LAB: Linear Regression on Whiteside data

M1 MIDS/MFA/LOGOS

Université Paris Cité

Année 2024

Course Homepage

Moodle

Introduction

The purpose of this lab is to introduce linear regression using base R and the tidyverse. We work on a dataset provided by the MASS package. This dataset is investigated in the book by Venables and Ripley. This discusssion is worth being read. Our aim is to relate regression as a tool for data exploration with regression as a method in statistical inference. To perform regression, we will rely on the base R function lm() and on the eponymous S3 class lm. We will spend time understanding how the formula argument can be used to construct a design matrix from a dataframe representing a dataset.

Packages installation and loading (again)

# We will use the following packages. 
# If needed, install them : pak::pkg_install(). 
stopifnot(
  require("magrittr"),
  require("lobstr"),
  require("ggforce"),
  require("patchwork"), 
  require("gt"),
  require("glue"),
  require("skimr"),
  require("corrr"),
  require("GGally"),
  require("broom"),
  require("tidyverse"),
  require("ggfortify"),
  require("autoplotly")

)

Besides the tidyverse, we rely on skimr to perform univariate analysis, GGally::ggpairs to perform pairwise (bivariate) analysis. Package corrr provide graphical tools to explore correlation matrices. At some point, we will showcase the exposing pipe %$% and the classical pipe %>% of magrittr. We use gt to display handy tables, patchwork to compose graphical objects. glue provides a kind of formatted strings. Package broom proves very useful when milking lienar models produced by lm() (and many other objects produced by estimators, tests, …)

Dataset

The dataset is available from package MASS. MASS can be downloaded from cran.

whiteside <- MASS::whiteside # no need to load the whole package

cur_dataset <- str_to_title(as.character(substitute(whiteside)))
# ?whiteside

The documentation of R tells us a little bit more about this data set.

Mr Derek Whiteside of the UK Building Research Station recorded the weekly gas consumption and average external temperature at his own house in south-east England for two heating seasons, one of 26 weeks before, and one of 30 weeks after cavity-wall insulation was installed. The object of the exercise was to assess the effect of the insulation on gas consumption.

This means that our sample is made of 56 observations. Each observation corresponds to a week during heating season. For each observation. We have the average external temperature Temp (in degrees Celsius) and the weekly gas consumption Gas. We also have Insul which tells us whether the observation has been recorded Before or After treatment.

Temperature is the explanatory variable or the covariate. The target/response is the weekly Gas Consumption. We aim to predict or to explain the variations of weekly gas consumption as a function average weekly temperature.

The question is wether the treatment (insulation) modifies the relation between gas consumption and external temperature, and if we conclude that the treatment modifies the relation, in which way?.

Have a glimpse at the data.

whiteside %>% 
  glimpse

Rows: 56
Columns: 3
$ Insul <fct> Before, Before, Before, Before, Before, Before, Before, Before, …
$ Temp  <dbl> -0.8, -0.7, 0.4, 2.5, 2.9, 3.2, 3.6, 3.9, 4.2, 4.3, 5.4, 6.0, 6.…
$ Gas   <dbl> 7.2, 6.9, 6.4, 6.0, 5.8, 5.8, 5.6, 4.7, 5.8, 5.2, 4.9, 4.9, 4.3,…

Even though the experimenter, Mr Whiteside, decided to apply a treatment to his house. This is not exactly what we call experimental data. Namely, the experimenter has no way to clamp the external temperature. With respect to the Temperature variable (the explanatory variable) we are facing observational data.

Columnwise exploration

Question

Before before proceeding to linear regressions of Gas with respect to Temp, perform univariate analysis on each variable.

• Compute summary statistics
• Build the corresponding plots

Solution

skimr does the job. There are no missing data, complete rate is always 1, we remove non-informative columns from the output.

whiteside |>
  skimr::skim() |>
  select(-n_missing, -complete_rate, -factor.ordered, - factor.n_unique) |>
  gt() |>
  gt::fmt_number(decimals=1) |>
  gt::tab_caption("Whiteside dataset. Columnwise summaries")

Whiteside dataset. Columnwise summaries

skim_type	skim_variable	factor.top_counts	numeric.mean	numeric.sd	numeric.p0	numeric.p25	numeric.p50	numeric.p75	numeric.p100	numeric.hist
factor	Insul	Aft: 30, Bef: 26	NA	NA	NA	NA	NA	NA	NA	NA
numeric	Temp	NA	4.9	2.7	−0.8	3.1	4.9	7.1	10.2	▃▅▇▇▃
numeric	Gas	NA	4.1	1.2	1.3	3.5	4.0	4.6	7.2	▁▆▇▂▁

An alternative way of doing this univariate analysis consists in separating categorical variables from numerical ones.

sk <- whiteside %>% 
  skimr::skim() %>% 
  select(-n_missing, - complete_rate, -factor.ordered, - factor.n_unique)

skimr::yank(sk, "factor") |> 
  gt() |>
  gt::tab_caption("Whiteside dataset. Categorical variables summaries")


skimr::yank(sk, "numeric") |> 
  gt() |>
  gt::fmt_number(decimals=1) |>
  gt::tab_caption("Whiteside dataset. Categorical variables summaries")

We may test the normality of the Gas and Temp distribution

Var	statistic	p.value	method
Temp	0.98	0.48	Shapiro-Wilk normality test
Gas	0.96	0.08	Shapiro-Wilk normality test

Both variables pass the Shapiro-Wilk test with flying colors.

We may use the Jarque-Bera test See R bloggers on this

Whiteside dataset

Var	statistic	p.value	parameter	method
Temp	1.37	0.50	2.00	Jarque Bera Test
Gas	2.74	0.25	2.00	Jarque Bera Test

Both variables also pass the Jarque-Bera test with flying colors. This is noteworthy since the Jarque-Bera compares empirical skewness and kurtosis to Gaussian skewness and kurtosis.

Pairwise exploration

Question

Compare distributions of numeric variables with respect to categorical variable Insul

Solution

We start by plotting histograms

To abide to the DRY principle, we take advantage of the fact that aesthetics and labels can be tuned incrementally.

p_xx <- whiteside |>
    ggplot() +
      geom_histogram(
        mapping=aes(fill=Insul, color=Insul, y=after_stat(density)),
        position="dodge",
        alpha=.1,
        bins=10) 

p_1 <- p_xx + 
  aes(x=Temp)  +
  theme(legend.position = "None")+
  labs(title="External Temperature",
       subtitle = "Weekly Average (Celsius)")

p_2 <- p_xx + 
  aes(x=Gas)  +
  labs(title="Gas Consumption" ,
       subtitle="Weekly Average")

# patchwork the two graphical objects 
(p_1 + p_2) +
  patchwork::plot_annotation(
    caption="Whiteside dataset from package MASS"
  )

Note the position parameter for geom_histogram(). Check the result for position="stack", position="identity". Make up your mind about the most convenient choice.

Gas Consumption distribution After looks shifted with respect to distribution Before.

Another visualization strategy consists in using the faceting mechanism

r <- whiteside %>% 
  pivot_longer(cols=c(Gas, Temp),
              names_to = "Vars",
              values_to = "Vals") %>% 
  ggplot() +
  aes(x=Vals)  +
  facet_wrap(~ Insul + Vars ) + 
  xlab("")

r +
  aes(y=after_stat(density)) +
  geom_histogram(alpha=.3, fill="white", color="black", bins=6) +
  ggtitle(glue("{cur_dataset} data"))

lm(Temp ~ Insul, data=whiteside) |>
  broom::glance()

# A tibble: 1 × 12
  r.squared adj.r.squared sigma statistic p.value    df logLik   AIC   BIC
      <dbl>         <dbl> <dbl>     <dbl>   <dbl> <dbl>  <dbl> <dbl> <dbl>
1    0.0263       0.00830  2.74      1.46   0.232     1  -135.  276.  282.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>

lm(Temp ~ Insul, data=whiteside) |>
  anova()  |>
  broom::tidy() |>
  gt::gt() |>
  gt::fmt_number(decimals = 3) |>
  gt::tab_caption("Testing temperature homogeneity over the seasons")

Testing temperature homogeneity over the seasons

term	df	sumsq	meansq	statistic	p.value
Insul	1.000	10.950	10.950	1.461	0.232
Residuals	54.000	404.855	7.497	NA	NA

The difference between temperature distributions over the two seasons is not deemed to be significant by ANOVA.

It is important that the grouping variable (here Insul) is a factor.

Covariance and correlation between Gas and Temp

Question

Compute the covariance matrix of Gas and Temp

Solution

# Compute the covariance matrix for Gas and Temp
C <- whiteside |>
  select(where(is.numeric)) |> 
  cov()

mu_n <- whiteside %>% 
  select(where(is.numeric)) %>% 
  colMeans()

matador::mat2latex(round(C,2))

\[ \begin{bmatrix} 7.56 & -2.19\\ -2.19 & 1.36 \end{bmatrix} \]

Question

• Compute correlations (Pearson, Kendall, Spearman) and correlations per group
• Comment

Solution

# use magrittr pipe %>%  to define a pipeline function
my_cor <-  . %>% 
  summarize(
    pearson= cor(Temp, Gas, method="pearson"),
    kendall=cor(Temp, Gas, method="kendall"),
    spearman=cor(Temp,Gas, method="spearman")
  ) 

# even better: pass column names as arguments

my_cor2 <-  function(df, col1, col2){
  df %>% 
  summarize(
    pearson= cor({{col1}}, {{col2}}, method="pearson"),
    kendall=cor({{col1}}, {{col2}}, method="kendall"),
    spearman=cor({{col1}}, {{col2}}, method="spearman")
  ) 
}

t1 <- whiteside |>
  group_by(Insul) |> 
  my_cor()

# group_by(whiteside, Insul) |> my_cor2(Gas, Temp)

t2 <- whiteside |>
  my_cor() |>
  mutate(Insul="pooled")

# whiteside |> my_cor2(Gas, Temp) |> mutate(Insul="pooled")

bind_rows(t1, t2)  |> 
    gt() |>
    gt::fmt_number(decimals=2) |>
    gt::tab_caption("Whiteside data: correlations between Gas and Temp")

Whiteside data: correlations between Gas and Temp

Insul	pearson	kendall	spearman
Before	−0.97	−0.85	−0.96
After	−0.90	−0.72	−0.86
pooled	−0.68	−0.47	−0.62

Note the sharp increase of all correlation coefficients we data are grouped according to the control/treatment variable (Insul).

Question

Use ggpairs from GGally to get a quick overview of the pairwise interactions.

Solution

theme_set(theme_minimal())

whiteside |>
  GGally::ggpairs()

Question

Build a scatterplot of the Whiteside dataset

Solution

p <- whiteside |> 
  ggplot() +
  aes(x=Temp, y=Gas) +
  geom_point(aes(shape=Insul)) +
  xlab("Average Weekly Temperature (Celsius)") +
  ylab("Average Weekly Gas Consumption 1000 cube feet") +
  labs(
    ## Use list unpacking
  )

p + 
  ggtitle(glue("{cur_dataset} dataset"))

Note that the dataset looks like the stacking of two bananas corresponding to the two heating seasons.

Question

Build boxplots of Temp and Gas versus Insul

Solution

q <- whiteside %>% 
  ggplot() +
  aes(x=Insul)

qt <- q + 
  geom_boxplot(aes(y=Temp))

qg <- q + 
  geom_boxplot(aes(y=Gas))

(qt + qg) +
  patchwork::plot_annotation(title = glue("{cur_dataset} dataset"))

Note the two low extremes/outliers for the Gas Consumption after Insulation.

Question

Build violine plots of Temp and Gas versus Insul

Solution

(q + 
  geom_violin(aes(y=Temp))) +
(q + 
  geom_violin(aes(y=Gas))) +
  patchwork::plot_annotation(title = glue("{cur_dataset} dataset"))

Question

Plot density estimates of Temp and Gas versus Insul.

Solution

r +
  stat_density(alpha=.3 , 
               fill="grey", 
               color="black", 
#               bw = "SJ",
               adjust = .5 ) +
  ggtitle(glue("{cur_dataset} data"))

Hand-made calculation of simple linear regression estimates for Gas versus Temp

Question

Compute slope and intercept using elementary computations

Solution

slope <- C[1,2] / C[1,1] # slope 

intercept <- whiteside %$%  # exposing pipe from magrittr
  (mean(Gas) - slope * mean(Temp)) # intercept

# with(whiteside,
#     mean(Gas) - b * mean(Temp)) 

In simple linear regression, the slope follows from the covariance matrix in a straightforward way. The slope can also be expressed as the linear correlation coefficient (Pearson) times the ration between the standard deviation of the response variable and the standard deviation of the explanatory variable.

Question

Overlay the scatterplot with the regression line.

Solution

p + 
  geom_abline(slope=slope, intercept = intercept) +
  ggtitle(glue("{cur_dataset} data"), subtitle = "Least square regression line")

Using lm()

lm stands for Linear Models. Function lm has a number of arguments, including:

• formula
• data

Question

Use lm() to compute slope and intercept. Denote the object created by constructor lm() as lm0.

• What is the class of lm0 ?

Solution

lm0 <- lm(Gas ~ Temp, data = whiteside)

The result is an object of class lm.

The generic function summary() has a method for class lm

lm0 %>% 
  summary()

Call:
lm(formula = Gas ~ Temp, data = whiteside)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.6324 -0.7119 -0.2047  0.8187  1.5327 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   5.4862     0.2357  23.275  < 2e-16 ***
Temp         -0.2902     0.0422  -6.876 6.55e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.8606 on 54 degrees of freedom
Multiple R-squared:  0.4668,    Adjusted R-squared:  0.457 
F-statistic: 47.28 on 1 and 54 DF,  p-value: 6.545e-09

The summary is made of four parts

• The call. Very useful if we handle many different models (corresponding to different formulae, or different datasets)
• A numerical summary of residuals
• A commented display of the estimated coefficients
• Estimate of noise scale (under Gaussian assumptions)
• Squared linear correlation coefficient between response variable \(Y\) (Gas) and predictions \(\widehat{Y}\)
• A test statistic (Fisher’s statistic) for assessing null hypothesis that slope is null, and corresponding \(p\)-value (under Gaussian assumptions).

Including a rough summary in a report is not always a good idea. It is easy to extract tabular versions of the summary using functions tidy() and glance() from package broom.

For html output gt::gt() allows us to polish the final output

Solution

We can use the exposing pipe from magrittr (or the with construct from base R) to build a function that extracts the coefficients estimates, standard error, \(t\)-statistic and associated p-values.

tidy_lm <- . %$% (    # <1>  The lhs is meant to be of class lm
  tidy(.)  %>%         # <2> . acts as a pronoun for magrittr pipes     
  gt::gt() %>% 
  gt::fmt_number(decimals=2) %>% 
  gt::tab_caption(glue("Linear regrression. Dataset: {call$data},  Formula: {deparse(call$formula)}"))  # <3> call is evaluated as a member of the pronoun `.`
)

tidy_lm(lm0)

Linear regrression. Dataset: whiteside, Formula: Gas ~ Temp

term	estimate	std.error	statistic	p.value
(Intercept)	5.49	0.24	23.28	0.00
Temp	−0.29	0.04	−6.88	0.00

deparse() is an important function from base R. It is very helpful when trying to take advantage of lazy evaluation mechanisms.

Question

Function glance() extract informations that can be helpful when performing model/variable selection.

Solution

The next chunk handles several other parts of the summary.

glance_lm <-  . %$% (
  glance(.) %>% 
  mutate(across(-c(p.value), 
                ~ round(.x, digits=2)),
         p.value=signif(p.value,3)) %>% 
  gt::gt() %>% 
  gt::tab_caption(glue("Dataset: {call$data},  Formula: {deparse(call$formula)}"))
)

glance_lm(lm0)

Dataset: whiteside, Formula: Gas ~ Temp

r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
0.47	0.46	0.86	47.28	6.55e-09	1	-70.04	146.07	152.15	39.99	54	56

• r.squared (and adj.r.squared)
• sigma estimates the noise standard deviation (under homoschedastic Gaussian noise assumption)
• statistic is the Fisher statistic used to assess the hypothesis that the slope (Temp coefficient) is zero. It is compared with quantiles of Fisher distribution with 1 and 54 degrees of freedom (check pf(47.28, df1=1, df2=54, lower.tail=F) or qf(6.55e-09, df1=1, df2=54, lower.tail=F)).

Question

R offers a function confint() that can be fed with objects of class lm. Explain the output of this function.

Solution

Under the Gaussian homoschedastic noise assumption, confint() produces confidence intervals for estimated coefficients. Using the union bound, we can derive a conservative confidence rectangle.

M <- confint(lm0, level=.95)  

as_tibble(M) |>
  mutate(Coeff=rownames(M)) |>
  relocate(Coeff) |>
  gt::gt() |>
  gt::fmt_number(decimals=2)

Coeff	2.5 %	97.5 %
(Intercept)	5.01	5.96
Temp	−0.37	−0.21

conf_levels <- c(.90, .95, .99)

purrr::map(conf_levels, \(x) confint(lm0, level=x)) |>
  purrr::map(as_tibble) |>
  purrr::map(\(x) mutate(x, coeffs=c("Intercept", "Temp")))|>
  purrr::map(\(x) pivot_longer(x, cols=ends_with("%"), names_to="conf", values_to="bound")) |>
  bind_rows() |>
  gt::gt() |>
  gt::fmt_number(decimals = 2)

coeffs	conf	bound
Intercept	5 %	5.09
Intercept	95 %	5.88
Temp	5 %	−0.36
Temp	95 %	−0.22
Intercept	2.5 %	5.01
Intercept	97.5 %	5.96
Temp	2.5 %	−0.37
Temp	97.5 %	−0.21
Intercept	0.5 %	4.86
Intercept	99.5 %	6.12
Temp	0.5 %	−0.40
Temp	99.5 %	−0.18

Question

Plot a \(95\%\) confidence region for the parameters (assuming homoschedastic Gaussian noise).

Solution

Assuming homoschedastic Gaussian noise, the distribution of \((X^T X)^{1/2}\frac{\widehat{\theta}- \theta}{\widehat{\sigma}}\) is \(\frac{\mathcal{N}(0, \text{Id}_p)}{\sqrt{\chi^2_{n-p}/(n-p)}}\) with \(p=2\) and \(n=56\), where the Gaussian vector and the \(\chi^2\) variable are independent. Hence \[\frac{1}{2}\left\Vert (X^T X)^{1/2}\frac{\widehat{\theta}- \theta}{\widehat{\sigma}} \right\Vert^2= \frac{(\widehat{\theta}-\theta)^T X^TX (\widehat{\theta}-\theta)}{2\widehat{\sigma}^2}\] is distributed accordding to a Fisher distribution with \(p=2\) and \(n-p=54\) degrees of freedom. Let \(f_{\alpha}\) denote the quantile of order \(1-\alpha\) of the Fisher distribution with \(p=2\) and \(n-p=54\) degrees of freedom, the following ellipsoid is a \(1-\alpha\) confidence region: \[\left\{ \theta : \frac{(\widehat{\theta}-\theta)^T X^TX (\widehat{\theta}-\theta)}{2\widehat{\sigma}^2} \leq f_{\alpha}\right\}\]

X <- model.matrix(lm0)
coefficients(lm0)["(Intercept)"]

(Intercept) 
   5.486193 

sum(residuals(lm0)^2)

[1] 39.99487

conf_region <- function(X, x0=0, y0=0, sigma=1, df1=2, df2=nrow(X)-2, alpha=.05, ...){ 

  falpha <- qf(alpha, df1, df2, lower.tail = F)

  specdec <- eigen(t(X) %*% X)
  angle <- acos(specdec$vectors[1,1])
  a <- sigma * sqrt(2*falpha/specdec$values[1])
  b <- sigma * sqrt(2*falpha/specdec$values[2])

  

  ggforce::geom_ellipse(
    aes(
      x0= x0,
      y0= y0,
      a = a,
      b = b,
      angle= angle
    ),
    ...
)
}

require(zeallot)  # experimental 

Loading required package: zeallot

c(x0,y0) %<-% coefficients(lm0)

ggplot() +
  conf_region(X, x0, y0, linetype='dashed') +
  conf_region(X, x0, y0, alpha=.01, linetype='dotted') +
  conf_region(X, x0, y0, alpha=.1, )

  coord_fixed()

<ggproto object: Class CoordFixed, CoordCartesian, Coord, gg>
    aspect: function
    backtransform_range: function
    clip: on
    default: FALSE
    distance: function
    expand: TRUE
    is_free: function
    is_linear: function
    labels: function
    limits: list
    modify_scales: function
    range: function
    ratio: 1
    render_axis_h: function
    render_axis_v: function
    render_bg: function
    render_fg: function
    setup_data: function
    setup_layout: function
    setup_panel_guides: function
    setup_panel_params: function
    setup_params: function
    train_panel_guides: function
    transform: function
    super:  <ggproto object: Class CoordFixed, CoordCartesian, Coord, gg>

Diagnostic plots

Method plot.lm() of generic S3 function plot from base R offers six diagnostic plots. By default it displays four of them.

In order to obtain diagnostic plots as ggplot objects, use package ggfortify which defines an S3 method for class ‘lm’ for generic function autoplot (defined in package ggplot).

Question

What are the diagnostic plots good for?

Solution

Diagnostic plots for linear regression serve several purposes:

• Visualization of possible dependencies between residuals and fitted values. OLS theory tells us that residuals and fitted values are (empirically) linearly uncorrelated. This does not preclude other kinds of dependencies.
• With some overplotting, it is also possible to visualize possible dependencies between residuals and variables that were not taken into account while computing OLS estimates.
• Assesment of possible heteroschedasticity, for example, dependence of noise variance on fitted values.
• Assessing departure of noise distribution from Gaussian setting.
• Spotting points with high leverage
• Spotting possible outliers.

Question

Load package ggfortify, and call autoplot() (from ggplot2) to build the diagnostic plots for lm0.

Generic function autoplot() called on on an object of class lm builds a collection of grapical objects that parallel the output of base R generic plot() function. The graphical objects output by autoplot() can be further tweaked as any ggplot object.

Solution

vanilla <- c(
  "title"= "Diagnostic plot for linear regression  Gas ~ Temp",
  "caption"= "Whiteside data from package MASS"
)

diag_plots <- autoplot(lm0, data=whiteside)

diag_plots is an instance of an S4 class

class(diag_plots)

[1] "ggmultiplot"
attr(,"package")
[1] "ggfortify"

diag_plots@plots[[1]] +
  labs(!!!vanilla)

This diagnostic plots would be much more helpful if we could map Insul on the shape aesthetics.

# diag_plots@plots[[1]]  + aes(shape=Insul)

lm0_augmented <- augment(lm0, data=whiteside) 

lm0_augmented |>
  ggplot() +
  aes(x=.fitted, y=.resid) +
  geom_point(aes(shape=Insul)) +
  geom_smooth(formula='y~x', 
              method="loess", 
              se=T, 
              color="black",
              linewidth=.5) +
  xlab("Fitted values") +
  ylab("Residuals") +
  labs(
    !!!vanilla,
    subtitle = "Residuals versus fitted values"
  )

Overplotting (mapping Insul over shape aesthetics) reveals that residuals do depend on Insul: residual signs are a function of Insul.

The smooth regression line (blue line) departs from the x axis, but not in a striking way.

patchwork::wrap_plots(
  (diag_plots@plots[[2]] + coord_fixed() + labs()),  
  (
  augment(lm0, whiteside) |>
  ggplot() +
    aes(y=sort(.std.resid), 
        x=qnorm(seq_along(.std.resid)/length(.std.resid))) +
    geom_point(aes(shape=Insul)) + 
    geom_abline(intercept = 0, slope=1.0111398) +
    geom_smooth(method="lm", se=F, linetype="dotted") +
    geom_abline(intercept=0, slope=4/3) +
    xlab("Normal quantiles") +
    ylab("Standardized residuals") +
    coord_fixed() +
    labs()
  )
) +
patchwork::plot_annotation(
  title =vanilla["title"],
  subtitle="QQ plots for linear regression",
  caption = vanilla["caption"]
  )

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

Warning: Removed 1 row containing non-finite outside the scale range
(`stat_smooth()`).

diag_plots@plots[[3]] +
  geom_smooth(method="lm", se=T, formula='y ~ x', linetype="dotted") +
  labs(
    !!!vanilla,
    subtitle = "Standardized residuals versus fitted values"
  )

The amplitude of standardized residuals seems to be an increasing function of fitted values. This suggests departure from homoschedasticity.

Two points are singled out (with row index)

diag_plots@plots[[4]] +
  
  labs(
    !!!vanilla,
    subtitle = ""
  )

Double check points with high leverage.

Why points with high leverage and large standardized residuals matter.

The motivation and usage of diagnostic plots is explained in detail in the book by Fox and Weisberg: An R companion to applied regression.

In words, the diagnostic plots serve to assess the validity of the Gaussian linear modelling assumptions on the dataset. The assumptions can be challenged for a number of reasons:

• The response variable may be a function of the covariates but not a lienar one.
• The noise may be heteroschedastic
• The noise may be non-Gaussian
• The noise may exhibit dependencies.

The diagnostic plots are built from the information gathered in the lm object returned by lm(...).

It is convenient to extract the required pieces of information using method augment.lm. of generic function augment() from package broom.

Solution

autoplot(lm0)

whiteside_aug <- lm0 %>% 
  augment(whiteside)

lm0 %$% ( # exposing pipe !!! 
  augment(., data=whiteside) %>% 
  mutate(across(!where(is.factor), ~ signif(.x, 3))) %>% 
  group_by(Insul) %>% 
  sample_n(5) %>% 
  ungroup() %>% 
  gt::gt() %>% 
  gt::tab_caption(glue("Dataset {call$data},  {deparse(call$formula)}"))
)

Dataset whiteside, Gas ~ Temp

Insul	Temp	Gas	.fitted	.resid	.hat	.sigma	.cooksd	.std.resid
Before	7.5	4.0	3.31	0.690	0.0344	0.863	0.01190	0.816
Before	2.9	5.8	4.64	1.160	0.0272	0.854	0.02590	1.360
Before	8.5	3.6	3.02	0.581	0.0495	0.865	0.01250	0.692
Before	7.4	4.2	3.34	0.861	0.0332	0.860	0.01780	1.020
Before	7.5	3.9	3.31	0.590	0.0344	0.865	0.00869	0.698
After	0.8	4.6	5.25	-0.654	0.0578	0.864	0.01880	-0.783
After	7.2	2.8	3.40	-0.597	0.0309	0.865	0.00790	-0.704
After	5.0	3.6	4.04	-0.435	0.0179	0.867	0.00237	-0.510
After	4.7	3.5	4.12	-0.622	0.0179	0.864	0.00486	-0.730
After	1.6	4.2	5.02	-0.822	0.0437	0.861	0.02180	-0.977

Recall that in the output of augment()

• .fitted: \(\widehat{Y} = H \times Y= X \times \widehat{\beta}\)
• .resid: \(\widehat{\epsilon} = Y - \widehat{Y}\) residuals, \(\sim (\text{Id}_n - H) \times \epsilon\)
• .hat: diagonal coefficients of Hat matrix \(H\)
• .sigma: is meant to be the estimated standard deviation of components of \(\widehat{Y}\)

Compute the share of explained variance

Solution

whiteside_aug %$% {
  1 - (var(.resid)/(var(Gas)))
}

[1] 0.4668366

# with(whiteside_aug,
#   1 - (var(.resid)/(var(Gas)))

Plot residuals against fitted values

Solution

diag_1 <- whiteside_aug %>% 
  ggplot() +
  aes(x=.fitted, y=.resid)+
  geom_point(aes(shape= Insul), size=1, color="black") +
  geom_smooth(formula = y ~ x,
              method="loess",
              se=F,
              linetype="dotted",
              linewidth=.5,
              color="black") +
  geom_hline(yintercept = 0, linetype="dashed") +
  xlab("Fitted values") +
  ylab("Residuals)") +
  labs(caption = "Residuals versus Fitted")

Question

Fitted against square root of standardized residuals.

Solution

diag_3 <- whiteside_aug %>%
  ggplot() +
  aes(x=.fitted, y=sqrt(abs(.std.resid))) +
  geom_smooth(formula = y ~ x,
              se=F,
              method="loess",
              linetype="dotted",
              linewidth=.5,
              color="black") +
  xlab("Fitted values") +
  ylab("sqrt(|standardized residuals|)") +
  geom_point(aes(shape=Insul), size=1, alpha=1) +
  labs(caption = "Scale location")

Question

Hand-made normal qqplot for lm0

Solution

diag_2 <- whiteside_aug %>% 
  ggplot() +
  aes(sample=.std.resid) +
  geom_qq(size=.5, alpha=.5) +
  stat_qq_line(linetype="dotted",
              linewidth=.5,
              color="black") +
#  coord_fixed() +
  labs(caption="Residuals qqplot") +
  xlab("Theoretical quantiles") +
  ylab("Empirical quantiles of standadrdized residuals")

Question

Solution

(diag_1 + diag_2 + diag_3 + guide_area()) + 
  plot_layout(guides="collect") +
  plot_annotation(title=glue("{cur_dataset} dataset"),
                  subtitle = glue("Regression diagnostic  {deparse(lm0$call$formula)}"), caption = 'The fact that the sign of residuals depend on Insul shows that our modeling is too naive.\n The qqplot suggests that the residuals are not collected from Gaussian homoschedastic noise.'
                  )

Taking into account Insulation

Question

Design a formula that allows us to take into account the possible impact of Insulation. Insulation may impact the relation between weekly Gas consumption and average external Temperature in two ways. Insulation may modify the Intercept, it may also modify the slope, that is the sensitivity of Gas consumption with respect to average external Temperature.

Have a look at formula documentation (?formula).

Solution

lm1 <- lm(Gas ~ Temp * Insul, data = whiteside)

vanilla_lm1 <- vanilla
vanilla_lm1["title"] <- "Diagnostic plot for linear regression  Gas ~ Temp*Insul"

Question

Check the design using function model.matrix(). How can you relate this augmented design and the one-hot encoding of variable Insul?

Solution

model.matrix(lm1) |>
  head()

  (Intercept) Temp InsulAfter Temp:InsulAfter
1           1 -0.8          0               0
2           1 -0.7          0               0
3           1  0.4          0               0
4           1  2.5          0               0
5           1  2.9          0               0
6           1  3.2          0               0

InsulAfter is the result of one-hot encoding of boolean column Insul: Before is encoded by 0 and After is encoded by 1.

Temp:InsulAfter is the elementwise product of Temp and InsulAfter.

In one-hot-encoding, a categorical columns with k levels is mapped to k 0-1 valued columns. An occurrence of level j : 1 ≤ j ≤ k is mapped to a sequence of j-1 0‘s, a 1, followed by k-j 0’s.
The resulting matrix has column rank at most ’k-1’, since the rowsums are all equal to 1. In order to eliminate this redundancy, one of the k columns is dropped. Here the column with 1’s for Before has been dropped.

Note that other encodings (called contrasts in the statistics literature) are possible.

Question

Display and comment the part of the summary of lm1 concerning estimated coefficients.

Solution

lm1 %>% 
  tidy_lm()

Linear regrression. Dataset: whiteside, Formula: Gas ~ Temp * Insul

term	estimate	std.error	statistic	p.value
(Intercept)	6.85	0.14	50.41	0.00
Temp	−0.39	0.02	−17.49	0.00
InsulAfter	−2.13	0.18	−11.83	0.00
Temp:InsulAfter	0.12	0.03	3.59	0.00

Solution

bind_rows(
  lm0 |>
    glance() |>
    mutate(model="Gas ~ Temp"),
  lm1 |>
    glance() |>
    mutate(model="Gas ~ Temp*Insul")
) |>
  relocate(model) |>
  gt::gt() |>
  gt::fmt_number(decimals=2) |>
  gt::tab_caption("Comparing two models for Whiteside dataset")

Comparing two models for Whiteside dataset

model	r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
Gas ~ Temp	0.47	0.46	0.86	47.28	0.00	1.00	−70.04	146.07	152.15	39.99	54.00	56.00
Gas ~ Temp*Insul	0.93	0.92	0.32	222.33	0.00	3.00	−14.10	38.20	48.33	5.43	52.00	56.00

Question

Comment the diagnostic plots built from the extended model using autoplot(). If possible, generate alternative diagnostic plots pipelining broom and ggplot2.

Solution

diag_plots_lm1 <- autoplot(lm1, data=whiteside)

diag_plots_lm1@plots[[1]] +
  labs(
    !!!vanilla_lm1,
    subtitle="Residuals versus fitted values"
  )

whiteside_aug1 <-  augment(lm1, whiteside)

whiteside_aug1 |>
  ggplot() + 
  aes(x=.fitted, y=.resid) +
  geom_point(aes(shape=Insul)) +
  geom_smooth(method="loess", formula='y ~ x') +
  labs(
    !!!vanilla_lm1,
    subtitle="Residuals versus fitted values"
  )

The sign of residuals does not look like a function of Insul anymore.

p +
  geom_smooth(formula='y ~ poly(x, 2)',linewidth=.5, color="black",linetype="dashed",  method="lm", se=FALSE)+
  aes(color=Insul) +
  geom_smooth(aes(linetype=Insul), 
              formula='y ~ x',linewidth=.5, color="black", method="lm", se=FALSE) +
  scale_color_manual(values= c("Before"="red", "After"="blue")) +
  geom_abline(intercept = 6.8538, slope=-.3932, color="red") +
  geom_abline(intercept = 6.8538 - 2.13, slope=-.3932 +.1153, color="blue") + labs(
    title=glue("{cur_dataset} dataset"),
    subtitle = glue("Regression: {deparse(lm1$call$formula)}")
    )

Solution

(diag_1 %+% whiteside_aug1) +
(diag_3 %+% whiteside_aug1) +  
(diag_2 %+% whiteside_aug1) +
  guide_area() +
  plot_layout(guides = "collect",) +
  plot_annotation(title=glue("{cur_dataset} dataset"),
                  subtitle = glue("Regression diagnostic  {deparse(lm1$call$formula)}"), caption = 'One possible outlier.\n Visible on all three plots.'
                  )

The formula argument defines the design matrix and the Least-Squares problem used to estimate the coefficients.

cowplot::plot_grid(
  (diag_1 %+% whiteside_aug1),
  (diag_3 %+% whiteside_aug1),
  (diag_2 %+% whiteside_aug1),
  align="none",
  nrow=3, 
  labels=c('A', 'B', 'C')
)

diag_plots_lm1@plots[[1]] + 
  diag_plots_lm1@plots[[3]] +
  patchwork::plot_annotation(
    title="Gas ~ Temp * Insul",
    # subtitle = "Residuals versus fitted",
    caption="Whiteside data"
  )

diag_plots_lm1@plots[[2]] +
  labs(!!!vanilla_lm1)

diag_plots@plots[[4]] +
  labs(
    !!! vanilla_lm1
  )

Function model.matrix() allows us to inspect the design matrix.

Solution

model.matrix(lm1) %>% 
  as_tibble() %>% 
  mutate(Insul=ifelse(InsulAfter,"After", "Before")) %>% 
  ungroup() %>% 
  slice_sample(n=5) |>
  gt::gt() |>
  gt::tab_caption(glue("Design matrix for {deparse(lm1$call$formula)}")) 

Design matrix for Gas ~ Temp * Insul

(Intercept)	Temp	InsulAfter	Temp:InsulAfter	Insul
1	8.8	1	8.8	After
1	-0.8	0	0.0	Before
1	7.5	1	7.5	After
1	6.3	0	0.0	Before
1	7.4	0	0.0	Before

X <- model.matrix(lm1)

In order to solve le Least-Square problems, we have to compute \[(X^T \times X)^{-1} \times X^T\] This can be done in several ways.

lm() uses QR factorization.

Solution

Q <- qr.Q(lm1$qr)
R <- qr.R(lm1$qr)  # R is upper triangular 

norm(X - Q %*% R, type="F") # QR Factorization

[1] 1.753321e-14

signif(t(Q) %*% Q, 2)      # Q's columns form an orthonormal family

         [,1]     [,2]     [,3]    [,4]
[1,]  1.0e+00 -1.4e-17  3.1e-17 1.7e-16
[2,] -1.4e-17  1.0e+00 -3.5e-17 1.4e-17
[3,]  3.1e-17 -3.5e-17  1.0e+00 0.0e+00
[4,]  1.7e-16  1.4e-17  0.0e+00 1.0e+00

H <- Q %*% t(Q)             # The Hat matrix 

norm(X - H %*% X, type="F") # H leaves X's columns invariant

[1] 1.758479e-14

norm(H - H %*% H, type="F") # H is idempotent

[1] 7.993681e-16

# eigen(H, symmetric = TRUE, only.values = TRUE)$values

sum((solve(t(X) %*% X) %*% t(X) %*% whiteside$Gas - lm1$coefficients)^2)

[1] 3.075652e-29

Once we have the QR factorization of \(X\), solving the normal equations boils down to inverting a triangular matrix.

sum((solve(R) %*% t(Q) %*% whiteside$Gas - lm1$coefficients)^2)

[1] 2.050287e-29

#matador::mat2latex(signif(solve(t(X) %*% X), 2))

\[ (X^T \times X)^{-1} = \begin{bmatrix} 0.18 & -0.026 & -0.18 & 0.026\\ -0.026 & 0.0048 & 0.026 & -0.0048\\ -0.18 & 0.026 & 0.31 & -0.048\\ 0.026 & -0.0048 & -0.048 & 0.0099 \end{bmatrix} \]

Solution

whiteside_aug1 %>% 
  glimpse()

Rows: 56
Columns: 9
$ Insul      <fct> Before, Before, Before, Before, Before, Before, Before, Bef…
$ Temp       <dbl> -0.8, -0.7, 0.4, 2.5, 2.9, 3.2, 3.6, 3.9, 4.2, 4.3, 5.4, 6.…
$ Gas        <dbl> 7.2, 6.9, 6.4, 6.0, 5.8, 5.8, 5.6, 4.7, 5.8, 5.2, 4.9, 4.9,…
$ .fitted    <dbl> 7.168419, 7.129095, 6.696532, 5.870731, 5.713435, 5.595463,…
$ .resid     <dbl> 0.031581243, -0.229094875, -0.296532170, 0.129269357, 0.086…
$ .hat       <dbl> 0.22177670, 0.21586370, 0.15721835, 0.07782904, 0.06755399,…
$ .sigma     <dbl> 0.3261170, 0.3241373, 0.3230041, 0.3256103, 0.3259138, 0.32…
$ .cooksd    <dbl> 0.0008751645, 0.0441520664, 0.0466380672, 0.0036646607, 0.0…
$ .std.resid <dbl> 0.11083298, -0.80096122, -1.00001423, 0.41675591, 0.2775375…

Understanding .fitted column

Solution

sum((predict(lm1, newdata = whiteside) - whiteside_aug1$.fitted)^2)

[1] 0

sum((H %*% whiteside_aug1$Gas - whiteside_aug1$.fitted)^2)

[1] 3.478877e-28

Question

Try understanding the computation of .resid in an lm object. Compare .resid with the projection of Gas on the linear subspace orthogonal to the columns of the design matrix.

Solution

sum((whiteside_aug1$.resid + H %*% whiteside_aug1$Gas - whiteside_aug1$Gas)^2)

[1] 3.461127e-28

Question

Understanding .hat

Solution

sum((whiteside_aug1$.hat - diag(H))^2)

[1] 0

Question

Understanding .std.resid

• Estimate noise intensity from residuals
• Compare with the output of glance()

Solution

sigma_hat <- sqrt(sum(lm1$residuals^2)/lm1$df.residual)

lm1 |>
  glance() |>
  gt::gt() |>
  gt::fmt_number(decimals=2) |>
  gt::tab_caption(glue("The hand-made estimate of sigma is {round(sigma_hat,2)}"))

The hand-made estimate of sigma is 0.32

r.squared	adj.r.squared	sigma	statistic	p.value	df	logLik	AIC	BIC	deviance	df.residual	nobs
0.93	0.92	0.32	222.33	0.00	3.00	−14.10	38.20	48.33	5.43	52.00	56.00

\[ \widehat{r}_i = \frac{\widehat{\epsilon}_i}{\widehat{\sigma} \sqrt{1 - H_{i,i}}} \]

sum((sigma_hat * sqrt(1 -whiteside_aug1$.hat) * whiteside_aug1$.std.resid - whiteside_aug1$.resid)^2)

[1] 4.471837e-28

Question

Understanding column .sigma

Solution

Column .sigma contains the leave-one-out estimates of \(\sigma\), that is whiteside_aug1$.sigma[i] is the estimate of \(\sigma\) you obtain by leaving out the i row of the dataframe.

There is no need to recompute everything for each sample element.

\[ \widehat{\sigma}^2_{(i)} = \widehat{\sigma}^2 \frac{n-p-1- \frac{\widehat{\epsilon}_i^2}{\widehat{\sigma}^2 {(1 - H_{i,i})}}\frac{}{}}{n-p-2} \]

Appendix

S3 classes in R

Generic functions for S3 classes

methods(autoplot) lists the S3 classes for which an autoplot method is defined. Some methods are defined in ggplot2, others like autoplot.lm are defined in extension packages like ggfortify.

S4 classes in R

The output of autoplot.lm is an instance of S4 class

tibbles with list columns