Confidence Interval Estimation Of Regression Parameters

 

Interval Estimation 

of

Regression Parameters

Confidence Interval about a Regression Line

A confidence interval estimate for a simple linear regression line is based on sample statistics and their accompanying sampling distributions, with a statement indicating how confident in terms of probability the interval contains the population linear regression line. The probability associated with a confidence interval is 1 - α  or (1 - α). Thus, the confidence interval is the distance between the two curves (dotted lines) and  (1 - α )% chance that the population linear regression line will lie within the space.

Confidence Level

The estimates are based on sample data and vary from one sample drawn from the same population, and these estimates produce slightly different intervals. The confidence coefficient or confidence level is the percentage (probability) of them that will contain the population linear regression line or parameters of the model. 

Confidence Interval for Intercept Parameter

Let a be the unbiased estimate of α computed from the values of a small random sample of size n. The sample is selected from a normal distribution with a mean α and an unknown standard deviation σα. The sampling distribution of an approach to t distribution with E(a) =  α and standard deviation sa.

where:

Confidence Interval for Slope Parameter

Let b be the unbiased estimate of β computed from the values of a small random sample of size n selected from a bivariate normal population with mean β and standard deviation σb. As σb is unknown, so replace it by its estimate sb.

where:

Confidence Interval for the Mean value of Response Variable

Let Y₀ be the individual value of the response variable at X = X₀, computed from the values of a small random sample of size n selected from a bivariate normal population having mean μY.X₀ and unknown standard deviation σY.X₀. 

where:

The sampling distribution of Y₀ approaches the t distribution with (n - 2) degrees of freedom.

Prediction Interval

Let Y₀ be the individual value at X = X₀, computed from the values of a small random. The sampling distribution of the Y0 approach to the t distribution with (n-2) degrees of freedom with mean μY0 = α + β X0 and standard deviation σY0.

Where:

As σY. X is unknown, so replace by its estimate sY.X. The estimate of σY0 is denoted by sY0, given by

Practice Question

The age and systolic blood pressure of 100 individuals gave the following information:

i.      Compute the regression line which is used to estimate the true value, i.e., μY. X

ii. Assume normality, construct a 95% confidence interval for Y and Y₀, and find the true value of blood pressure for the age 50 years.

iii.       Predict blood pressure for the age 50 years and compute a 95% confidence interval for this estimate.

Solution: The OLS method is using to estimate the parameters

i.                    Estimation of Regression Line