@@ -14,6 +14,12 @@ topic: "Analysis of variance"
1414theorem : " Ordinary least squares for two-way ANOVA"
1515
1616sources :
17+ - authors : " Olbricht, Gayla R."
18+ year : 2011
19+ title : " Two-Way ANOVA: Interaction"
20+ in : " Stat 512: Applied Regression Analysis"
21+ pages : " Purdue University, Spring 2011, Lect. 27"
22+ url : " https://www.stat.purdue.edu/~ghobbs/STAT_512/Lecture_Notes/ANOVA/Topic_27.pdf"
1723
1824proof_id : " P371"
1925shortcut : " anova2-ols"
@@ -122,33 +128,37 @@ $$ \label{eq:rss-der-mu-zero}
122128\begin{split}
1231290 &= 2 n \hat{\mu} + 2 \left( \sum_{i=1}^{a} n_{i \bullet} \alpha_i + \sum_{j=1}^{b} n_{\bullet j} \beta_j + \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij} \gamma_{ij} \right) - 2 \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\
124130\hat{\mu} &= \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \sum_{i=1}^{a} \frac{n_{i \bullet}}{n} \alpha_i - \sum_{j=1}^{b} \frac{n_{\bullet j}}{n} \beta_j - \sum_{i=1}^{a} \sum_{j=1}^{b} \frac{n_{ij}}{n} \gamma_{ij} \\
125- \hat{\mu} &\overset{\eqref{eq:samp-size}}{=} \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \sum_{j=1}^{b} \sum_{i=1}^{a} \frac{n_{ij}}{n} \alpha_i - \sum_{i=1}^{a} \sum_{j=1}^{b} \frac{n_{ij}}{n} \beta_j - \sum_{i=1}^{a} \sum_{j=1}^{b} \frac{n_{ij}}{n} \gamma_{ij} \\
126- \hat{\mu} &\overset{\eqref{eq:anova2-cons}}{=} \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk}
131+ &\overset{\eqref{eq:samp-size}}{=} \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \sum_{j=1}^{b} \sum_{i=1}^{a} \frac{n_{ij}}{n} \alpha_i - \sum_{i=1}^{a} \sum_{j=1}^{b} \frac{n_{ij}}{n} \beta_j - \sum_{i=1}^{a} \sum_{j=1}^{b} \frac{n_{ij}}{n} \gamma_{ij} \\
132+ &\overset{\eqref{eq:anova2-cons}}{=} \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\
133+ &\overset{\eqref{eq:mean-samp}}{=} \bar{y}_{\bullet \bullet \bullet}
127134\end{split}
128135$$
129136
130137$$ \label{eq:rss-der-alpha-zero}
131138\begin{split}
1321390 &= 2 n_{i \bullet} \hat{\mu} + 2 n_{i \bullet} \hat{\alpha}_i + 2 \left( \sum_{j=1}^{b} n_{ij} \beta_j + \sum_{j=1}^{b} n_{ij} \gamma_{ij} \right) - 2 \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\
133140\hat{\alpha}_i &= \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \hat{\mu} - \sum_{j=1}^{b} \frac{n_{ij}}{n_{i \bullet}} \beta_j - \sum_{j=1}^{b} \frac{n_{ij}}{n_{i \bullet}} \gamma_{ij} \\
134- \hat{\alpha}_i &= \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \hat{\mu} - \frac{n}{n_{i \bullet}} \sum_{j=1}^{b} \frac{n_{ij}}{n} \beta_j - \frac{n}{n_{i \bullet}} \sum_{j=1}^{b} \frac{n_{ij}}{n} \gamma_{ij} \\
135- \hat{\alpha}_i &\overset{\eqref{eq:anova2-cons}}{=} \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk}
141+ &= \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \hat{\mu} - \frac{n}{n_{i \bullet}} \sum_{j=1}^{b} \frac{n_{ij}}{n} \beta_j - \frac{n}{n_{i \bullet}} \sum_{j=1}^{b} \frac{n_{ij}}{n} \gamma_{ij} \\
142+ &\overset{\eqref{eq:anova2-cons}}{=} \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\
143+ &\overset{\eqref{eq:mean-samp}}{=} \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet \bullet \bullet}
136144\end{split}
137145$$
138146
139147$$ \label{eq:rss-der-beta-zero}
140148\begin{split}
1411490 &= 2 n_{\bullet j} \hat{\mu} + 2 n_{\bullet j} \hat{\beta}_j + 2 \left( \sum_{i=1}^{a} n_{ij} \alpha_i + \sum_{i=1}^{a} n_{ij} \gamma_{ij} \right) - 2 \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} \\
142150\hat{\beta}_j &= \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} - \hat{\mu} - \sum_{i=1}^{a} \frac{n_{ij}}{n_{\bullet j}} \alpha_i - \sum_{i=1}^{a} \frac{n_{ij}}{n_{\bullet j}} \gamma_{ij} \\
143- \hat{\beta}_j &= \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} - \hat{\mu} - \frac{n}{n_{\bullet j}} \sum_{i=1}^{a} \frac{n_{ij}}{n} \alpha_i - \frac{n}{n_{\bullet j}} \sum_{i=1}^{a} \frac{n_{ij}}{n} \gamma_{ij} \\
144- \hat{\beta}_j &\overset{\eqref{eq:anova2-cons}}{=} \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk}
151+ &= \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} - \hat{\mu} - \frac{n}{n_{\bullet j}} \sum_{i=1}^{a} \frac{n_{ij}}{n} \alpha_i - \frac{n}{n_{\bullet j}} \sum_{i=1}^{a} \frac{n_{ij}}{n} \gamma_{ij} \\
152+ &\overset{\eqref{eq:anova2-cons}}{=} \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\
153+ &\overset{\eqref{eq:mean-samp}}{=} \bar{y}_{\bullet j \bullet} - \bar{y}_{\bullet \bullet \bullet}
145154\end{split}
146155$$
147156
148157$$ \label{eq:rss-der-gamma-zero}
149158\begin{split}
1501590 &= 2 n_{ij} (\hat{\mu} + \hat{\alpha}_i + \hat{\beta}_j + \hat{\gamma_{ij}}) - 2 \sum_{k=1}^{n_{ij}} y_{ijk} \\
151160\hat{\gamma_{ij}} &= \frac{1}{n_{ij}} \sum_{k=1}^{n_{ij}} y_{ijk} - \hat{\alpha}_i - \hat{\beta}_j - \hat{\mu} \\
152- \hat{\gamma_{ij}} &= \frac{1}{n_{ij}} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} + \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \; .
161+ &= \frac{1}{n_{ij}} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} - \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} + \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\
162+ &\overset{\eqref{eq:mean-samp}}{=} \bar{y}_{i j \bullet} - \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet j \bullet} + \bar{y}_{\bullet \bullet \bullet} \; .
153163\end{split}
154164$$
0 commit comments