#####  Copter ANOVAs I
# Experimental design and analysis must match. 
# If they don't match we risk making wrong conclusions about the results. 
# This occurs more often in science than we wish to admit. 
# Today we run various analyses that do not fully match the complex experimental design for the helicopters data.
# Next week, we will analyze the results in the full, correct way and compare those results to today's results.

# So save your results today – you will need to look at them again next week.

# A reminder: the experiment used three treatments to potentially affect helicopter flight times: 
#    a. wing length 
#    b. body width
#    c. whether wings included a fold or not.
# Today we treat each of these as separate treatments – in reality the experiment investigated the interactions of the 3 treatments. 
# We ignore interactions today !warning! !warning!; there is already a mismatch between design and analyses - do you see why?

##### R Steps to follow: 
# 1. Import and attach the copter data from 
# https://sciences.ucf.edu/biology/d4lab/wp-content/uploads/sites/23/2021/09/helicopter-data.csv
# Below I assume you called this "data" for simplicity.

# 2. In case there are any NAs, it may be helpful to remove them with a command like this:
data <- na.omit(data)
attach(data)

# You have already examined normality of these data. To save time right now, let's *assume* that
# normality is not too far out of whack. But you also should already expect the other key
# assumption (homogeneous variance) to be a big problem. This is more critical for ANOVAs than normality so let’s revisit:

# 3. First make a factor for each of ID, Wing Length (WL) and Body Width (BW):
fid <- factor(ID) # where ID indicates a unique design - i.e., a unique WL & BW and fold combination.
fwl <- factor(WL)
fbw <- factor(BW)
ffold <- factor(Fold)

# Then run a Levene's test by first loading the package “car” and then running:
leveneTest(Time~fid)

# NOTE: This is a shortcut – using IDs for all combinations of wing length, fold, and body width. 
# Are the variances among each design about the same? 
# If so, then we can use ANOVA with confidence. 
# If not, then all we do below is on shaky ground. 
# And iterative attempts to transform data may not help. 
# This SHOULD be a big red flag to you! Beware of analyses with bad assumptions!

# 4. Let's stumble ahead anyway just so you can see how to conduct an ANOVA. We will first use Wing Length (WL), in two ways:
# The first reports results as a simple ANOVA, with the expectation that treatments are categorical.
# The second produces regression output, with the expectation that treatments are quantitative.

model1 <- aov(Time ~ fwl) # an ANOVA on factored WL - to make WL categories
model2 <- lm(Time ~ WL) # a linear model on quantitative WL
summary(model1)
summary(model2)

# What does the aov output show you that the lm command does not? 
# Vice versa - What does the lm output show you that the aov command does not?
# Do you get different answers? [Hint: look at F statistics]

# 5. Now rename and repeat those analyses but use Fold in place of WL 
# (I assume you call those model3 & model4). How does this compare to WL outcomes?
# Now you are starting to compare the relative effects of folds and wing lengths.

# 6. Now rename and repeat those analyses but use BW in place of Fold 
# (I assume you call those model5 & model6). 
# How does this compare to above outcomes? 

# Do you see a problem with evaluating these effects (WL, Fold, BW) *separately*? You should!

# 8. Now let's get into details. How much of the variation in flight times is explained by WL, Fold, or BW? 
# How can you tell? Is this satisfactory?

# 9. Now let's try a factorial statement that starts to represent the complex experiment.
# So far we have only analyzed pieces that do not fully represent the whole. A factorial design
# allows us to examine the effects of one factor on the effects of a second factor. Our
# factorial experiment answers the question: “Does the effect of wing length on flight time
# depend on whether wings are folded and body width?” 
# Alternatively: “Does the effect of folded wings on flight time depend on wing length and folds?”

model7 <- aov(Time ~ fwl*fbw*ffold)
model8 <- lm(Time ~ WL*BW*ffold) # why use quantitative WL and BW here but use the factored ffold?
summary(model7)
summary(model8)

# NOTE: R knows when you enter A*B that you mean this: [A + B + A X B]. 
# So you will be able to evaluate each effect and the interaction.
# Look at the ANOVA results (model 7) first: Did treatments clearly affect flight time, and did
# treatments affect each other's flight time? Which effect seems most important?
# If there are significant interaction effects, that is the focus; separate effects of each treatment
# in an interaction are no longer solo. Instead, the effect of one on the other must be considered.

# Now let's examine the linear model (lm) results: notice that the rows are named
# differently? This takes a moment's thought because now we see an intercept – what is this?
# And what do the something:something lines tell you?

# 10. Let's make an interaction graph to help interpret this. Enter this command:

interaction.plot(Fold, trace.factor=fwl, response=Time, fun=mean, type="b")

# Do longer Wings make copters stay in the air longer? How do you know?
# Do folded Wings alter that effect of Wings? How do you know?
# How would you explain this in a few sentences that would convince your advisor?

# So what have we accomplished? 
# We computed a factorial ANOVA to find out if there was an interaction between three experimental treatments. 
# We can compare SS & MS in the ANOVA table to see which treatment had a greater effect. 
# But that basic ANOVA leaves us wishing for more details. And lm output gave us that.
# The lm output and the interaction plot allow us to tease apart what happened, and to
# quantify those effects. We will refine this later.

# One final question: how much variation in helicopter flight times did we
# account for with our most recent model? Does this seem like enough?
# Next week we fully account for Group effects and the covariate of Stairs.