9 Multiple linear regression

Learning outcomes

Questions

How do I use the linear model framework with three predictor variables?

Objectives

Be able to expand the linear model framework with three predictor variables
Define the equation for the line of best fit for each categorical variable
Be able to construct and analyse any possible combination of predictor variables in the data

9.1 Libraries and functions

Click to expand

R
Python

9.1.1 Libraries

# A collection of R packages designed for data science
library(tidyverse)

# Converts stats functions to a tidyverse-friendly format
library(rstatix)

# Creates diagnostic plots using ggplot2
library(ggResidpanel)

# Helper functions for tidying data
library(broom)

9.1.2 Functions

# Gets underlying data out of model object
broom::augment()

# Creates diagnostic plots
ggResidpanel::resid_panel()

# Performs an analysis of variance
stats::anova()

# Creates a linear model
stats::lm()

9.1.3 Libraries

# A fundamental package for scientific computing in Python
import numpy as np

# A Python data analysis and manipulation tool
import pandas as pd

# Simple yet exhaustive stats functions.
import pingouin as pg

# Python equivalent of `ggplot2`
from plotnine import *

# Statistical models, conducting tests and statistical data exploration
import statsmodels.api as sm

# Convenience interface for specifying models using formula strings and DataFrames
import statsmodels.formula.api as smf

9.1.4 Functions

# Reads in a .csv file
pandas.read_csv()

# Creates a model from a formula and data frame
statsmodels.formula.api.ols()

9.2 Purpose and aim

Revisiting the linear model framework and expanding to systems with three predictor variables.

9.3 Data and hypotheses

The data set we’ll be using is located in data/CS5-pm2_5.csv. It contains data on air pollution levels measured in London, in 2019. It also contains several meteorological measurements. Each variable was recorded on a daily basis.

Note: some of the variables are based on simulations.

It contains the following variables:

variable	explanation
`avg_temp`	average daily temperature (\(^\circ C\))
`date`	date of record
`location`	location in London (`inner` or `outer`)
`pm2_5`	concentration of PM2.5 (\(\mu g / m^3\))
`rain_mm`	daily rainfall in mm (same across both locations)
`wind_m_s`	wind speed in \(m/s\)

Let’s first load the data:

pm2_5 <- read_csv("data/CS5-pm2_5.csv")

Rows: 730 Columns: 6
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (2): date, location
dbl (4): avg_temp, pm2_5, rain_mm, wind_m_s

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

head(pm2_5)

# A tibble: 6 × 6
  avg_temp date       location pm2_5 rain_mm wind_m_s
     <dbl> <chr>      <chr>    <dbl>   <dbl>    <dbl>
1      4.5 01/01/2019 inner     17.1     2.3     3.87
2      4.9 01/01/2019 outer     10.8     2.3     5.84
3      4.3 02/01/2019 inner     14.9     2.3     3.76
4      4.8 02/01/2019 outer     11.4     2.3     6   
5      4   03/01/2019 inner     18.5     1.4     2.13
6      4.5 03/01/2019 outer     15.0     1.4     2.57

It’s the pm2_5 response variable we’re interested in here. Let’s start by checking if there might be a difference between PM2.5 level in inner and outer London:

ggplot(pm2_5,
       aes(x = location, y = pm2_5)) +
    geom_boxplot() +
    geom_jitter(width = 0.1, alpha = 0.3)

I’ve added the (jittered) data to the plot, with some transparency (alpha = 0.3). It’s always good to look at the actual data and not just summary statistics (which is what the box plot is).

There seems to be quite a difference between the PM2.5 levels in the two London areas, with the levels in inner London being markedly higher. I’m not surprised by this! So when we do our statistical testing, I would expect to find a clear difference between the locations.

Apart from the location, there are quite a few numerical descriptor variables. We could plot them one-by-one, but that’s a bit tedious. So instead we use the pairs() function again. This only works on numerical data, so we select all the columns that are numeric with select_if(is.numeric):

pm2_5 %>% 
    select_if(is.numeric) %>% 
    pairs(lower.panel = NULL)

We can see that there is not much of a correlation between pm2_5 and avg_temp or rain_mm, whereas there might be something going on in relation to wind_m_s.

Other notable things include that rainfall seems completely independent of wind speed (rain fall seems pretty constant). Nor does the average temperature seem in any way related to wind speed (it looks like a random collection of data points!).

We can visualise the relationship between pm2_5 and wind_m_s in a bit more detail, by plotting the data against each other and colouring by location:

ggplot(pm2_5,
       aes(x = wind_m_s, y = pm2_5,
           colour = location)) +
    geom_point()

This seems to show that there might be some linear relationship between PM2.5 levels and wind speed.

Another way of looking at this would be to create a correlation matrix, like we did before in the correlations chapter:

pm2_5 %>% 
    select_if(is.numeric) %>% 
    cor()

            avg_temp       pm2_5     rain_mm    wind_m_s
avg_temp  1.00000000  0.03349457  0.03149221 -0.01107855
pm2_5     0.03349457  1.00000000 -0.02184951 -0.41733945
rain_mm   0.03149221 -0.02184951  1.00000000  0.04882097
wind_m_s -0.01107855 -0.41733945  0.04882097  1.00000000

This confirms what we saw in the plots, there aren’t any very strong correlations between the different (numerical) variables, apart from a negative correlation between pm2_5 and wind_m_s, which has a Pearson’s r of \(r\) = -0.42.

Let’s first load the data:

pm2_5_py = pd.read_csv("data/CS5-pm2_5.csv")

pm2_5_py.head()

   avg_temp        date location   pm2_5  rain_mm  wind_m_s
0       4.5  01/01/2019    inner  17.126      2.3      3.87
1       4.9  01/01/2019    outer  10.821      2.3      5.84
2       4.3  02/01/2019    inner  14.884      2.3      3.76
3       4.8  02/01/2019    outer  11.416      2.3      6.00
4       4.0  03/01/2019    inner  18.471      1.4      2.13

It’s the pm2_5 response variable we’re interested in here. Let’s start by checking if there might be a difference between PM2.5 level in inner and outer London:

(ggplot(pm2_5_py, aes(x = "location", y = "pm2_5")) +
    geom_boxplot() +
    geom_jitter(width = 0.1, alpha = 0.7))

I’ve added the (jittered) data to the plot, with some transparency (alpha = 0.7). It’s always good to look at the actual data and not just summary statistics (which is what the box plot is).

Apart from the location, there are quite a few numerical descriptor variables. At this point I should probably bite the bullet and install seaborn, so I can use the pairplot() function.

But I’m not going to ;-)

I’ll just tell you that there is not much of a correlation between pm2_5 and avg_temp or rain_mm, whereas there might be something going on in relation to wind_m_s. So I plot that instead and colour it by location:

(ggplot(pm2_5_py,
        aes(x = "wind_m_s",
            y = "pm2_5",
            colour = "location")) +
     geom_point())

This seems to show that there might be some linear relationship between PM2.5 levels and wind speed.

If I would plot all the other variables against each other, then I would spot that rainfall seems completely independent of wind speed (rain fall seems pretty constant). Nor does the average temperature seem in any way related to wind speed (it looks like a random collection of data points!). You can check this yourself!

Another way of looking at this would be to create a correlation matrix, like we did before in the correlations chapter:

pm2_5_py.corr()

          avg_temp     pm2_5   rain_mm  wind_m_s
avg_temp  1.000000  0.033495  0.031492 -0.011079
pm2_5     0.033495  1.000000 -0.021850 -0.417339
rain_mm   0.031492 -0.021850  1.000000  0.048821
wind_m_s -0.011079 -0.417339  0.048821  1.000000

9.5 Implement and interpret the test

From our initial observations we derived that there might be some relationship between PM2.5 levels and wind speed. We also noticed that this is likely to be different between inner and outer London.

If we would want to test for every variable and interaction, then we would end up with a rather huge model, which would even include 3-way and a 4-way interaction! To illustrate the point that the process of model testing applies to as many variables as you like, we’re adding the avg_temp and rain_mm variables to our model.

So in this case we create a model that takes into account all of the main effects (avg_temp, location, rain_mm, wind_m_s). We also include a potential two-way interaction (location:wind_m_s). The two-way interaction may be of interest since the PM2.5 levels in response to wind speed seem to differ between the two locations.

Our model is then as follows:

pm2_5 ~ avg_temp + location + rain_mm + wind_m_s + wind_m_s:location

So let’s define and explore it!

R
Python

We write the model as follows:

lm_pm2_5_full <- lm(pm2_5 ~ avg_temp + location +
                            rain_mm + wind_m_s +
                            wind_m_s:location,
                    data = pm2_5)

Let’s look at the coefficients:

lm_pm2_5_full


Call:
lm(formula = pm2_5 ~ avg_temp + location + rain_mm + wind_m_s + 
    wind_m_s:location, data = pm2_5)

Coefficients:
           (Intercept)                avg_temp           locationouter  
              18.18286                 0.01045                -2.07084  
               rain_mm                wind_m_s  locationouter:wind_m_s  
              -0.02788                -0.28545                -0.42945

Extracting coefficients with tidy()

This will give us quite a few coefficients, so instead of just calling the lm object, I’m restructuring the output using the tidy() function from the broom package. It’s installed with tidyverse but you have to load it separately using library(broom).

lm_pm2_5_full %>%
    tidy() %>% 
    select(term, estimate)

# A tibble: 6 × 2
  term                   estimate
  <chr>                     <dbl>
1 (Intercept)             18.2   
2 avg_temp                 0.0105
3 locationouter           -2.07  
4 rain_mm                 -0.0279
5 wind_m_s                -0.285 
6 locationouter:wind_m_s  -0.429

The question is, are all of these terms statistically significant? To find out we perform an ANOVA:

anova(lm_pm2_5_full)

Analysis of Variance Table

Response: pm2_5
                   Df  Sum Sq Mean Sq   F value  Pr(>F)    
avg_temp            1    5.29    5.29    5.0422 0.02504 *  
location            1 3085.37 3085.37 2940.5300 < 2e-16 ***
rain_mm             1    2.48    2.48    2.3644 0.12457    
wind_m_s            1  728.13  728.13  693.9481 < 2e-16 ***
location:wind_m_s   1  134.82  134.82  128.4912 < 2e-16 ***
Residuals         724  759.66    1.05                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From this we can see that the interaction between location and wind_m_s is statistically significant. Which means that we can’t just talk about the effect of location or wind_m_s on PM2.5 levels, without taking the other variable into account!

The p-value for the avg_temp is significant, whereas the rain_mm main effect is not. This means that rain fall is not contributing much to model’s ability to explain our data. This matches what we already saw when we visualised the data.

What to do? We’ll explore this in more detail in the chapter on model comparisons, but for now the most sensible option would be to redefine the model, but exclude the rain_mm variable. Here I have rewritten the model and named it lm_pm2_5_red to indicate it is a reduced model (with fewer variables than our original full model):

lm_pm2_5_red <- lm(pm2_5 ~ avg_temp + location + wind_m_s + location:wind_m_s, data = pm2_5)

Let’s look at the new model coefficients:

lm_pm2_5_red


Call:
lm(formula = pm2_5 ~ avg_temp + location + wind_m_s + location:wind_m_s, 
    data = pm2_5)

Coefficients:
           (Intercept)                avg_temp           locationouter  
              18.13930                 0.01031                -2.07380  
              wind_m_s  locationouter:wind_m_s  
              -0.28631                -0.42880

As we did in the linear regression on grouped data, we end up with two linear equations, one for inner London and one for outer London.

Our reference group is inner (remember, it takes a reference group in alphabetical order and we can see outer in the output).

So we end up with:

\(PM2.5_{inner} = 18.14 + 0.01 \times avg\_temp - 0.29 \times wind\_m\_s\)

\(PM2.5_{outer} = (18.14 - 2.07) + 0.01 \times avg\_temp + (-0.29 - 0.43) \times wind\_m\_s\)

which gives

\(PM2.5_{outer} = 16.07 + 0.01 \times avg\_temp - 0.72 \times wind\_m\_s\)

We still need to check the assumptions of the model:

resid_panel(lm_pm2_5_red,
            plots = c("resid", "qq", "ls", "cookd"),
            smoother = TRUE)

`geom_smooth()` using formula = 'y ~ x'
`geom_smooth()` using formula = 'y ~ x'

They all look pretty good, with the only weird thing being a small empty zone of predicted values just under 16. Nothing that is getting me worried though.

It’d be useful to visualise the model. We can take the model and use the augment() function to extract the fitted values (.fitted). These are the values for pm2_5 that the model is predicting. We can then plot these against the wind_m_s measurements, colouring by location:

lm_pm2_5_red %>% 
  augment() %>% 
  ggplot(aes(x = wind_m_s,
             y = pm2_5, colour = location)) +
  geom_point() +
  geom_smooth(aes(y = .fitted))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

We write the model as follows:

# create a linear model
model = smf.ols(formula = "pm2_5 ~ avg_temp + C(location) + rain_mm + wind_m_s + wind_m_s:location", data = pm2_5_py)
# and get the fitted parameters of the model
lm_pm2_5_full_py = model.fit()

This will give us quite a few coefficients, so instead of just printing the entire summary table, we’re extracting the parameters with .params:

lm_pm2_5_full_py.params

Intercept                     18.182858
C(location)[T.outer]          -2.070843
avg_temp                       0.010451
rain_mm                       -0.027880
wind_m_s                      -0.285450
wind_m_s:location[T.outer]    -0.429455
dtype: float64

The question is, are all of these terms statistically significant? To find out we perform an ANOVA:

sm.stats.anova_lm(lm_pm2_5_full_py, typ = 2)

                       sum_sq     df           F         PR(>F)
C(location)        123.803830    1.0  117.991919   1.441933e-25
avg_temp             1.803685    1.0    1.719012   1.902360e-01
rain_mm              0.350639    1.0    0.334179   5.633886e-01
wind_m_s           728.129799    1.0  693.948101  8.928636e-108
wind_m_s:location  134.820234    1.0  128.491164   1.567268e-27
Residual           759.661960  724.0         NaN            NaN

# create a linear model
model = smf.ols(formula = "pm2_5 ~ avg_temp + C(location) * wind_m_s", data = pm2_5_py)
# and get the fitted parameters of the model
lm_pm2_5_red_py = model.fit()

Let’s look at the new model coefficients:

lm_pm2_5_red_py.params

Intercept                        18.139300
C(location)[T.outer]             -2.073802
avg_temp                          0.010312
wind_m_s                         -0.286313
C(location)[T.outer]:wind_m_s    -0.428800
dtype: float64

As we did in the linear regression on grouped data, we end up with two linear equations, one for inner London and one for outer London.

Our reference group is inner (remember, it takes a reference group in alphabetical order and we can see outer in the output).

So we end up with:

\(PM2.5_{inner} = 18.14 + 0.01 \times avg\_temp - 0.29 \times wind\_m\_s\)

\(PM2.5_{outer} = (18.14 - 2.07) + 0.01 \times avg\_temp + (-0.29 - 0.43) \times wind\_m\_s\)

which gives

\(PM2.5_{outer} = 16.07 + 0.01 \times avg\_temp - 0.72 \times wind\_m\_s\)

We still need to check the assumptions of the model:

dgplots(lm_pm2_5_red_py)

They all look pretty good, with the only weird thing being a small empty zone of predicted values just under 16. Nothing that is getting me worried though.

It’d be useful to visualise the model. We can take the model and extract the fitted values (.fittedvalues). These are the pm2_5 that the model is predicting. We can then plot these against the wind_m_s measurements, colouring by location. We’re also adding the original values to the plot with geom_point():

(ggplot(pm2_5_py, aes(x = "wind_m_s",
                     y = "pm2_5",
                     colour = "location")) +
    geom_point() +
    geom_smooth(aes(y = lm_pm2_5_red_py.fittedvalues)))

9.6 Exploring models

Rather than stop here however, we will use the concept of the linear model to its full potential and show that we can construct and analyse any possible combination of predictor variables for this data set. Namely we will consider the following four extra models, where reduce the complexity to the model, step-by-step:

Model	Description
1. `pm2_5 ~ wind_m_s + location`	An additive model
2. `pm2_5 ~ wind_m_s`	Equivalent to a simple linear regression
3. `pm2_5 ~ location`	Equivalent to a one-way ANOVA
4. `pm2_5 ~ 1`	The null model, where we have no predictors

9.6.1 Additive model

To create the additive model, we drop the interaction term (keep in mind, this is to demonstrate the process - we would normally not do this because the interaction term is significant!).

R
Python

First, we define the model:

lm_pm2_5_add <- lm(pm2_5 ~ avg_temp + location + wind_m_s,
                   data = pm2_5)

We can visualise this as follows:

lm_pm2_5_add %>% 
    augment() %>% 
    ggplot(aes(x = wind_m_s, y = pm2_5,
               colour = location)) +
    geom_point() +
    geom_smooth(aes(y = .fitted))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

Next, we extract the coefficient estimates:

lm_pm2_5_add


Call:
lm(formula = pm2_5 ~ avg_temp + location + wind_m_s, data = pm2_5)

Coefficients:
  (Intercept)       avg_temp  locationouter       wind_m_s  
     19.04867        0.01587       -4.05339       -0.49868

First, we define the model

# create a linear model
model = smf.ols(formula = "pm2_5 ~ avg_temp + C(location) + wind_m_s",
                data = pm2_5_py)
                
# and get the fitted parameters of the model
lm_pm2_5_add_py = model.fit()

We can visualise this as follows:

(ggplot(pm2_5_py, aes(x = "wind_m_s",
                      y = "pm2_5",
                      colour = "location")) +
    geom_point() +
    geom_smooth(aes(y = lm_pm2_5_add_py.fittedvalues)))

Next, we extract the coefficient estimates:

lm_pm2_5_add_py.params

Intercept               19.048672
C(location)[T.outer]    -4.053394
avg_temp                 0.015872
wind_m_s                -0.498683
dtype: float64

So our two equations would be as follows:

\(PM2.5_{inner} = 19.04 + 0.016 \times avg\_temp - 0.50 \times wind\_m\_s\)

\(PM2.5_{outer} = (19.22 - 4.05) + 0.016 \times avg\_temp - 0.50 \times wind\_m\_s\)

gives

\(PM2.5_{outer} = 15.17 + 0.016 \times avg\_temp - 0.50 \times wind\_m\_s\)

9.6.2 Revisiting linear regression

R
Python

First, we define the model:

lm_pm2_5_wind <- lm(pm2_5 ~ wind_m_s,
                   data = pm2_5)

We can visualise this as follows:

lm_pm2_5_wind %>% 
    augment() %>% 
    ggplot(aes(x = wind_m_s, y = pm2_5)) +
    geom_point() +
    geom_smooth(aes(y = .fitted))

`geom_smooth()` using method = 'loess' and formula = 'y ~ x'

Alternative using geom_smooth()

ggplot(pm2_5, aes(x = wind_m_s, y = pm2_5)) +
    geom_point() +
    geom_smooth(method = "lm", se = FALSE)

`geom_smooth()` using formula = 'y ~ x'

Next, we extract the coefficient estimates:

lm_pm2_5_wind


Call:
lm(formula = pm2_5 ~ wind_m_s, data = pm2_5)

Coefficients:
(Intercept)     wind_m_s  
    17.3267      -0.5285

First, we define the model

# create a linear model
model = smf.ols(formula = "pm2_5 ~ wind_m_s",
                data = pm2_5_py)
                
# and get the fitted parameters of the model
lm_pm2_5_wind_py = model.fit()

We can visualise this as follows:

(ggplot(pm2_5_py, aes(x = "wind_m_s",
                      y = "pm2_5")) +
    geom_point() +
    geom_smooth(aes(y = lm_pm2_5_wind_py.fittedvalues), colour = "blue"))

Alternative using geom_smooth()

(ggplot(pm2_5_py, aes(x = "wind_m_s", y = "pm2_5")) +
    geom_point() +
    geom_smooth(method = "lm", se = False, colour = "blue"))

Next, we extract the coefficient estimates:

lm_pm2_5_wind_py.params

Intercept    17.326708
wind_m_s     -0.528510
dtype: float64

This gives us the following equation:

\(PM2.5 = 17.33 - 0.53 \times wind\_m\_s\)

9.6.3 Revisiting ANOVA

If we’re just looking at the effect of location, then we’re essentially doing a one-way ANOVA.

R
Python

First, we define the model:

lm_pm2_5_loc <- lm(pm2_5 ~ location,
                   data = pm2_5)

We can visualise this as follows:

lm_pm2_5_loc %>% 
    augment() %>% 
    ggplot(aes(x = location, y = pm2_5)) +
    geom_jitter(alpha = 0.3, width = 0.1) +
    geom_point(aes(y = .fitted), colour = "blue", size = 3)

OK, what’s going on here? I’ve plotted the .fitted values (the values predicted by the model) in blue and overlaid the original (with a little bit of jitter to avoid overplotting). However, there are only two predicted values!

We can check this and see that each unique fitted value occurs 365 times:

lm_pm2_5_loc %>% 
    augment() %>% 
    count(location, .fitted)

# A tibble: 2 × 3
  location .fitted     n
  <chr>      <dbl> <int>
1 inner       16.9   365
2 outer       12.8   365

This makes sense if we think back to our original ANOVA exploration. There we established that an ANOVA is just a special case of a linear model, where the fitted values are equal to the mean of each group.

We could even check this:

pm2_5 %>% 
    group_by(location) %>% 
    summarise(mean_pm2_5 = mean(pm2_5))

# A tibble: 2 × 2
  location mean_pm2_5
  <chr>         <dbl>
1 inner          16.9
2 outer          12.8

So, that matches. We move on and extract the coefficient estimates:

lm_pm2_5_loc


Call:
lm(formula = pm2_5 ~ location, data = pm2_5)

Coefficients:
  (Intercept)  locationouter  
       16.943         -4.112

These values match up exactly with the predicted values for each individual location.

First, we define the model:

# create a linear model
model = smf.ols(formula = "pm2_5 ~ C(location)", data = pm2_5_py)
# and get the fitted parameters of the model
lm_pm2_5_loc_py = model.fit()

We can visualise this as follows:

(ggplot(pm2_5_py, aes(x = "location",
                     y = "pm2_5")) +
    geom_jitter(alpha = 0.3, width = 0.1) +
    geom_point(aes(y = lm_pm2_5_loc_py.fittedvalues), colour = "blue", size = 3))

OK, what’s going on here? I’ve plotted the fittedvalues (the values predicted by the model) in blue and overlaid the original (with a little bit of jitter to avoid overplotting). However, there are only two predicted values!

We can check this and see that each unique fitted value occurs 365 times, using the value_counts() function on the fitted values:

lm_pm2_5_loc_py.fittedvalues.value_counts()

16.942926    365
12.831356    365
dtype: int64

We could even check this:

pm2_5_py.groupby("location")["pm2_5"].mean()

location
inner    16.942926
outer    12.831356
Name: pm2_5, dtype: float64

So, that matches. We move on and extract the coefficient estimates:

lm_pm2_5_loc_py.params

Intercept               16.942926
C(location)[T.outer]    -4.111570
dtype: float64

These values match up exactly with the predicted values for each individual location.

This gives us the following equation:

\(\bar{PM2.5_{inner}} = 16.94\)

\(\bar{PM2.5_{outer}} = 16.94 - 4.11 = 12.83\)

9.6.4 The null model

The null model by itself is rarely analysed for its own sake but is instead used a reference point for more sophisticated model selection techniques. It represents your data as an overal average value.

R
Python

We define the null model as follows:

lm_pm2_5_null <- lm(pm2_5 ~ 1, data = pm2_5)

We can just view the model:

lm_pm2_5_null


Call:
lm(formula = pm2_5 ~ 1, data = pm2_5)

Coefficients:
(Intercept)  
      14.89

We define the null model as follows:

# create a linear model
model = smf.ols(formula = "pm2_5 ~ 1", data = pm2_5_py)
# and get the fitted parameters of the model
lm_pm2_5_null_py = model.fit()

We can just view the model parameters:

lm_pm2_5_null_py.params

Intercept    14.887141
dtype: float64

This shows us that there is just one value: 14.89. This is the average across all the PM2.5 values in the data set.

Here we’d predict the PM2.5 values as follows:

\(PM2.5 = 14.89\)

9.7 Exercises

9.7.1 Trees

Exercise

Level:

Trees: an example with only continuous variables

Use the data/CS5-trees.csv data set. This is a data frame with 31 observations of 3 continuous variables. The variables are the height height, diameter girth and timber volume volume of 31 felled black cherry trees.

Investigate the relationship between volume (as a dependent variable) and height and girth (as predictor variables).

Here all variables are continuous and so there isn’t a way of producing a 2D plot of all three variables for visualisation purposes using R’s standard plotting functions.
construct four linear models
- Assume volume depends on height, girth and an interaction between girth and height
- Assume volume depends on height and girth but that there isn’t any interaction between them.
- Assume volume only depends on girth (plot the result, with the regression line).
- Assume volume only depends on height (plot the result, with the regression line).
For each linear model write down the algebraic equation that the linear model produces that relates volume to the two continuous predictor variables.
Check the assumptions of each model. Do you have any concerns?

NB: For two continuous predictors, the interaction term is simply the two values multiplied together (so girth:height means girth x height)

Use the equations to calculate the predicted volume of a tree that has a diameter of 20 inches and a height of 67 feet in each case.

Answer

Let’s construct the four linear models in turn.

Full model

R
Python

First, we read in the data:

trees <- read_csv("data/CS5-trees.csv")

# define the model
lm_trees_full <- lm(volume ~ height * girth,
                   data = trees)

# view the model
lm_trees_full


Call:
lm(formula = volume ~ height * girth, data = trees)

Coefficients:
 (Intercept)        height         girth  height:girth  
     69.3963       -1.2971       -5.8558        0.1347

First, we read in the data:

trees_py = pd.read_csv("data/CS5-trees.csv")

# create a linear model
model = smf.ols(formula = "volume ~ height * girth",
                data = trees_py)
# and get the fitted parameters of the model
lm_trees_full_py = model.fit()

Extract the parameters:

lm_trees_full_py.params

Intercept       69.396316
height          -1.297083
girth           -5.855848
height:girth     0.134654
dtype: float64

We can use this output to get the following equation:

volume = 69.40 + -1.30 \(\times\) height + -5.86 \(\times\) girth + 0.13 \(\times\) height \(\times\) girth

If we stick the numbers in (girth = 20 and height = 67) we get the following equation:

volume = 69.40 + -1.30 \(\times\) 67 + -5.86 \(\times\) 20 + 0.13 \(\times\) 67 \(\times\) 20

volume = 45.81

Here we note that the interaction term just requires us to multiple the three numbers together (we haven’t looked at continuous predictors before in the examples and this exercise was included as a check to see if this whole process was making sense).

If we look at the diagnostic plots for the model using the following commands we get:

R
Python

resid_panel(lm_trees_full,
            plots = c("resid", "qq", "ls", "cookd"),
            smoother = TRUE)

dgplots(lm_trees_full_py)

All assumptions are OK.

There is some suggestion of heterogeneity of variance (with the variance being lower for small and large fitted (i.e. predicted volume) values), but that can be attributed to there only being a small number of data points at the edges, so I’m not overly concerned.
Similarly, there is a suggestion of snaking in the Q-Q plot (suggesting some lack of normality) but this is mainly due to the inclusion of one data point and overall the plot looks acceptable.
There are no highly influential points

Additive model

R
Python

# define the model
lm_trees_add <- lm(volume ~ height + girth,
                   data = trees)

# view the model
lm_trees_add


Call:
lm(formula = volume ~ height + girth, data = trees)

Coefficients:
(Intercept)       height        girth  
   -57.9877       0.3393       4.7082

# create a linear model
model = smf.ols(formula = "volume ~ height + girth",
                data = trees_py)
# and get the fitted parameters of the model
lm_trees_add_py = model.fit()

Extract the parameters:

lm_trees_add_py.params

Intercept   -57.987659
height        0.339251
girth         4.708161
dtype: float64

We can use this output to get the following equation:

volume = -57.99 + 0.34 \(\times\) height + 4.71 \(\times\) girth

If we stick the numbers in (girth = 20 and height = 67) we get the following equation:

volume = -57.99 + 0.34 \(\times\) 67 + 4.71 \(\times\) 20

volume = 58.91

If we look at the diagnostic plots for the model using the following commands we get the following:

R
Python

resid_panel(lm_trees_add,
            plots = c("resid", "qq", "ls", "cookd"),
            smoother = TRUE)

dgplots(lm_trees_add_py)

This model isn’t great.

There is a worrying lack of linearity exhibited in the Residuals plot suggesting that this linear model isn’t appropriate.
Assumptions of Normality seem OK
Equality of variance is harder to interpret. Given the lack of linearity in the data it isn’t really sensible to interpret the Location-Scale plot as it stands (since the plot is generated assuming that we’ve fitted a straight line through the data), but for the sake of practising interpretation we’ll have a go. There is definitely suggestions of heterogeneity of variance here with a cluster of points with fitted values of around 20 having noticeably lower variance than the rest of the dataset.
One point is influential and if there weren’t issues with the linearity of the model I would remove this point and repeat the analysis. As it stands there isn’t much point.

Height-only model

R
Python

# define the model
lm_height <- lm(volume ~ height,
              data = trees)

# view the model
lm_height


Call:
lm(formula = volume ~ height, data = trees)

Coefficients:
(Intercept)       height  
    -87.124        1.543

# create a linear model
model = smf.ols(formula = "volume ~ height",
                data = trees_py)
# and get the fitted parameters of the model
lm_height_py = model.fit()

Extract the parameters:

lm_height_py.params

Intercept   -87.123614
height        1.543350
dtype: float64

We can use this output to get the following equation:

volume = -87.12 + 1.54 \(\times\) height

If we stick the numbers in (girth = 20 and height = 67) we get the following equation:

volume = -87.12 + 1.54 \(\times\) 67

volume = 16.28

If we look at the diagnostic plots for the model using the following commands we get the following:

R
Python

resid_panel(lm_height,
            plots = c("resid", "qq", "ls", "cookd"),
            smoother = TRUE)

dgplots(lm_height_py)

This model also isn’t great.

The main issue here is the clear heterogeneity of variance. For trees with bigger volumes the data are much more spread out than for trees with smaller volumes (as can be seen clearly from the Location-Scale plot).
Apart from that, the assumption of Normality seems OK
And there aren’t any hugely influential points in this model

Girth-only model

R
Python

# define the model
lm_girth <- lm(volume ~ girth,
               data = trees)

# view the model
lm_girth


Call:
lm(formula = volume ~ girth, data = trees)

Coefficients:
(Intercept)        girth  
    -36.943        5.066

# create a linear model
model = smf.ols(formula = "volume ~ girth",
                data = trees_py)
# and get the fitted parameters of the model
lm_girth_py = model.fit()

Extract the parameters:

lm_girth_py.params

Intercept   -36.943459
girth         5.065856
dtype: float64

We can use this output to get the following equation:

volume = -36.94 + 5.07 \(\times\) girth

If we stick the numbers in (girth = 20 and height = 67) we get the following equation:

volume = -36.94 + 5.07 \(\times\) 20

volume = 64.37

If we look at the diagnostic plots for the model using the following commands we get the following:

R
Python

resid_panel(lm_girth,
            plots = c("resid", "qq", "ls", "cookd"),
            smoother = TRUE)

dgplots(lm_girth_py)

The diagnostic plots here look rather similar to the ones we generated for the additive model and we have the same issue with a lack of linearity, heterogeneity of variance and one of the data points being influential.

9.8 Summary

Key points

We can define a linear model with any combination of categorical and continuous predictor variables
Using the coefficients of the model we can construct the linear model equation
The underlying assumptions of a linear model with three (or more) predictor variables are the same as those of a two-way ANOVA