The linear model object

I think it is worth taking a few minutes to discuss the linear model object (child_height in our case) which we just created. A model object is fundamentally a list so it can be subsetted using the $ operator. We can extract a lot of information from our model object including the coefficients as below for example.

#a model object is fundamentally a list
#we can use $ to subset model objects in the same way we can dataframes (or named lists)
child_height$coefficients 

## (Intercept)      Height 
## 138.5397222   0.1298447

It is well worth taking some time to explore all the outputs you can get from the linear model object as they provide so much more information than what is contained in the model summary. This additional information can be extremely useful when dealing with more complex models or model applications.

Advancing the model

#adding in another term into our model
child_height_2 <- lm(Height ~ Mother + Father, data = galton)
summary(child_height_2)

## 
## Call:
## lm(formula = Height ~ Mother + Father, data = galton)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -22.8389  -6.7505  -0.4524   6.9210  29.2214 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 55.77426   10.76724   5.180 2.74e-07 ***
## Mother       0.28321    0.04914   5.764 1.13e-08 ***
## Father       0.37990    0.04589   8.278 4.52e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.465 on 895 degrees of freedom
## Multiple R-squared:  0.1089, Adjusted R-squared:  0.1069 
## F-statistic: 54.69 on 2 and 895 DF,  p-value: < 2.2e-16

We can see in the code above, we use the + operator to add another term into our model. So our model now includes both the height of a childs mother and their father. When examining our beta values, we can see that a 1cm increase in the height of a child’s father appears to be associated with a greater increase in a childs height compared to a 1cm increase in their mothers height (0.38cm vs 0.28cm). We can also see that both our model terms are highly significant. This model also explains significantly more variation in child’s height (10.7%) than our original model.

Beautifying our output

Whilst the above summary outputs are useful for use to explore our models they may not be as useful if we want to communicate our data with others. To help with this, we can use the jtools package. This package generates much more attractive model outputs.

library(jtools)
#we can use the jtools package to generate a much more attractive output
#summ is the jtools version of the summarise function
summ(child_height_2) 

This is a huge improvement over our previous output which is much more understandable to less technically focused audiences. We can also use the export_summs function from jtools to summarise two models side by side.

#export_summs function from jtools can be used to summarise two models side by side
export_summs(child_height, child_height_2, scale = TRUE)

Checking model assumptions

There are a number of neat little tricks that can be performed with a model object. One of my favourites is to put the model into the base R plot function.

#checking the assumptions of the model using the plot function
plot(child_height_2)

Figure 1: Diagnostic plots for a linear model

Figure 2: Diagnostic plots for a linear model

Figure 3: Diagnostic plots for a linear model

Figure 4: Diagnostic plots for a linear model

This simple line of code creates four graphs by default (however this can be increased to six by setting the which option to c(1,2,3,4,5,6)) which can be used to check the assumptions of a linear model. A complete description of this is outside the scope of this post but can be found here.

Extending our model

The R syntax to define models allows us to do a lot more than just add new model terms. We are also able to add interaction terms using the * operator.

#adding interaction terms into the model
child_height_3 <- lm(Height ~ Mother * Father, data = galton)
summ(child_height_3)

Interaction effects in a linear model indicate that the two variables are not independent. This would mean the value of one variable (mothers height) is depends on the value of another variable (fathers height). In our example, this is not the case, however there are really world examples where interaction terms are very useful. We have included this example simply to illustrate their syntax.

Why R makes your life easier

Above we have created linear models using a relatively basic syntax. A major benefit of modelling in R is that the same syntax can be applied to a range of other models. Let’s use the generalised linear model as an example. Such models allow us to predict categorical response variables rather than numeric response variables.

#calculating a generalised linear model using the same syntax
#glm is a base R function for a general linear model 
#Before the model is run, the Gender variable needs to be recoded into 0 or 1
#summ from the jtools package creates a visually appealing output
galton <- galton %>%
  mutate(
    Gender = if_else(Gender == "M", 1, 0)
  )
gender <- glm(Gender ~ Height, data = galton, family = "binomial") 
summ(gender)

First, we need to recode our Gender variable to be 0 or 1. In our example, we set male (M) to 1 and female (F) to 0. We can then enter this into the glm function and set the model family to binomial meaning our response variable is either one value or the other. We can see in our output that height does have a strong relationship with children’s gender suggesting that males are on average taller than females.

The reason for including this example is to highlight one key point, the R language has a very consistent method to define formulas in a model. This means that once you can create a linear model, you are capable of programming a whole range of other models. This is particularly useful for beginners as you can very rapidly develop a range of modelling techniques.