Section 31 Prediction & Residuals

31.1 Model

SBP = overall mean + sampling variability

\[ \large y_{j} = \mu + \epsilon_{j} \]

SBP = overall mean + Group effect + sampling variability

\[ \large y_{ij} = \mu + \tau_{i} + \epsilon_{ij} \]

\[ \large y_{ij} = \mu_{i} + \epsilon_{ij} \]

Prediction

\[ \large \hat{y_{ij}} = \hat\mu_{i} \]

Residual

\[ \large \epsilon_{ij} = y_{ij} - \hat\mu_{i} \]

\(\large i\) = treatment index; i: 1 to g

\(\large j\) = observation index within treatment; j: 1 to n

\(\large y_{ij}\) = j -th observation (replicate) in the i -th treatment

\(\large \mu\) = overall mean effect

\(\large \tau_{i}\) = effect of treatment group i

\(\large \epsilon_{ij} \sim NID(0, \sigma^2)\)