The Simple Regression Model
The simple regression model (SRM) is model for association in the population between an explanatory variable X and response Y. The SRM states that these averages align on a line with intercept β0 and slope β1: µy|x = E(Y|X = x) = β0 + β1x
Deviation from the Mean The deviation of observed responses around the conditional means µy|x are called errors (ε). The error’s equation: ε = y - µy|x
Errors can be positive or negative, depending on whether data lie above (positive) or below the conditional means (negative). Because the errors are not observed, the SRM makes three assumptions about them: * Independent. The error for one observation is independent of the error for any other observation. * Equal variance. All errors have the same variance, Var(ε) = σε2. * Normal. The errors are normally distributed.
If these assumptions hold, then the collection of all possible errors forms a normal population with mean 0 and variance σε2, abbreviated ε ̴̴ N (0, σε2). Simple Regression Model (SRM) observed values of the response Y are linearly related to values of the explanatory variable X by the equation: y = β0 + β1x + ε, ε ̴̴ N (0, σε2)
The observations: 1. are independent of one another, 2. have equal variance σε2 around the regression line, and 3. are normally distributed around the regression line.
21.2 Conditions for the SRM ( Simple Regression Model )
Instead of checking for random residual variation, we have three specific conditions. Checklist for the simple regression model * Is the association between y and x linear? * Have we ruled out obvious lurking variables? Errors appears to be a sample from a normal population. | * Are the errors evidently independent? * Are the variances of the residuals similar? * Are the residuals nearly normal?
21.3 INTERFERENCE IN REGRESSION
Confidence intervals and hypothesis tests work as in inferences for the mean of a population: * The 95% confidence
Deviation from the Mean The deviation of observed responses around the conditional means µy|x are called errors (ε). The error’s equation: ε = y - µy|x
Errors can be positive or negative, depending on whether data lie above (positive) or below the conditional means (negative). Because the errors are not observed, the SRM makes three assumptions about them: * Independent. The error for one observation is independent of the error for any other observation. * Equal variance. All errors have the same variance, Var(ε) = σε2. * Normal. The errors are normally distributed.
If these assumptions hold, then the collection of all possible errors forms a normal population with mean 0 and variance σε2, abbreviated ε ̴̴ N (0, σε2). Simple Regression Model (SRM) observed values of the response Y are linearly related to values of the explanatory variable X by the equation: y = β0 + β1x + ε, ε ̴̴ N (0, σε2)
The observations: 1. are independent of one another, 2. have equal variance σε2 around the regression line, and 3. are normally distributed around the regression line.
21.2 Conditions for the SRM ( Simple Regression Model )
Instead of checking for random residual variation, we have three specific conditions. Checklist for the simple regression model * Is the association between y and x linear? * Have we ruled out obvious lurking variables? Errors appears to be a sample from a normal population. | * Are the errors evidently independent? * Are the variances of the residuals similar? * Are the residuals nearly normal?
21.3 INTERFERENCE IN REGRESSION
Confidence intervals and hypothesis tests work as in inferences for the mean of a population: * The 95% confidence