Fluent Essays

Simple Regression Pre-analysis (include ALL charts, graphs and tables you generate formatted to class expectations). Assume all variables (with the exception of the subject variable) are ratio level. YOU MUST HAVE the Data Analysis Tool Pack add-in installed to complete this assignment.  Please review the notes for the week regarding how to use the Regression tool within the Data Analysis Tool Pack. Open the dataset associated with this assignment. Use New Capital Expense as the explanatory variable and End of Year Inventory as your response variable  Generate a single Descriptive summary table in Excel of the explanatory and response variables. That is, simultaneously enter both variables into the Descriptive Statistic tool. Discuss the mean, median, standard deviation, range, kurtosis, and skewness for EACH variable. Please DO NOT JUST LIST THE numerical summaries.  Discussion involves interpretation and should result in you writing complete sentences.  See the exemplar, post to Moodle, for a sample writeup. Use Excel to calculate and the report the correlation number for these two variables. Interpret the result of the correlation number by discussing the linear strength and direction of the relation between the two variables. Use Excel to generate and report a scatter plot for the 2 variables. The Explanatory variable should be on the horizontal axis.  Interpret the form and direction of the plot. Format plot to class expectations. Using only the visual inspection of the scatter plot, is a transformation warranted? Please review course notes and use what you notice about the dot pattern within the scatter plot as evidence to support your answer.  Use the Data Analysis Tool pack to run the regression analysis to generate a residual plot for these variables. DO NOT INTERPRET NOR REPORT THE NUMERICAL SUMMARIES generated in the regression tables. Copy and Paste ONLY the residual plot into your report.  Using only the visual inspection of the residual plot, is a transformation warranted? Review course notes and use what you notice about the dot pattern within the residual plot to support your answer. Check the 4 simple regression assumptions. Reference class notes so that you are clear what the checks should be and how to perform them in Excel.  Create 4 subheadings to discuss EACH of the 4 assumptions indicating if the assumptions were met or were violated.  Under each subheading, also include any tables or charts you’ve generated as evidence for your conclusions. Format all charts and tables to class expectations (see exemplar). Your claims regarding the assumptions must be based on content contained within the tables, numerical summaries , or charts that you generated. Please reference them by name in your write up.  For instance “  It can be seen in Figure 1 ….” Etc … If any of the simple regression assumptions were violated, attempt to fix them using any combination of the 8 basic mathematical transformations (applied to either the explanatory or response variable or both) as presented in the notes for this week.  Discuss in detail which transformation approach you decided to take and why.  DO NOT SUBMIT the transformed dataset for grading.  But,  please report all charts or tables generated in Excel that you used to support your transformation efforts. Homework – Week 5 Please submit your assignment to Moodle in PDF format only. Tables, charts, and graphs should be formatted as expected for the class. Font size of 11 pt or higher.  DO NOT SUBMIT YOUR EXCEL DATASET.  Copy and paste the numerical prompts below into your submission document and record your responses , as applicable, directly beneath. Include the following in your submission: Simple Regression Pre-analysis (include ALL charts, graphs and tables you generate formatted to class expectations). Assume all variables (with the exception of the subject variable) are ratio level. 1. YOU MUST HAVE the Data Analysis Tool Pack add-in installed to complete this assignment. Please review the notes for the week regarding how to use the Regression tool within the Data Analysis Tool Pack. 2. Open the dataset associated with this assignment. Use New Capital Expense as the explanatory variable and End of Year Inventory as your response variable 3. Generate a single Descriptive summary table in Excel of the explanatory and response variables. That is, simultaneously enter both variables into the Descriptive Statistic tool. Discuss the mean, median, standard deviation, range, kurtosis, and skewness for EACH variable. Please DO NOT JUST LIST THE numerical summaries. Discussion involves interpretation and should result in you writing complete sentences. See the exemplar, post to Moodle, for a sample writeup. 4. Use Excel to calculate and the report the correlation number for these two variables. Interpret the result of the correlation number by discussing the linear strength and direction of the relation between the two variables. 5. Use Excel to generate and report a scatter plot for the 2 variables. The Explanatory variable should be on the horizontal axis. Interpret the form and direction of the plot. Format plot to class expectations. 6. Using only the visual inspection of the scatter plot, is a transformation warranted? Please review course notes and use what you notice about the dot pattern within the scatter plot as evidence to support your answer. 7. Use the Data Analysis Tool pack to run the regression analysis to generate a residual plot for these variables. DO NOT INTERPRET NOR REPORT THE NUMERICAL SUMMARIES generated in the regression tables. Copy and Paste ONLY the residual plot into your report. Using only the visual inspection of the residual plot, is a transformation warranted? Review course notes and use what you notice about the dot pattern within the residual plot to support your answer. 8. Check the 4 simple regression assumptions. Reference class notes so that you are clear what the checks should be and how to perform them in Excel. Create 4 subheadings to discuss EACH of the 4 assumptions indicating if the assumptions were met or were violated. Under each subheading, also include any tables or charts you’ve generated as evidence for your conclusions. Format all charts and tables to class expectations (see exemplar). Your claims regarding the assumptions must be based on content contained within the tables, numerical summaries , or charts that you generated. Please reference them by name in your write up. For instance “ It can be seen in Figure 1 ….” Etc … 9. If any of the simple regression assumptions were violated, attempt to fix them using any combination of the 8 basic mathematical transformations (applied to either the explanatory or response variable or both) as presented in the notes for this week. Discuss in detail which transformation approach you decided to take and why. DO NOT SUBMIT the transformed dataset for grading. But, please report all charts or tables generated in Excel that you used to support your transformation efforts. Exemplar: Homework 1 Quant 530 1 Homework 1 : Exemplar 1. Generate descriptive statistics tables for each variable. Discuss measures of center, variability, skewness, kurtosis, and maximum and minimum values for each variable individually (5pts). Discuss a comparison of the mean and variability for both variables. (5 pts) Table 1 Descriptive statistic tables Approach 1 Approach 2 Mean 14.4 Mean 12.3 Standard Error 1.377259 Standard Error 1.452946 Median 14 Median 12 Mode 12 Mode 10 Standard Deviation 6.159289 Standard Deviation 6.497773 Sample Variance 37.93684 Sample Variance 42.22105 Kurtosis 0.110085 Kurtosis -0.74375 Skewness 0.337807 Skewness -0.08879 Range 24 Range 24 Minimum 4 Minimum 0 Maximum 28 Maximum 24 Sum 288 Sum 246 Count 20 Count 20 Measures of center Measures of the center provide an indication of what is likely the most typical number of subscriptions sold when either sales approach is used. It can be seen from the Descriptive Statistics tables that for both approach 1 and approach 2, the mean and median are close in value. This suggests that the distribution of subscription sales under both approaches are near bell in shape with most subscription sales remaining close to the typical value regardless of approach used. Even so, the tables suggests that when approach 1 is applied, on average a larger number of subscriptions will be sold than when approach 2 is applied. Exemplar: Homework 1 Quant 530 2 Measures of variability Given the near bell shape of both distributions, the Empirical Rule can be applied to provide a range estimate for the percentage of subscriptions sold. That is for approach 1 and approach 2, approximately 68\% of the number of subscriptions sold are between (8.2, 20.6) and (5.8, 18.8) respectively. Agents that are applying approach 1 are more likely and at a more consistent rate sale a higher number of subscriptions that those who applied approach 2. Furthermore, it can be seen from the Descriptive Statistics table that there have been agents who have not been successful selling any subscriptions while applying approach 2. This has not occurred, as of yet, when approach 1 has been applied. In fact, the highest selling agent applied approach 1 when accomplishing this goal. Kurtosis and Skewness The reported skewness numbers indicate that distributions of subscription sales under both approaches are fairly symmetric in shape suggesting a balance in the number of agents who sell subscriptions over and below the expected number. In fact, the reported kurtosis values suggest that it is not very likely that large percentages of agents will sell extremely poorly or well, beyond what is considered typical, regardless of which approach is used. 2. Generate a histogram for each variable. Discuss the shape of the histogram. (5pts) Figure 1: Number of subscriptions sold using Approach 1 0 1 2 3 4 5 6 7 8 9 5 10 15 20 25 30 S u b sc ri p ti o n s so ld Sale Categories Exemplar: Homework 1 Quant 530 3 Figure 2: Number of subscriptions sold using Approach 2 Shape of histograms Although the reported mean and median numerical summaries suggested that both approaches resulted in near bell shaped subscription sales distributions, closer to a bi-modal shaped distribution can be seen in Figure 2 for subscriptions sold under the method of approach 2. This suggests that the reported mean value of approximately 12 subscriptions sold for approach 2 may not be the best representation of the number of subscriptions that are likely to be sold by agents using this approach. If fact, it might be worth examining under which conditions are agents more likely to sell the high of 20 as opposed to the high of 10 subscriptions under this method. If these conditions are easily replicated, then it is possible that approach 2 may be considered as viable (or even more so) an approach to the sell of subscriptions as approach 1 is suggested to be. 3. Calculate the correlation coefficient between the two variables. Interpret the results (5 pts) A correlation of approximately .7 suggest a moderately positive relation between the subscription sales performance of an agent applying approach 2 and their performance when applying approach 1. That is, in general we can expect that if an agent does well selling subscriptions using approach 1, then there is a likely chance that they will also do well when using approach 2. It’s possible that this may mean that it is not necessarily the talent of the sales agent that is generating the sales differences observed but the actual use of the particular approach. More investigation is needed. 0 1 2 3 4 5 6 7 5 10 15 20 25 S u b sc ri p ti o n s so ld Sales Categories Exemplar: Homework 1 Quant 530 4 4. Select one of the variables to be explanatory and the other to be response. Indicate your selection. Generate a scatter plot of the two variables. Interpret the direction, form, and strength observed through examination of the data pattern (5 pts) Figure 3: Scatterplot of the relation between the number of subscriptions sold under approach 1 and approach 2 Figure 3 shows the dot pattern resulting when an agent’s sales performance using approach 1 (explanatory) is being evaluated for impact on their performance when using approach 2 (response). It appears from Figure 3 that a positive upward trending dot pattern is most evident for low and high selling agents. That is, if agents sell subscriptions well or badly under approach 1 they appear to be likely to also sell well or badly under approach 2. However that same positive linear trend doesn’t appear to be as evident for those agents who tend to sell near the mid-range of subscription levels around the reported mean value of 14.5 for approach 1. The dot pattern in that area appears more scattered and less systematically predictable. 0 5 10 15 20 25 30 0 5 10 15 20 25 30 A p p ro a ch 2 s a le s Approach 1 Sales Exemplar: Homework 1 Quant 530 5 5. On the scatter plot in Excel, generate the regression line and the regression equation. Re-Write the regression equation using the variable names. Write an interpretation of the regression equation using the variable names and the regression coefficient values. (5 pts) Figure 4: Scatterplot of the relation between the number of subscriptions sold under approach 1 and approach 2 with regression equation included The regression equation, (Approach 2 Sales) = .7(Approach 1 sales) + 2.3, suggests that for the typical agent the change in the rate of a sales made under approach 1 is lower than under approach 2. In other words, this suggests that more effort may be needed to generate a comparable number of subscription sales using approach 1 than when using approach 2. Moreover, the equation suggests that even the worst agent, that doesn’t successfully sale any subscriptions under approach 1 is likely on average to sell approximately two subscriptions using approach 2. y = 0.6959x + 2.2791 0 5 10 15 20 25 30 0 5 10 15 20 25 30 A p p ro a ch 2 s a le s Approach 1 Sales Simple Regression Learning Objectives • Conduct a descriptive statistics investigation prior to running a simple regression analysis • Check simple regression usage assumptions. Address violations to assumptions • Run a simple regression analysis. Interpret simple regression table output • Generate resulting simple regression equation from regression table output Simple Regression • Examines relation between 2 variables (explanatory and response variable) • Begin analysis with a descriptive statistic analysis (numerical and graphical) of each variable • Ensure the simple regression usage assumptions are met prior to running analysis in Excel • It is possible that a transformation to one or both variables may need to be applied prior to running the analysis Simple Regression • Examines relation between 2 variables (explanatory(x) and response variable(y)) • Explanatory variable is examined to determine how well it serves as a predictor for values of the response variable • Used to qualify the strength of the linear relation between the two variables • The sample regression equation, generated from the analysis, is dependent on the sample used and is an estimate of the true population regression equation The equation summarizes the overall dot pattern observed in a bivariate scatter plot of the two variables ෢(𝐵0 𝑖𝑠 𝑠𝑜𝑚𝑒𝑡𝑖𝑚𝑒𝑠 𝑢𝑠𝑒𝑑) ෢(𝐵1𝑖𝑠 𝑠𝑜𝑚𝑒𝑡𝑖𝑚𝑒𝑠 𝑢𝑠𝑒𝑑) Sample Simple Regression Line Sample Simple Linear Regression Coefficients Intercept b0 • It is the value where the regression line cross the y-axis on the graph • Value of the sample regression equation when the explanatory variable x = 0 • It is the expected mean value of the response variable when x=0 • If in practice the explanatory variable never has a value of zero, then the intercept is not interpreted • If x=0 is outside of the range of observed data used in the sample, do not interpret the intercept Example interpretation of the regression intercept (number units is 1k) 𝑆𝑎𝑙𝑒𝑠 𝑑𝑜𝑙𝑙𝑎𝑟𝑠 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ = 3.7 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑠 + 203.5 Interpretation : If there are no employees, the equation predicts that sales for the month will be on average $203,500. Intercept b0 Slope b1 • The regression slope coefficient is the mean amount of change in the response variable that the equation predicts as the explanatory variable is increased by 1 unit • The sign of the regression slope coefficient indicates the direction (positive or negative) of the linear relation between the 2 variables of the line is positive • If the regression slope coefficient is very near 0, then as one variable increases, the other remains fairly constant. Therefore, there is no meaningful predictive relationship present Example interpretation of the regression slope (number units is 1k) 𝑆𝑎𝑙𝑒𝑠 𝑑𝑜𝑙𝑙𝑎𝑟𝑠 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ = 3.7 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑠 + 203.5 Interpretation : If the number of employees increases by 1, the equation predicts the sales dollars per month on average will increase by approximately $3700. Intercept b1 • Results can be read in the simple regression Excel output table Simple Regression Usage Assumptions • The relation between the explanatory and response variables is linear • Check the scatter plot. Should have a linear dot pattern • The response variable does not have a relation the residuals (independent of the errors in prediction) • Check a scatterplot of residuals versus predicted values (residual plot). Should have a scattered dot pattern. Check the correlation between these values. Correlation should be very near zero • Residuals are normally distributed • Check a histogram of the residuals. Check skewness numbers. Check normal probability plots • Variance of residuals should be the statistically the same across all values of the explanatory variable • Check a scatterplot of residuals versus predicted values (residual plot). Should have a scattered dot pattern where the variance across values is consistent Residuals The estimates for the response variable made through the regression line are called the predicted values • To find residuals, subtract the predicted value (calculated with the regression equation) from the observed value • negative residual: the regression equation is overestimating • postive residual: the regression equation is underestimating • Best fitting line means the sum of its least squared residuals is smallest Scatter plot check for linearity Residual Plot check Assumptions met Assumptions not met Assumptions not met Scattered dot pattern and equal variance Normality Plot check Normal Skewed Left Skewed Right Thick tailed Thin tailed Assumptions are met if the residual dot pattern closely follows the line Histogram Normality check Near Normal Right Skewed Left Skewed Options to Fix linearity problems • Increase the sample size • Check for outliers and remove them if warranted or assign new variables through imputation (beyond the scope of the course) • Apply a transformation to the entire data set for one or both variables • Trying transformations may be an iterative process of ‘try and check’ in order to meet assumptions. Try fixing the linearity challenge first and then proceed on to the other fixes as needed • As transformations are applied, be sure to re-check all assumptions to ensure that they all hold after application of the transformation Applying a transformation Often trial and error. No one technique works all the time • Transform the explanatory (x) values only. • Try if linearity is the only assumption violation • Transform the response (y) values only. • Try when non-normality and/or unequal variances are the assumption violations • Transform both sets of values. • Try when linearity, non-normality , and unequal variances are the assumption violations Frequently used Transformations • In simple regression, try looking at the scatter plot dot pattern for hints on which transformation to try Try LN transformation Try reciprocal (1/x) or exponential 𝑒−𝑥 transformation Other Frequently used Transformations • Power transformations transform the response variable to some power (usually between -1 and 2) • Try if the residual variances are unequal and/or residuals are not normal • A square root or reciprocal transformation can be applied to the response variable • Try if the residual variances are unequal Which tried transformation should you use? • Run the simple regression analysis, in Excel, to generate the regression output table for each transformation • Examine the adjusted R squared value for each transformation and for the original data • Select the data set to use with the largest adjusted r squared value (or the smallest standard error value) What to check in the regression output table • The overall significance of the regression equation (in the ANOVA table) • less than .05 for the significance and/or p-values suggests the model is a good fit to the data • The overall significance of the regression coefficients (in the coefficient table) • less than .05 for the significance and/or p-values the explanatory variable is a good predictor of the response variable Approach we will use in this class • If regression equation is not significant, the model is not a good fit to the data • If the regression model is significant but the coefficient is not, the model provides improved fit over using the expected value of the response variable as the estimated prediction • If the regression model is significant and the coefficient is significant, the model is a good fit for the data and the explanatory variable is contributing significantly towards the quality of prediction What to check in the regression output table Interpretation: Results suggest that because the p- value .09>.05, the presence of East variable in the regression equation is not contributing significantly to the prediction of the response variable. Interpretation: Results suggest that because the significance F value .0003<.05, the regression equation is a good fit to the data and provides meaningful prediction of the response variables given the presence of the explanatory variables in the model Other numerical summaries in the regression output table to check • Correlation Coefficient: Strength of Linear Relation • Coefficient of Determination (r^2) - Use Adjusted R^2 • fraction of the variation in the data accounted for by the regression equation . Values are between 0 and 1 in the table but are sometimes reported as percentages • SEE (Standard Error of the Estimate) • standard deviation of the residuals • how spread out the observations are from the regression line Writing the Simple Regression Equation Report the regression equation ONLY if all simple regression assumptions are met Use the values in the coefficients column 𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 = −1.76 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑅𝑜𝑠𝑒𝑠 + 183475.43 Simple Regression Equation Writing the Simple Regression Equation Be sure to name the transformation variable, if needed, when writing the regression equation 𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 = −3.67 𝐿𝑁 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑅𝑜𝑠𝑒𝑠 + 32.58 Simple Regression Equation with LN transformation on the explanatory variable Introduction Ideally the goal is to select explanatory variables to build the simplest regression equation possible that will produce a good estimate of the response variable (parsimony) this to simplify interpretation of the regression coefficients selecting good variables for use in a regression analysis usually involves initially engaging in some research To use linear simple linear regression analysis: There needs to be a linear relation between: 1) the response variable and the explanatory variable Investigate the linear relations by examining: 1) Correlation: the numerical measure of strength in the relation between the explanatory and response variables 2) scatter plot of the response variable against the independent variable the plot can show linear dot pattern , indicating likely need for the explanatory variable to be included in the regression equation the plot can show non-linear dot pattern suggesting a transformation of an explanatory variable may be needed the plot can show a horizontal dot pattern suggesting the variable should not be used in the regression equation 3) residual plots looking for a scattered, cloudy dot pattern with consistent variance at each level of the explanatory variable to help confirm linear relation likely exists if a pattern is observed, this suggests transformations many need to be applied correlation Response variables (variable of interest) : typically placed on y-axis explanatory variables: typically placed on x-axis Choosing the role of the variables depends on how you think the variables are related Correlation coefficient (r): permits you to quantify the strength of the linear relation between the 2 variables In Excel use =CORREL feature properties of correlation coefficient range from -1 to 1 sign gives the direction of the association unitless measure not affected by changes to the center or scale of the variables depends only on the z-scores sensative to outliers does not imply causation (could be lurking variables standing behind the association) check the scatter plot for straightness even if the value of r is high scatter Scatterplots displays relation (association) between 2 quantitative variables investigor can see patterns, trends, and outliers 1. Direction of the association negative positive 2. Form of the association nonlinear vs linear 3. Strength of association How discernable is the data pattern or trend? weak, moderate, strong residual Question: What line can be used to model the linear data pattern displayed in the scatterplot? the equation of the best fit line is the regression line (model) the estimates for the response variable made through the regression line are called the predicted values residuals To find residuals, subtract the predicted value (calculated with the regression equation) from the observed value negative residual: the regression equation is overestimating postive residual: the regression equation is underestimating Best fitting line means the sum of its least squared residuals is smallest Example1A Question: Is there a linear relation between the response variable (salary) and housing price? 1) correlation (CORELL feature in Excel) 2) scatter plot 3) residual plot 1) Correlation 0.9086846363 3) Residual Plot (Run the regression in Excel, using the Data Analysis Toolbar. Select Residual plot A correlation of approximately .91 implies that salary and housing price have a strong, positive, linear relation where salary increase ad housing price increases in a systematically predictable way Salary($10,000) Housing Price ($10,000) 1 45 2) Scatter Plot (select variables and use the Insert .. Charts menu) 2 47 3 57 4.5 59 5.2 60 8.4 65 11 66 12.4 67 12.5 69 13.1 70 15 70 17 72 22.2 73 22.6 74 There is a pattern (dot pattern is not scattered) evident in the residual plot. This suggests that some other relation, beyond a linear relation 25.8 76 would likely better represent the relation between salary and housing price. Hence a transformation should be performed From the scatter plot is can be seen that the relation appears to be positive and strong. However there also appears to be a slight bend in the dot pattern. Further investigation is needed in order to determine if the trend is linear Housing Price ($10,000) 1 2 3 4.5 5.2 8.4 11 12.4 12.5 13.1 15 17 22.2 22.6 25.8 45 47 57 59 60 65 66 67 69 70 70 72 73 74 76 Salary Housing Price ($10,000) Transform1 Goal: Make the dot pattern of a scatterplot more nearly linear calculate transform for the explanatory and the response variables look at scatterplots of combinations of transformed and/or non-transformed values Can use Adjusted R-Squared to help chose between candidate linear equation models that result Non-linear dot patterns in scatterplots and possible transformations to straighten year salary (in $10,000) LN(salary) 1980 1 1982 2 1990 3 1990 4.5 1991 5.2 1996 8.4 1997 11 1998 12.4 1999 12.5 1999 13.1 1999 15 2001 17 2001 22.2 2004 22.6 2005 25.8 (LN(x), Y) (x, y2) (LN(y), LN(x)) (LN(x), Y) (x, 1/y) (x, SQRT(y)) (x, LN(y)) (LN(y), LN(x)) Significance What else to check in the Regression Output The overall significance of the regression equation (in the ANOVA table) The regression coefficent is the average change in y expected per unit change in the explanatory variable when all other explanatory variables are held constant How significant are the regression equation and/or the coefficients? less than .05 for the significance and/or p-values , 1 or more of the explanatory variables are good predictors of the response variable also, check R-squared (use adjusted R-squared) Standard Approach we will use (1 of the following will be the investigation conclusion) If regression equation is not significant, the model is not a good fit to the data If the regression model is significant but the coefficient is not, the model provides improved fit over using the expected value of the response variable as the estimated prediction If the regression model is significant and the coefficient is significant, the model is a good fit for the data and the explanatory variable is contributing significantly towards the quality of prediction Coefficient significance Model significance Example2 Is the regression model with the Square Root Transformed response variable salary the better model? Yes according to the analysis (scatter plots, residual plots, adjusted r-squared, and standard error comparisons) year (actual) year (Explanatory) salary (in $10,000) (Response) year (Explanatory) SQRT(Salary) Compare Scatter Plots : Which looks visually straighter ? 1980 80 1 80 1 1982 82 2 82 1.4142135624 1990 90 3 90 1.7320508076 1990 90 4.5 90 2.1213203436 1991 91 5.2 91 2.2803508502 1996 96 8.4 96 2.8982753492 1997 97 11 97 3.3166247904 1998 98 12.4 98 3.5213633723 1999 99 12.5 99 3.5355339059 1999 99 13.1 99 3.6193922142 1999 99 15 99 3.8729833462 2001 101 17 101 4.1231056256 2001 101 22.2 101 4.7116875958 2004 104 22.6 104 4.7539457296 2005 105 25.8 105 5.0793700397 Compare Residual Plots : Which has a cloudy, scattered dot pattern ? Response: Salary Response : Square Root of Salary Compare Adjusted R-squared values : Which is largest ? Compare Standard Error: Which is smallest? Both models are significant The regression model with the transformed response variable has the largest Adjusted R-squared value and smallest standard error Response Variable : Salary Response Variable : Square Root Of Salary Regression Statistics Regression Statistics Multiple R 0.9150264296 Multiple R 0.9668776273 R Square 0.8372733668 R Square 0.9348523462 Adjusted R Square 0.8247559335 Adjusted R Square 0.9298409882 Standard Error 3.28038926 Standard Error 0.3337444496 Observations 15 Observations 15 ANOVA ANOVA df SS MS F Significance F df SS MS F Significance F Regression 1 719.7849352684 719.7849352684 66.8885821383 0.0000017547 Regression 1 20.7785720521 20.7785720521 186.5467103404 0.0000000044 Residual 13 139.892398065 10.7609536973 Residual 13 1.4480096496 0.1113853577 Total 14 859.6773333333 Total 14 22.2265817017 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Lower 95.0\% Upper 95.0\% Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Lower 95.0\% Upper 95.0\% Intercept -80.4842432619 11.3048781078 -7.1194260119 0.0000078229 -104.906947591 -56.0615389328 -104.906947591 -56.0615389328 Intercept -12.4661593835 1.1501501874 -10.8387230813 0.0000000701 -14.9509077987 -9.9814109684 -14.9509077987 -9.9814109684 year (Explanatory) 0.9657567381 0.1180841892 8.1785440109 0.0000017547 0.710651357 1.2208621192 0.710651357 1.2208621192 year (Explanatory) 0.164087017 0.0120138007 13.6582103637 0.0000000044 0.1381327785 0.1900412554 0.1381327785 0.1900412554 salary (in $10,000) (Response) 80 82 90 90 91 96 97 98 99 99 99 101 101 104 105 1 2 3 4.5 5.2 8.4 11 12.4 12.5 13.1 15 17 22.2 22.6 25.8 SQRT(Salary) 80 82 90 90 91 96 97 98 99 99 99 101 101 104 105 1 1.4142135623730951 1.7320508075688772 2.1213203435596424 2.2803508501982761 2.8982753492378879 3.3166247903553998 3.5213633723318019 3.5355339059327378 3.6193922141707713 3.872983346207417 4.1231056256176606 4.7116875957558984 4.7539457296018854 5.0793700396801178 regression equation after trans To undo the LN transformation to determine a regression equation for the original salary units, take the exponential (e) of both sides of the equation To Undo the LN transformations (x, LN(Y)) LN(Y) = ax + b y = e(ax + b) y = eax (eb) (LN(x), y) y = aLN(x) + b (LN(x), LN(y)) LN(y) = aLN(x) + b LN(y) = aLN(x) + b LN(y) = LN(xa) + b y = xa + eb numerical summaries Numerical Summaries: Excel Regression Output Table (color coded) Regression Coefficients Excel Regression Output Table y-intercept: average value of the response variable when all explanatory variables are 0 Regression Statistics impact of explanatory variables are not considered Multiple R 0.9150264296 R Square 0.8372733668 regression coefficients: the average predicted change in the response variable per unit change in the explanatory variable Adjusted R Square 0.8247559335 Standard Error 3.28038926 Correlation Coefficient: Strength of Linear Relation Observations 15 Coefficient of Determination (r^2) - Use Adjusted R^2 ANOVA fraction of the variation in the data accounted for by the regression equation df SS MS F Significance F sometimes reported as a percentage Regression 1 719.7849352684 719.7849352684 66.8885821383 0.0000017547 between 0 and 1 Residual 13 139.892398065 10.7609536973 Total 14 859.6773333333 SEE (Standard Error of the Estimate) standard deviation of the residuals Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Lower 95.0\% Upper 95.0\% how spread out the observations are from the regression line Intercept -80.4842432619 11.3048781078 -7.1194260119 0.0000078229 -104.906947591 -56.0615389328 -104.906947591 -56.0615389328 year (Explanatory) 0.9657567381 0.1180841892 8.1785440109 0.0000017547 0.710651357 1.2208621192 0.710651357 1.2208621192 Confidence Intervals : Range, withing specified level of confidence, that the population parameter resides Influential Points Influential Points Does the regression equation change significantly when the isolated points is removed? Would a different result from the analysis be concluded? If yes, then the point is influential In the regression output look at: significance of model or coefficients numerical summaries Example with influencial point without influential point recode Recode multiple category variables (3 or more categories) to 2 category variable Gender: male (1) female (0) Political Party (0) republican, (1) democrat, (2) libertarian, (3) green Student IQ Study hours Gender Political Party Test score 1 110 40 1 0 100 2 110 40 0 2 95 3 120 30 1 1 90 4 110 40 1 3 85 5 100 20 0 3 80 6 110 40 1 1 75 7 90 0 0 0 70 8 110 40 0 2 65 9 80 10 1 3 60 10 80 10 0 1 55 The dummy variables are 1 when (zero otherwise): x1: republican x2: democrat x3: libertarian Student IQ Study hours Gender x1 x2 x3 Test score 1 110 40 1 1 0 0 100 2 110 40 0 0 0 1 95 3 120 30 1 0 1 0 90 4 110 40 1 0 0 0 85 5 100 20 0 80 6 110 40 1 75 7 90 0 0 70 8 110 40 0 65 9 80 10 1 60 10 80 10 0 55