Basic Factorial Design, BF[1]

1. What is the critical F-value used for in ANOVA? (What is the p-value for an F statistic?)

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• ANOVA for the BF[1] Design
  

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Barley seeds are divided into 30 lots of 100 seeds each. The 30 lots are then divided at random into ten different groups of three lots each, with each group receiving a different treatment combination. The amount of water given per day is varied (4 ml or 8 ml). Also, the number of seeds sprouted is measured at different lengths of time randomly assigned to the groups. Thus, the researchers can understand the effect of the age of the seeds (1 week, 3 weeks, 6 weeks, 9 weeks, and 12 weeks), as well as water, on growth.

control	sucrose	glucose	fructose
2.3	3.6	3.0	2.1
1.7	4.0	2.8	2.3

ANOVA measures variability in treatment effects with the sum of squares (SS) divided by the number of units of unique information (df). For the BF[1] design,

\[ {SS}_{Treatments} = n\sum_{i=1}^{a}(\bar{y}_{i.}-\bar{y}_{..})^{2} \]

\[ {SS}_{E} = \sum_{i=1}^{a}\sum_{j=1}^{n}({y}_{ij}-\bar{y}_{i.})^{2} \]

\[ {SS}_{Total} = {SS}_{Treatments} + {SS}_{E} \]

where \( n \) is the group size, and \( a \) is the number of treatments.

The df for a table equals the number of free numbers, the number of slots in the table you can fill in before the pattern of repetitions and adding to zero tell you what the remaining numbers have to be.

\[ {df}_{Treatments}=a-1 \]

\[ {df}_{E}=N-a \]

The ultimate statistic we want to calculate is Variability in treatment effects/Variability in residuals.

Variability in treatment effects: \[ {MS}_{Treatments}=\frac{{SS}_{Treatments}}{{df}_{Treatments}} \]

Variability in residuals \[ {MS}_{E}=\frac{{SS}_{E}}{{df}_{E}} \]

The ratio of these two MS's is called the F ratio. The following quantity is our test statistic for the null hypothesis that there are no treatment effects.

\[ F = \frac{{MS}_{Treatments}}{{MS}_{E}} \]

If the null hypothesis is true, then F is a random variable \( \sim F({df}_{Treatments}, {df}_{E}) \). The F-distribution.

qplot(x = rf(500, 3, 4), geom = "density")

\( {H}_{0}:\tau_1=\tau_2=\tau_3=\tau_4 \)

We can find the p-value for our F calculation with the following code

pf(17.67, 3, 4, lower.tail = FALSE)

• Analyzing the leafhopper data in R. code here

• We cannot ALWAYS use the same formula for the treatment effects. It depends on inside and outside factors.
• We do not ALWAYS divide the \( {MS}_{factor} \) by \( {MS}_{E} \). To test some effects in some designs we will need a different denominator.

• One factor is inside another if each group of the first (inside) fits completely inside some group of the second (outside) factor.
• Estimated effect for a factor = Average for the factor - sum of estimated effects for all outside factors.
• The sum of estimated effects for all outside factors is called the partial fit.
• Thus, the general rule is:

\[ Effect = Average - Partial Fit \]