Formal ANOVA II

1. What kinds of values are contained in an ANOVA table (like the one on p. 76)? From where do we get these different values? (you might not know all of the details yet–get to know your own blind spots!)

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• CA-SINZ Assumptions in assembly line code
• Formally present Analysis of variance (ANOVA)

In a randomized double-blind study (n = 127), science faculty from research-intensive universities rated the application materials of an applicant who was randomly assigned either a male or female name for a laboratory manager position. Faculty participants rated the male applicant as significantly more competent than the (identical) female applicant. These participants also selected a higher starting salary and offered more hypothetical career mentoring to the male applicant. See materials here

We assume every observation in a similar condition is affected exactly the same. (Gets the same true score).

For example,

animals_sim <- animals %>%
  mutate(benchmark = mean(calm)) %>%
  group_by(animal) %>%
  mutate(animal_mean = mean(calm),
         aminal_effect = animal_mean - benchmark) #every "dog" observation gets the same effect

We add the effects as we go down the assembly line.

All effects are added on.

calm_sim = benchmark 
         + aminal_effect 
         + cue_effect 
         + interaction_effect 
         + student_effect 

The piece of code for adding error is not dependent on which condition the observations is in. Every condition gets the same standard deviation, here 0.65.

 + rnorm(68, 0, 0.68) #rnorm(n, mean, sd = 0.65)

Takes 68 independent draws from a normal distribution.

 + rnorm(68, 0, 0.68) #rnorm function assumes independence

It's rnorm(), and not rbinom() or rpois()…

 + rnorm(64, 0, 0.68)

The second argument is the mean.

 + rnorm(64, 0, 0.68) #rnorm(n, mean = 0, sd)

It is reasonable to assume that the structure of a sugar molecule has something to do with its food value. An experiment was conducted to compare the effects of four sugar diets on the survival of leafhoppers. The four diets were glucose and fructose (6-carbon atoms), sucrose (12-carbon), and a control (2% agar). The experimenter prepared two dishes with each diet, divided the leafhoppers into eight groups of equal size, and then randomly assigned them to dishes. Then she counted the number of days until half the insects had died in each group.

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• Draw the factor diagram, including the benchmark and residuals.

Formal ANOVA starts with the simple idea that we can compare our estimate of treatment effect variability to our estimate of chance error variability to measure how large our treatment effect is.

Variability in treatment effects = True Effect Differences + Error

Variability in residuals = Error

Variability in treatment effects/Variability in residuals

• If our null hypothesis is, \( {H}_{0} \): True Effect Differences \( =0 \), then what would we expect the following ratio to equal?

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)

• We cannot always use the same formula for the treatment effects. It depends on inside and outside factors
• Estimated effect for a factor = Average for the factor - sum of estimated effects for all outside factors. That is,

\[ Effect = Average - Partial Fit \]

• One factor is inside another if each group of the first (inside) fits completely inside some group of the second (outside) factor.