The article “Anodic Fenton Treatment of Treflan MTF” describes a two-factor experiment designed to study the sorption of the herbicide trifluralin.

The article “Anodic Fenton Treatment of Treflan MTF” describes a two-factor experiment designed to study the sorption of the herbicide trifluralin.

Question
Study design
asked 2021-01-15

The article “Anodic Fenton Treatment of Treflan MTF” describes a two-factor experiment designed to study the sorption of the herbicide trifluralin. The factors are the initial trifluralin concentration and the \(\displaystyle{F}{e}^{{{2}}}\ :\ {H}_{{{2}}}\ {O}_{{{2}}}\) delivery ratio. There were three replications for each treatment. The results presented in the following table are consistent with the means and standard deviations reported in the article. \(\begin{array}{|c|c|}\hline \text{Initial Concentration (M)} & \text{Delivery Ratio} & \text{Sorption (%)} \\ \hline 15 & 1:0 & 10.90 \quad 8.47 \quad 12.43 \\ \hline 15 & 1:1 & 3.33 \quad 2.40 \quad 2.67 \\ \hline 15 & 1:5 & 0.79 \quad 0.76 \quad 0.84 \\ \hline 15 & 1:10 & 0.54 \quad 0.69 \quad 0.57 \\ \hline 40 & 1:0 & 6.84 \quad 7.68 \quad 6.79 \\ \hline 40 & 1:1 & 1.72 \quad 1.55 \quad 1.82 \\ \hline 40 & 1:5 & 0.68 \quad 0.83 \quad 0.89 \\ \hline 40 & 1:10 & 0.58 \quad 1.13 \quad 1.28 \\ \hline 100 & 1:0 & 6.61 \quad 6.66 \quad 7.43 \\ \hline 100 & 1:1 & 1.25 \quad 1.46 \quad 1.49 \\ \hline 100 & 1:5 & 1.17 \quad 1.27 \quad 1.16 \\ \hline 100 & 1:10 & 0.93 \quad 0.67 \quad 0.80\\ \hline \end{array}\) a) Estimate all main effects and interactions. b) Construct an ANOVA table. You may give ranges for the P-values. c) Is the additive model plausible? Provide the value of the test statistic, its null distribution, and the P-value.

Answers (1)

2021-01-16

Step 1 Given: \(\displaystyle{I}={3}\)
\(\displaystyle{J}={4}\)
\(\displaystyle{K}={3}\) a) The mean is the sum of all values divided by the number of data values: \(\displaystyle\overline{{{X}}}_{{{1}.}}={3.6992}\)
\(\displaystyle\overline{{{X}}}_{{{2}.}}={2.6492}\)
\(\displaystyle\overline{{{X}}}_{{{3}.}}={2.575}\)
\(\displaystyle\overline{{{X}}}_{{{.1}}}={8.2011}\)
\(\displaystyle\overline{{{X}}}_{{{.2}}}={1.9656}\)
\(\displaystyle\overline{{{X}}}_{{{.3}}}={0.9322}\)
\(\displaystyle\overline{{{X}}}_{{{.4}}}={0.7989}\)
\(\displaystyle\overline{{{X}}}_{{{11}}}={10.6}\)
\(\displaystyle\overline{{{X}}}_{{{12}}}={2.8}\)
\(\displaystyle\overline{{{X}}}_{{{13}}}={0.7967}\)
\(\displaystyle\overline{{{X}}}_{{{14}}}={0.6}\)
\(\displaystyle\overline{{{X}}}_{{{21}}}={7.1033}\)
\(\displaystyle\overline{{{X}}}_{{{22}}}={1.6967}\)
\(\displaystyle\overline{{{X}}}_{{{23}}}={0.8}\)
\(\displaystyle\overline{{{X}}}_{{{24}}}={0.9967}\)
\(\displaystyle\overline{{{X}}}_{{{31}}}={6.9}\)
\(\displaystyle\overline{{{X}}}_{{{32}}}={1.4}\)
\(\displaystyle\overline{{{X}}}_{{{33}}}={1.2}\)
\(\displaystyle\overline{{{X}}}_{{{34}}}={0.8}\)
\(\displaystyle\overline{{{X}}}={2.9744}\) Step 2 The main effects are the difference of the mean of the values in the category decreased by the overall mean. \(\displaystyle{\color{blue}{\text{Main effects concentration}}}\)
\(\displaystyle\overline{{{X}}}_{{{1}.}}\ -\ \overline{{{X}}}={3.6992}\ -\ {2.9744}={0.7248}\)
\(\displaystyle\overline{{{X}}}_{{{2}.}}\ -\ \overline{{{X}}}={2.6492}\ -\ {2.9744}=\ -{0.3252}\)
\(\displaystyle\overline{{{X}}}_{{{3}.}}\ -\ \overline{{{X}}}={2.575}\ -\ {2.9744}=\ -{0.3994}\)
\(\displaystyle{\color{blue}{\text{Main effects delivery ratio}}}\)
\(\displaystyle\overline{{{X}}}_{{{.1}}}\ -\ \overline{{{X}}}={8.2011}\ -\ {2.9744}={5.2267}\)
\(\displaystyle\overline{{{X}}}_{{{.2}}}\ -\ \overline{{{X}}}={1.9656}\ -\ {2.9744}=\ -{1.0088}\)
\(\displaystyle\overline{{{X}}}_{{{.3}}}\ -\ \overline{{{X}}}={0.9322}\ -\ {2.9744}=\ -{2.0422}\)
\(\displaystyle\overline{{{X}}}_{{{.4}}}\ -\ \overline{{{X}}}={0.7989}\ -\ {2.9744}=\ -{2.1755}\) The main effects are difference of the mean of the values in the category decreased by the mean of the row and the mean of the column, and increased by the overall mean. \(\displaystyle\overline{{{X}}}_{{{11}}}\ -\ \overline{{{X}}}_{{{1}.}}\ -\ \overline{{{X}}}_{{{.1}}}\ +\ \overline{{{X}}}={10.6}\ -\ {3.6992}\ -\ {8.2011}\ +\ {2.9744}={1.6741}\)
\(\displaystyle\overline{{{X}}}_{{{12}}}\ -\ \overline{{{X}}}_{{{1}.}}\ -\ \overline{{{X}}}_{{{.2}}}\ +\ \overline{{{X}}}={2.8}\ -\ {3.6992}\ -\ {1.9656}\ +\ {2.9744}={0.1096}\)
\(\displaystyle\overline{{{X}}}_{{{13}}}\ -\ \overline{{{X}}}_{{{1}.}}\ -\ \overline{{{X}}}_{{{.3}}}\ +\ \overline{{{X}}}={0.7967}\ -\ {3.6992}\ -\ {0.9322}\ +\ {2.9744}=\ -{0.8603}\)
\(\displaystyle\overline{{{X}}}_{{{14}}}\ -\ \overline{{{X}}}_{{{1}.}}\ -\ \overline{{{X}}}_{{{.4}}}\ +\ \overline{{{X}}}={0.6}\ -\ {3.6992}\ -\ {0.7989}\ +\ {2.9744}=\ -{0.9237}\)
\(\displaystyle\overline{{{X}}}_{{{21}}}\ -\ \overline{{{X}}}_{{{2}.}}\ -\ \overline{{{X}}}_{{{.1}}}\ +\ \overline{{{X}}}={7.1033}\ -\ {2.6492}\ -\ {8.2011}\ +\ {2.9744}=\ -{0.7726}\)
\(\displaystyle\overline{{{X}}}_{{{22}}}\ -\ \overline{{{X}}}_{{{2}.}}\ -\ \overline{{{X}}}_{{{.2}}}\ +\ \overline{{{X}}}={1.6967}\ -\ {2.6492}\ -\ {1.9656}\ +\ {2.9744}={0.0563}\)
\(\displaystyle\overline{{{X}}}_{{{23}}}\ -\ \overline{{{X}}}_{{{2}.}}\ -\ \overline{{{X}}}_{{{.3}}}\ +\ \overline{{{X}}}={0.8}\ -\ {2.6492}\ -\ {0.9322}\ +\ {2.9744}={0.1930}\)
\(\displaystyle\overline{{{X}}}_{{{24}}}\ -\ \overline{{{X}}}_{{{2}.}}\ -\ \overline{{{X}}}_{{{.4}}}\ +\ \overline{{{X}}}={0.9967}\ -\ {2.6492}\ -\ {0.7989}\ +\ {2.9744}={0.5230}\)
\(\displaystyle\overline{{{X}}}_{{{31}}}\ -\ \overline{{{X}}}_{{{3}.}}\ -\ \overline{{{X}}}_{{{.1}}}\ +\ \overline{{{X}}}={6.9}\ -\ {2.575}\ -\ {8.2011}\ +\ {2.9744}=\ -{0.9017}\)
\(\displaystyle\overline{{{X}}}_{{{32}}}\ -\ \overline{{{X}}}_{{{3}.}}\ -\ \overline{{{X}}}_{{{.2}}}\ +\ \overline{{{X}}}={1.4}\ -\ {2.575}\ -\ {1.9656}\ +\ {2.9744}=\ -{0.1662}\)
\(\displaystyle\overline{{{X}}}_{{{33}}}\ -\ \overline{{{X}}}_{{{3}.}}\ -\ \overline{{{X}}}_{{{.3}}}\ +\ \overline{{{X}}}={1.2}\ -\ {2.575}\ -\ {0.9322}\ +\ {2.9744}={0.6672}\)
\(\displaystyle\overline{{{X}}}_{{{34}}}\ -\ \overline{{{X}}}_{{{3}.}}\ -\ \overline{{{X}}}_{{{.4}}}\ +\ \overline{{{X}}}={0.8}\ -\ {2.575}\ -\ {0.7989}\ +\ {2.9744}={0.4005}\) Step 3 b) Let us first determine the sum of squares: \(\displaystyle{S}{S}{A}={J}{K}\ {\sum_{{{i}={1}}}^{{{I}}}}\ {\overline{{{X}}}_{{{i}.}}^{{{2}}}}\ -\ {I}{J}{K}\ \overline{{{X}}}^{{{2}}}\)
\(\displaystyle={4}{\left({3}\right)}{\left({3.6992}\right)}^{{{2}}}\ +\ {4}{\left({3}\right)}{\left({2.6492}\right)}^{{{2}}}\ +\ {4}{\left({3}\right)}{\left({2.575}\right)}^{{{2}}}\ -\ {3}{\left({4}\right)}{\left({3}\right)}{\left({2.9744}\right)}^{{{2}}}\)
\(\displaystyle\approx\ {9.49}\)
\(\displaystyle{S}{S}{B}={I}{K}\ {\sum_{{{j}={1}}}^{{{J}}}}\ {\overline{{{X}}}_{{.{j}}}^{{{2}}}}\ -\ {I}{J}{K}\ \overline{{{X}}}^{{{2}}}\)
\(\displaystyle={3}{\left({3}\right)}{\left({8.2011}\right)}^{{{2}}}\ +\ {3}{\left({3}\right)}{\left({1.9656}\right)}^{{{2}}}\ +\ {3}{\left({3}\right)}{\left({0.9322}\right)}^{{{2}}}\)
\(\displaystyle+\ {3}{\left({3}\right)}{\left({0.7989}\right)}^{{{2}}}\ -\ {3}{\left({4}\right)}{\left({3}\right)}{\left({2.9744}\right)}^{{{2}}}\)
\(\displaystyle\approx\ {335.16}\)
\(\displaystyle{S}{S}{A}{B}={K}\ {\sum_{{{i}={1}}}^{{{I}}}}\ {\sum_{{{j}={1}}}^{{{j}}}}\ {\overline{{{X}}}_{{{i}{j}}}^{{{2}}}}\ -\ {J}{K}\ {\sum_{{{i}={1}}}^{{{I}}}}\ {\overline{{{X}}}_{{{i}}}^{{{2}}}}\)
\(\displaystyle-\ {I}{K}\ {\sum_{{{j}={1}}}^{{{J}}}}\ {\overline{{{X}}}_{{.{j}}}^{{{2}}}}\ +\ {I}{J}{K}\ \overline{{{X}}}^{{{2}}}\)
\(\displaystyle={3}{\left({10.6}\right)}^{{{2}}}\ +\ {3}{\left({2.8}\right)}^{{{2}}}\ +\ {3}{\left({0.7967}\right)}^{{{2}}}\ +\ {3}{\left({0.6}\right)}^{{{2}}}\)
\(\displaystyle+\ {3}{\left({7.1033}\right)}^{{{2}}}\ +\ {3}{\left({1.6967}\right)}^{{{2}}}\ +\ {3}{\left({0.8}\right)}\ +\ {3}{\left({0.9967}\right)}^{{{2}}}\)
\(\displaystyle+\ {3}{\left({6.9}\right)}^{{{2}}}\ +\ {3}{\left({1.4}\right)}^{{{2}}}\ +\ {3}{\left({1.2}\right)}^{{{2}}}\ +\ {3}{\left({0.8}\right)}^{{{2}}}\)
\(\displaystyle-\ {4}{\left({3}\right)}{\left({3.6992}\right)}^{{{2}}}\ -\ {4}{\left({3}\right)}{\left({2.6492}\right)}^{{{2}}}\ -\ {4}{\left({3}\right)}{\left({2.575}\right)}^{{{2}}}\)
\(\displaystyle-\ {3}{\left({3}\right)}{\left({8.2011}\right)}^{{{2}}}\ -\ {3}{\left({3}\right)}{\left({1.9656}\right)}^{{{2}}}\ -\ {3}{\left({3}\right)}{\left({0.9322}\right)}^{{{2}}}\ -\ {3}{\left({3}\right)}{\left({0.7989}\right)}^{{{2}}}\)
\(\displaystyle+\ {3}{\left({4}\right)}{\left({3}\right)}{\left({2.9744}\right)}^{{{2}}}\)
\(\displaystyle\approx\ {20.3}\)
\(\displaystyle{S}{S}{E}=\ {\sum_{{{i}={1}}}^{{{I}}}}\ {\sum_{{{j}={1}}}^{{{J}}}}\ {\sum_{{{k}={1}}}^{{{K}}}}\ {\overline{{{X}}}_{{{i}{j}{k}}}^{{{2}}}}\ -\ {K}\ {\sum_{{{i}={1}}}^{{{I}}}}\ {\sum_{{{j}={1}}}^{{{J}}}}\ {\overline{{{X}}}_{{{i}{j}}}^{{{2}}}}\)
\(\displaystyle={10.90}^{{{2}}}\ +\ {8.47}^{{{2}}}\ +\ {12.43}^{{{2}}}\ +\ {3.33}^{{{2}}}\ +\ {2.40}^{{{2}}}\ +\ {2.67}^{{{2}}}\)
\(\displaystyle+\ {0.79}^{{{2}}}\ +\ {0.76}^{{{2}}}\ +\ {0.84}^{{{2}}}\ +\ {0.54}^{{{2}}}\ +\ {0.69}^{{{2}}}\ +\ {0.57}^{{{2}}}\)
\(\displaystyle+\ {6.84}^{{{2}}}\ +\ {7.68}^{{{2}}}\ {6.79}^{{{2}}}\ +\ {1.72}^{{{2}}}\ +\ {1.55}^{{{2}}}\ +\ {1.82}^{{{2}}}\)
\(\displaystyle+\ {0.68}^{{{2}}}\ +\ {0.83}^{{{2}}}\ +\ {0.89}^{{{2}}}\ +\ {0.58}^{{{2}}}\ +\ {1.13}^{{{2}}}\ +\ {1.28}^{{{2}}}\)
\(\displaystyle+\ {6.61}^{{{2}}}\ +\ {6.66}^{{{2}}}\ +\ {7.43}^{{{2}}}\ +\ {1.25}^{{{2}}}\ +\ {1.46}^{{{2}}}\ +\ {1.49}^{{{2}}}\)
\(\displaystyle+\ {1.17}^{{{2}}}\ +\ {1.27}^{{{2}}}\ +\ {1.16}^{{{2}}}\ +\ {0.93}^{{{2}}}\ +\ {0.67}^{{{2}}}\ +\ {0.80}^{{{2}}}\)
\(\displaystyle-\ {3}{\left({10.6}\right)}^{{{2}}}\ -\ {3}{\left({2.8}\right)}^{{{2}}}\ -\ {3}{\left({0.7967}\right)}^{{{2}}}\ -\ {3}{\left({0.6}\right)}^{{{2}}}\)
\(\displaystyle-\ {3}{\left({7.1033}\right)}^{{{2}}}\ -\ {3}{\left({1.6967}\right)}^{{{2}}}\ -\ {3}{\left({0.8}\right)}^{{{2}}}\ -\ {3}{\left({0.9967}\right)}^{{{2}}}\)
\(\displaystyle-\ {3}{\left({6.9}\right)}^{{{2}}}\ -\ {3}{\left({1.4}\right)}^{{{2}}}\ -\ {3}{\left({1.2}\right)}^{{{2}}}\ -\ {3}{\left({0.8}\right)}^{{{2}}}\)
\(\displaystyle={9.78}\) Step 4 The degrees of freedom for the effect of a variable is the number of categories of the variable decreased by 1. \(\displaystyle{d}{f}_{{{A}}}={I}\ -\ {1}={3}\ -\ {1}={2}\) The degrees of freedom for the effect of a variable is the number of categories of the variable decreased by 1. \(\displaystyle{d}{f}_{{{B}}}={J}\ -\ {1}={4}\ -\ {1}={3}\) The degrees of freedom for the interactions is the product of degrees of freedom for the effect of each variable. \(\displaystyle{d}{f}_{{{A}{B}}}={d}{f}_{{{A}}}\ \cdot\ {d}{f}_{{{B}}}={2}\ \cdot\ {3}={6}\) The degrees of freedom for error is the product of the number of categories for each variable, and the number of replicates per treatment decreased by 1. \(\displaystyle{d}{f}_{{{E}}}={I}{J}{\left({K}\ -\ {1}\right)}={3}\ \cdot\ {4}\ \cdot\ {\left({3}\ -\ {1}\right)}={12}\ \cdot\ {2}={24}\) Step 5 \(\displaystyle{M}{S}{A}{i}{s}{S}{S}{A}\div{d}{b}{y}{d}{f}_{{{A}}}:\)
\(\displaystyle{M}{S}{A}=\ {\frac{{{S}{S}{A}}}{{{d}{f}_{{{A}}}}}}=\ {\frac{{{9.49}}}{{{2}}}}={4.74}\) MSB is SSB divided by \(\displaystyle{d}{f}_{{{B}}}:\)
\(\displaystyle{M}{S}{B}=\ {\frac{{{S}{S}{B}}}{{{d}{f}_{{{B}}}}}}=\ {\frac{{{335.16}}}{{{3}}}}={111.72}\) MSAB is SSAB divided by \(\displaystyle{d}{f}_{{{A}{B}}}:\)
\(\displaystyle{M}{S}{A}{B}=\ {\frac{{{S}{S}{A}{B}}}{{{d}{f}_{{{A}{B}}}}}}=\ {\frac{{{20.3}}}{{{6}}}}={3.38}\) MSE is SSE divided by \(\displaystyle{d}{f}_{{{E}}}:\)
\(\displaystyle{M}{S}{E}=\ {\frac{{{S}{S}{E}}}{{{d}{f}_{{{E}}}}}}=\ {\frac{{{9.78}}}{{{24}}}}={0.41}\) The value of the test statistic \(\displaystyle{F}_{{{A}}}\) is then MSA divided by MSE: \(\displaystyle{F}_{{{A}}}=\ {\frac{{{M}{S}{A}}}{{{M}{S}{E}}}}=\ {\frac{{{4.74}}}{{{0.41}}}}\ \approx\ {11.64}\) The value of the test statistic \(\displaystyle{F}_{{{B}}}\) is then MSB divided by MSE: \(\displaystyle{F}_{{{B}}}=\ {\frac{{{M}{S}{B}}}{{{M}{S}{E}}}}=\ {\frac{{{111.72}}}{{{0.41}}}}\ \approx\ {274.16}\) The value of the test statistic \(\displaystyle{F}_{{{A}{B}}}\) is then MSAB divided by MSE: \(\displaystyle{F}_{{{A}{B}}}=\ {\frac{{{M}{S}{A}{B}}}{{{M}{S}{E}}}}=\ {\frac{{{3.38}}}{{{0.41}}}}\ \approx\ {8.3}\) Step 6 The p-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value corresponding to the rows is the number (or interval) in the row title of the F-distribution table in the appendix containing the F-value in the row \(\displaystyle{d}{f}_{{{2}}}={d}{f}_{{{E}}}={24}\ \text{and in the column}\ {d}{f}_{{{1}}}={d}{f}_{{{A}}}={2}:\)
\(\displaystyle{P}\ {<}\ {0.001}\) The P-value corresponding to the columns is the number (or interval) in the row title of the F-distribution table in the appendix containing the F-value in the row \(\displaystyle{d}{f}_{{{2}}}={d}{f}_{{{E}}}={24}\ \text{and in the column}\ {d}{f}_{{{1}}}={d}{f}_{{{B}}}={3}:\)
\(\displaystyle{P}\ {<}\ {0.001}\) The P-value corresponding to the interaction is the number (or interval) in the row title of the F-distribution table in the appendix containing the F-value in the row \(\displaystyle{d}{f}_{{{2}}}={d}{f}_{{{E}}}={24}\ \text{and in the column}\ {d}{f}_{{{1}}}={d}{f}_{{{A}{B}}}={6}:\)
\(\displaystyle{P}\ {<}\ {0.001}\) Combine the information in an ANOVA table: \(\begin{array}{c} \text{Sourse} & df & SS & MS & F & P \\ \hline \text{Rows} & 2 & 9.49 & 4.74 & 11.64 & P<0.001 \\ \text{Columns} & 3 & 335.16 & 111.72 & 274.16 & P<0.001 \\ \text{Interaction} & 6 & 20.3 & 3.38 & 8.3 & P<0.001 \\ \hline \text{Total} & 35 & 374.72 \\ \end{array}\)

Step 7 c) By part (b): \(\displaystyle{F}_{{{A}{B}}}={8.3}\ {\quad\text{and}\quad}\ {P}\ {<}\ {0.001}\) If the P-value is less than the significance level, then reject the null hypothesis. \(\displaystyle{P}\ {<}\ {0.05}\ \Rightarrow\ {R}{e}{j}{e}{c}{t}\ {H}_{{{0}}}\) There is sufficient evidence the claim that the additive model is plausible.

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