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Type 2 Modified Augmented Design: Rows, Columns, Whole Plots, Subplots Primary Check The field is divided into rows and columns, which form a grid of whole plots (i.e., incomplete blocks). The experimental units (plots) within each whole plot are referred to as subplots. At the center of each whole plot, there is a subplot of the primary check cultivar. Column 1 1 2 4 3 5 6 WholePlot (wp) 1 wp2 wp3 wp4 wp5 wp6 wp7 wp8 wp9 wp10 wp11 wp12 wp13 wp14 wp15 wp16 wp17 wp18 wp19 wp20 wp21 wp22 wp23 wp24 Row 2 3 4 = Primary Check (Check1) Type 2 Modified Augmented Design: Secondary Checks An arbitrary number of secondary checks is selected (must be >1); here, I selected 2. An arbitrary number of whole plots is selected to be assigned secondary checks; here, I selected 4. One subplot of each secondary check is randomly located within each of the selected whole plots. Column 1 1 2 4 3 5 6 WholePlot (wp) 1 wp2 wp3 wp4 wp5 wp6 wp7 wp8 wp9 wp10 wp11 wp12 wp13 wp14 wp15 wp16 wp17 wp18 wp19 wp20 wp21 wp22 wp23 wp24 Row 2 3 4 = Primary Check (Check1) = Secondary Check A (Check2) = Secondary Check B (Check3) Type 2 Modified Augmented Design: Adjustments – Method 1 There are two ways to make adjustments, Method 1 and Method 3. Method 1 ignores the secondary checks and adjusts experimental lines based on row and column averages for the primary check alone. To obtain Method 1 adjustments, first run an ANOVA with Check1 plots as the dependent variable and Row and Column as class effects. Column 1 1 2 4 3 5 6 WholePlot (wp) 1 wp2 wp3 wp4 wp5 wp6 wp7 wp8 wp9 wp10 wp11 wp12 wp13 wp14 wp15 wp16 wp17 wp18 wp19 wp20 wp21 wp22 wp23 wp24 Row 2 3 4 = Primary Check (Check1) = Secondary Check A (Check2) = Secondary Check B (Check3) Type 2 Modified Augmented Design: 1/18/13 Adjustments - Method 1 - ANOVA SAS Output Method 1 ANOVA - Spike Length (cm) The GLM Procedure Dependent Variable: SpkLngth Source DF Sum of Squares Mean Square F Value Pr > F 8 1.39740741 0.17467593 0.79 0.6212 Error 15 3.32685185 0.22179012 Corrected Total 23 4.72425926 Model R-Square Coeff Var Root MSE SpkLngth Mean 0.295794 8.619245 0.470946 5.463889 DF Type I SS Mean Square F Value Pr > F Row 3 0.74537037 0.24845679 1.12 0.3721 Column 5 0.65203704 0.13040741 0.59 0.7093 Source DF Type III SS Mean Square F Value Pr > F Row 3 0.74537037 0.24845679 1.12 0.3721 Column 5 0.65203704 0.13040741 0.59 0.7093 Source “whole plot error”= 1) error for comparing plots in different whole plots 2) Error used to calculate standard error for T3 Use F test results for Row and Column effects to help you determine Method 1 appropriateness Method1AdjustedLine102Row(R)2 Column(C)3 = RawLine102R2C3 – (Check1AvgR2 – CheckAvg) - (Check1AvgC3 – Check1Avg) Line “MALT-102” from data in corresponding Excel file = RawLine102R2C3 – Check1AvgR2 – Check1AvgC3 + 2*Check1Avg = 8.47 – 5.86 – 5.58 + 2*5.46 = 7.95 Type 2 Modified Augmented Design: Adjustments – Method 3 Method 3 uses data from all checks to calculate adjustments. Adjustments are based on the slope from regressing the average of all secondary check subplots within each whole plot upon the primary check subplot within each whole plot. Thus, the adjustment parameter is estimated using only check subplots in wholeplots with secondary checks. Column 1 1 2 4 3 5 6 WholePlot (wp) 1 wp2 wp3 wp4 wp5 wp6 wp7 wp8 wp9 wp10 wp11 wp12 wp13 wp14 wp15 wp16 wp17 wp18 wp19 wp20 wp21 wp22 wp23 wp24 Row 2 3 4 = Primary Check (Check1) = Secondary Check A (Check2) = Secondary Check B (Check3) Type 2 Modified Augmented Design: Adjustments – Method 3 - Regression Row 2 3 3 4 Column 1 2 4 4 WholePlot 7 14 16 22 Check1 5.50 5.47 5.73 4.53 Check2 4.83 4.83 5.00 5.17 Check3 8.17 7.83 10.00 7.33 AvgCheck2,3 vs Check1 8 y = 0.7026x + 2.916 R² = 0.41406 7 6 Check23Avg 6.50 6.33 7.50 6.25 Use slope of regression to adjust experimental lines 5 4 4 5 6 7 8 Method3AdjustedLine102WholePlot(wp)9 = RawLine102wp9 – slope*(Check1wp9 – Check1Avg) Line “MALT-102” from data in corresponding Excel file = 8.47 – 0.7026*(5.5 – 5.46) = 8.44 NOTE: Method1AdjustedEntryX = 7.95 Type 2 Modified Augmented Design: Adjustments – Method 3 - Regression NOTE: if the data is missing for any of the checks in a whole plot, then you must exclude the remaining check subplots in that whole plot from the regression. However, you may not want to exclude those other subplots from the dataset as a whole, because they can be used in other calculations, just not calculation of the regression coefficient. Row 2 3 3 4 Column 1 2 4 4 WholePlot 7 14 16 22 Check1 5.50 5.47 5.73 4.53 Check2 4.83 4.83 5.00 . Check3 8.17 7.83 10.00 7.33 Check23Avg 6.50 6.33 7.50 . Type 2 Modified Augmented Design: Adjustments – Method 1 vs Method 3 - Relative Efficiency Prior to adjusting the values for the experimental subplots, you will probably want to decide whether you want to use a Method 1 adjustment, a Method 3 adjustment, or no adjustment. The first step in comparing the adjustment methods is to look at the relative efficiencies for Method 1 and Method 3. As you can see by the equation below, a higher relative efficiency means a greater reduction in experimental error after adjustments: Relative Efficiency = Intra-WholePlotErrorWithoutAdjustment Intra-WholePlotErrorWithAdjustment x 100% Type 2 Modified Augmented Design: Adjustments – Method 1 vs Method 3 - Relative Efficiency The relative efficiency of Method 1 or Method 3 compares the experimental error within whole plots for unadjusted versus adjusted data. The intra-whole plot errors used to calculate relative efficiency are based on secondary check subplots only. Column 1 1 2 4 3 5 6 WholePlot (wp) 1 wp2 wp3 wp4 wp5 wp6 wp7 wp8 wp9 wp10 wp11 wp12 wp13 wp14 wp15 wp16 wp17 wp18 wp19 wp20 wp21 wp22 wp23 wp24 Row 2 3 4 = Primary Check (Check1) = Secondary Check A (Check2) = Secondary Check B (Check3) Type 2 Modified Augmented Design: Adjustments – Method 1 vs Method 3 - Relative Efficiency Relative Efficiency = Intra-WholePlotErrorWithoutAdjustment Intra-WholePlotErrorWithAdjustment i Intra-WholePlotErrorWithoutAdjustment = ΣΣ(Yini -Yi)2 i Σ(ni -1) i Intra-WholePlotErrorWithAdjustment = ni ni ΣΣ(Yin Adj-YiAdj)2 i i Σ(ni -1) x 100% i = number of secondary checks ni = number of subplots of each secondary check Yini = value of subplot of secondary check i in subplot ni Yi = mean of secondary check i across all subplots ni NOTE: The ni term in the numerator may be a bit confusing, as the design demands planting the same number of subplots for each secondary check (i.e., n instead of ni). However, you may have missing data for some secondary check plots, which is fine. The important concept here is that the Intra-WholePlotError is basically a weighted average variance of the secondary checks. When there is no missing data from the secondary checks, then the Intra-WholePlot Error is simply the average variance of the secondary checks. 1/18/13 SAS Output Type 2 Modified Augmented Design: Method 1 ANOVA - Spike Length (cm) The GLM Procedure Dependent Variable: SpkLngth Adjustments – Method 1 vs Method 3 Source DF Sum of Squares Mean Square F Value Pr > F 8 1.39740741 0.17467593 0.79 0.6212 Error 15 3.32685185 0.22179012 Corrected Total 23 4.72425926 Model Example Dataset Results (see corresponding Excel file for calculations): 1/18/13 R-Square Coeff Var Root MSE SpkLngth Mean 0.295794 8.619245 0.470946 SAS Output 5.463889 Source DF Type I SS Mean Square F Value Pr > F 0.74537037 0.24845679 The GLM Procedure 0.65203704 0.13040741 1.12 0.3721 0.59 0.7093 Method 1 ANOVA - Spike Length (cm) Row 3 Column 5 Dependent Variable: SpkLngth Source Model Error Source DF Type III SS Mean Square F Value Pr > F DF Sum of Squares Mean Square F Value Pr > F Row Column Corrected Total 3 8 0.74537037 0.24845679 1.39740741 0.17467593 1.12 0.3721 0.79 0.6212 5 15 0.65203704 0.13040741 3.32685185 0.22179012 0.59 23 4.72425926 Relative Efficiency 0.295794 8.619245 0.470946 R-Square Source Coeff Var Root MSE 0.7093 SpkLngth Mean 5.463889 DF Type I SS Mean Square 3 0.74537037 0.24845679 Column 5 0.65203704 0.13040741 Source DF Type III SS Mean Square Row 3 0.74537037 124. F Value Pr > F 7 0.24845679 1.12 0.3721 Column 5 0.65203704 0.13040741 Method Row 1 95.0 F Value Pr > F 0.59 0.7093 0.59 0.7093 Method 3 1.12 0.3721 Type 2 Modified Augmented Design: Adjustments – Method 1 vs Method 3 It is important to note that the F-tests from the ANOVA results apply to Row and Column effects, which can be accounted for by Method 1 adjustments. The Row and Column effects generally are best at accounting for gradients that stretch across a substantial portion of the field. Method 3 does not necessarily require gradients or any other type of pattern in the field to account for field effects; it only requires that the secondary checks are affected by the field in a way similar to how the primary checks are affected. That being said, it is common to find no significant effects in the ANOVA but still find that a Method 3 adjustment is appropriate. Type 2 Modified Augmented Design: Adjustments – Method 1 vs Method 3 The type-2 modified augmented design intentionally makes the final selection of an adjustment method to be somewhat subjective, based on the user’s understanding of the biological system. Evidence on the appropriateness of each method includes: • Relative efficiency of Method 1 vs Method 3 • ANOVA results for Row and Column effects • Biological meaning of analysis parameters - I once had a relative efficiency of 112% for Method 3 but a negative Method 3 regression coefficient • Heat maps or other semi-quantitative/qualitative evaluations of field effects • Knowledge of the field or fieldbook notes from the experiment • Knowledge of the lines used as checks - e.g., you may have more confidence in a marginal Method 1 relative efficiency when you know that your primary check used to calculate Method 1 adjustments is much less sensitive to field effects than most of the other lines in the experiment. MAD Type 2 References • Lin, CS and Poushinsky, G (1983). A modified augmented design for an early stage of plant selection involving a large number of test lines without replication. Biometrics 39:553-561. • Lin, CS, Poushinsky, G and Jui, PY (1983). Simulation study of three adjustment methods for the modified augmented design and comparison with the balanced lattice square design. The Journal of Agricultural Science 100:527-534. • Lin, CS and Poushinsky, G (1985). A modified augmented design (type 2) for rectangular plots. Canadian Journal of Plant Science 65:743-749. • May, KW, Kozub, GC and Schaalje, GB (1989). Field evaluation of a modified augmented design (type 2) for screening barley lines. Canadian Journal of Plant Science 69:1-15.