Confidence Interval Estimates of Regression Coefficients.

 

Confidence Interval Estimation

Of

Regression Parameters

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 samples 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 α^ be the estimate of α computed from the values of a small random sample of size n selected from a bivariate normal population having mean "μα" and standard deviation "σα." The population mean and standard deviation are unknown, so replace them with their estimates.

The sampling distribution of α^ approaches the t-distribution with mean "μα^" and standard deviation "sα^"

α^ ~ t(α, sα^)

Where:

Thus, a (1 - α) % confidence interval estimate for α is given by

Confidence Interval for Slope Parameter

Let β^ be the estimate of β computed from the values of a small random sample of size n selected from a bivariate normal population having mean "μβ" and standard deviation "σβ." The population mean and standard deviation are unknown, so replace them with their estimates.

The sampling distribution of α^ approaches the t-distribution with mean "μβ^" and standard deviation "sβ^".

β^ ~ t(β, sβ^)

Where:

Thus, a (1 - α) % confidence interval estimate for β is given by

Confidence Interval for the Mean value of Response Variable

Let Y^ = α^ + β^ X0 be the estimate of Y = α + β X0 + ϵ at X = X0 computed from the values of a small random sample of size n. The sampling distribution of Y^ = α^ + β^ X₀ approaches the t-distribution with μY.X = α + β X and standard deviation σy.x.

Where:

The population standard deviation is unknown, so replace it with its estimate given below:

The test statistic:

A 100(1-α)% confidence interval is given by:

Practice Question 1.5

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

∑X=4421, ∑Y=12130, ∑XY=542735, ∑X²=208349, ∑Y²=1498976

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

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

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

Solution: The OLS method is using to estimate the parameters

i.                    Estimation of Regression Line

The estimated regression line is given by:

Y^ = α^ + β^ X

Y^ = 98.97 + 0.5015X

95% Confidence interval for α, β

1 - α = 0.95

α = 0.05

tα/2(n-2) = t 0.025(98) = 1.96

95% confidence interval for α

 

α^ ± tα/2(n-2) sα^

98.97 ± 1.96 (6.337)

98.97 ± 12.547

86.423 < α < 111.517

95% confidence interval for β

 β^ ±  tα/2(n-2) sβ^

 0.5015 ± 1.96 (0.138)

 0.5015 ± 0.271

0.2305 <  β < 0.7725

iii. Prediction of blood pressure for the age 50 years, i.e., X0 = 50

Y^ = α^ + β^ X

Y^0 = 98.97 + 0.5015 x 50

Y^ = 124.23

95% prediction interval for true value

Y0 ± tα/2(n-2) sY^

124.23 ± 1.96 x 15.96

124.23 ± 31.40

92.83 ≤ μY ≤ 155.63

• Read More: Goodness of Fit Test