero. The probability of regression coefficient being due to chance alone is very small. 
  • Which one of the regression models should we use? We use this model as it is the best among the ones we have. Specifically, it has a high R Square, all coefficients are significant and there are no multicollinearity problems. By contrast, the model 1 has a variables which is likely to be correlatied with X2, resulting in incoherent results, such as a negative coefficient for X.
  • Interpret the regression coefficients Let the Stock Price Y be the dependent variable, and let equite X1 and dividend X2 be the independent variables. Then, the general multiple regression equation is given as: . Here, out estmated multiple regression equation is given as: Predicted ... = error. - The intercept indicated that i both equity and dividends are zero, then the predicte dstock price would be ... - The slope coefficient implies that if dividends are constant, as the equity increases by one unti, the predicted stock price increasses by .. Similarily,  - The Interaction coefficient implies that, the stock price inceeases by ..$ as the return increases with dividend.
  • Interaction Variables, i.e. Is there an extra premium for experience of men, in addition to the usual premium for experience? Interpretation: we want to see whether the effect of years on salary is different across the two genders Interaction of Dummy The interaction is the penalty per extra year of experience for females relative to males, in one category.  Interaction of Variable The coefficient interaction implies that, the stock price increases by Y as the return increases with annual dividend rate.
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  • Standard Error of Estimate The Standard error of estimaes (std or residuals) is the magnitude of error when the explanatory variables are taken together. If they are ignored, then the Std. of observed Y gives the magnitude of error.  The standard error of estimates in the simple linear regression is fiven as s1 66.78 and s2 135.91. The former tells us the magnitude of error when explanatory varibale Age of Truck if ignored. The latter tells us the magnitude of error when the explanatory varibale Miles Driven are ignored. Since the SE is less than the Std. of Observed Stock Prices, we can conclude that togehter, they are good predictore of stock price.  SE can increase as more "bad" variables are added (when Adj. R Square decreases) SE of multiple regression with two explanatory variables is less that SE of two simple linear regressions. This implies that when both explanatory variables are taken together, the prediction of Y is better.
  • Should you keep all coefficients in the model? Probably Rush should be elimnated. It lies on the margin of significance of 5.7%. This implies that we cannot be sure whether the variable for RUSH is significantly different from zero. It might be due to chance alone that is is included in the regression.  Depending on our desired level of confidence, we should eliminate the variable
  • What do we have to assume for testing purposes? The population model if given as: Y = β0 + β1X1 + ξ The Error component is assumed to have mean 0 and a variance and prbabilstically independent. For testing purposes we have to assume the errors are normally distributed.
  • Interpret the Confidence Interval The above Confidence Interval tells us that with 95% Confidence, the population regression coefficient will never be equal to zero. That means the dependent variable is always positively affected by the variable of rooms in the population with 95% Confidence. That is, some part of variability of the dependent variable is always explained by this variable.
  • Why do we standardize? Measure variables with different means on a single scale NORMSDIST(x) NORMSINV(p) + Mathematically convenient representation of the variable.  + Dimensionless  + Useful to evaluate different types of random variable with one yardstick. + Allows us to use one distribution to compare apples to oranges  Standardized scores standardize two things: center and spread. The reason that the standard deviation is changed to one specifically is that now our z-score is a measure of how many standard deviations we are from the mean, regardless of the size of that deviation.Now that we have adjusted for the spread of our distribution, we can use a single model to look at the likelihood of any observation (as long as the data is approximately normal).
  • Estimate required Sample Size answer If you wanted to be 95% condifent that the error in the estimate of p would be no more than 5%,then the std sample size formula required n to be 300
  • Interpretation Confidence Interval Reasoning  If we had enough resources to select 100 random samples of size 200 from the popluation of X, and then computed the 100 sample means with CIs, about 95 out of 100 CI would contain the unknown true population mean.Hence, we can be 95% certain of capturing the true population mean
  • Interpretation of the correlation coefficient. Correlation is the defree of linear relationship between two variables. The table shows that the correlation coefficient is .. This figure indicates that there is a positive association between the variables that is not very strong. The positive sign of the correlation coefficient means that the higher rainfall, the higher the yield is (and vice versa)
  • Residual Analysis in Regression The difference between the observed value of the dependent variable (y) and the predicted value (ŷ) is called the residual (e). Both the sum and the mean of the residuals are equal to zero: Σ e = 0 and e = 0. Each data point has one residual. Residual = Observed value - Predicted value Standardized residuals = (observed count – expected count) / √expected count --> measure of the strength of the difference between observed and expected values.  Residual Plot: Graph that shows the residuals on the Y axis and the Predictor variable on the X axis. The residuals should be randomly distributed around the horizontal line. Random pattern indicates = linear model provides a decent fit to the data. Non-random U-shaped = non-linear model

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