Getting Started

1. Load house_subset.csv data.
2. Load tidyverse

library('tidyverse')
theme_set(theme_minimal())

houses <- read.csv('../Lab 3/house_subset.csv') # choose your directory!
houses$zipcode <- as.factor(houses$zipcode)

Two way ANOVA

Above we calculated one way ANOVA, that is the standard ANOVA. We compared population means using a sample continuous variable, sqft_living, and one categorical variable, zipcode. But as you can see above, that magical F-distribution is capable of testing many things at the same time. So what about comparing populaton means of a continuous variable based on the variances changing among two categorical variables, e.g. zipcode and whether it is multistorey (more than one floors):

temp <- subset(houses, zipcode %in% c(98122, 98027, 98116, 98072))
temp$multistorey <- temp$floors > 1 

ggplot(temp, aes(x=price, y=zipcode, colour=multistorey)) + 
  geom_boxplot() 

It is obvious that whether or not it is multistorey effects the house price (compare the blue and red). It is also seen that zipcode effects the house prices, e.g. when we comparey only the reds we see that they are varying, but when we compare the blue boxplots they don’t seem very different. Notice also that the dealbreaker is the zipcode == 98072 & multistorey == F.

To carry out ANOVA in R, we don’t need to know more. We will use the same function:

temp <- subset(houses, zipcode %in% c(98122, 98027, 98116, 98072))
temp$multistorey <- temp$floors > 1 
anova(lm(price ~ zipcode + multistorey,data = temp))

## Analysis of Variance Table
## 
## Response: price
##               Df     Sum Sq    Mean Sq  F value  Pr(>F)    
## zipcode        3 6.5188e+11 2.1729e+11   3.3584 0.01823 *  
## multistorey    1 7.7201e+12 7.7201e+12 119.3173 < 2e-16 ***
## Residuals   1300 8.4112e+13 6.4702e+10                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We can see that both zipcode and multistorey are significantly changing the variability. Thus we reject the hypothesis that the population means among different zipcodes and house types are equal. In other words, where your house is and whether or not it is multistorey are significant factors to effect house prices.

As we discussed above, the dealbreaker (combo-killer, killjoy, blacksheep) is one region, 98072 (see the above boxplot). If we drop that region and compare the rest the test results might be different:

temp <- subset(houses, zipcode %in% c(98122, 98027, 98116))
temp$multistorey <- temp$floors > 1 
anova(lm(price ~ zipcode + multistorey,data = temp))

## Analysis of Variance Table
## 
## Response: price
##               Df     Sum Sq    Mean Sq F value Pr(>F)    
## zipcode        2 5.8219e+10 2.9110e+10   0.406 0.6664    
## multistorey    1 5.3697e+12 5.3697e+12  74.897 <2e-16 ***
## Residuals   1028 7.3702e+13 7.1695e+10                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Now, we cannot reject the hypothesis that average houseprices among different regions are equal, but we can still reject that average house prices are equal when the house is one floor or more than one floor.

3. Two Way ANOVA

Choose one continuous variable and two categorical variables

• if there are many categories in the categorical variables, you may find it useful to filter out some categories from both variables.

• You may create a binary variable and use it as the second categorical variable (see the above examples where we created multistorey).

1. Plot the boxplot or jitter plot or density ringes or violin plot of the continuous variable.

• Map the continuous variable to x axis and one categorical variable to y axis. Don’t make your plots simply fancier, but map the other categorical variable to colour (see the last boxplot above). Make sure that your categorical variables are treated as categorical.

• Interpret whether the population means can be equal. You can simply compare the medians and tell what you see.

2. Carry out two way ANOVA

• Carry out two way ANOVA to test whether population means are equal.
• Interpret the results. According to the p-values, does any of the categorical variables significantly changing the population mean?

One Way ANOVA (EXERCISE, NOT GRADED, NOT TO BE SUBMITTED)

Choose one continuous variable and one categorical variable

• if there are many categories in the categorical variable, you may find it useful to filter out some categories using subset(DATA, CATGVAR %in% c(V1,V2,...,VN)). See the above examples where we subseted to the data to compare only four regions.

1. Plot the boxplot or jitter plot or density ringes or violin plot of the continuous variable.

• Map the continuous variable to x axis and categorical variable to y axis. Map categorical variable also to the colour or fill to make your plot fancier. Make sure that the categorical variable is treated as categorical (if it is string, you don’t need to to anything but if it is numeric then you may convert it. See the above examples where I converted zipcodes using as.factor).

• Interpret whether the population means can be equal. You can simply compare the medians and tell what you see.

2. Carry out one way ANOVA:

• If there are too many categories in the variable, you can compare only, e.g. four categories. (see the first bulletpoint)
• Carry out the ANOVA to test whether population means are equal
• Interpret the results. According to the p-values, can they be equal?