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| MSD | SE | \(SE = \frac{MSD*n}{t_{1-\alpha, 2n-2}*\sqrt{2}}\) | `msd*n/(qt(1-P/2,2*n-2)*sqrt(2))` | |  See related questions on Stats.SE: http://stats.stackexchange.com/q/2917/1381 and http://stats.stackexchange.com/q/4485/1381 #### Calculating \(MSE\) given \(F\), \(df_{\text{group}}\), and \(SS\)  Given:  \(\label{eq:f}  F = MS_g/MS_e\)  Where \(g\) indicates the group, or treatment. Rearranging this equation  gives: \(MS_e=MS_g/F\)  Given  \(MS_x = SS_x/df_x\)  Substitute \(MS_e/df_e\) for \(SS_e\) in the first equation  \(F=\frac{SS_g/df_g}{MS_e}\)  Then solve for \(MS_e\)  \(\label{eq:mse}  MS_e = \frac{SS_g}{df_g\times F}\)  \(\label{eq:dft}  df_{\text{total}}=(df_a+1)\times(df_b+1)...\times(n)-1\)  Which depends on the experimental design:  For factors a, b... (usually 1 or 2, sometimes 3) where \(n\) is the  number of replicates within each treatment combination.  - One-way anova \(df_{\text{total}}=an-1\); where \(a\) is the number of  treatments  - Two-way anova without replication \(df_{\text{total}}=(a+1)(b+1)-1\)  also known as ’’randomized complete block design’’ (RCBD)  - Two-way anova with \(n\) replicates  \(df_{\text{total}}=(a+1)(b+1)(n)-1\) aka ’’RCBD with replication’’  #### Example  An example application of this is in Starr et al. [2008] table 3 [Figure 11] (Figure 11).  The results are from one (two?) factor ANOVA with repeated measures,  with treatment and week as the factors and no replication.  We will calculate MSE from the \(SS_{\text{treatment}}\)  \(df_{\text{treatment}}\), and \(F\)-value given in the table; these are  \(109.58\), \(2\), and \(0.570\), respectively; \(df_{\text{weeks}}\) is given  as \(10\).  For the 1997 *Eriphorium vaginatum*, the mean \(A_{max}\) in table 4 is  \(13.49\).  Calculate \(MS_e\):  \(MS_e = \frac{109.58}{0.57 \times 2} = 96.12\)