advertisement

Computational formulae for simple linear regression Linear model : Yi = α + βXi + εi a is the estimate of the true intercept, α, and b is the estimate of the true slope β n is the sample size (number of pairs of points) εi are random elements (residuals) sampled from a normal distribution N(0, σε2) 𝑏= ∑𝑛𝑖=1 𝑋𝑖𝑌𝑖 − (∑𝑛𝑖=1 𝑋𝑖 ∑𝑛𝑖=1 𝑌𝑖) /𝑛 2 ∑𝑛𝑖=1 𝑋𝑖 2 − (∑𝑛𝑖=1 𝑋𝑖 ) /𝑛 𝑎 = 𝑌̅ − 𝑏𝑋̅ To carry out a t-test for the slope, we need to estimate the standard error of the slope, Sb as follows: 𝑛 𝑆𝑆𝑒𝑟𝑟𝑜𝑟 = ∑ 𝑌𝑖 2 − 𝑖=1 𝑀𝑆𝑒𝑟𝑟𝑜𝑟 = (∑𝑛𝑖=1 𝑌𝑖 )2 (∑𝑛𝑖=1 𝑋𝑖𝑌𝑖 − (∑𝑛𝑖=1 𝑋𝑖 ∑𝑛𝑖=1 𝑌𝑖) /𝑛 )2 − 2 𝑛 ∑𝑛𝑖=1 𝑋𝑖 2 − (∑𝑛𝑖=1 𝑋𝑖 ) /𝑛 𝑆𝑆𝑒𝑟𝑟𝑜𝑟 𝑛−2 𝑀𝑆𝑒𝑟𝑟𝑜𝑟 𝑆𝑏 = √ 2 𝑛 ∑𝑖=1 𝑋𝑖 2 − (∑𝑛𝑖=1 𝑋𝑖 ) /𝑛 Finally the test statistic, t is calculated as: 𝑡= 𝑏−𝛽 , 𝑆𝑏 Here β is the slope as specified in the null Hypothesis (most commonly β = 0) The critical value of Student’s t-distribution is obtained with degrees of freedom, DF = n-2. Note that 1 or 2-tailed tests can be conducted.