Applied Statistics Module 3- Interpreting Fitting Models

Applied Statistics Module 3: Interpreting Fitting Models 3.1 Interpreting fitted models Fitted models A population linear regression line has the form E(Y)=β0+β1X, where β0 is the intercept parameter and β1 is the slope parameter. Since β0 and β1 are rarely known, estimators b0 and b1 are needed to build linear models used for predicting the value of the response variable Y. The sample linear regression line is Y^=b0+b1X, which gives the fitted values for Y. The animation below gives the reaction times (in milliseconds) of 6 adult drivers after consuming alcohol (in standard drink units). Reaction time is the response variable and alcohol consumption is the explanatory variable. Interpreting regression parameter estimates The estimates for the population regression parameters, b0 and b1, can be interpreted as follows:   b0 represents the estimated simple linear regression Y-intercept, which is the fitted value of Y when X=0. b1 represents the estimated simple linear regression slope, which is the change in the fitted value of Y per unit change in X. The Y-intercept, b0, has a practical meaning for datasets in which the predictor variable X can take on a value of 0 or values near 0. The measurement unit of b0 is the same as the measurement unit of the response Y variable. The slope, b1, always has a practical meaning. The measurement unit of b1 is the same as the measurement unit of Y per measurement unit of X. Example 3.1.1: Interpreting regression parameter estimates. The regression line Y^=4+6X predicts a driver's reaction time (in milliseconds) using the amount of alcohol consumed (in standard alcohol units). 1. 2. 3. 4. What is the interpretation for the intercept estimate? What is the measurement unit for the intercept estimate? What is the interpretation for the slope estimate? What is the measurement unit for the slope estimate? Solution 1. The interpretation of the intercept estimate is the fitted value of the driver's reaction time when the driver has had 0 standard alcohol units. 2. The measurement unit for the intercept estimate is milliseconds, the same units as the reaction time. 3. The interpretation of the slope estimate is the change in the fitted value of the driver's reaction time per unit increase in alcohol consumption. 4. The measurement unit for the slope estimate is milliseconds per standard alcohol unit. 3.2 Linear regression assumptions Linear regression assumptions Not all data can be modeled using linear regression. Certain assumptions need to be met in order to determine whether the relationship between an explanatory variable and a response variable is linear. The following assumptions or conditions should be met before creating a linear regression model:     the true relationship is linear errors have equal variance around the line errors are normally distributed the observations are independent The details of each condition are beyond the scope of the material. The first step, naturally, is to create a scatter plot. Sometimes, looking at the scatter plot is not sufficient, so a residual plot is needed. A residual plot graphs the relationship between the difference in the actual and predicted value, and the explanatory variable. For the assumptions to be satisfied, the residuals should have no discernable pattern. That is, the residuals should have a mean of zero, constant variance, and independent of each other. 3.3 Correlation and coefficient of determination Correlation Correlation describes the association or dependence between two variables. A positive correlation between two variables means that as one variable increases, the other variable increases as well. A negative correlation between two variables means that as one variable increases, the other variable decreases. The strength of correlation between a predictor variable and a response variable can be measured by the correlation coefficient. The population correlation coefficient is denoted by ρ and the sample correlation coefficient is denoted by R. The strength of correlation can be described by the absolute value of R. The table below gives a rough guideline. Table 3.3.1: Strength of correlation. Value of |R| Strength of correlation 0<|R|≤0.40 Weak 0.40<|R|≤0.80 Moderate 0.80<|R|≤1.00 Strong Correlation matrix Often, applications deal with more than one variable. A correlation matrix is a table that shows the correlation coefficients between each pair of variables. In the participation activity below, the correlation coefficient for Exam1 and Exam2 can be found in row 1, column 2 of the table. Use the coefficient matrix below to answer the questions below about the correlation between exam scores in the Exam Scores dataset. Link to spreadsheet 1) What is the correlation coefficient between Exam1 and Exam4? 0.277656 0.261306 0.078613 Correct The correlation coefficient between Exam1 and Exam4 (0.261306) can be obtained from the table by looking at row 1 and column 4 of the matrix. Alternatively, the same correlation coefficient can be obtained by looking at row 4 and column 1 of the matrix. 2) What is the sign or direction of the correlation between Exam1 and Exam4? Positive Negative Correct Since 0.261306>0, the correlation is positive. That is, as Exam1 increases, Exam4 increases as well. Similarly, as Exam4 increases, Exam1 increases as well. 3) What is the strength of the correlation between Exam1 and Exam4? Weak Moderate Strong Correct The correlation is weak because the correlation coefficient (0.261306) is between 0 and 0.40. Feedback? Coefficient of determination Another quantity that shows how well a regression equation represents the data is the square of the correlation coefficient, known as the coefficient of determination. The coefficient of determination, denoted by R2, gives the ratio of the variance in the response variable explained by the predictor variable. Conceptually, the coefficient of determination gives the percentage of data that is closest to the regression line. The farther the actual data points are from the regression line, the less useful the line actually is in predicting the value of the response variable.