## YOUR TURN #1

```{r}
# submit the following code to simulate some data
set.seed(2)
x1 <- sample(1:5, size = 1000, replace = TRUE, 
             prob = c(0.1,0.2,0.3,0.3,0.1))
x2 <- rnorm(n = 1000, mean = 12, sd = 2)
noise <- rnorm(n = 1000, mean = 0, sd = 4)
y <- 5 + 10*x1 + -4*x2 + noise
df <- data.frame(y, x1, x2)
head(df)

# Use lm() as we did above to attempt to recover the "true" values that were
# used to generate y.

m <- lm(y ~ x1 + x2, data = df)
summary(m)

# No interaction when we simulated y, so just use x1 + x2, not x1 * x2.

```




## YOUR TURN #2

Update model `m2` to include the lotsize variable, which is the size of the lot in acres on which the house is built. Call the model `m3`. What is the interpretation of lotsize? Generate diagnostic plots. What does the last plot (Residuals vs Leverage) reveal?

Challenge: simulate data from the model and compare to the observed totalvalue. 

```{r}
# enter your code below
m3 <- lm(log(totalvalue) ~ finsqft + lotsize, data = homes)
summary(m3)
exp(coef(m3)) %>% round(4)

# Intreptation of lotsize coefficient: Every additional acre increases home value by about 0.35%.

# check diagnostics
plot(m3)

# row 4624 appears to be an influential observation. It is a huge home on a huge
# lot with a very high value!
homes[4624, c("totalvalue", "finsqft", "lotsize")]

# simulate data
sim_tv <- simulate(m3, nsim = 50)
plot(density(log(homes$totalvalue)))
for(i in 1:50)lines(density(sim_tv[[i]]), col = "red")

# if we like, we can fit the model without row 4624 as follows:
m3 <- lm(log(totalvalue) ~ finsqft + lotsize, data = homes, subset = -4624)
summary(m3)


```

## YOUR TURN #3

Add cooling to our model and save it as `m5`. Does it help our model? What is the interpretation of the coefficient?

```{r}
m5 <- lm(log(totalvalue) ~ finsqft + lotsize + censustract + cooling, data = homes)
summary(m5)
anova(m5)

exp(coef(m5)) %>% round(4)
exp(confint(m5)) %>% round(4)

# totalvalue of home with central air is expected to be about 31% to 34% higher than a home without central air.

```

### YOUR TURN #4

```{r}
# Add an interaction for cooling and finsqft to `m6` and save as `m7`.
m7 <- lm(log(totalvalue) ~ finsqft + lotsize + censustract + 
    cooling + finsqft:censustract + finsqft:lotsize + finsqft:cooling, 
    data = homes)

# Is the interaction significant? 
summary(m7)

# Yes, it seems significant

# Create an effect plot to visualize the interaction of cooling and finsqft (or
# lack thereof).
plot(ggpredict(m7, terms = c("finsqft", "cooling")))

# The interaction is significant but extremely weak. Notice how the lines look almost parallel.

plot(ggpredict(m7, terms = c("finsqft[1000,2000,3000,4000,5000,6000]", "cooling")))


```



### YOUR TURN #5

```{r eval=FALSE}
# Update model `m9` to include a non-linear effect for lotsize using a natural
# spline with 5 DF and save the new model as `m10`.
m10 <- lm(log(totalvalue) ~ ns(finsqft, df = 3) + ns(lotsize, df = 5) + 
            censustract + cooling,
          data = homes) 

# Generate an effect plot for the non-linear lotsize effect.
plot(ggpredict(m10, terms = ~ns(lotsize, df = 5)))

# Check the partial-residual plot for lotsize in model `m10`.
crPlots(m10, ~ns(lotsize, df = 5), smooth=list(smoother=gamLine))

```


## YOUR TURN #6


Fit the following three models, each with increasing levels of complexity in the splines. Which one is "better" as indicated by `anova`, `AIC` and `BIC`?

```{r}
# Fit the following three models, each with increasing levels of complexity in
# the splines.
mod_a <- lm(log(totalvalue) ~ ns(finsqft, df = 3) * ns(lotsize, df = 3) + 
            censustract + cooling,
          data = homes) 
mod_b <- lm(log(totalvalue) ~ ns(finsqft, df = 4) * ns(lotsize, df = 4) + 
            censustract + cooling,
          data = homes) 
mod_c <- lm(log(totalvalue) ~ ns(finsqft, df = 5) * ns(lotsize, df = 5) + 
            censustract + cooling,
          data = homes) 

# Compare the models using `anova`, `AIC` and `BIC`. Which is better?
AIC(mod_a, mod_b, mod_c)
BIC(mod_a, mod_b, mod_c)
anova(mod_a, mod_b, mod_c)

# mod_c appears to be better despite its complexity.

```



## YOUR TURN #7

Modify the code below to produce an effect plot for finsqft (with values set to 1500, 2000, 2500, 3000) and lotsize set to 0.5 acre. Leave cooling and censustract as is.

```{r}
eff_out <- ggpredict(m12, terms = "lotsize[1:10]", 
                    condition = c(finsqft = 2500, 
                                  cooling = "Central Air",
                                  censustract = "101"))
plot(eff_out)

eff_out <- ggpredict(m12, terms = "finsqft[1500, 2000, 2500, 3000]", 
                    condition = c(lotsize = 0.5, 
                                  cooling = "Central Air",
                                  censustract = "101"))
plot(eff_out)

```