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**Regression Concept & Examples, Latent Variables, & Partial Least Squares (PLS)**

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**Simple Regression Model**

Make prediction about the starting salary of a current college graduate Data set of starting salaries of recent college graduates Data Set Compute Average Salary How certain are of this prediction? There is variability in the data.

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**Simple Regression Model**

Use total variation as an index of uncertainty about our prediction Compute Total Variation The smaller the amount of total variation the more accurate (certain) will be our prediction.

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**Simple Regression Model**

How “explain” the variability - Perhaps it depends on the student’s GPA Salary GPA

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**Simple Regression Model**

Find a linear relationship between GPA and starting salary As GPA increases/decreases starting salary increases/decreases

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**Simple Regression Model**

Least Squares Method to find regression model Choose a and b in regression model (equation) so that it minimizes the sum of the squared deviations – actual Y value minus predicted Y value (Y-hat)

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**Simple Regression Model**

How good is the model? a= 4,779 & b = 5,370 A computer program computed these values u-hat is a “residual” value The sum of all u-hats is zero The sum of all u-hats squared is the total variance not explained by the model “unexplained variance” is 7,425,926

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**Simple Regression Model**

Total Variation = 23,000,000

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**Simple Regression Model**

Total Unexplained Variation = 7,425,726

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**Simple Regression Model**

Relative Goodness of Fit Summarize the improvement in prediction using regression model Compute R2 – coefficient of determination Regression Model (equation) a better predictor than guessing the average salary The GPA is a more accurate predictor of starting salary than guessing the average R2 is the “performance measure“ for the model. Predicted Starting Salary = 4, ,370 * GPA

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**Detailed Regression Example**

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**Data Set Obs # Salary GPA Months Work 1 20000 2.8 48 2 24500 3.4 24 3**

23000 3.2 4 25000 3.8 5 6 22500 36 7 27500 4.0 20 8 19000 2.6 9 24000 10 28500 12

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**Scatter Plot - GPA vs Salary**

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**Scatter Plot - Work vs Salary**

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**Pearson Correlation Coefficients -1 <= r <= 1**

Salary GPA Months Work 1

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**Three Regressions Salary = f(GPA) Salary = f(Work)**

Salary = f(GPA, Work) Interpret Excel Output

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**Interpreting Results Regression Statistics Statistical Significance**

Multiple R, R2, R2adj Standard Error Sy Statistical Significance t-test p-value F test

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**Regression Statistics Table**

Multiple R R = square root of R2 R2 Coefficient of Determination R2adj used if more than one x variable Standard Error Sy This is the sample estimate of the standard deviation of the error (actual – predicted)

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**ANOVA Table Table 1 gives the F statistic Tests the claim**

there is no significant relationship between your all of your independent and dependent variables The significance F value is a p-value should reject the claim: Of NO significant relationship between your independent and dependent variables if p< Generally = 0.05

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**Regression Coefficients Table**

Coefficients Column gives b0 , b1, ,b2 , … , bn values for the regression equation. The b0 is the intercept b1value is next to your independent variable x1 b2 is next to your independent variable x2. b3 is next to your independent variable x3

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**Regression Coefficients Table**

p values for individual t tests each independent variables t test - tests the claim that there is no relationship between the independent variable (in the corresponding row) and your dependent variable. Should reject the claim Of NO significant relationship between your independent variable (in the corresponding row) and dependent variable if p<.

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**Regression Statistics**

Salary = f(GPA) Regression Statistics f(GPA) Multiple R R Square Adjusted R Square Standard Error Observations 10 ANOVA df SS MS F Significance F Regression 1 Residual 8 Total 9 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept GPA

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**Regression Statistics**

Salary = f(Work) Regression Statistics f(Work) Multiple R R Square Adjusted R Square Standard Error Observations 10 ANOVA df SS MS F Significance F Regression 1 E-05 Residual 8 Total 9 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.49E-09 Months Work 5.53E-05

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**Regression Statistics**

Salary = f(GPA, Work) Regression Statistics f(GPA,Work) Multiple R R Square Adjusted R Square Standard Error Observations 10 ANOVA df SS MS F Significance F Regression 2 Residual 7 Total 9 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept GPA Months Work

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**Compare Three “Models”**

Regression Statistics f(GPA) Multiple R R Square Adjusted R Square Standard Error Observations 10 Regression Statistics f(Work) Multiple R R Square Adjusted R Square Standard Error Observations 10 Regression Statistics f(GPA,Work) Multiple R R Square Adjusted R Square Standard Error Observations 10

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**Latent Variables (Theoretical Entities)**

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**Latent Variables Latent Variables**

Explanatory Variables that are not directly measured Identified by “Exploratory Factor Analysis” Confirmed by “Confirmatory Factor Analysis” Statistical Methods for Latent Variables Principles Components Analysis PLS SEM

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**Example: Confirmatory Factor Analysis Intention to Use Travelocity Website**

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Research Instrument

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