In charlotte-ngs/lbgfs2020: Livestock Breeding and Genomics (Fall 2020)

cat(cnt$out(ps_suffix = "Variance Components"), "\n")

We are given the following dataset for the trait live weight (LiveWeight) for cattle.

\textit{Der folgende Datensatz umfasst das Merkmal Lebendgewicht (LiveWeight) von Rindern.}

knitr::kable(tbl_data_anova,
             booktabs = TRUE,
             escape = FALSE,
             format = 'latex')

\begin{enumerate} \item[a)] Compute the estimate of the error variance $\sigma_e^2$ from the residuals of the fixed linear model specified below.

\textit{Schätzen Sie die Fehlervarianz $\sigma_e^2$ basierend auf den Residuen des folgenden fixen Modells.} \points{r lPointsQ5$TaskA} \end{enumerate}

The fixed linear model that is used is

$$y = Xb + e$$ where $y$ is the vector of all live weight values, $b$ is the vector of the effects caused by the different farms. The fixed linear model is specified in R by

tbl_data_anova$Farm <- as.factor(tbl_data_anova$Farm)
lm_lweight <- lm(LiveWeight ~ 0 + Farm, data = tbl_data_anova)

The resulting effects of the farms are

(vec_coef_lweight <- coefficients(lm_lweight))

\solstart

The esimate of the error variance is computed based on the resiudals. The residuals can be obtained by the function residuals() in R.

vec_res <- residuals(lm_lweight)
ssq_res <- crossprod(vec_res)
(n_est_res_var <- ssq_res / (nrow(tbl_data_anova)-length(vec_coef_lweight)))

The error standard deviation is

(n_est_res_sd <- sqrt(n_est_res_var))

\solend

\clearpage \pagebreak

\begin{enumerate} \item[b)] Verify your result from task a) with the output of the summary()-function applied to the result of the lm()-function

\textit{Verifizieren Sie das Resultat aus Aufgabe a) anhand des Outputs der summary-Funktion angewendet auf das Resultat der lm()-Funktion} \points{r lPointsQ5$TaskB} \end{enumerate}

\solstart

From the task, we have the result object of the lm()-function which is called lm_lweight. Applying the summary()-method leads to